0 Introduction: The Kantian Schema and the Two Needs of Pure Reason
[1]Before properly introducing the topic of this paper, which is Kant’s postulates of practical reason, and in order to lay the minimal but necessary ground for that introduction, I would like to present a truly oversimplified schematic[2] of Kant’s architectonic for the human mind – an oversimplified schematic that is also selectively tailored to reflect the essential concerns of this paper, rather than attempting to be a comprehensive picture. To that end, the following should be read intuitively, i.e. top-down, with the text in brackets indicating the directed relation between the item preceding it and the item following it.
Pure Reason Pure Reason
[gives analytically] [gives a priori as an empirical fact of having-reason, and also indirectly but analytically, because freedom is a necessary condition of the moral law]
The Moral Law Freedom
[which both, as pure ideas, subtend]
The Concepts of the Understanding[3]
[which are appended to]
Image Representations
[that are furnished by]
The Imagination
[and which together, i.e. concept of understanding and image representation, provide]
A Determinate Object
[for]
A determinate maxim of the understanding
[in order to, with the purpose of determining an action, give a rule to]
The Will
We can easily see that there are two sides to the metatheoretical or philosophical motivation for Kant here, essentially corresponding to the speculative (top) and empirical (bottom) poles of human reason, as well as to the distinction between theoretical and practical reason. Namely, on one end, there is pure speculative reason, and on the other, the practical faculties of imagination and understanding in their working-together to determine the will, or the faculty of action. That is, Kant wants both (1) to keep strictly to the formal conditions and limitations of pure reason as his subjective ground of orientation (his “rock” or point of absolute philosophical reference, as it were), and (2) to adequately account for the effective determination of the will by practical reason. His concern with the practical is also, obviously, a concern with morality; but morality actually derives analytically from pure reason for Kant. This is in fact crucial, as the postulates of practical reason must, for Kant, evolve out of the concept of morality, rather than constitute its foundation.[4]
Now Kant argues that the practical postulates respond to and are justified by a need of pure reason. This need is doubly articulated, like pure reason itself, in both theoretical and practical forms.[5] The critical difference in the two cases for Kant is that in the one (namely the theoretical), positing the postulates would transgress the proper limits or bounds of reason and is thus strictly prohibited by Kant’s own critical method, whereas this positing is precisely what Kant argues is justified in the other case (namely the practical). As the infamous phrase – now long since reified into a cliché – attached to this complexly architectonic argument goes, Kant found it necessary to “deny knowledge in order to make room for faith.”[i] The original German is actually illuminating here:
Ich musste das Wissen aufheben, um zum Glauben Platz zu bekommen.
The word commonly translated as “to deny” is aufheben, whose noun form is much more well-known by non-German-speaking philosophers thanks to Hegel, namely Aufhebung, always marked as “a technical term” of philosophy and typically translated as “sublation”. Now we certainly do not mean to retroactively project a Hegelian concept into the Kantian corpus, but we do not need to – Aufhebung was an established technical term for Kant as well.[6] [ii]
It is not, therefore, unreasonable to think that Kant saw the making-room-for (we might think here of Heidegger’s Lichtung and “clearing”) faith by practical reason as the Aufhebung of theoretical reason – indeed, this exactly corresponds to the function of the practical postulates in the resolution of the antinomy of pure reason, where the dialectical opposition mutually negates or cancels itself out on the theoretical side, opening the door to a joint affirmation on the practical side.
The essential movement of the Kantian Aufhebung, then, is this: from the unconditional negative limits of theoretical reason to the conditional but (it is argued) necessary positive projection of the postulates of practical reason, the former being the condition (of possibility) of the latter. The critical discipline of the first renders the paradoxical questions of pure reason undecidable, while this theoretical undecidability – in conjunction with the necessity for practical determination of the will (i.e., thanks to the necessity of acting) – not only permits but forces a practical decision on the theoretically undecidable questions of pure reason. That is, on Kant’s view, practical necessity forces us to go beyond the proper bounds of pure reason – out into the “dark night”[7] where our theoretical reason is both absolutely freed and simultaneously utterly confounded, and where only a subjective ground of orientation remains (like an absolute compass, indicating the direction of some absolute reference point, but no more), the clear path before us having vanished, swallowed into the sublime darkness of the supersensible into which our understanding cannot penetrate. Of course, as Nietzsche infamously will not fail to note, this subjective ground itself, being the absolute point of reference for the fundamentally important Kantian operation of “orientation” in thinking, is open to question and critique – so that the battle over the postulates begins even deeper, in the foundations of the system and on the field of morality, even of reason, itself.
The two needs of pure reason are, then, intimately intertwined (indeed, I would say a double articulation of one need). The practical need could not arise were it not for the theoretical need, but only the former can be satisfied, this satisfaction being the controversy under examination here. There are thus two key loci for a critical analysis of the Kantian account in this regard:
(1) The justification of the theoretical need
(2) The justification of the practical need, and the justification of the practical solution (namely, the postulates)
We can also frame this in terms of the distinction in epistemic status between the propositions attached to the two needs:
(1) Analytic necessity (the step from reason to morality)
(2) Synthetic a priori (the step from morality to the highest good / God)[8]
These should be the two major hinges in the structure of any critique of Kant’s postulates of practical reason. That is, a foundational critical analysis of the roles of reason, freedom, and morality in the Kantian account, on the one hand, and on the other, a critical analysis of the consequent practical postulates, insofar as they derive from and are justified by the former and foundational component of the account.
1 The Theoretical Need and Antinomy of Pure Reason
The situation is then this: human reason strays into the dark night of pure reason, where no empirical objects are given for intuition and instead there is merely or simply space for intuition[9] – that is, no objective grounds for cognition are given, i.e. the rules of the understanding (always conforming to possible experience) yield no determinate maxim for judgment because the understanding always works in conjunction with the imagination, and since the imagination supplies no possible determinate object of experience (not only does it receive no sense-data or intuition for that, but it would have to furnish a determinate and empirically possible object from nothing, which Kant of course thinks is impossible[10]) to the understanding, the latter can yield no determinate concept-object for judgment or for the formation of a maxim. The understanding thus cannot produce a determinate maxim in this case because it is given no a priori object for that maxim –– save perhaps (1) the very lack of such an object, i.e. the “nothing” or “void”; or (2) a self-reflective maxim which would actually therefore not be a maxim at all but rather a law of pure reason, i.e. a subjective ground of orientation for navigating the void, where no objective determinations are possible.[11]
Kant of course chooses the second of these options, the self-reflexive law of pure reason. We will examine the first option, which he rejects, later, for its exclusion actually plays a quite important if indirect role in the consequent Kantian account (not to mention in post-Kantian thought – one instinctively associates this void as the only possible ground of orientation in pure reason with Nietzsche, and what Kant would surely think of as pure moral relativism and groundlessness of reason in the chaotic void of the will to power (and hence, one presumes, Nietzsche’s work as – and this is actually perhaps not as an unfair a hypothetical view, as far as it goes, of course – by turns enthusiastic and delirious)).
Now this law can be framed (1) formally, as the universalizability criterion for all propositions, and (2) as an operative rule or condition of possibility for all determinate maxims specifying “how to go on” for a given synthesis of the understanding. The first of these is quite familiar, of course – Kant interprets it as the moral law, expressed by the Categorical Imperative. Nietzsche, for instance, though he will even more deeply question the conceptualization of reason as universalizability (indeed, this concept of reason itself) in the first place, will also famously attack this Kantian interpretation of the more fundamental Kantian assertion – i.e., this interpretation that introduces morality into the heart of reason, or as analytically derivable from the very form of reason.
Now Kant argues that (1), the very form of reason, also entails (2), insofar as the possibility of a rule telling any given synthesis “how to keep going” is entailed by the universalizability criterion. That is, the universalizability criterion of the form of reason entails that a determinate maxim of the understanding specifying the operative form of a synthesis be grounded in a principle or rule of pure reason as its condition of possibility; but pure reason is of course not bound by the empirical constraints, e.g. temporal, that bind the understanding – it moves in a logical temporality[12] where there are not particular maxims for particular syntheses, but only principles for such maxims of synthesis in general or as such. Thus the principles of reason, or the form of universalizability, precisely do not concern limited or particular syntheses, but rather specify the unconditional form of any possible synthesis, a therefore conceptually complete (and completely non-empirical) idea of pure reason. This pure idea, though in a sense the extension-to-unconditionality of the limited concepts of the synthetic understanding, is actually totally independent and indeed is the condition of possibility of the latter.
The point being: the pure idea of synthesis, which is necessarily presupposed by the fact of particular syntheses of the understanding (an empirical fact), must itself be the idea of a total or absolute synthesis – in short, a complete(d) synthesis (or: “synthesis” as such). If we were talking blithely in Hegelese (which I am glad that, in general, we are not), we might say that the logical temporality here is that of the dialectic, so that any possible actual synthesis of the understanding carried to infinity or completion would be captured not at the end (or final moment/movement of the synthesis) but as a whole by the “absolute (or pure) idea” of synthesis as such, from its birth and beginning to its sublation/completion and final end, via its logical parts – i.e. what empirically would have been experienced by an immortal being in time as stages are viewed as logical states or phases in a static logical picture by this same immortal being outside of time (i.e., in logical temporality and outside of empirical or chronological temporality).
Therefore, the argument goes: since it is given that there are, empirically, syntheses of the understanding, a principle (of synthesis) of pure reason must be necessarily presupposed. And this seems right – it is like saying you cannot apply a template if you do not have the template. And there would probably not be a problem if these temporalities were kept separate, at least, there would be different problems. But the problem for Kant as it stands lies in the crossing of these two temporalities in practical reason.
Here the fundamental question is essentially this: given (1) that “knowing (understanding, via a rule) how to go on” presupposes as ground in pure reason an idea of synthesis as a ruled or ordered “going-on” as such, and (2) that such an idea of pure reason necessarily includes the end of synthesis or the exhaustion or completion of “going-on”, insofar as its universal form entails a universal rule that unconditionally drives this activity of “going on” in the first place, and (3) that despite having the rule to go on “to the end”, the understanding can never actually complete an empirical synthesis, then: (4) is the existence of an end to synthesis as such, a final or ultimate end as it were (and, crucially, an “end” in the sense of logical temporality, rather than empirical/chronological temporality), required, in order for the actual syntheses to have the force or “motive” to keep going on to its particular, empirically unattainable end? If we accept with Kant that it is, then the existence of God will be a necessary consequence to secure the real possibility of this end (which is of course the Kantian “highest good” of universally practiced morality).
That this end is possible is entailed by the very form of synthesis as such, which is precisely an operation of determinate “end-seeking” (or condition-seeking, or cause-seeking, etc.); and that limited particular empirical syntheses can be completed is also given (e.g., if I specify from the outset an initial and final bound for a causal or otherwise empirical chain, and the bounded part lies within my possible experience or better my actual experience, I can plausibly “go through” the entire synthesis with my limited mind and time); but that any unlimited empirical synthesis can be completed, e.g. that a given causal chain could be traced all the way down to the origin of all causal chains, while entailed by the form of reason, can never be verified by a finite human intellect. Thus the possibility of a “divine reason”, or more strictly, an unbounded reason, is entailed by the very form of bounded, human reason – but to posit that this reason exists is another claim altogether[13], indeed, it is “a claim” and thus not analytically entailed by pure reason;– in fact it is not even fully accurate to characterize it as a claim, in truth it is a decision (or perhaps, thinking again of Heidegger, a challenging-forth-to-decision). The question here, the practical question and the question of positing existence, concerns the necessity of hope for human reason in its actual and practical synthetic activity, including the formulation of maxims for action. It is a question of a decision – of the necessity of a decision – for action, a decision for action (thus a practical decision) in response to the fundamental boundedness of human reason, its own response to its own inability, felt as a need of pure reason, to decide on its own fundamental questions. As a question of the necessity of a decision, this is therefore also a justificatory question, asking after the legitimacy of this decision.
This is our question, also that of the antinomy of pure reason, hinging on the theoretical undecidability by human, bounded reason of certain irreducible and inherent questions[14] of pure reason – and the Kantian solution to this antinomy via a practical decision to act, i.e. to posit the postulates.
Now the antinomy of pure reason takes the form of four dyadic “antithetics”[15], forming two couples or classes; the first two antithetics constitute the “mathematical” antinomies[16], while the third and fourth constitute the “dynamical” antinomies. In brief, here is Kant’s bare statement of them:
First Antinomy (spatiotemporal finitude of the world vs spatiotemporal infinity of the world]
Thesis: The world has a beginning in time, and in space it is also enclosed in boundaries.[iii]
Antithesis: The world has no beginning and no bounds in space, but is infinite with regard to both time and space.[iv]
Second Antinomy (finite divisibility of composite substances in the world vs infinite divisibility of substances in the world)
Thesis: Every composite substance in the world consists of simple parts, and nothing exists anywhere except the simple or what is composed of simples.[v]
Antithesis: No composite thing in the world consists of simple parts, and nowhere in it does there exist anything simple.[vi]
Third Antinomy (necessity of transcendental freedom in addition to natural causality vs natural causality being the only causality)
Thesis: Causality in accordance with laws of nature is not the only one from which all appaerances of the world can be derived. It is also necessary to assume another causality through freedom in order to explain them. (484)
Antithesis: There is no freedom, but everything in the world happens solely in accordance with laws of nature.[vii]
Fourth Antinomy (the necessary existence of a necessary being vs the necessary non-existence of a necessary being)
Thesis: To the world there belongs something that, either as a part of it or as its cause, is an absolutely necessary being.[viii]
Antithesis: There is no absolutely necessary being existing anywhere, either in the world or outside the world as its cause.[ix]
We will examine the mathematical antinomies and Kant’s conclusion regarding them first; and we will defer a consideration of the “drive for cognition” that generates the theoretical as well as the practical need until the practical section, insofar as we can accept it prima facie for the purposes of our analysis of the mathematical antinomies, even though it will become a critically problematic notion once we progress to the practical.
2 On the Antinomy of Pure Reason: The Mathematical
We will present Kant’s formal solution, the “transcendental idealist” solution, to the problem in due course. For now let us simply say that Kant’s ultimate conclusion regarding the mathematical antinomies is a negative one – a joint negation not of the theses but of the antithetics themselves, insofar as they rest upon an unacknowledged critical transgression of the proper bounds of human reason. It is a “negative” solution in the sense that the limits of human reason are “negative”; what Kant asserts is precisely not a negative position on any of the particular theses or antitheses, but the negative limit of human reason insofar as this latter renders the former (i.e. theses/antitheses) undecidable. Not a negation, but something more primary – a negative condition of possibility of all possible particular theoretical negations, one which renders both negation and affirmation strictly impossible in this case by halting the process at the beginning, as it were, i.e. by precluding not an affirmative, not a negative answer, but by precluding the very propriety of a decision to answer as such (this is of course the Kantian Aufhebung from the introduction to this paper).
It should be noted that if we accept Kant’s solution to the first antinomy, the only really critical obstacle remaining for the Kantian account is on the practical side, i.e. accounting for the divergence in (re)solution in the case of the dynamical antinomies as opposed to the mathematical, including the justification via practical necessity for the positing of the postulates of practical reason.
Now there are two core problem here; the first concerns series that are infinite (or, at this point in the analysis, I suppose I should strictly speaking say “series whose finitude or infinity is undecidable for human reason”). There are, naturally, infinitely many examples: any given causal chain in the empirical order of nature; any given moment in a temporal sequence of the world[17]; the progressive division of composite material substances into more elementary substances; any given segment or section of space extended sequentially; and perhaps most exemplarily of all, perhaps so exemplary or even fundamental for all the other examples that Kant neglects to explicitly mention it in the section of the Critique on the antinomies for this very reason – the series of natural numbers, or the natural series of number.
The second core problem concerns the notions of an original, primordial, or transcendent ground on the one hand, and a transcendently-necessary (not “transcendentally-necessary) immanent element (an original or irreducible element) on the other. These are two potential solutions to the problem of serial unconditionality (or infinity), in which every element of the series is conditioned by a predecessor and conditions a successor in turn, but which therefore must as a whole have either (1) a primordial or “first” element that has no prior condition but does condition a successor, or (2) the series must be conditioned as a totality, by a condition outside of the series (i.e., by a condition that is not itself one of the conditional elements of the series) that conditions the series as a whole rather than by individual part.[18]
We might try to sketch an illustration of these concepts by means of the aforementioned example of the natural series of number. In as oversimplified of terms as possible, we could identify the function of the irreducible element in zero, which is the only natural number with no predecessor, and which establishes the synthetic rule for constructing more, indeed the rest, of the natural numbers (termed, in mathematics, the successor function – in essence, given any n, S(n) = n + 1 and S(n) is a number) (we are – and uncontroversially, or without violating the proper bounds of explicative rigor, I might add – deferring considerations of, e.g., negative numbers for now for purposes of simplicity). The successor function says that given a number (and zero is a number – though a number unlike any other natural number, since its immanent status as number is established by a transcendent axiom rather than the successor function, which is of course crucial), and given an operation equivalent to the successor function, we can construct all natural numbers, indeed, “to infinity”. The way I have framed this example, both of Kant’s alternatives are present – there is a transcendently-immanent element of the series that also, in its transcendence, serves as the transcendent ground for the series (namely, zero), conditioning the series as a whole by providing the rule that constitutes its form or structure, namely some transcendental rule (transcendental for the series) like the successor function, which requires only that a first element of the series be given in order to effect an infinite synthesis.
There is another way we can mathematically think about what is at play here (don’t worry, I will eventually present Kant’s own actual moves, my hope is simply that the work we do here will minimalize the explication and interpretation necessary there). This second number-theoretical thought experiment will bear more on second antinomy. Every natural number can be thought of as a particular aggregate of units, or of 1s (this stems from the transcendental rule of operation, which produces the first non-zero number, namely 1, by adding 1 to its predecessor, namely 0, and all subsequent natural numbers in the same fashion, recursively). At this point we slightly broaden the complexity of the example by no longer dealing with just whole numbers: any number can be decomposed into some aggregate of digits, which themselves signify the numerical place of aggregates of units in the number. That is, even numbers greater than a natural number but less than its successor ultimately reduce to units.
But here is the problem. We attain numbers between the naturals by division – we assert the naturals as initial and final bounds and call the number constituted by the bounded part a fractional, or rational number. It is presumably rational because it is a rational way of extending the natural number line or series to include more (or more precise) numbers. What is going on here, a method sourcing back to Richard Dedekind, is that a gap is posited between two consecutive natural numbers, in other words the continuous fabric of the natural number line is here cut (the cut is the positing of the gap), and the number that fills the gap is nothing other than the cut or gap itself. The cut creates the gap which becomes the new kind of number – and it is quite poetic and fitting, not to mention psychoanalytically and literarily intriguing, that this is called the rational method of number extension or even number construction (additionally, the contemporary method of John H. Conway for the construction of number, which is conservative over the advances of previous methods and which also generates a much larger and more comprehensive multiplicity of numbers (thought confidently to be all possible numbers), called in general Conway numbers or surreal numbers (encompassing all the familiar classes of number, from natural and integer and rational to real, imaginary, complex, and more), employs a single, fundamental operational device very close to Dedekindian cutting, a closeness Conway is well aware of). So why does this rational method propose a problem for pure reason?
Because its inherent formal condition of possibility, as a rule for how to go on, logically entails that it (can) go to infinity. How many rational numbers are there between any two given consecutive naturals? Infinitely many, or as many as the real number continuum, the inclusive bound of the “first” mathematical infinity (strictly speaking, it contains more numbers than the infinity of the natural number line, since it includes the natural numbers and indeed infinitely many numbers between them, but in terms of size it is a famously open question as to whether the cardinality of the real number continuum is greater than that of the natural series). Of course, a real human being, actually performing this synthetic operation of division, say with paper and pencil – beginning with two natural numbers as bounds, then the first rational number produced by this cut or bounding as initial bound and the second natural number as final bound, then the rational number produced by this cut and the second natural number, and so on to infinity – will never be able to empirically verify this infinity, or in other words, whether the division does not halt at some irreducible element it encounters somewhere deep in the recursive operation.
That is the general structure of the problem, framed number-theoretically: given a synthetic series, does the synthesis have an original step / does the series have a first member; and given a starting-point of a synthetic series, will the given synthesis ever meet an empirical stopping-point, or continue forever (i.e., without encountering a final member of the series)? Kant frames the problem, and his solution, in terms of series that go on “to infinity” and that go on “indefinitely”:
Mathematicians speak solely of a progressus in infinitum. But those who study concepts (philosophers) want, in place of this, to make the expression progressus in indefinitum the only valid one. […] One can rightly say of a straight line that it could be extended to infinity, and here the distinction between the infinite [vs] a progress of indeterminate length (progressus in indefinitum) would be an empty subtlety [… for here] the [second] means no more than ‘Extend it as far as you want,’ but the [first] means ‘You ought never to stop extending it’ […] yet if we are talking only about what can be done, then the [second] expression is entirely correct, for you could always make it greater, to infinity. And this is also the situation in all cases where one is speaking only of a forward progress, i.e., of a progress from the condition to the conditioned; this possible progress in the series of appearances goes to infinity. […] For here reason never needs an absolute totality in the series, because it is not presupposed as a condition as given (datum), but it is only added on as something conditioned, which is capable of being given (dabile), and this without end. It is entirely otherwise with the problem how far does the regress extend when it ascends from the given conditioned to its conditions in the series: whether I can say here that there is a regress to infinity or only a regress extending indeterminately far (in indefinitum) […] To this I say: If the whole was given in empirical intuition, then the regress in the series of its inner conditions goes to infinity. But if only one member of the series is given, from which the regress to an absolute totality is first of all to proceed, then only an indeterminate kind of regress (in indefinitum) takes place. […] In neither of these two cases, that of the regressus in infinitum as well as in that of the in indefinitum, is the series of conditions regarded as being given as infinite in the object. It is not things in themselves that are given, but only appearances, which, as conditions of one another, are given only in the regress itself Thus the question is no longer how big this series of conditions is in itself – whether it is finite or infinite – for it is nothing in itself; rather, the question is how we are to institute the empirical regress and how far we are to continue it.[x]
Kant’s answer has two basic parts, which elaborate the essential argument at work in the above passage, which is the famous transcendental idealist (re)solution. The first part is to dissolve the mathematical antinomies as merely dialectical oppositions, rather than genuinely contradictory or analytical opposites[19], insofar as they depend upon “an illusion arising from the fact that one has applied the idea of absolute totality, which is valid only as a condition of things in themselves, to appearances that exist only in representation, and that, if they constitute a series, exist in the successive regress but otherwise do not exist at all.”[xi] The second part is a consequent specification of the status of the principle or rule of pure reason which underlies both the antinomy and the confusions over it. That is, the critical or transcendental idealist solution necessitates a reevaluation of the proper role of the principle of pure reason that drives the synthetic quest for the unconditioned. No longer can it be assumed as an axiomatic principle for thinking the totality of an object as given in itself, since this object is only given as a problem i.e. in the synthesis and as a mere representation, hence Kant will term the principle a regulative rule of reason – in essence, what we have been calling a rule that (merely) specifies “how to go on”.
Kant’s first move is to argue that whichever side of a mathematical antinomy is taken, to affirm that thesis or antithesis already transgresses the proper bounds of reason, because no concept of the understanding can be adequate to the pure idea of unbounded reason that is being asserted (e.g., the spatiotemporal infinity/finitude of the world):
Accordingly, if I could antecedently see about a cosmological idea that whatever side of the unconditioned in the regressive synthesis of appearance it might come down on, it would be either too big or too small for every concept of the understanding, then I would comprehend that since it has to do with an object of experience, which should conform to a possible concept of the understanding, this idea must be entirely empty and without significance because the object does not fit it no matter how I may accommodate the one to the other. And this is actually the case with all the world-concepts, which is why reason, as long as it holds to them, is involved in an unavoidable antinomy. For assume:
First, that the world has no beginning; then it is too big for your concept; for this concept, which consists in a successive regress, can never reach the whole eternity that has elapsed. Suppose it has a beginning, then once again it is too small for your concept of understanding in the necessary empirical regress. For since the beginning always presupposes a preceding time, it is still not unconditioned, and the law of the empirical use of the understanding obliges you to ask for a still higher temporal condition, and the world is obviously too small for this law.
It is exactly the same with the two answers to the question about the magnitude of the world in space.[xii]
Thus we have been brought at least to the well-grounded suspicion that the cosmological ideas, and all the sophistical assertions about them that have come into conflict with one another, are perhaps grounded on an empty and merely imagined concept of the way the object of these ideas is given to us; and this suspicion may already have put us on the right track for exposing the semblance that has so long misled us.[xiii]
That is, the antithetic rests upon a strictly meaningless or simply confused idea that can have no possible adequate conception in the understanding, since it takes something that is only empirically given as an object or totality in itself (as though it were not merely given empirically, i.e. as a sensible representation, but as a thing in itself and independent of its representation in/by the mind). The very infinity of space-time cannot be considered as a property of space-time in itself (if it is indeed infinite, which is of course the whole point), since the extension of space-time is only given by and in empirical synthesis.[20] As Kant says, “The entire antinomy of pure reason rests on this dialectical argument: If the conditioned is given, then the whole series of all conditions for it is also given; now objects of the senses are given as conditioned; consequently, etc.”[xiv] – but in fact “The series of appearances is to be encountered only in the regressive synthesis itself, but is not encountered in itself in appearance, as a thing on its own given prior to every regress.”[xv] Hence Kant writes:
Accordingly, the antinomy of pure reason in its cosmological ideas is removed by showing that it is merely dialectical and a conflict due to an illusion arising from the fact that one has applied the idea of absolute totality, which is valid only as a condition of things in themselves, to appearances that exist only in representation, and that, if they constitute a series, exist in the successive regress but otherwise do not exist at all.[xvi]
Now if we accept this resolution, there remains for Kant to explain a new interpretation of the root of the problem in the first place, namely the universal form of reason that unconditionally drives all synthetic activity toward completion. The reinterpretation is essentially: this rule does not say that there is a completion of synthesis, though it appears to say what that completion would be, rather it merely says how to synthesize – and since the form of synthesis is a movement to completion, this rule can mistakenly be interpreted so that the completion inherently contained in the rule is descriptive rather than prescriptive. That is, here Kant rules off-limits or out-of-bounds precisely what he will do on the basis of practical reason – “namely, the ascription of objective reality to an idea that merely serves as a rule”:
Since through the cosmological principle of totality no maximum in the series of conditions in a world of sense, as a thing in itself, is given, but rather this maximum can merely be given as a problem in the regress of this series, the principle of pure reason we are thinking of retains its genuine validity only in a corrected significance: not indeed as an axiom for thinking the totality in the object as real, but as a problem for the understanding, thus for the subject in initiating and continuing, in accordance with the completeness of the idea, the regress in the series of conditions for a given conditioned. […] Thus the principle of reason is only a rule, prescribing a regress in the series of conditions for given appearances, in which regress it is never allowed to stop with an absolutely unconditioned. Thus it is not a principle of the possibility of experience and of the empirical cognition of objects of sense, hence not a principle of the understanding, for every experience is enclosed within its boundaries (conforming to the intuition in which it is given); nor is it a constitutive principle of reason for extending the concept of the world of sense beyond all possible experience; rather it is a principle of the greatest possible continuation and extension of experience, in accordance with which no empirical boundary would hold as an absolute boundary; thus it is a principle of reason which, as a rule, postulates what should be effected by us in the regress, but does not anticipate what is given in itself in the object prior to any regress. Hence I call it a regulative principle of reason, whereas the principle of the absolute totality of the series of conditions, as given in itself in the object (in the appearances), would be a constitutive cosmological principle, the nullity of which I have tried to show through just this distinction, thereby preventing – what would otherwise unavoidably happen (through a transcendental subreption) – the ascription of objective reality to an idea that merely serves as a rule. Now in order to determine the sense of this rule of pure reason appropriately, it must first be noted that it cannot say what the object is, but only how the empirical regress is to be instituted so as to attain to the complete concept of the object.[xvii]
It remains to compare Kant’s resolution of the problem to the actual mathematical answer – for while I may have presented the example of the natural series as one example among others, it actually has much more far-ranging implications, insofar as the problems there are in fact not only foundational problems for pure reason for Kant, but for number theory in mathematics as well. And insofar as all of higher mathematics can be analytically derived from basic number theory (something not proven until well after Kant of course, in fact it constituted the great project of the founders of both modern mathematics and modern analytic philosophy, e.g. Frege and Russell), this problem of pure reason therefore also concerns the possibility of all of pure mathematics, which Kant avers to be of synthetic a priori status. Indeed, the example we have been sketching is perhaps the exemplar of the synthetic a priori. And we recall that Kant will argue such a status for the postulates of practical reason, as well, so that this issue is relevant on many fronts.
But as it happens, this task more suitably lands towards the end of the paper, after our explication of Kant’s account of the practical need and postulates, since it will depend upon the general anti-Kantian alternative we will have established by then, what might be called a kind of cognitive constructivism. So bravely forward, to the realm where courage is most needed, since the familiar is most receded.
3 The Practical Need and Postulates of Pure Reason
We have laid out much of the practical side of the issue already. The broad argument is roughly: reason analytically yields the moral law as a universal rule for moral synthetic activity, the universal form of the moral law necessarily contains a formally ultimate end, i.e. the highest good or universally practiced morality, and the possibility of this ultimate end (which, recall, unites in itself the conditions of all particular moral ends) is then secured by the posited (i.e. synthetically a priori necessary) existence of God, i.e. a necessary, omnipotent, original, etc. being. The possibility of this end is entailed by the very form of the moral law, but it is only secured via God (since what is logically entailed, as we well know by now, may well be empirically impossible, and thus strictly theoretically undecidable). And the question is why is this necessary, since even all that is logically entailed is the possibility of the ultimate end as an operational goal, not its existence, which is critically and irrevocably outside the range of our empirical verifiability (and existence is an empirical predicate, of course).As Kant himself says, what is at issue is positing the empirical object of “a being whose concept (if it is not to be vaguely determined and hence might be subject to association with every possible delusion) demands that it be of infinite magnitude as distinguished from everything created; but no experience or intuition at all can be adequate to that concept, hence none can unambiguously prove the existence of such a being.”[xviii]
Kant essentially argues (unsurprisingly at this point) that the necessity of positing the postulates derives from a practical need of reason. Recall that the theoretical need was the empirically impossible need to complete syntheses, to reach the unconditional in a series (whether a transcendent ground or a transcendently-immanent element). There Kant judged the need to exceed the capacity or proper bounds of human reason, and thus interpreted it immanently or operationally, prescriptively rather than descriptively, so that it only “instruct[s] as to how to operate but not as to whither” – whereas now Kant says the following:
But although on its own behalf morality does not need the representation of an end which would have to precede the determination of the will, it may well be that it has a necessary reference to such an end, not as the ground of its maxims but as a necessary consequence accepted in conformity to them. – For in the absence of all reference to an end no determination of the will can take place in human beings at all, since no such determination can occur without an effect, and its representation, though not as the determining ground of the power of choice nor as an end that comes first in intention, must nonetheless be admissible as the consequence of that power’s determination to an end through the law (finis in consequentiam veniens); without this end, a power of choice which does not [thus] add to a contemplated action the thought of either an objectively or subjectively determined object (which it has or should have), instructed indeed as to how to operate but not as to whither, can itself obtain no satisfaction.[xix]
The argument here resembles the description of how image-representations formed by the imagination subtend pure concepts of the understanding so as to bring them “down to earth” and make them, well, understandable (and/i.e. applicable):
However exalted the application of our concepts, and however far up from sensibility we may abstract them, still they will always be appended to image representations, whose proper function is to make these concepts which are not otherwise derived from experience, serviceable for experiential use. For how would we procure sense and significance for our concepts if we did not underpin them with some intuition (which ultimately must always be an example from some possible experience)?[xx]
Thus the idea is that while only the formal possibility of an ultimate end to synthesis is logically entailed by the form of pure reason, so that no determinate object, i.e. representable by the imagination in conjunction with an adequate concept of the understanding, is given or required at all by pure reason in and of itself, practical reason – reason insofar as it is to effectively determine action – requires some sensible intuition of how to determinately proceed, i.e. apply the rule; the fear here is one inherited much later by John Rawls, namely of the formal procedure of reason being unable to yield a determinate rule for action when such action is not optional or a matter of curiosity, but rather a matter of urgent practical necessity. In essence: how could we procure “sense and significance” for the abstract moral law as a rule for conduct if we did not have some minimal but ideally exemplary illustration of its effective application, i.e. what its result would look like if it worked properly (since we are essentially Humeans for Kant in regards to empirical reality, i.e. we have no insight into things in themselves [hence into their transcendental causality, if we with Kant postulate transcendental freedom], but only into how they appear to us as cognitive representations, i.e. into their external effects, e.g. causal effects on other empirical objects in that vicinity of the causal nexus or order of nature)?
But there is a second level to the argument. And that is: even if we formed such a determinate final end for ourselves so as to bring the abstract concept (really principle) down to earth and “sense”, and so as to provide an imaginable goal to direct our actual synthetic moral activity, we could simply acknowledge its merely operational status as an immanent rule for use (or synthesis) – we would not thereby theoretically need to posit its reality or secure this reality through a transcendent ground, namely the postulate of God.
Now Kant argues as to the first that freedom is the “keystone” to the architectonic edifice under examination here, because (a) it is presupposed by the moral law, which is itself analytically derived from the form of reason, and which gives a rule for activity[/action], thus presupposing the practical ability to determine the will in the effecting of this activity; (b) is also given analytically by reason itself, insofar as the experience of having reason or being a creature with reason factically includes the experience of willing, i.e. of beginning causal series of effects solely from one’s own will; so that (c) all other such concepts such as God and immortality of the soul become appended to it, much as the pure concepts of the understanding are appended to image-representations of the imagination to render them graspable – i.e., so that these concepts “get stability and objective reality” via it, “that is, their possibility is proved by this: that freedom is real”[xxi]:
Now, the concept of freedom, insofar as its reality is proved by an apodictic law of practical reason, constitutes the keystone of the whole structure of a system of pure reason, even of speculative reason; and all other concepts (like those of God and immortality), which as mere ideas remain without support in the latter, now attach themselves to this concept and with it and by means of it get stability and objective reality, that is, their possibility is proved by this: that freedom is real, for this idea reveals itself through the moral law.[xxii]
Perhaps oddly or perhaps strategically, Kant focuses on the possibility of the pure idea of freedom as given a priori by the moral law, since it is a condition of the moral law (that is, the moral law for conduct presupposes that the agent subject to it has the freedom to follow it) – rather than the factical experience of it as noted in the first Critique in the sections on the antinomy (i.e., the factical power to begin causal or phenomenal series from no prior ground other than one’s will). In fact, he says that “For, had not the moral law already been distinctly thought in our reason, we should never consider ourselves justified in assuming such a thing as freedom (even though it is not self-contradictory). But were there no freedom, the moral law would not be encountered at all in ourselves.”[xxiii] We can interpret this choice in the following manner.
The facticity of our power to start causal series is not enough, on its own, to assume that we have extra-natural or transcendental freedom – e.g., a hardcore materialist-expressivist could argue that all consciousness is epiphenomenal, and in fact our “will” that we think produces actions ex nihilo is a mere metaphysical specter, and all there is in reality is natural and deterministic causality in its infinite (but immanent and non-mysterious) complexity (e.g. from the level of atoms to the level of thoughts and desires). Here the problem is, in essence, that we are in the causal series, so that whether our apparently free actions are in fact transcendentally free or merely naturally determined is strictly undecidable for our limited reason – if we could step outside the intrinsic structural bounds of the mind, if we could look upon the entire serial causal nexus as a whole and simultaneously with reference to its parts, then we could judge definitively. And in fact this is precisely the point: only a transcendent ground can function as the unconditioned condition of completeness for an immanent series for Kant (since he rules out the empirical verifiability of an irreducible element, and which being an immanent element in the series would be precisely empirical, whereas a transcendent ground would be outside the legitimate range of such empirical verifiability in the first place or a priori)[21]. So that given that we can never empirically verify an irreducible/transcendent element of the series, and given that the series exists (as a series, i.e. a universally conditioned or ordered structure), we must (out of practical necessity) posit the theoretical existence of a transcendent ground.[22]
Kant distinguishes this necessity from the arbitrary or unregulated positing of transcendently-immanent elements, such as spiritual beings with a noumenal causality floating in a plane parallel to the order of natural empirical causality, invisibly influencing it with their wills much as actual human beings are claimed to be able to influence it:
Many supersensible things may be thought (for objects of sense do not fill up the whole field of possibility) to which, however, reason feels no need to extend itself, much less to assume their existence. […] the assumption of [such] spiritual beings would rather be disadvantageous to the use of reason. […] Thus that is not a need at all, but merely impertinent inquisitiveness straying into empty dreaming to investigate them – or play with such figments of the brain. It is quite otherwise with the concept of a first original being as a supreme intelligence and at the same time as the highest good. For not only does our reason already feel a need to take the concept of the unlimited as the ground of the concepts of all limited beings – hence of all other things – , but this need even goes as far as the presupposition of its existence, without which one can provide no satisfactory ground at all for the contingency of the existence of things in the world, let alone for the purposiveness and order which is encountered everywhere[…][xxiv]
And the critical key is what follows upon this:
Without assuming an intelligent author we cannot give any intelligible ground of it without falling into plain absurdities; and although we cannot prove the impossibility of such a purposiveness apart from an intelligent cause (for then we would have sufficient objective grounds for asserting it and would not need to appeal to subjective ones), given our lack of insight there yet remains a sufficient ground for assuming such a cause in reason’s need to presuppose something intelligible in order to explain this given appearance, since nothing else with which reason can combine any concept provides a remedy for this need.[xxv]
What is the justification? (1) We cannot ground the purposiveness or directed-conditionality of synthetic series in an intelligible ground unless it is transcendent (i.e. an intelligible “meta-subject” acting upon our unlimited causal nexus from the outside the way we can act upon limited causal series from their “outside”), (2) since otherwise, i.e. if we were to attempt to ground it in a transcendently-immanent element, we would fall into “plain absurdities” – like, we may hypothesize, grounding it in the void; and since (3) though we cannot prove such an alternative ground impossible (for then the issue would be decided and there would be no question of a need), we can also a priori never verify it (we have already given the argument for this multiple times now), then (4) the inherent formal need of reason to presuppose some unconditional ground provides a “sufficient ground”, i.e. a subjective ground of orientation, for positing it as a “postulate” of practical reason, which Kant opposes to a “rational hypothesis” of merely theoretical reason:
A need of reason to be used in a way which satisfies it theoretically would be nothing other than a pure rational hypothesis, i.e. an opinion sufficient to hold something true on subjective grounds simply because one can never expect to find grounds other than these on which to explain certain given effects, and because reason needs a ground of explanation. By contrast, rational faith, which rests on a need of reason’s use with a practical intent, could be called a postulate of reason – not as if it were an insight which did justice to all the logical demands for certainty, but because this holding true (if only the person is morally good) is not inferior in degree to knowing, even though it is completely different from it in kind.
A pure rational faith is therefore the signpost or compass by means of which the speculative thinker orients himself in his rational excursions into the field of supersensible objects[.][xxvi]
Now to examine an obvious question we keep deferring, namely: Kant says that reason feels its own need, feels its inherent limitations – but how can reason feel? Kant’s answer in Religion:
Reason does not feel; it has insight into its lack and through the drive for cognition it effects the feeling of a need. It is the same way with moral feeling, which does not cause any moral law, for this arises wholly from reason; rather, it is caused or effected by moral laws, hence by reason, because the active yet free will needs determinate grounds.[xxvii]
And in the first Critique:
Reason is driven by a propensity of its nature to go beyond its use in experience, to venture to the outermost bounds of all cognition by means of mere ideas in a pure use, and to find peace only in the completion of its circle in a self-subsisting systematic whole. Now is this striving grounded merely in its speculative interest, or rather uniquely and solely in its practical interest?[xxviii]
The drive for cognition. This is the really fundamental thing, the source of the needs of pure reason – the truly universal operational form of reason as such, and an active or dynamical form – i.e. an unconditional drive to “keep going on” with the synthetic activity of thinking, of cognition (so not merely conscious contemplation and thought, but the subsurface gears and functions grinding away beneath the epiphenomenal veil of conscious awareness as well). Now I believe that in the third Critique Kant essentially identifies this drive, as the motor force of cognitive activity, as spirit – but that is a consideration for a different paper, since what suffices for our argument here is simply this: that the drive for cognition goes critically unquestioned by Kant. This is how, on Nietzsche’s view, morality sneaks its way into the foundation of Kantian reason: a drive (to pursue the synthetic activity of cognition) is interpreted morally by Kant, who avers that the operational rule for how to go on to infinity – because analytically derived from the form of reason itself – not only gives the gift of a “how” but also imposes the demand or imperative of an “ought”. This drive of pure reason which is even more generally the drive of reason (or cognition) as such and in all its systematically-interrelated functional parts, is analytically moral for Kant, the quintessential rationalist.
And indeed, insofar as morality concerns the practical problems of interacting with other agents of human reason, it is an a priori problem for human activity; but morality, as the solution to the moral problem (even social problem), is only analytically the solution of reason if the form of reason is itself moral, i.e. if the drive for cognition not only gifts a practical rule for moral synthesis but demands this moral synthesis as an unconditional duty.
This brings us to the concluding sections of this paper, where an alternative account is provided that is meant to refute certain parts of the Kantian account, but more importantly, to account for the gaps in the Kantian account (much as a rational number accounts for the gap between two naturals – though only via the critical cut that opens this gap, as a wound but also as a passageway, in the first place).
4 Toward a Critique of the Critique
We can begin by more closely examining a notion we have taken for granted thus far though frequently employing it throughout this paper, namely synthesis. Kant first uses this technical term in a pre-critical essay with regards to the difference between mathematics and philosophy (difference in method and difference in the kind of certainty attainable by each). Here he characterizes mathematical activity as a constructive synthesis, as Wood and Guyer note in the introduction to the Critique of Pure Reason:
This essay takes major steps toward the position of the Critique of Pure Reason, although crucial differences still remain. Kant’s most radical departure from prevailing orthodoxy and his biggest step toward the Critique comes in his account of mathematical certainty. Instead of holding that mathematics proceeds by the two-front process of analyzing concepts on the one hand and confirming the results of those analyses by comparison with our experience on the other hand, Kant argues that in mathematics definitions of concepts, no matter how similar they may seem to those current in ordinary use, are artificially constructed by a process which he for the first time calls “synthesis”, and that mathematical thinking gives itself objects “in concreto” for these definitions, or constructs objects for its own concepts from their definitions. […] Thus, we can have certain knowledge of the definition because we ourselves construct it; and we can have certain knowledge that the definition correctly applies to its objects because the true objects of mathematics are nothing but objects constructed, however that may be, in accordance with the definitions that we ourselves have constructed.[xxix]
At this point Kant still defined philosophy as solely analytical, in contradistinction to mathematics, but this is of course a view he will abandon by the time of his critical works:
Before Kant’s mature work could be written, he would have to discover a philosophical method that could yield “material” or synthetic judgments. This would be the philosophical work of the 1770s that would finally pave the way for the Critique of Pure Reason.
Once Kant takes this further step, however, the contrast between mathematics and philosophy provided in the Inquiry will have to be revised. The difference between mathematics and philosophy will no longer simply be that the former uses the synthetic method and the latter the analytical method. On Kant’s mature account, both mathematics and philosophy must use a synthetic method. This [means] that the difference between the concrete constructions of mathematics and the abstract results of philosophy will have to be recast as a difference within the synthetic method: The use of the synthetic method in mathematics will yield synthetic yet certain results about determinate objects, whereas the use of the synthetic method in philosophy will yield synthetic yet certain principles for the experience of objects, or what Kant will call “schemata” of the pure concepts of the understanding, “the true and sole conditions for providing [these concepts] with a relation to objects”.[xxx]
This parallels the distinction we have already met with, namely Kant’s re-evaluation of the principle of reason as a regulative rule following the dissolution of the mathematical antinomies. Finally, Kant’s own definition of synthesis in the first Critique:
Now space and time contain a manifold of pure a priori intuition, but belong nevertheless among the conditions of the receptivity of our mind, under which alone it can receive representations of objects, and thus they must always also effect the concept of these objects. Only the spontaneity of our thought requires that this manifold first be gone through, taken up, and combined in a certain way in order for a cognition to be made out of it. I call this action synthesis.
By synthesis in the most general sense, however, I understand the action of putting different representations together with each other and comprehending their manifoldness in one cognition. […]
Synthesis in general is, as we shall subsequently see, the mere effect of the imagination, of a blind though indispensable function of the soul, without which we would have no cognition at all, but of which we are seldom even conscious. Yet to bring this synthesis to concepts is a function that pertains to the understanding, and by means of which it first provides cognition in the proper sense.
Now pure synthesis, generally represented, yields the pure concept of the understanding. By this synthesis, however, I understand that which rests on a ground of synthetic unity a priori; thus our counting (as is especially noticeable in the case of larger numbers) is a synthesis in accordance with concepts, since it takes place in accordance with a common ground of unity (e.g., the decad). Under this concept, therefore, the synthesis of the manifold becomes necessary.[xxxi]
Synthesis, in short, can be thought of as an operation of totalization. It is what “counts-as-one” a manifold or multiplicity of intuition, yielding a determinate representation. Kant says that “only the spontaneity of our thought” requires such synthesis – so can we then not identify this “cognitive spontaneity” with the drive for cognition – so that the drive is in itself spontaneous, unconditioned (and/or grounded in the void)?
Now what of the synthetic a priori, such as the truths of mathematics?
Kant derives the a priori status of synthetic propositions from the ground of synthetic unity that is the pure concept of the understanding. In the empirical case, this essentially means the spatiotemporal causal nexus or framework conditioning all possible objects of experience. And in the case of the synthetic activity of counting, for instance, it is grounded in a “common ground of unity”, for example the decad (i.e., a standard unit). Recall the stage in our number-theoretical thought experiment in which we asserted that since all numbers reduce to collections of units, there must either be a transcendently-immanent unit that generates the series of units, or a transcendent ground of unity that conditions them as a whole. Kant’s choice is the second – so that his appeal to the “decad” here is not unlike his appeal to God (where the decad serves as a transcendent number grounding the counting of all immanent numbers, securing them a determinate a priori place, God serves as the transcendent ground of existence grounding all immanent/contingent existence in the world). As Guyer and Wood further explain,
[W]hat Kant is saying is that judgments that are synthetic but also genuinely universal, that is, a priori, can be grounded in one of two ways: in the case of mathematics, such judgments are grounded in the construction of a mathematical object; in the other case, such judgments are grounded in the condition of determining the relative position of one object in space and time to others. […] Kant’s argument is that although all particular representations are given to the mind in temporal form, and all representations of outer objects are given to the mind as spatial representations, these representations cannot be linked to each other in the kind of unified order the mind demands, in which each object in space and time has a determinate relation to any other, except by means of certain principles that are inherent in the mind and that the mind brings to bear on the appearances it experiences. These principles will be, or be derived from, the pure concepts of the understanding that have a subjective origin yet necessarily apply to all the objects of our experience, and those concepts will not have any determinate use except in the exposition of appearances.[xxxii]
That is, in mathematics, synthetic a priori propositions are a priori because they are grounded in axiomatic definitions (i.e. definitions instituted by we ourselves, by fiat as it were), but are synthetic because they produce determinate results (e.g., “7+5=12” is a priori true because of, e.g., axioms about numbers and the addition operation, while it is synthetic because “12” cannot be analytically derived from either “7” or “5”). Whereas in the second kind of grounding, it is not the construction of an object in the order of the series, but the conditions of possibility for any such constructible object, e.g. an object synthesized in perception that entail the a priori status. The critical Kantian distinction here is that in the first case, reason immanently gives itself its own object, while in the second, reason requires that an object be given by a transcendent condition or ground.
Our core critical thesis contra Kant, then, is that in fact the boundary drawn here does not exist.
5 The Constructivist Account[23]
Epistemologically, all judgments are synthetic. Analyticity is a pole of synthesis, an extreme form or an outer bound perhaps, yet still within the synthetic continuum (Kant would want us to say here, then, that analyticity is the outermost form preceding only the outer bound of the void). (Essentially, to satisfy this point, we can adopt Quine’s holistic epistemological model of the “web of knowledge/belief”, which though not essential for our account, is not incompatible with anything we will aver; furthermore, in this regard, we can think of analytic propositions as being like Wittgensteinian tautology and contradiction, i.e. a kind of more purely formal or even skeletal instance of proposition and precisely what remains structurally invariable as a kind of ur-frame in the variation of all other aspects, producing all possible synthetic statements that are therefore bounded by these minimalistic forms at either end of the continuum of variation, and which nevertheless remain immanently within it.)
All cognition is synthetic in the way that Kant characterizes the synthetic a priori construction of the objects of mathematics – or (and not quite the same thing), all cognition begins with nothing but the pure forms of intuition (space and time) and a drive for cognition, or a drive to synthesize. Logically begins – as Kant famously notes, there must be some actual empirical intuition (some received material) to get the whole process started chronologically (or in time, existing in experience). But after that, once the process is running, it is independent or autonomous from intuition (but not the pure forms of intuition, of course, which are also its conditions) – so that, e.g., once I have seen red but once, I can forever picture it in my imagination. Which is also to say: synthesis does not require a determinate intuition, but only the fact or the given that there was at least once at least one intuition, i.e., it is a fact that at some point something has been received, which secures the link to reality or the outside world, grounding the entire system and at least putting the onus on solipsism. Given that empirical fact as necessary presupposition, all that synthesis requires are the pure forms of intuition – because it can start from nothing, it can synthesize the void.
Charles Sanders Peirce, the independent co-founder of modern mathematical logic (though usually Frege is the one remembered, as with so many such pairs, e.g. Leibniz/Newton), read Kant at a relatively early age. In fact,
During his freshman year at college (Harvard), in 1855, when he was 16 years old, [Peirce] began private study of philosophy in general, starting with Schiller’s Letters on the Aesthetic Education of Man and continuing with Kant’s Critique of Pure Reason. After three years of intense study of Kant, Peirce concluded that Kant’s system was vitiated by what he called its “puerile logic,” and about the age of 19 he formed the fixed intention of devoting his life to study of and research in logic.[xxxiii]
He nevertheless had a similar conception of mathematics at the theoretical level – only he went in a different direction with it. That is, Peirce too thought that all the objects of mathematics were merely mental constructs of our own construction, so that where his contemporaries Russell and Frege wished to ground mathematics in logic, Peirce thought that mathematics was even more independent of empirical reality than logic and thus reversed the terms, i.e. grounding logic in mathematics. But Peirce saw the synthetic activity of mathematics as a model for the activity of cognition as such – seeing an underlying structural unity in the methods of geometry and algebra, for instance (famously opposed in this regard, even by Kant[24]), as both forms of what he called diagramming.
This notion of cognitive diagramming aligns quite neatly in most respects with Wittgenstein’s Tractarian view of thought as a picturing. This picturing or diagramming, it is crucial to note, is strictly structural, or in more Kantian terms, purely formal – it is totally independent of empirical reality, requires no empirical intuition for material, and requires only certain structural conditions like the Kantian pure forms of pure intuition (we might say: the pure forms of receptivity as such). Thus far, meaning at least the last two paragraphs, we have not necessarily strayed that far from the Kantian account.
Now, Kant projects a transcendent ground of unity for all mathematical units, which for him is the One (and in his example, the decad); this unity is simply a formal condition for all objects of possible intuition whatsoever, even of pure intuition (so even the pure constructs of mathematics). He takes this view, I think, because he cannot countenance the prospect of mathematics and moreover transcendental unity being founded on the void – “founded on the void”, as in founded on a transcendent ground. But here we discover Kant’s own dialectical confusion – for to be immanently grounded in the void is not the same as Kant’s dialectical interpretation of the void as a transcendent ground that is merely absent. This foundation in the void, as a matter of fact, is something else entirely, which will dissolve the Kantian problem as a problem.
Mathematically/historically, we see that the long-standing prejudice for the transcendent One (quite prominent in ancient Greek mathematics and philosophy for instance) only gives way very late, though with the most significant of consequences (i.e., modern mathematics and logic, which are very plausibly conditions and often causes of the development of modern technology). Specifically, the singular foundation of all mathematics that comes to replace the postulate of a transcendent One is the synthesizing or totalization of the empty set (insofar as all of mathematics can be derived from or reduced to basic number theory, which can be more-than-adequately modeled according to the theory/constructive method of John H. Conway, which starts with this – as do all forms of set theory, another and more well-known foundational mathematics that can capture the required basic number theory). All numbers are not units – particular instances of a transcendent universal – but immanent unfoldings or elaborations of the empty set, nested enframings of nothing as such, immanent syntheses of the void.
What, then, is the meaning of the void? We might invoke the Heideggerian distinction between Being and beings here – the void is the fact of space rather than the determinate beings which can fill it, but we know that for Kant this pure or mere space is really just the pure forms of intuition – it’s not like intuition is in a room with no objects it can seize upon, but rather there is no intuition at all, only the structural potential for intuition – only the given conditions of possibility necessary for any determinate intuition to take place at all. That is, to talk of cognition as though it were language, here nothing is said, rather here language resides, creating a place where saying can take place. And to complete the metaphor, let’s put it in terms of mathematics-as-diagramming as language as cognition: then we could say that anything “said” or written or represented in, e.g., mathematics, by virtue of its very fact of being captured in a symbol, implies that it exists for the language (of mathematics) – so that if I write a variable x, we all understand that x is given as existing for the universe of mathematical discourse (and so that I do not have to more formally write “Ǝ[x]” every time I want to use the symbol). And that here where nothing is said – where nothing is said – there is only “Ǝ[ ]”. That is, the void would be the sheer facticity of immanent existence – and in fact the other symbol for the empty set in mathematics, besides “”, is “[ ]”. This symbol, which lies as the transcendently-immanent singular ground of mathematics, symbolizes the capturing or totalization of the void.
It symbolizes the facticity of language, the sheer power of existence that language has, so that even nothing can be given a determinate existence, a fixed representation, a unique symbol in it – i.e., “can be said”. But whence the “transcendently”, then? The empty set, as symbol of the void, is also the symbolic mark of the subject of language in language. Insofar as it is the unique mark of totalization, or the very operation of cognition (naming in language, counting in mathematics, synthesizing sensible objects of perception, etc.), it is the proper name of the transcendental or structural subject. To complete the picture and elaborate these claims, I will draw on the psychoanalytic theory of Jacques Lacan.[25] [xxxiv]
In essence, we can take the constructive corpus of mathematics and of language generally to constitute a single constructive corpus (again, a somewhat Quinean idea) that can simply be defined as structure, or as symbolic structure, i.e. what Lacan names the Symbolic Order. The symbolic order would therefore essentially constitute all the structural conditions of cognition, acting like the Kantian transcendental conditions of possibility (of reason) in giving the formal space for cognitive activity. This transcendental structure would likewise constitute a place for the subject, and this place (occupy-able by empirical subjects) would be the transcendental ego (boundary-shell/membrane delimiting interiority and exteriority). And since a given empirical subject would be, as part of reality or the Lacanian Real, strictly transcendent to the immanent order of the Symbolic (or of Kantian pure reason) – the structural mark of the transcendental ego names the transcendent empirical subject which it in no way depends on, but rather opens up a space for – i.e., creates a structure place where an empirical subject can come to be. This mark of the void can stand for the absent empirical subject because it names the transcendental place of subjectivity.
Notes
[1] In this paper I will confine mere source-citations to the endnotes, while utilizing the footnotes to deliver commentary and/or provide citations/excerpts of text.
[2] To see an unsimplified schematic, a truly impressive feat of documentation, Google image search “Kant-cpr1”, and it should be the first result. The original can be found on the author’s, i.e. Andrew Stephenson’s, web site @ url http://oxford.academia.edu/AndrewStephenson
[3] Here specifically moral ones.
[4] To get ahead of ourselves quite a bit, cf. Religion p. 34: “Yet an end proceeds from morality just the same; for it cannot possibly be a matter of indifference to reason how to answer the question, What is then the result of this right conduct of ours? nor to what we are to direct our doings or nondoings, even granted this is not fully in our control, at least as something with which they are to harmonize. And this is indeed only the idea of an object that unites within itself the formal condition of all such ends as we ought to have (duty) with everything which is conditional upon ends we have and which conforms to duty (happiness proportioned to its observance), that is, the idea of a highest good in the world, for whose possibility we must assume a higher, moral, most holy, and omnipotent being who alone can unite the two elements of this good. This idea is not (practically considered) an empty one; for it meets our natural need, which would otherwise be a hindrance to moral resolve, to think for all our doings and nondoings taken as a whole some sort of ultimate end which reason can justify. What is most important here, however, is that this idea rises out of morality and is not its foundation.”
[5] Cf Religion p.6-8: “But one can regard the need of reason as twofold: first in its theoretical, second in its practical use. The first need I have just mentioned; but one sees very well that it is only conditioned, i.e. we must assume the existence of God if we want to judge about the first causes of everything contingent, chiefly in the order of ends which is actually present in the world. Far more important is the need of reason in its practical use, because it is unconditioned, and we are necessitated to presuppose the existence of God not only if we want to judge, but because we have to judge.”
[6] For Kant, as we will see later, there are two kinds of oppositions – genuinely or analytical contradictory opposition (affirmation vs denial of a single subject) and dialectical opposition (opposing two predicates of one subject, which while opposed do not technically i.e. logically contradict each other). Now Aufhebung for Kant occurs not in denial in the first sense (in which the denial is the contradictory opposite of the affirmation), but rather in the dialectical sense, in which neither particular predicate is denied, but rather they cancel each other out (it is the dialectical opposition itself which is aufgehoben, negatively transformed into a truth which dissolves what it had formerly constituted as problematic. It is like this: you are struggling with a stuck door, and you posit that only either pushing or pulling will open it, but no matter how much force you exert in either direction, it remains stuck, so that you cannot verify which of the opposed alternatives is the case – and then you realize that the door has simply been locked all along, and in fact slides into the doorframe rather than swinging inwards or outwards – your problem dissolves as a problem).
[7] Cf. Religion p. 6: “Yet through this, namely through the mere concept, nothing is settled in respect of the existence of this [supersensible] object and its actual connection with the world (the sum total of all objects of possible experience). But now there enters the right of reason’s need, as a subjective ground for presupposing and assuming something which reason may not presume to know through objective grounds; and consequently for orienting itself in thinking, solely through reason’s own need, in that immeasurable space of the supersensible, which for us is filled with dark [dicker] night.”
[8] Cf. Religion p.35: “The proposition, ‘There is a God, hence there is a highest good in the world,’ if it is to proceed (as proposition of faith) simply from morality, is a synthetic a priori proposition; for although accepted only in a practical context, it yet exceeds the concept of duty that morality contains (and which does not presuppose any matter of the power of choice, but only this power’s formal laws), and hence cannot be analytically evolved out of morality. But how is such a proposition a priori possible?”
[9] Cf. Religion p. 5: “By analogy, one can easily guess that it will be a concern of pure reason to guide its use when it wants to leave familiar objects (of experience) behind, extending itself beyond all bounds of experience and finding no object of intuition at all, but merely space for intuition; for then it is no longer in a position to bring its judgments under a determinate maxim according to objective grounds of cognition, but solely to bring its judgments under a determinate maxim according to a subjective ground of differentiation in the determination of its own faculty of judgment. [Kant’s footnote: Thus to orient oneself in thinking in general means: when objective principles of reason are insufficient for holding something true, to determine the matter according to a subjective principle.]”
[10] This point will be of critical importance later in the paper, as we present an alternative to the Kantian account.
[11] Note that Kant here seems to perhaps imply an identification of three items: the dark night of the supersensible; the space for/of intuition as such, i.e. space-of-intuition solely qua space; and finally, the void.
[12] A logical temporality, as opposed to chronological or empirical temporality, as per the Kantian inner form of intuition that structures all experience as serially successive.
[13] Again, Religion p. 35: “Agreement with the mere idea of a moral lawgiver for all human beings is indeed identical with the moral concept of duty in general, and to this extent the proposition commanding the agreement would be analytic. But the acceptance of the existence of this lawgiver means more than the mere possibility of such an object.”
[14] Cf. The Critique of Pure Reason p.467-8: “A dialectical theorem of pure reason [i.e. the thesis or antithesis of an antinomy] must accordingly have the following feature, distinguishing it from all sophistical propositions: it does not concern an arbitrary question that one might raise only at one’s option, but one that every human reason must necessarily come up against in the course of its progress; and second, this proposition and its opposite must carry with them not merely an artificial illusion that disappears as soon as someone has insight into it, but rather a natural and unavoidable illusion, which even if one is no longer fooled by it, still deceives though it does not defraud and which thus can be rendered harmless but never destroyed.”
[15] Cf. The Critique of Pure Reason p.467: “If any sum total of dogmatic doctrines is a ‘thetic’, then by ‘antithetic’ I understand not the dogmatic assertion of the opposite but rather the conflict between what seem to be dogmatic cognitions (thesin cum antithesi), without the ascription of a preeminent claim to approval of one side or the other. Thus an antithetic does not concern itself with one-sided assertions, but considers only the conflict between general cognitions of reason and the causes of this conflict. The transcendental antithetic is an investigation into the antinomy of pure reason, its causes and its result.”
[16] [on the singular and plural uses of the word “antinomy”]
[17] Cf. The Critique of Pure Reason p.466-7: “We have two expressions, world and nature, which are sometimes run together. The first signifies the mathematical whole of all appearances and the totality of their synthesis in the great as well as in the small, i.e. in their progress through composition as well as through division. But the very same world is called nature insofar as it is considered as a dynamic whole and one does not look at the aggregation in space or time so as to bring about a quantity, but looks instead at the unity in the existence of appearances.
[…] In regard to the distinction between the mathematically and the dynamically unconditioned toward which the regress aims, I would call the first two world-concepts in a narrower sense (the world in great and small), but the remaining two transcendent concepts of nature.”
[18] Cf. The Critique of Pure Reason p.465: “Now one can think of this unconditioned either as subsisting merely in the whole series, in which thus every member without exception is conditioned, and only their whole is absolutely unconditioned, or else the absolutely unconditioned is only a part of the series, to which the remaining members of the series are subordinated but that itself stands under no other condition.”
[19] In essence, an analytical opposition is symmetrical (“a” vs “ ~a”) while a dialectical opposition contains more than the mere, symmetrical negation (“a” vs “ ~a & b”); Cf. The Critique of Pure Reason p.517-18: “If someone said that every body either smells good or smells not good, then there is a third possibility, namely that a body has no smell (aroma) at all, and thus both conflicting propositions can be false. […]
Accordingly, if I say that as regards space either the world is infinite or it is not infinite (non est infinitus), then if the first proposition is false, its contradictory opposite, ‘the world is not infinite’ must be true. Through it I would rule out only an infinite world, without positing another one, namely a finite one. But if it is said that the world is either infinite or finite (not-infinite), then both propositions could be false. For then I regard the world as determined in itself regarding its magnitude, since in the opposition I not only rule out its infinitude, and with it, the whole separate existence of the world, but I also add a determination of the world, as a thing active in itself, which might likewise be false, if, namely, the world were not given at all as a thing in itself, and hence, as regards its magnitude, neither as infinite nor as finite. Permit me to call such an opposition a dialectical opposition, but the contradictory one an analytical opposition. Thus two judgments dialectically opposed to one another could both be false, because one does not merely contradict the other, but says something more than is required for a contradiction. (CPR p. 517-518)
[20]Cf. The Critique of Pure Reason p. 513: “If, accordingly, I represent all together all existing objects of sense in all time and all spaces, I do not posit them as being there in space and time prior to experience, but rather this representation is nothing other than the thought of a possible experience in its absolute completeness. In it alone are those objects (which are nothing but mere representations) given.”
[21] Again, the rejection or exclusion referenced in this parenthetical will become critical very soon now.
[22] Cf. Kant’s footnote on Religion p. 7: “Since reason needs to presuppose reality as given for the possibility of all things, and considers the differences between things only as limitations arising through the negations attaching to them, it sees itself necessitated to take as a ground one single possibility, namely that of an unlimited being, to consider it as original and all others as derived. Since also the thoroughgoing possibility of every thing must be encountered within existence as a whole – or at least since this is the only way in which the principle of thoroughgoing determination makes it possible for our reason to distinguish between the possible and the actual – we find a subjective ground of necessity, i.e. a need in our reason itself to take the existence of a most real (highest) being as the ground of all possibility.”
[23] I have not explicitly extended the consequences of this account to the preclusion of the postulates of pure practical reason, but the extension should be obvious – once the legitimacy of founding the drive for cognition, and thus the need of reason, in the void is established, there is no longer an epistemic distinction between the practical postulates, which are synthetic, and the derivation of morality from the form of reason, which is also synthetic. And if we thus take the form of reason to be the totalization of the void, rather than the universality of a transcendent ground, we no longer have Kantian analytical morality. Now the reader might not ask if this itself is not a major problem for the constructivist account – must we not provide some alternative firm ground for morality, or explain how the void constitutes this? I do not think so – what we have shown is the theoretical necessity of this picture, and a reinterpretation that dissolves the practical need; rather than providing an alternative normative theory, hopefully we have opened up a space for the construction of such theories (beginning immediately, obviously, with Schophenhauer and then exemplarily Nietzsche).
[25] For purposes of brevity (though that might be laughable at this point), I have chosen to merely provide the relevant citation concerning Lacanian psychoanalytic theory by Bruce Fink, leaving it to the reader to draw the connections – which hopefully by now will be quite obvious. Hence see the citation corresponding to endnote xxxiv.
[i] Cf., e.g., the first paragraph of the introduction to Religion Within the Boundaries of Mere Reason, p. vii
[iii] The Critique of Pure Reason p.470
[iv] The Critique of Pure Reason p. 471
[v] The Critique of Pure Reason p. 476
[vi] The Critique of Pure Reason p. 477
[vii] The Critique of Pure Reason p. 485
[viii] The Critique of Pure Reason p. 490
[ix] The Critique of Pure Reason p. 491
[x] The Critique of Pure Reason 521-23
[xi] The Critique of Pure Reason p. 519
[xii] The Critique of Pure Reason p. 508-9
[xiii] The Critique of Pure Reason p. 510
[xiv] The Critique of Pure Reason p. 514
[xv] The Critique of Pure Reason p. 518
[xvi] The Critique of Pure Reason p. 519
[xvii] The Critique of Pure Reason p. 520-1
[xxi] The Critique of Practical Reason p. 3
[xxii] The Critique of Practical Reason p. 3
[xxiii] The Critique of Practical Reason p. 4
[xxvii] Kant’s footnote on Religion p. 8
[xxviii] The Critique of Pure Reason p. 673
[xxix] The Critique of Pure Reason p. 32
[xxx] The Critique of Pure Reason p. 34
[xxxi] The Critique of Pure Reason p. 210-11
[xxxii] The Critique of Pure Reason p. 51-3
[xxxiv] Fink, The Lacanian Subject: Between Language and Jouissance p. 5-6:
“Lacan accounts for the foreignness [of the discourse of the Other in the self] as follows: we are born into a world of discourse, a discourse or language that precedes our birth and that will live on after our death. Long before a child is born, a place is prepared for it in its parents’ linguistic universe […] one cannot even say that a child knows what it wants prior to the assimilation of language: when a baby cries, the meaning of that act is provided by the parents or caretakers who attempt to name the pain the child seems to be expressing [this is also how the ego originally forms in the mirror-stage – it is not just that the infant recognizes its image in a moment of jouissance, but that the parent-figure approbates/reinforces this joy-of-recognition.]”
The Lacanian Subject p. 51-3:
“The parties to the vel of alienation that concern us here are not, however, your money and your life, but the subject and the Other, the subject being assigned the losing position (that of the money in the previous example, which you had no choice but to lose). In Lacan’s vel, the sides are by no means even: in his or her confrontation with the Other, the subject immediately drops out of the picture. While alienation is the necessary ‘first step’ in acceding to subjectivity, this step involves choosing ‘one’s own’ disappearance.
Lacan’s concept of the subject as manqué-à-être is useful here: the subject fails to come forth as a someone, as a particular being; in the most radical sense, he or she is not, he or she has no being. The subject exists – insofar as the word has wrought him or her from nothingness, and he or she can be spoken of, talked about, and discoursed upon – yet remains beingless. […] Alienation gives rise to a pure possibility of being, a place where one might expect to find a subject, but which nevertheless remains empty. Alienation engenders, in a sense, a place in which it is clear that there is, as of yet, no subject: a place where something is conspicuously lacking. The subject’s first guise is this very lack.
Lack in Lacan’s work has, to a certain extent, an ontological status: it is the first step beyond nothingness. To qualify something as empty is to use a spatial metaphor implying that it could alternatively be full, that it has some sort of existence above and beyond its being full or empty. A metaphor often used by Lacan is that of something qui manque à sa place, which is out of place, not where it should be or usually is; in other words, something which is missing. Now for something to be missing, it must first have been present and localized; it must first have had a place. And something only has a place within an ordered system – space-time coordinates or a Dewey decimal book classification, for example – in other words, within some sort of symbolic structure.
Alienation represents the instituting of the symbolic order – which must be realized anew for each subject – and the subject’s assignation of a place therein. A place he or she does not ‘hold’ as of yet, but a place designated for him or her, and for him or her alone. When Lacan says (in Seminar XI) that the subject’s being is eclipsed by language, that the subject here slips under or behind the signifier, it is in part because the subject is completely submerged by language, his or her only trace being a place-marker or place-holder in the symbolic order.
The process of alienation may, as J.-A. Miller suggests, be viewed as yielding the subject as empty set, {}, in other words, a set which has no elements, a symbolic which transforms nothingness into something by marking or representing it. Set theory generates its whole domain on the basis of this one symbol and a certain number of axioms. Lacan’s subject, analogously, is grounded in the naming of the void. The signifier is what founds the subject; the signifier is what wields ontic clout, wresting existence from the real that it marks and annuls. What it forges is, however, in no sense substantial or material.
The empty set as the subject’s place-holder within the symbolic order is not unrelated to the subject’s proper name. That name is often selected long before the child’s birth, and it inscribes the child in the symbolic. A priori, this name has absolutely nothing to do with the subject; it is as foreign to him or her as any other signifier. But in time this signifier – more, perhaps, than any other – will go to the root of his or her being and become inextricably tied to his or her subjectivity. It will become the signifier of his or her very absence as subject, standing in for him or her.”
0 Introduction: The Kantian Schema and the Two Needs of Pure Reason
Before properly introducing the topic of this paper, which is Kant’s postulates of practical reason, and in order to lay the minimal but necessary ground for that introduction, I would like to present a truly oversimplified schematic of Kant’s architectonic for the human mind – an oversimplified schematic that is also selectively tailored to reflect the essential concerns of this paper, rather than attempting to be a comprehensive picture. To that end, the following should be read intuitively, i.e. top-down, with the text in brackets indicating the directed relation between the item preceding it and the item following it.
Pure Reason Pure Reason
[gives analytically] [gives a priori as an empirical fact of having-reason, and also indirectly but analytically, because freedom is a necessary condition of the moral law]
The Moral Law Freedom
[which both, as pure ideas, subtend]
The Concepts of the Understanding
[which are appended to]
Image Representations
[that are furnished by]
The Imagination
[and which together, i.e. concept of understanding and image representation, provide]
A Determinate Object
[for]
A determinate maxim of the understanding
[in order to, with the purpose of determining an action, give a rule to]
The Will
We can easily see that there are two sides to the metatheoretical or philosophical motivation for Kant here, essentially corresponding to the speculative (top) and empirical (bottom) poles of human reason, as well as to the distinction between theoretical and practical reason. Namely, on one end, there is pure speculative reason, and on the other, the practical faculties of imagination and understanding in their working-together to determine the will, or the faculty of action. That is, Kant wants both (1) to keep strictly to the formal conditions and limitations of pure reason as his subjective ground of orientation (his “rock” or point of absolute philosophical reference, as it were), and (2) to adequately account for the effective determination of the will by practical reason. His concern with the practical is also, obviously, a concern with morality; but morality actually derives analytically from pure reason for Kant. This is in fact crucial, as the postulates of practical reason must, for Kant, evolve out of the concept of morality, rather than constitute its foundation.
Now Kant argues that the practical postulates respond to and are justified by a need of pure reason. This need is doubly articulated, like pure reason itself, in both theoretical and practical forms. The critical difference in the two cases for Kant is that in the one (namely the theoretical), positing the postulates would transgress the proper limits or bounds of reason and is thus strictly prohibited by Kant’s own critical method, whereas this positing is precisely what Kant argues is justified in the other case (namely the practical). As the infamous phrase – now long since reified into a cliché – attached to this complexly architectonic argument goes, Kant found it necessary to “deny knowledge in order to make room for faith.”[i] The original German is actually illuminating here:
Ich musste das Wissen aufheben, um zum Glauben Platz zu bekommen.
The word commonly translated as “to deny” is aufheben, whose noun form is much more well-known by non-German-speaking philosophers thanks to Hegel, namely Aufhebung, always marked as “a technical term” of philosophy and typically translated as “sublation”. Now we certainly do not mean to retroactively project a Hegelian concept into the Kantian corpus, but we do not need to – Aufhebung was an established technical term for Kant as well. [ii]
It is not, therefore, unreasonable to think that Kant saw the making-room-for (we might think here of Heidegger’s Lichtung and “clearing”) faith by practical reason as the Aufhebung of theoretical reason – indeed, this exactly corresponds to the function of the practical postulates in the resolution of the antinomy of pure reason, where the dialectical opposition mutually negates or cancels itself out on the theoretical side, opening the door to a joint affirmation on the practical side.
The essential movement of the Kantian Aufhebung, then, is this: from the unconditional negative limits of theoretical reason to the conditional but (it is argued) necessary positive projection of the postulates of practical reason, the former being the condition (of possibility) of the latter. The critical discipline of the first renders the paradoxical questions of pure reason undecidable, while this theoretical undecidability – in conjunction with the necessity for practical determination of the will (i.e., thanks to the necessity of acting) – not only permits but forces a practical decision on the theoretically undecidable questions of pure reason. That is, on Kant’s view, practical necessity forces us to go beyond the proper bounds of pure reason – out into the “dark night” where our theoretical reason is both absolutely freed and simultaneously utterly confounded, and where only a subjective ground of orientation remains (like an absolute compass, indicating the direction of some absolute reference point, but no more), the clear path before us having vanished, swallowed into the sublime darkness of the supersensible into which our understanding cannot penetrate. Of course, as Nietzsche infamously will not fail to note, this subjective ground itself, being the absolute point of reference for the fundamentally important Kantian operation of “orientation” in thinking, is open to question and critique – so that the battle over the postulates begins even deeper, in the foundations of the system and on the field of morality, even of reason, itself.
The two needs of pure reason are, then, intimately intertwined (indeed, I would say a double articulation of one need). The practical need could not arise were it not for the theoretical need, but only the former can be satisfied, this satisfaction being the controversy under examination here. There are thus two key loci for a critical analysis of the Kantian account in this regard:
(1) The justification of the theoretical need
(2) The justification of the practical need, and the justification of the practical solution (namely, the postulates)
We can also frame this in terms of the distinction in epistemic status between the propositions attached to the two needs:
(1) Analytic necessity (the step from reason to morality)
(2) Synthetic a priori (the step from morality to the highest good / God)
These should be the two major hinges in the structure of any critique of Kant’s postulates of practical reason. That is, a foundational critical analysis of the roles of reason, freedom, and morality in the Kantian account, on the one hand, and on the other, a critical analysis of the consequent practical postulates, insofar as they derive from and are justified by the former and foundational component of the account.
1 The Theoretical Need and Antinomy of Pure Reason
The situation is then this: human reason strays into the dark night of pure reason, where no empirical objects are given for intuition and instead there is merely or simply space for intuition – that is, no objective grounds for cognition are given, i.e. the rules of the understanding (always conforming to possible experience) yield no determinate maxim for judgment because the understanding always works in conjunction with the imagination, and since the imagination supplies no possible determinate object of experience (not only does it receive no sense-data or intuition for that, but it would have to furnish a determinate and empirically possible object from nothing, which Kant of course thinks is impossible) to the understanding, the latter can yield no determinate concept-object for judgment or for the formation of a maxim. The understanding thus cannot produce a determinate maxim in this case because it is given no a priori object for that maxim –– save perhaps (1) the very lack of such an object, i.e. the “nothing” or “void”; or (2) a self-reflective maxim which would actually therefore not be a maxim at all but rather a law of pure reason, i.e. a subjective ground of orientation for navigating the void, where no objective determinations are possible.
Kant of course chooses the second of these options, the self-reflexive law of pure reason. We will examine the first option, which he rejects, later, for its exclusion actually plays a quite important if indirect role in the consequent Kantian account (not to mention in post-Kantian thought – one instinctively associates this void as the only possible ground of orientation in pure reason with Nietzsche, and what Kant would surely think of as pure moral relativism and groundlessness of reason in the chaotic void of the will to power (and hence, one presumes, Nietzsche’s work as – and this is actually perhaps not as an unfair a hypothetical view, as far as it goes, of course – by turns enthusiastic and delirious)).
Now this law can be framed (1) formally, as the universalizability criterion for all propositions, and (2) as an operative rule or condition of possibility for all determinate maxims specifying “how to go on” for a given synthesis of the understanding. The first of these is quite familiar, of course – Kant interprets it as the moral law, expressed by the Categorical Imperative. Nietzsche, for instance, though he will even more deeply question the conceptualization of reason as universalizability (indeed, this concept of reason itself) in the first place, will also famously attack this Kantian interpretation of the more fundamental Kantian assertion – i.e., this interpretation that introduces morality into the heart of reason, or as analytically derivable from the very form of reason.
Now Kant argues that (1), the very form of reason, also entails (2), insofar as the possibility of a rule telling any given synthesis “how to keep going” is entailed by the universalizability criterion. That is, the universalizability criterion of the form of reason entails that a determinate maxim of the understanding specifying the operative form of a synthesis be grounded in a principle or rule of pure reason as its condition of possibility; but pure reason is of course not bound by the empirical constraints, e.g. temporal, that bind the understanding – it moves in a logical temporality where there are not particular maxims for particular syntheses, but only principles for such maxims of synthesis in general or as such. Thus the principles of reason, or the form of universalizability, precisely do not concern limited or particular syntheses, but rather specify the unconditional form of any possible synthesis, a therefore conceptually complete (and completely non-empirical) idea of pure reason. This pure idea, though in a sense the extension-to-unconditionality of the limited concepts of the synthetic understanding, is actually totally independent and indeed is the condition of possibility of the latter.
The point being: the pure idea of synthesis, which is necessarily presupposed by the fact of particular syntheses of the understanding (an empirical fact), must itself be the idea of a total or absolute synthesis – in short, a complete(d) synthesis (or: “synthesis” as such). If we were talking blithely in Hegelese (which I am glad that, in general, we are not), we might say that the logical temporality here is that of the dialectic, so that any possible actual synthesis of the understanding carried to infinity or completion would be captured not at the end (or final moment/movement of the synthesis) but as a whole by the “absolute (or pure) idea” of synthesis as such, from its birth and beginning to its sublation/completion and final end, via its logical parts – i.e. what empirically would have been experienced by an immortal being in time as stages are viewed as logical states or phases in a static logical picture by this same immortal being outside of time (i.e., in logical temporality and outside of empirical or chronological temporality).
Therefore, the argument goes: since it is given that there are, empirically, syntheses of the understanding, a principle (of synthesis) of pure reason must be necessarily presupposed. And this seems right – it is like saying you cannot apply a template if you do not have the template. And there would probably not be a problem if these temporalities were kept separate, at least, there would be different problems. But the problem for Kant as it stands lies in the crossing of these two temporalities in practical reason.
Here the fundamental question is essentially this: given (1) that “knowing (understanding, via a rule) how to go on” presupposes as ground in pure reason an idea of synthesis as a ruled or ordered “going-on” as such, and (2) that such an idea of pure reason necessarily includes the end of synthesis or the exhaustion or completion of “going-on”, insofar as its universal form entails a universal rule that unconditionally drives this activity of “going on” in the first place, and (3) that despite having the rule to go on “to the end”, the understanding can never actually complete an empirical synthesis, then: (4) is the existence of an end to synthesis as such, a final or ultimate end as it were (and, crucially, an “end” in the sense of logical temporality, rather than empirical/chronological temporality), required, in order for the actual syntheses to have the force or “motive” to keep going on to its particular, empirically unattainable end? If we accept with Kant that it is, then the existence of God will be a necessary consequence to secure the real possibility of this end (which is of course the Kantian “highest good” of universally practiced morality).
That this end is possible is entailed by the very form of synthesis as such, which is precisely an operation of determinate “end-seeking” (or condition-seeking, or cause-seeking, etc.); and that limited particular empirical syntheses can be completed is also given (e.g., if I specify from the outset an initial and final bound for a causal or otherwise empirical chain, and the bounded part lies within my possible experience or better my actual experience, I can plausibly “go through” the entire synthesis with my limited mind and time); but that any unlimited empirical synthesis can be completed, e.g. that a given causal chain could be traced all the way down to the origin of all causal chains, while entailed by the form of reason, can never be verified by a finite human intellect. Thus the possibility of a “divine reason”, or more strictly, an unbounded reason, is entailed by the very form of bounded, human reason – but to posit that this reason exists is another claim altogether, indeed, it is “a claim” and thus not analytically entailed by pure reason;– in fact it is not even fully accurate to characterize it as a claim, in truth it is a decision (or perhaps, thinking again of Heidegger, a challenging-forth-to-decision). The question here, the practical question and the question of positing existence, concerns the necessity of hope for human reason in its actual and practical synthetic activity, including the formulation of maxims for action. It is a question of a decision – of the necessity of a decision – for action, a decision for action (thus a practical decision) in response to the fundamental boundedness of human reason, its own response to its own inability, felt as a need of pure reason, to decide on its own fundamental questions. As a question of the necessity of a decision, this is therefore also a justificatory question, asking after the legitimacy of this decision.
This is our question, also that of the antinomy of pure reason, hinging on the theoretical undecidability by human, bounded reason of certain irreducible and inherent questions of pure reason – and the Kantian solution to this antinomy via a practical decision to act, i.e. to posit the postulates.
Now the antinomy of pure reason takes the form of four dyadic “antithetics”, forming two couples or classes; the first two antithetics constitute the “mathematical” antinomies, while the third and fourth constitute the “dynamical” antinomies. In brief, here is Kant’s bare statement of them:
First Antinomy (spatiotemporal finitude of the world vs spatiotemporal infinity of the world]
Thesis: The world has a beginning in time, and in space it is also enclosed in boundaries.[iii]
Antithesis: The world has no beginning and no bounds in space, but is infinite with regard to both time and space.[iv]
Second Antinomy (finite divisibility of composite substances in the world vs infinite divisibility of substances in the world)
Thesis: Every composite substance in the world consists of simple parts, and nothing exists anywhere except the simple or what is composed of simples.[v]
Antithesis: No composite thing in the world consists of simple parts, and nowhere in it does there exist anything simple.[vi]
Third Antinomy (necessity of transcendental freedom in addition to natural causality vs natural causality being the only causality)
Thesis: Causality in accordance with laws of nature is not the only one from which all appaerances of the world can be derived. It is also necessary to assume another causality through freedom in order to explain them. (484)
Antithesis: There is no freedom, but everything in the world happens solely in accordance with laws of nature.[vii]
Fourth Antinomy (the necessary existence of a necessary being vs the necessary non-existence of a necessary being)
Thesis: To the world there belongs something that, either as a part of it or as its cause, is an absolutely necessary being.[viii]
Antithesis: There is no absolutely necessary being existing anywhere, either in the world or outside the world as its cause.[ix]
We will examine the mathematical antinomies and Kant’s conclusion regarding them first; and we will defer a consideration of the “drive for cognition” that generates the theoretical as well as the practical need until the practical section, insofar as we can accept it prima facie for the purposes of our analysis of the mathematical antinomies, even though it will become a critically problematic notion once we progress to the practical.
2 On the Antinomy of Pure Reason: The Mathematical
We will present Kant’s formal solution, the “transcendental idealist” solution, to the problem in due course. For now let us simply say that Kant’s ultimate conclusion regarding the mathematical antinomies is a negative one – a joint negation not of the theses but of the antithetics themselves, insofar as they rest upon an unacknowledged critical transgression of the proper bounds of human reason. It is a “negative” solution in the sense that the limits of human reason are “negative”; what Kant asserts is precisely not a negative position on any of the particular theses or antitheses, but the negative limit of human reason insofar as this latter renders the former (i.e. theses/antitheses) undecidable. Not a negation, but something more primary – a negative condition of possibility of all possible particular theoretical negations, one which renders both negation and affirmation strictly impossible in this case by halting the process at the beginning, as it were, i.e. by precluding not an affirmative, not a negative answer, but by precluding the very propriety of a decision to answer as such (this is of course the Kantian Aufhebung from the introduction to this paper).
It should be noted that if we accept Kant’s solution to the first antinomy, the only really critical obstacle remaining for the Kantian account is on the practical side, i.e. accounting for the divergence in (re)solution in the case of the dynamical antinomies as opposed to the mathematical, including the justification via practical necessity for the positing of the postulates of practical reason.
Now there are two core problem here; the first concerns series that are infinite (or, at this point in the analysis, I suppose I should strictly speaking say “series whose finitude or infinity is undecidable for human reason”). There are, naturally, infinitely many examples: any given causal chain in the empirical order of nature; any given moment in a temporal sequence of the world; the progressive division of composite material substances into more elementary substances; any given segment or section of space extended sequentially; and perhaps most exemplarily of all, perhaps so exemplary or even fundamental for all the other examples that Kant neglects to explicitly mention it in the section of the Critique on the antinomies for this very reason – the series of natural numbers, or the natural series of number.
The second core problem concerns the notions of an original, primordial, or transcendent ground on the one hand, and a transcendently-necessary (not “transcendentally-necessary) immanent element (an original or irreducible element) on the other. These are two potential solutions to the problem of serial unconditionality (or infinity), in which every element of the series is conditioned by a predecessor and conditions a successor in turn, but which therefore must as a whole have either (1) a primordial or “first” element that has no prior condition but does condition a successor, or (2) the series must be conditioned as a totality, by a condition outside of the series (i.e., by a condition that is not itself one of the conditional elements of the series) that conditions the series as a whole rather than by individual part.
We might try to sketch an illustration of these concepts by means of the aforementioned example of the natural series of number. In as oversimplified of terms as possible, we could identify the function of the irreducible element in zero, which is the only natural number with no predecessor, and which establishes the synthetic rule for constructing more, indeed the rest, of the natural numbers (termed, in mathematics, the successor function – in essence, given any n, S(n) = n + 1 and S(n) is a number) (we are – and uncontroversially, or without violating the proper bounds of explicative rigor, I might add – deferring considerations of, e.g., negative numbers for now for purposes of simplicity). The successor function says that given a number (and zero is a number – though a number unlike any other natural number, since its immanent status as number is established by a transcendent axiom rather than the successor function, which is of course crucial), and given an operation equivalent to the successor function, we can construct all natural numbers, indeed, “to infinity”. The way I have framed this example, both of Kant’s alternatives are present – there is a transcendently-immanent element of the series that also, in its transcendence, serves as the transcendent ground for the series (namely, zero), conditioning the series as a whole by providing the rule that constitutes its form or structure, namely some transcendental rule (transcendental for the series) like the successor function, which requires only that a first element of the series be given in order to effect an infinite synthesis.
There is another way we can mathematically think about what is at play here (don’t worry, I will eventually present Kant’s own actual moves, my hope is simply that the work we do here will minimalize the explication and interpretation necessary there). This second number-theoretical thought experiment will bear more on second antinomy. Every natural number can be thought of as a particular aggregate of units, or of 1s (this stems from the transcendental rule of operation, which produces the first non-zero number, namely 1, by adding 1 to its predecessor, namely 0, and all subsequent natural numbers in the same fashion, recursively). At this point we slightly broaden the complexity of the example by no longer dealing with just whole numbers: any number can be decomposed into some aggregate of digits, which themselves signify the numerical place of aggregates of units in the number. That is, even numbers greater than a natural number but less than its successor ultimately reduce to units.
But here is the problem. We attain numbers between the naturals by division – we assert the naturals as initial and final bounds and call the number constituted by the bounded part a fractional, or rational number. It is presumably rational because it is a rational way of extending the natural number line or series to include more (or more precise) numbers. What is going on here, a method sourcing back to Richard Dedekind, is that a gap is posited between two consecutive natural numbers, in other words the continuous fabric of the natural number line is here cut (the cut is the positing of the gap), and the number that fills the gap is nothing other than the cut or gap itself. The cut creates the gap which becomes the new kind of number – and it is quite poetic and fitting, not to mention psychoanalytically and literarily intriguing, that this is called the rational method of number extension or even number construction (additionally, the contemporary method of John H. Conway for the construction of number, which is conservative over the advances of previous methods and which also generates a much larger and more comprehensive multiplicity of numbers (thought confidently to be all possible numbers), called in general Conway numbers or surreal numbers (encompassing all the familiar classes of number, from natural and integer and rational to real, imaginary, complex, and more), employs a single, fundamental operational device very close to Dedekindian cutting, a closeness Conway is well aware of). So why does this rational method propose a problem for pure reason?
Because its inherent formal condition of possibility, as a rule for how to go on, logically entails that it (can) go to infinity. How many rational numbers are there between any two given consecutive naturals? Infinitely many, or as many as the real number continuum, the inclusive bound of the “first” mathematical infinity (strictly speaking, it contains more numbers than the infinity of the natural number line, since it includes the natural numbers and indeed infinitely many numbers between them, but in terms of size it is a famously open question as to whether the cardinality of the real number continuum is greater than that of the natural series). Of course, a real human being, actually performing this synthetic operation of division, say with paper and pencil – beginning with two natural numbers as bounds, then the first rational number produced by this cut or bounding as initial bound and the second natural number as final bound, then the rational number produced by this cut and the second natural number, and so on to infinity – will never be able to empirically verify this infinity, or in other words, whether the division does not halt at some irreducible element it encounters somewhere deep in the recursive operation.
That is the general structure of the problem, framed number-theoretically: given a synthetic series, does the synthesis have an original step / does the series have a first member; and given a starting-point of a synthetic series, will the given synthesis ever meet an empirical stopping-point, or continue forever (i.e., without encountering a final member of the series)? Kant frames the problem, and his solution, in terms of series that go on “to infinity” and that go on “indefinitely”:
Mathematicians speak solely of a progressus in infinitum. But those who study concepts (philosophers) want, in place of this, to make the expression progressus in indefinitum the only valid one. […] One can rightly say of a straight line that it could be extended to infinity, and here the distinction between the infinite [vs] a progress of indeterminate length (progressus in indefinitum) would be an empty subtlety [… for here] the [second] means no more than ‘Extend it as far as you want,’ but the [first] means ‘You ought never to stop extending it’ […] yet if we are talking only about what can be done, then the [second] expression is entirely correct, for you could always make it greater, to infinity. And this is also the situation in all cases where one is speaking only of a forward progress, i.e., of a progress from the condition to the conditioned; this possible progress in the series of appearances goes to infinity. […] For here reason never needs an absolute totality in the series, because it is not presupposed as a condition as given (datum), but it is only added on as something conditioned, which is capable of being given (dabile), and this without end. It is entirely otherwise with the problem how far does the regress extend when it ascends from the given conditioned to its conditions in the series: whether I can say here that there is a regress to infinity or only a regress extending indeterminately far (in indefinitum) […] To this I say: If the whole was given in empirical intuition, then the regress in the series of its inner conditions goes to infinity. But if only one member of the series is given, from which the regress to an absolute totality is first of all to proceed, then only an indeterminate kind of regress (in indefinitum) takes place. […] In neither of these two cases, that of the regressus in infinitum as well as in that of the in indefinitum, is the series of conditions regarded as being given as infinite in the object. It is not things in themselves that are given, but only appearances, which, as conditions of one another, are given only in the regress itself Thus the question is no longer how big this series of conditions is in itself – whether it is finite or infinite – for it is nothing in itself; rather, the question is how we are to institute the empirical regress and how far we are to continue it.[x]
Kant’s answer has two basic parts, which elaborate the essential argument at work in the above passage, which is the famous transcendental idealist (re)solution. The first part is to dissolve the mathematical antinomies as merely dialectical oppositions, rather than genuinely contradictory or analytical opposites, insofar as they depend upon “an illusion arising from the fact that one has applied the idea of absolute totality, which is valid only as a condition of things in themselves, to appearances that exist only in representation, and that, if they constitute a series, exist in the successive regress but otherwise do not exist at all.”[xi] The second part is a consequent specification of the status of the principle or rule of pure reason which underlies both the antinomy and the confusions over it. That is, the critical or transcendental idealist solution necessitates a reevaluation of the proper role of the principle of pure reason that drives the synthetic quest for the unconditioned. No longer can it be assumed as an axiomatic principle for thinking the totality of an object as given in itself, since this object is only given as a problem i.e. in the synthesis and as a mere representation, hence Kant will term the principle a regulative rule of reason – in essence, what we have been calling a rule that (merely) specifies “how to go on”.
Kant’s first move is to argue that whichever side of a mathematical antinomy is taken, to affirm that thesis or antithesis already transgresses the proper bounds of reason, because no concept of the understanding can be adequate to the pure idea of unbounded reason that is being asserted (e.g., the spatiotemporal infinity/finitude of the world):
Accordingly, if I could antecedently see about a cosmological idea that whatever side of the unconditioned in the regressive synthesis of appearance it might come down on, it would be either too big or too small for every concept of the understanding, then I would comprehend that since it has to do with an object of experience, which should conform to a possible concept of the understanding, this idea must be entirely empty and without significance because the object does not fit it no matter how I may accommodate the one to the other. And this is actually the case with all the world-concepts, which is why reason, as long as it holds to them, is involved in an unavoidable antinomy. For assume:
First, that the world has no beginning; then it is too big for your concept; for this concept, which consists in a successive regress, can never reach the whole eternity that has elapsed. Suppose it has a beginning, then once again it is too small for your concept of understanding in the necessary empirical regress. For since the beginning always presupposes a preceding time, it is still not unconditioned, and the law of the empirical use of the understanding obliges you to ask for a still higher temporal condition, and the world is obviously too small for this law.
It is exactly the same with the two answers to the question about the magnitude of the world in space.[xii]
Thus we have been brought at least to the well-grounded suspicion that the cosmological ideas, and all the sophistical assertions about them that have come into conflict with one another, are perhaps grounded on an empty and merely imagined concept of the way the object of these ideas is given to us; and this suspicion may already have put us on the right track for exposing the semblance that has so long misled us.[xiii]
That is, the antithetic rests upon a strictly meaningless or simply confused idea that can have no possible adequate conception in the understanding, since it takes something that is only empirically given as an object or totality in itself (as though it were not merely given empirically, i.e. as a sensible representation, but as a thing in itself and independent of its representation in/by the mind). The very infinity of space-time cannot be considered as a property of space-time in itself (if it is indeed infinite, which is of course the whole point), since the extension of space-time is only given by and in empirical synthesis. As Kant says, “The entire antinomy of pure reason rests on this dialectical argument: If the conditioned is given, then the whole series of all conditions for it is also given; now objects of the senses are given as conditioned; consequently, etc.”[xiv] – but in fact “The series of appearances is to be encountered only in the regressive synthesis itself, but is not encountered in itself in appearance, as a thing on its own given prior to every regress.”[xv] Hence Kant writes:
Accordingly, the antinomy of pure reason in its cosmological ideas is removed by showing that it is merely dialectical and a conflict due to an illusion arising from the fact that one has applied the idea of absolute totality, which is valid only as a condition of things in themselves, to appearances that exist only in representation, and that, if they constitute a series, exist in the successive regress but otherwise do not exist at all.[xvi]
Now if we accept this resolution, there remains for Kant to explain a new interpretation of the root of the problem in the first place, namely the universal form of reason that unconditionally drives all synthetic activity toward completion. The reinterpretation is essentially: this rule does not say that there is a completion of synthesis, though it appears to say what that completion would be, rather it merely says how to synthesize – and since the form of synthesis is a movement to completion, this rule can mistakenly be interpreted so that the completion inherently contained in the rule is descriptive rather than prescriptive. That is, here Kant rules off-limits or out-of-bounds precisely what he will do on the basis of practical reason – “namely, the ascription of objective reality to an idea that merely serves as a rule”:
Since through the cosmological principle of totality no maximum in the series of conditions in a world of sense, as a thing in itself, is given, but rather this maximum can merely be given as a problem in the regress of this series, the principle of pure reason we are thinking of retains its genuine validity only in a corrected significance: not indeed as an axiom for thinking the totality in the object as real, but as a problem for the understanding, thus for the subject in initiating and continuing, in accordance with the completeness of the idea, the regress in the series of conditions for a given conditioned. […] Thus the principle of reason is only a rule, prescribing a regress in the series of conditions for given appearances, in which regress it is never allowed to stop with an absolutely unconditioned. Thus it is not a principle of the possibility of experience and of the empirical cognition of objects of sense, hence not a principle of the understanding, for every experience is enclosed within its boundaries (conforming to the intuition in which it is given); nor is it a constitutive principle of reason for extending the concept of the world of sense beyond all possible experience; rather it is a principle of the greatest possible continuation and extension of experience, in accordance with which no empirical boundary would hold as an absolute boundary; thus it is a principle of reason which, as a rule, postulates what should be effected by us in the regress, but does not anticipate what is given in itself in the object prior to any regress. Hence I call it a regulative principle of reason, whereas the principle of the absolute totality of the series of conditions, as given in itself in the object (in the appearances), would be a constitutive cosmological principle, the nullity of which I have tried to show through just this distinction, thereby preventing – what would otherwise unavoidably happen (through a transcendental subreption) – the ascription of objective reality to an idea that merely serves as a rule. Now in order to determine the sense of this rule of pure reason appropriately, it must first be noted that it cannot say what the object is, but only how the empirical regress is to be instituted so as to attain to the complete concept of the object.[xvii]
It remains to compare Kant’s resolution of the problem to the actual mathematical answer – for while I may have presented the example of the natural series as one example among others, it actually has much more far-ranging implications, insofar as the problems there are in fact not only foundational problems for pure reason for Kant, but for number theory in mathematics as well. And insofar as all of higher mathematics can be analytically derived from basic number theory (something not proven until well after Kant of course, in fact it constituted the great project of the founders of both modern mathematics and modern analytic philosophy, e.g. Frege and Russell), this problem of pure reason therefore also concerns the possibility of all of pure mathematics, which Kant avers to be of synthetic a priori status. Indeed, the example we have been sketching is perhaps the exemplar of the synthetic a priori. And we recall that Kant will argue such a status for the postulates of practical reason, as well, so that this issue is relevant on many fronts.
But as it happens, this task more suitably lands towards the end of the paper, after our explication of Kant’s account of the practical need and postulates, since it will depend upon the general anti-Kantian alternative we will have established by then, what might be called a kind of cognitive constructivism. So bravely forward, to the realm where courage is most needed, since the familiar is most receded.
3 The Practical Need and Postulates of Pure Reason
We have laid out much of the practical side of the issue already. The broad argument is roughly: reason analytically yields the moral law as a universal rule for moral synthetic activity, the universal form of the moral law necessarily contains a formally ultimate end, i.e. the highest good or universally practiced morality, and the possibility of this ultimate end (which, recall, unites in itself the conditions of all particular moral ends) is then secured by the posited (i.e. synthetically a priori necessary) existence of God, i.e. a necessary, omnipotent, original, etc. being. The possibility of this end is entailed by the very form of the moral law, but it is only secured via God (since what is logically entailed, as we well know by now, may well be empirically impossible, and thus strictly theoretically undecidable). And the question is why is this necessary, since even all that is logically entailed is the possibility of the ultimate end as an operational goal, not its existence, which is critically and irrevocably outside the range of our empirical verifiability (and existence is an empirical predicate, of course).As Kant himself says, what is at issue is positing the empirical object of “a being whose concept (if it is not to be vaguely determined and hence might be subject to association with every possible delusion) demands that it be of infinite magnitude as distinguished from everything created; but no experience or intuition at all can be adequate to that concept, hence none can unambiguously prove the existence of such a being.”[xviii]
Kant essentially argues (unsurprisingly at this point) that the necessity of positing the postulates derives from a practical need of reason. Recall that the theoretical need was the empirically impossible need to complete syntheses, to reach the unconditional in a series (whether a transcendent ground or a transcendently-immanent element). There Kant judged the need to exceed the capacity or proper bounds of human reason, and thus interpreted it immanently or operationally, prescriptively rather than descriptively, so that it only “instruct[s] as to how to operate but not as to whither” – whereas now Kant says the following:
But although on its own behalf morality does not need the representation of an end which would have to precede the determination of the will, it may well be that it has a necessary reference to such an end, not as the ground of its maxims but as a necessary consequence accepted in conformity to them. – For in the absence of all reference to an end no determination of the will can take place in human beings at all, since no such determination can occur without an effect, and its representation, though not as the determining ground of the power of choice nor as an end that comes first in intention, must nonetheless be admissible as the consequence of that power’s determination to an end through the law (finis in consequentiam veniens); without this end, a power of choice which does not [thus] add to a contemplated action the thought of either an objectively or subjectively determined object (which it has or should have), instructed indeed as to how to operate but not as to whither, can itself obtain no satisfaction.[xix]
The argument here resembles the description of how image-representations formed by the imagination subtend pure concepts of the understanding so as to bring them “down to earth” and make them, well, understandable (and/i.e. applicable):
However exalted the application of our concepts, and however far up from sensibility we may abstract them, still they will always be appended to image representations, whose proper function is to make these concepts which are not otherwise derived from experience, serviceable for experiential use. For how would we procure sense and significance for our concepts if we did not underpin them with some intuition (which ultimately must always be an example from some possible experience)?[xx]
Thus the idea is that while only the formal possibility of an ultimate end to synthesis is logically entailed by the form of pure reason, so that no determinate object, i.e. representable by the imagination in conjunction with an adequate concept of the understanding, is given or required at all by pure reason in and of itself, practical reason – reason insofar as it is to effectively determine action – requires some sensible intuition of how to determinately proceed, i.e. apply the rule; the fear here is one inherited much later by John Rawls, namely of the formal procedure of reason being unable to yield a determinate rule for action when such action is not optional or a matter of curiosity, but rather a matter of urgent practical necessity. In essence: how could we procure “sense and significance” for the abstract moral law as a rule for conduct if we did not have some minimal but ideally exemplary illustration of its effective application, i.e. what its result would look like if it worked properly (since we are essentially Humeans for Kant in regards to empirical reality, i.e. we have no insight into things in themselves [hence into their transcendental causality, if we with Kant postulate transcendental freedom], but only into how they appear to us as cognitive representations, i.e. into their external effects, e.g. causal effects on other empirical objects in that vicinity of the causal nexus or order of nature)?
But there is a second level to the argument. And that is: even if we formed such a determinate final end for ourselves so as to bring the abstract concept (really principle) down to earth and “sense”, and so as to provide an imaginable goal to direct our actual synthetic moral activity, we could simply acknowledge its merely operational status as an immanent rule for use (or synthesis) – we would not thereby theoretically need to posit its reality or secure this reality through a transcendent ground, namely the postulate of God.
Now Kant argues as to the first that freedom is the “keystone” to the architectonic edifice under examination here, because (a) it is presupposed by the moral law, which is itself analytically derived from the form of reason, and which gives a rule for activity[/action], thus presupposing the practical ability to determine the will in the effecting of this activity; (b) is also given analytically by reason itself, insofar as the experience of having reason or being a creature with reason factically includes the experience of willing, i.e. of beginning causal series of effects solely from one’s own will; so that (c) all other such concepts such as God and immortality of the soul become appended to it, much as the pure concepts of the understanding are appended to image-representations of the imagination to render them graspable – i.e., so that these concepts “get stability and objective reality” via it, “that is, their possibility is proved by this: that freedom is real”[xxi]:
Now, the concept of freedom, insofar as its reality is proved by an apodictic law of practical reason, constitutes the keystone of the whole structure of a system of pure reason, even of speculative reason; and all other concepts (like those of God and immortality), which as mere ideas remain without support in the latter, now attach themselves to this concept and with it and by means of it get stability and objective reality, that is, their possibility is proved by this: that freedom is real, for this idea reveals itself through the moral law.[xxii]
Perhaps oddly or perhaps strategically, Kant focuses on the possibility of the pure idea of freedom as given a priori by the moral law, since it is a condition of the moral law (that is, the moral law for conduct presupposes that the agent subject to it has the freedom to follow it) – rather than the factical experience of it as noted in the first Critique in the sections on the antinomy (i.e., the factical power to begin causal or phenomenal series from no prior ground other than one’s will). In fact, he says that “For, had not the moral law already been distinctly thought in our reason, we should never consider ourselves justified in assuming such a thing as freedom (even though it is not self-contradictory). But were there no freedom, the moral law would not be encountered at all in ourselves.”[xxiii] We can interpret this choice in the following manner.
The facticity of our power to start causal series is not enough, on its own, to assume that we have extra-natural or transcendental freedom – e.g., a hardcore materialist-expressivist could argue that all consciousness is epiphenomenal, and in fact our “will” that we think produces actions ex nihilo is a mere metaphysical specter, and all there is in reality is natural and deterministic causality in its infinite (but immanent and non-mysterious) complexity (e.g. from the level of atoms to the level of thoughts and desires). Here the problem is, in essence, that we are in the causal series, so that whether our apparently free actions are in fact transcendentally free or merely naturally determined is strictly undecidable for our limited reason – if we could step outside the intrinsic structural bounds of the mind, if we could look upon the entire serial causal nexus as a whole and simultaneously with reference to its parts, then we could judge definitively. And in fact this is precisely the point: only a transcendent ground can function as the unconditioned condition of completeness for an immanent series for Kant (since he rules out the empirical verifiability of an irreducible element, and which being an immanent element in the series would be precisely empirical, whereas a transcendent ground would be outside the legitimate range of such empirical verifiability in the first place or a priori). So that given that we can never empirically verify an irreducible/transcendent element of the series, and given that the series exists (as a series, i.e. a universally conditioned or ordered structure), we must (out of practical necessity) posit the theoretical existence of a transcendent ground.
Kant distinguishes this necessity from the arbitrary or unregulated positing of transcendently-immanent elements, such as spiritual beings with a noumenal causality floating in a plane parallel to the order of natural empirical causality, invisibly influencing it with their wills much as actual human beings are claimed to be able to influence it:
Many supersensible things may be thought (for objects of sense do not fill up the whole field of possibility) to which, however, reason feels no need to extend itself, much less to assume their existence. […] the assumption of [such] spiritual beings would rather be disadvantageous to the use of reason. […] Thus that is not a need at all, but merely impertinent inquisitiveness straying into empty dreaming to investigate them – or play with such figments of the brain. It is quite otherwise with the concept of a first original being as a supreme intelligence and at the same time as the highest good. For not only does our reason already feel a need to take the concept of the unlimited as the ground of the concepts of all limited beings – hence of all other things – , but this need even goes as far as the presupposition of its existence, without which one can provide no satisfactory ground at all for the contingency of the existence of things in the world, let alone for the purposiveness and order which is encountered everywhere[…][xxiv]
And the critical key is what follows upon this:
Without assuming an intelligent author we cannot give any intelligible ground of it without falling into plain absurdities; and although we cannot prove the impossibility of such a purposiveness apart from an intelligent cause (for then we would have sufficient objective grounds for asserting it and would not need to appeal to subjective ones), given our lack of insight there yet remains a sufficient ground for assuming such a cause in reason’s need to presuppose something intelligible in order to explain this given appearance, since nothing else with which reason can combine any concept provides a remedy for this need.[xxv]
What is the justification? (1) We cannot ground the purposiveness or directed-conditionality of synthetic series in an intelligible ground unless it is transcendent (i.e. an intelligible “meta-subject” acting upon our unlimited causal nexus from the outside the way we can act upon limited causal series from their “outside”), (2) since otherwise, i.e. if we were to attempt to ground it in a transcendently-immanent element, we would fall into “plain absurdities” – like, we may hypothesize, grounding it in the void; and since (3) though we cannot prove such an alternative ground impossible (for then the issue would be decided and there would be no question of a need), we can also a priori never verify it (we have already given the argument for this multiple times now), then (4) the inherent formal need of reason to presuppose some unconditional ground provides a “sufficient ground”, i.e. a subjective ground of orientation, for positing it as a “postulate” of practical reason, which Kant opposes to a “rational hypothesis” of merely theoretical reason:
A need of reason to be used in a way which satisfies it theoretically would be nothing other than a pure rational hypothesis, i.e. an opinion sufficient to hold something true on subjective grounds simply because one can never expect to find grounds other than these on which to explain certain given effects, and because reason needs a ground of explanation. By contrast, rational faith, which rests on a need of reason’s use with a practical intent, could be called a postulate of reason – not as if it were an insight which did justice to all the logical demands for certainty, but because this holding true (if only the person is morally good) is not inferior in degree to knowing, even though it is completely different from it in kind.
A pure rational faith is therefore the signpost or compass by means of which the speculative thinker orients himself in his rational excursions into the field of supersensible objects[.][xxvi]
Now to examine an obvious question we keep deferring, namely: Kant says that reason feels its own need, feels its inherent limitations – but how can reason feel? Kant’s answer in Religion:
Reason does not feel; it has insight into its lack and through the drive for cognition it effects the feeling of a need. It is the same way with moral feeling, which does not cause any moral law, for this arises wholly from reason; rather, it is caused or effected by moral laws, hence by reason, because the active yet free will needs determinate grounds.[xxvii]
And in the first Critique:
Reason is driven by a propensity of its nature to go beyond its use in experience, to venture to the outermost bounds of all cognition by means of mere ideas in a pure use, and to find peace only in the completion of its circle in a self-subsisting systematic whole. Now is this striving grounded merely in its speculative interest, or rather uniquely and solely in its practical interest?[xxviii]
The drive for cognition. This is the really fundamental thing, the source of the needs of pure reason – the truly universal operational form of reason as such, and an active or dynamical form – i.e. an unconditional drive to “keep going on” with the synthetic activity of thinking, of cognition (so not merely conscious contemplation and thought, but the subsurface gears and functions grinding away beneath the epiphenomenal veil of conscious awareness as well). Now I believe that in the third Critique Kant essentially identifies this drive, as the motor force of cognitive activity, as spirit – but that is a consideration for a different paper, since what suffices for our argument here is simply this: that the drive for cognition goes critically unquestioned by Kant. This is how, on Nietzsche’s view, morality sneaks its way into the foundation of Kantian reason: a drive (to pursue the synthetic activity of cognition) is interpreted morally by Kant, who avers that the operational rule for how to go on to infinity – because analytically derived from the form of reason itself – not only gives the gift of a “how” but also imposes the demand or imperative of an “ought”. This drive of pure reason which is even more generally the drive of reason (or cognition) as such and in all its systematically-interrelated functional parts, is analytically moral for Kant, the quintessential rationalist.
And indeed, insofar as morality concerns the practical problems of interacting with other agents of human reason, it is an a priori problem for human activity; but morality, as the solution to the moral problem (even social problem), is only analytically the solution of reason if the form of reason is itself moral, i.e. if the drive for cognition not only gifts a practical rule for moral synthesis but demands this moral synthesis as an unconditional duty.
This brings us to the concluding sections of this paper, where an alternative account is provided that is meant to refute certain parts of the Kantian account, but more importantly, to account for the gaps in the Kantian account (much as a rational number accounts for the gap between two naturals – though only via the critical cut that opens this gap, as a wound but also as a passageway, in the first place).
4 Toward a Critique of the Critique
We can begin by more closely examining a notion we have taken for granted thus far though frequently employing it throughout this paper, namely synthesis. Kant first uses this technical term in a pre-critical essay with regards to the difference between mathematics and philosophy (difference in method and difference in the kind of certainty attainable by each). Here he characterizes mathematical activity as a constructive synthesis, as Wood and Guyer note in the introduction to the Critique of Pure Reason:
This essay takes major steps toward the position of the Critique of Pure Reason, although crucial differences still remain. Kant’s most radical departure from prevailing orthodoxy and his biggest step toward the Critique comes in his account of mathematical certainty. Instead of holding that mathematics proceeds by the two-front process of analyzing concepts on the one hand and confirming the results of those analyses by comparison with our experience on the other hand, Kant argues that in mathematics definitions of concepts, no matter how similar they may seem to those current in ordinary use, are artificially constructed by a process which he for the first time calls “synthesis”, and that mathematical thinking gives itself objects “in concreto” for these definitions, or constructs objects for its own concepts from their definitions. […] Thus, we can have certain knowledge of the definition because we ourselves construct it; and we can have certain knowledge that the definition correctly applies to its objects because the true objects of mathematics are nothing but objects constructed, however that may be, in accordance with the definitions that we ourselves have constructed.[xxix]
At this point Kant still defined philosophy as solely analytical, in contradistinction to mathematics, but this is of course a view he will abandon by the time of his critical works:
Before Kant’s mature work could be written, he would have to discover a philosophical method that could yield “material” or synthetic judgments. This would be the philosophical work of the 1770s that would finally pave the way for the Critique of Pure Reason.
Once Kant takes this further step, however, the contrast between mathematics and philosophy provided in the Inquiry will have to be revised. The difference between mathematics and philosophy will no longer simply be that the former uses the synthetic method and the latter the analytical method. On Kant’s mature account, both mathematics and philosophy must use a synthetic method. This [means] that the difference between the concrete constructions of mathematics and the abstract results of philosophy will have to be recast as a difference within the synthetic method: The use of the synthetic method in mathematics will yield synthetic yet certain results about determinate objects, whereas the use of the synthetic method in philosophy will yield synthetic yet certain principles for the experience of objects, or what Kant will call “schemata” of the pure concepts of the understanding, “the true and sole conditions for providing [these concepts] with a relation to objects”.[xxx]
This parallels the distinction we have already met with, namely Kant’s re-evaluation of the principle of reason as a regulative rule following the dissolution of the mathematical antinomies. Finally, Kant’s own definition of synthesis in the first Critique:
Now space and time contain a manifold of pure a priori intuition, but belong nevertheless among the conditions of the receptivity of our mind, under which alone it can receive representations of objects, and thus they must always also effect the concept of these objects. Only the spontaneity of our thought requires that this manifold first be gone through, taken up, and combined in a certain way in order for a cognition to be made out of it. I call this action synthesis.
By synthesis in the most general sense, however, I understand the action of putting different representations together with each other and comprehending their manifoldness in one cognition. […]
Synthesis in general is, as we shall subsequently see, the mere effect of the imagination, of a blind though indispensable function of the soul, without which we would have no cognition at all, but of which we are seldom even conscious. Yet to bring this synthesis to concepts is a function that pertains to the understanding, and by means of which it first provides cognition in the proper sense.
Now pure synthesis, generally represented, yields the pure concept of the understanding. By this synthesis, however, I understand that which rests on a ground of synthetic unity a priori; thus our counting (as is especially noticeable in the case of larger numbers) is a synthesis in accordance with concepts, since it takes place in accordance with a common ground of unity (e.g., the decad). Under this concept, therefore, the synthesis of the manifold becomes necessary.[xxxi]
Synthesis, in short, can be thought of as an operation of totalization. It is what “counts-as-one” a manifold or multiplicity of intuition, yielding a determinate representation. Kant says that “only the spontaneity of our thought” requires such synthesis – so can we then not identify this “cognitive spontaneity” with the drive for cognition – so that the drive is in itself spontaneous, unconditioned (and/or grounded in the void)?
Now what of the synthetic a priori, such as the truths of mathematics?
Kant derives the a priori status of synthetic propositions from the ground of synthetic unity that is the pure concept of the understanding. In the empirical case, this essentially means the spatiotemporal causal nexus or framework conditioning all possible objects of experience. And in the case of the synthetic activity of counting, for instance, it is grounded in a “common ground of unity”, for example the decad (i.e., a standard unit). Recall the stage in our number-theoretical thought experiment in which we asserted that since all numbers reduce to collections of units, there must either be a transcendently-immanent unit that generates the series of units, or a transcendent ground of unity that conditions them as a whole. Kant’s choice is the second – so that his appeal to the “decad” here is not unlike his appeal to God (where the decad serves as a transcendent number grounding the counting of all immanent numbers, securing them a determinate a priori place, God serves as the transcendent ground of existence grounding all immanent/contingent existence in the world). As Guyer and Wood further explain,
[W]hat Kant is saying is that judgments that are synthetic but also genuinely universal, that is, a priori, can be grounded in one of two ways: in the case of mathematics, such judgments are grounded in the construction of a mathematical object; in the other case, such judgments are grounded in the condition of determining the relative position of one object in space and time to others. […] Kant’s argument is that although all particular representations are given to the mind in temporal form, and all representations of outer objects are given to the mind as spatial representations, these representations cannot be linked to each other in the kind of unified order the mind demands, in which each object in space and time has a determinate relation to any other, except by means of certain principles that are inherent in the mind and that the mind brings to bear on the appearances it experiences. These principles will be, or be derived from, the pure concepts of the understanding that have a subjective origin yet necessarily apply to all the objects of our experience, and those concepts will not have any determinate use except in the exposition of appearances.[xxxii]
That is, in mathematics, synthetic a priori propositions are a priori because they are grounded in axiomatic definitions (i.e. definitions instituted by we ourselves, by fiat as it were), but are synthetic because they produce determinate results (e.g., “7+5=12” is a priori true because of, e.g., axioms about numbers and the addition operation, while it is synthetic because “12” cannot be analytically derived from either “7” or “5”). Whereas in the second kind of grounding, it is not the construction of an object in the order of the series, but the conditions of possibility for any such constructible object, e.g. an object synthesized in perception that entail the a priori status. The critical Kantian distinction here is that in the first case, reason immanently gives itself its own object, while in the second, reason requires that an object be given by a transcendent condition or ground.
Our core critical thesis contra Kant, then, is that in fact the boundary drawn here does not exist.
5 The Constructivist Account
Epistemologically, all judgments are synthetic. Analyticity is a pole of synthesis, an extreme form or an outer bound perhaps, yet still within the synthetic continuum (Kant would want us to say here, then, that analyticity is the outermost form preceding only the outer bound of the void). (Essentially, to satisfy this point, we can adopt Quine’s holistic epistemological model of the “web of knowledge/belief”, which though not essential for our account, is not incompatible with anything we will aver; furthermore, in this regard, we can think of analytic propositions as being like Wittgensteinian tautology and contradiction, i.e. a kind of more purely formal or even skeletal instance of proposition and precisely what remains structurally invariable as a kind of ur-frame in the variation of all other aspects, producing all possible synthetic statements that are therefore bounded by these minimalistic forms at either end of the continuum of variation, and which nevertheless remain immanently within it.)
All cognition is synthetic in the way that Kant characterizes the synthetic a priori construction of the objects of mathematics – or (and not quite the same thing), all cognition begins with nothing but the pure forms of intuition (space and time) and a drive for cognition, or a drive to synthesize. Logically begins – as Kant famously notes, there must be some actual empirical intuition (some received material) to get the whole process started chronologically (or in time, existing in experience). But after that, once the process is running, it is independent or autonomous from intuition (but not the pure forms of intuition, of course, which are also its conditions) – so that, e.g., once I have seen red but once, I can forever picture it in my imagination. Which is also to say: synthesis does not require a determinate intuition, but only the fact or the given that there was at least once at least one intuition, i.e., it is a fact that at some point something has been received, which secures the link to reality or the outside world, grounding the entire system and at least putting the onus on solipsism. Given that empirical fact as necessary presupposition, all that synthesis requires are the pure forms of intuition – because it can start from nothing, it can synthesize the void.
Charles Sanders Peirce, the independent co-founder of modern mathematical logic (though usually Frege is the one remembered, as with so many such pairs, e.g. Leibniz/Newton), read Kant at a relatively early age. In fact,
During his freshman year at college (Harvard), in 1855, when he was 16 years old, [Peirce] began private study of philosophy in general, starting with Schiller’s Letters on the Aesthetic Education of Man and continuing with Kant’s Critique of Pure Reason. After three years of intense study of Kant, Peirce concluded that Kant’s system was vitiated by what he called its “puerile logic,” and about the age of 19 he formed the fixed intention of devoting his life to study of and research in logic.[xxxiii]
He nevertheless had a similar conception of mathematics at the theoretical level – only he went in a different direction with it. That is, Peirce too thought that all the objects of mathematics were merely mental constructs of our own construction, so that where his contemporaries Russell and Frege wished to ground mathematics in logic, Peirce thought that mathematics was even more independent of empirical reality than logic and thus reversed the terms, i.e. grounding logic in mathematics. But Peirce saw the synthetic activity of mathematics as a model for the activity of cognition as such – seeing an underlying structural unity in the methods of geometry and algebra, for instance (famously opposed in this regard, even by Kant), as both forms of what he called diagramming.
This notion of cognitive diagramming aligns quite neatly in most respects with Wittgenstein’s Tractarian view of thought as a picturing. This picturing or diagramming, it is crucial to note, is strictly structural, or in more Kantian terms, purely formal – it is totally independent of empirical reality, requires no empirical intuition for material, and requires only certain structural conditions like the Kantian pure forms of pure intuition (we might say: the pure forms of receptivity as such). Thus far, meaning at least the last two paragraphs, we have not necessarily strayed that far from the Kantian account.
Now, Kant projects a transcendent ground of unity for all mathematical units, which for him is the One (and in his example, the decad); this unity is simply a formal condition for all objects of possible intuition whatsoever, even of pure intuition (so even the pure constructs of mathematics). He takes this view, I think, because he cannot countenance the prospect of mathematics and moreover transcendental unity being founded on the void – “founded on the void”, as in founded on a transcendent ground. But here we discover Kant’s own dialectical confusion – for to be immanently grounded in the void is not the same as Kant’s dialectical interpretation of the void as a transcendent ground that is merely absent. This foundation in the void, as a matter of fact, is something else entirely, which will dissolve the Kantian problem as a problem.
Mathematically/historically, we see that the long-standing prejudice for the transcendent One (quite prominent in ancient Greek mathematics and philosophy for instance) only gives way very late, though with the most significant of consequences (i.e., modern mathematics and logic, which are very plausibly conditions and often causes of the development of modern technology). Specifically, the singular foundation of all mathematics that comes to replace the postulate of a transcendent One is the synthesizing or totalization of the empty set (insofar as all of mathematics can be derived from or reduced to basic number theory, which can be more-than-adequately modeled according to the theory/constructive method of John H. Conway, which starts with this – as do all forms of set theory, another and more well-known foundational mathematics that can capture the required basic number theory). All numbers are not units – particular instances of a transcendent universal – but immanent unfoldings or elaborations of the empty set, nested enframings of nothing as such, immanent syntheses of the void.
What, then, is the meaning of the void? We might invoke the Heideggerian distinction between Being and beings here – the void is the fact of space rather than the determinate beings which can fill it, but we know that for Kant this pure or mere space is really just the pure forms of intuition – it’s not like intuition is in a room with no objects it can seize upon, but rather there is no intuition at all, only the structural potential for intuition – only the given conditions of possibility necessary for any determinate intuition to take place at all. That is, to talk of cognition as though it were language, here nothing is said, rather here language resides, creating a place where saying can take place. And to complete the metaphor, let’s put it in terms of mathematics-as-diagramming as language as cognition: then we could say that anything “said” or written or represented in, e.g., mathematics, by virtue of its very fact of being captured in a symbol, implies that it exists for the language (of mathematics) – so that if I write a variable x, we all understand that x is given as existing for the universe of mathematical discourse (and so that I do not have to more formally write “Ǝ[x]” every time I want to use the symbol). And that here where nothing is said – where nothing is said – there is only “Ǝ[ ]”. That is, the void would be the sheer facticity of immanent existence – and in fact the other symbol for the empty set in mathematics, besides “ QUOTE
”, is “[ ]”. This symbol, which lies as the transcendently-immanent singular ground of mathematics, symbolizes the capturing or totalization of the void.
It symbolizes the facticity of language, the sheer power of existence that language has, so that even nothing can be given a determinate existence, a fixed representation, a unique symbol in it – i.e., “can be said”. But whence the “transcendently”, then? The empty set, as symbol of the void, is also the symbolic mark of the subject of language in language. Insofar as it is the unique mark of totalization, or the very operation of cognition (naming in language, counting in mathematics, synthesizing sensible objects of perception, etc.), it is the proper name of the transcendental or structural subject. To complete the picture and elaborate these claims, I will draw on the psychoanalytic theory of Jacques Lacan. [xxxiv]
In essence, we can take the constructive corpus of mathematics and of language generally to constitute a single constructive corpus (again, a somewhat Quinean idea) that can simply be defined as structure, or as symbolic structure, i.e. what Lacan names the Symbolic Order. The symbolic order would therefore essentially constitute all the structural conditions of cognition, acting like the Kantian transcendental conditions of possibility (of reason) in giving the formal space for cognitive activity. This transcendental structure would likewise constitute a place for the subject, and this place (occupy-able by empirical subjects) would be the transcendental ego (boundary-shell/membrane delimiting interiority and exteriority). And since a given empirical subject would be, as part of reality or the Lacanian Real, strictly transcendent to the immanent order of the Symbolic (or of Kantian pure reason) – the structural mark of the transcendental ego names the transcendent empirical subject which it in no way depends on, but rather opens up a space for – i.e., creates a structure place where an empirical subject can come to be. This mark of the void can stand for the absent empirical subject because it names the transcendental place of subjectivity.
Notes
Note that Kant here seems to perhaps imply an identification of three items: the dark night of the supersensible; the space for/of intuition as such, i.e. space-of-intuition solely qua space; and finally, the void.
Cf. The Critique of Pure Reason p.466-7: “We have two expressions, world and nature, which are sometimes run together. The first signifies the mathematical whole of all appearances and the totality of their synthesis in the great as well as in the small, i.e. in their progress through composition as well as through division. But the very same world is called nature insofar as it is considered as a dynamic whole and one does not look at the aggregation in space or time so as to bring about a quantity, but looks instead at the unity in the existence of appearances.
[…] In regard to the distinction between the mathematically and the dynamically unconditioned toward which the regress aims, I would call the first two world-concepts in a narrower sense (the world in great and small), but the remaining two transcendent concepts of nature.”
Cf. The Critique of Pure Reason p.465: “Now one can think of this unconditioned either as subsisting merely in the whole series, in which thus every member without exception is conditioned, and only their whole is absolutely unconditioned, or else the absolutely unconditioned is only a part of the series, to which the remaining members of the series are subordinated but that itself stands under no other condition.”
In essence, an analytical opposition is symmetrical (“a” vs “ ~a”) while a dialectical opposition contains more than the mere, symmetrical negation (“a” vs “ ~a & b”); Cf. The Critique of Pure Reason p.517-18: “If someone said that every body either smells good or smells not good, then there is a third possibility, namely that a body has no smell (aroma) at all, and thus both conflicting propositions can be false. […]
Accordingly, if I say that as regards space either the world is infinite or it is not infinite (non est infinitus), then if the first proposition is false, its contradictory opposite, ‘the world is not infinite’ must be true. Through it I would rule out only an infinite world, without positing another one, namely a finite one. But if it is said that the world is either infinite or finite (not-infinite), then both propositions could be false. For then I regard the world as determined in itself regarding its magnitude, since in the opposition I not only rule out its infinitude, and with it, the whole separate existence of the world, but I also add a determination of the world, as a thing active in itself, which might likewise be false, if, namely, the world were not given at all as a thing in itself, and hence, as regards its magnitude, neither as infinite nor as finite. Permit me to call such an opposition a dialectical opposition, but the contradictory one an analytical opposition. Thus two judgments dialectically opposed to one another could both be false, because one does not merely contradict the other, but says something more than is required for a contradiction. (CPR p. 517-518)
Cf. The Critique of Pure Reason p. 513: “If, accordingly, I represent all together all existing objects of sense in all time and all spaces, I do not posit them as being there in space and time prior to experience, but rather this representation is nothing other than the thought of a possible experience in its absolute completeness. In it alone are those objects (which are nothing but mere representations) given.”
Cf. Kant’s footnote on Religion p. 7: “Since reason needs to presuppose reality as given for the possibility of all things, and considers the differences between things only as limitations arising through the negations attaching to them, it sees itself necessitated to take as a ground one single possibility, namely that of an unlimited being, to consider it as original and all others as derived. Since also the thoroughgoing possibility of every thing must be encountered within existence as a whole – or at least since this is the only way in which the principle of thoroughgoing determination makes it possible for our reason to distinguish between the possible and the actual – we find a subjective ground of necessity, i.e. a need in our reason itself to take the existence of a most real (highest) being as the ground of all possibility.”
[i] Cf., e.g., the first paragraph of the introduction to Religion Within the Boundaries of Mere Reason, p. vii
[iii] The Critique of Pure Reason p.470
[iv] The Critique of Pure Reason p. 471
[v] The Critique of Pure Reason p. 476
[vi] The Critique of Pure Reason p. 477
[vii] The Critique of Pure Reason p. 485
[viii] The Critique of Pure Reason p. 490
[ix] The Critique of Pure Reason p. 491
[x] The Critique of Pure Reason 521-23
[xi] The Critique of Pure Reason p. 519
[xii] The Critique of Pure Reason p. 508-9
[xiii] The Critique of Pure Reason p. 510
[xiv] The Critique of Pure Reason p. 514
[xv] The Critique of Pure Reason p. 518
[xvi] The Critique of Pure Reason p. 519
[xvii] The Critique of Pure Reason p. 520-1
[xxi] The Critique of Practical Reason p. 3
[xxii] The Critique of Practical Reason p. 3
[xxiii] The Critique of Practical Reason p. 4
[xxvii] Kant’s footnote on Religion p. 8
[xxviii] The Critique of Pure Reason p. 673
[xxix] The Critique of Pure Reason p. 32
[xxx] The Critique of Pure Reason p. 34
[xxxi] The Critique of Pure Reason p. 210-11
[xxxii] The Critique of Pure Reason p. 51-3
[xxxiv] Fink, The Lacanian Subject: Between Language and Jouissance p. 5-6:
“Lacan accounts for the foreignness [of the discourse of the Other in the self] as follows: we are born into a world of discourse, a discourse or language that precedes our birth and that will live on after our death. Long before a child is born, a place is prepared for it in its parents’ linguistic universe […] one cannot even say that a child knows what it wants prior to the assimilation of language: when a baby cries, the meaning of that act is provided by the parents or caretakers who attempt to name the pain the child seems to be expressing [this is also how the ego originally forms in the mirror-stage – it is not just that the infant recognizes its image in a moment of jouissance, but that the parent-figure approbates/reinforces this joy-of-recognition.]”
The Lacanian Subject p. 51-3:
“The parties to the vel of alienation that concern us here are not, however, your money and your life, but the subject and the Other, the subject being assigned the losing position (that of the money in the previous example, which you had no choice but to lose). In Lacan’s vel, the sides are by no means even: in his or her confrontation with the Other, the subject immediately drops out of the picture. While alienation is the necessary ‘first step’ in acceding to subjectivity, this step involves choosing ‘one’s own’ disappearance.
Lacan’s concept of the subject as manqué-à-être is useful here: the subject fails to come forth as a someone, as a particular being; in the most radical sense, he or she is not, he or she has no being. The subject exists – insofar as the word has wrought him or her from nothingness, and he or she can be spoken of, talked about, and discoursed upon – yet remains beingless. […] Alienation gives rise to a pure possibility of being, a place where one might expect to find a subject, but which nevertheless remains empty. Alienation engenders, in a sense, a place in which it is clear that there is, as of yet, no subject: a place where something is conspicuously lacking. The subject’s first guise is this very lack.
Lack in Lacan’s work has, to a certain extent, an ontological status: it is the first step beyond nothingness. To qualify something as empty is to use a spatial metaphor implying that it could alternatively be full, that it has some sort of existence above and beyond its being full or empty. A metaphor often used by Lacan is that of something qui manque à sa place, which is out of place, not where it should be or usually is; in other words, something which is missing. Now for something to be missing, it must first have been present and localized; it must first have had a place. And something only has a place within an ordered system – space-time coordinates or a Dewey decimal book classification, for example – in other words, within some sort of symbolic structure.
Alienation represents the instituting of the symbolic order – which must be realized anew for each subject – and the subject’s assignation of a place therein. A place he or she does not ‘hold’ as of yet, but a place designated for him or her, and for him or her alone. When Lacan says (in Seminar XI) that the subject’s being is eclipsed by language, that the subject here slips under or behind the signifier, it is in part because the subject is completely submerged by language, his or her only trace being a place-marker or place-holder in the symbolic order.
The process of alienation may, as J.-A. Miller suggests, be viewed as yielding the subject as empty set, { QUOTE
}, in other words, a set which has no elements, a symbolic which transforms nothingness into something by marking or representing it. Set theory generates its whole domain on the basis of this one symbol and a certain number of axioms. Lacan’s subject, analogously, is grounded in the naming of the void. The signifier is what founds the subject; the signifier is what wields ontic clout, wresting existence from the real that it marks and annuls. What it forges is, however, in no sense substantial or material.
The empty set as the subject’s place-holder within the symbolic order is not unrelated to the subject’s proper name. That name is often selected long before the child’s birth, and it inscribes the child in the symbolic. A priori, this name has absolutely nothing to do with the subject; it is as foreign to him or her as any other signifier. But in time this signifier – more, perhaps, than any other – will go to the root of his or her being and become inextricably tied to his or her subjectivity. It will become the signifier of his or her very absence as subject, standing in for him or her.”
