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Kant defines noumena like this:

if, however, I suppose that there be things that are merely objects of the understanding and that, nevertheless, can be given to an intuition, although not to sensible intuition (as coram intuiti intellectuali), then such things would be called noumena (intelligibilia). (A249)

(copied from SEP).

Kant considers mathematical objects to be objects, but he considers them to be cognized at least in part through sensibility. He doesn't seem to make Frege's ontological distinction between objects connected to the senses in some way (such as physical objects and geometrical objects) and objects that are not, such as numbers. Frege wants to prove that numbers can be known through pure logic (that is, without an intuition), but this is problematic, and an alternative view is that numbers are known through an entirely non-sensible intuition. Kant doesn't have a form of intuition that would work for this--at least not a form of intuition available to mortals, but it seems like a fairly straightforward extension to add a form of intuition connected to the understanding instead of to sensibility. And in this case, it seems to me that those abstract objects would be noumena.

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  • Kant does use phrases like "intuitive understanding" and "intellectual intuition" a few times (I used to assume these were the same in meaning, but some Förster fellow has a robust interpretation to the contrary). There is also a subtlety in his distinction between forms of intuition and formal intuition: so that he indirectly describes space as a "pure object," and, "Phenomena as objects of perception are not pure, that is, merely formal intuitions, like space and time, for they cannot be perceived in themselves," so he seems to have space and time "in themselves" (as some sort of object). Commented May 22 at 12:06
  • No. Kant did construe mathematical objects as objects and not mere concepts in CPR. They are attached to pure intuitions of productive imagination. He specifically emphasized that one cannot prove theorems about them otherwise in his examples with isosceles triangle and 7+5=12. That is how mathematics is "synthetic a priori". Frege's abstract objects, on the other hand, do not need intuitions, at least, not in arithmetic (he hesitated on geometry). But mathematical objects are the closest thing we have to noumena as cognized by "intellectual intuition" of their creator denied to us in CJ.
    – Conifold
    Commented May 22 at 12:11
  • @david, what do you mean by "you need a form of intuition to cognize( become aware of) their representation?"
    – lee pappas
    Commented May 22 at 12:17
  • @leepappas, I mean that Kant believed mortals can only be aware of an object through an intuition that represents the object under the categories, so if this were true for abstract objects, you would need a form of intuition that is not connected to sensibility--or so I thought, but Conifold seems to disagree. Commented May 22 at 12:54
  • @Conifold, yes, Frege thought we could become aware of objects through pure logic, but I was applying Kantian epistemology to Fregean ontology. So, if Kant thinks mathematical objects are objects, doesn't he also say that the only way to become aware of an object is by intuition, and that all intuition is connected to sensibility? It seems like a contradiction. Commented May 22 at 12:58

3 Answers 3

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At one point in the first Critique, Kant says:

For if I cogitate an understanding which was itself intuitive (as, for example, a divine understanding which should not represent given objects, but by whose representation the objects themselves should be given or produced), the categories would possess no significance in relation to such a faculty of cognition. They are merely rules for an understanding, whose whole power consists in thought, that is, in the act of submitting the synthesis of the manifold which is presented to it in intuition from a very different quarter, to the unity of apperception; a faculty, therefore, which cognizes nothing per se, but only connects and arranges the material of cognition, the intuition, namely, which must be presented to it by means of the object. But to show reasons for this peculiar character of our understanding, that it produces unity of apperception a priori only by means of categories, and a certain kind and number thereof, is as impossible as to explain why we are endowed with precisely so many functions of judgement and no more, or why time and space are the only forms of our intuition.

And then elsewhere: "If, by the term noumenon, we understand a thing so far as it is not an object of our sensuous intuition, thus making abstraction of our mode of intuiting it, this is a noumenon in the negative sense of the word. But if we understand by it an object of a non-sensuous intuition, we in this case assume a peculiar mode of intuition, an intellectual intuition, to wit, which does not, however, belong to us, of the very possibility of which we have no notion—and this is a noumenon in the positive sense."

So the equation of a faculty for apprehending Fregean logical objects with a faculty of intuitive understanding à la Kant appears plausible (as an extension of either philosopher's musings). In Brewer[18] we have:

According to a received view, Kant maintains that only God could possess an intuitive intellect, i.e., a mind with an ‘intellectual intuition’ or, equivalently, an ‘intuitive understanding’ of objects. Such a view finds support in the first Critique, for instance, where Kant writes, “intellectual intuition […] seems to pertain only to the original being” and in lecture transcripts where we read, “[o]nly the understanding of God is called intuition.” However, such a view is also brought into question by neglected texts suggesting that it involves a considerable oversimplification of his position. The second essay of this dissertation examines these texts in an effort to illuminate Kant’s conceptions of the intuitive and human intellects.

A review of Förster[12] offers that a complicating factor, here, is Kant's theory of moral knowledge:

Yet Förster contends that Kant changed his initial characterization of his project in light of the Göttingen review and his subsequent discussion with Garve. First, in attempting to distance himself from Berkeley by arguing that our representation of space must be a priori, rather than empirically derived, Kant came to realize that he needed to provide a schematism of space to supplement the schematism of time and attempted to work this out in The Metaphysical Foundations of Natural Science. Second, after facing Garve’s objection, raised in private correspondence, that he had failed to justify his claims about our cognition of the moral law, Kant set out to account for how specifically moral knowledge is possible, a project he attempted to carry out in The Groundwork for the Metaphysics of Morals. Förster argues that this shift signals a fundamental alteration in the critical project itself: it is now no longer a question of how a priori reference to non-empirical objects is possible but rather a question about the possibility of synthetic a priori cognitions such as knowledge of the categorical imperative.

Carrying out this project in the Second and Third Critiques, Kant maintains that to hope for the highest good, i.e. a world in which happiness is proportioned according to virtue, reason must believe in a supersensible ground to unify appearance and reality. Though reason is forced, on pain of internal contradiction, to believe in such a supersensible ground, Kant insists that this ground cannot be cognized by discursive beings like us. Förster argues that in sections 76 and 77 of the Third Critique, Kant then develops two contrasts with our discursive understanding: intellectual intuition and intuitive understanding. On this reading, an intellectual intuition is an intuition capable of grasping things in themselves, whereas an intuitive understanding is something which cognizes a whole and, by cognizing this whole, cognizes also the parts constituted by it. Kant conceives of both of these non-discursive ways of knowing as belonging to divine cognition, which, through intellectual intuition, grasps all actuality via an awareness of which possibilities it has chosen to actualize in creation, and which, through intuitive understanding, grasps the totality of possibility by grasping its own essence.

But so Kant in the second Critique will say, "We may call the consciousness of this fundamental law a fact of reason, because we cannot reason it out from antecedent data of reason, e.g., the consciousness of freedom (for this is not antecedently given), but it forces itself on us as a synthetic a priori proposition, which is not based on any intuition, either pure or empirical. It would, indeed, be analytical if the freedom of the will were presupposed, but to presuppose freedom as a positive concept would require an intellectual intuition, which cannot here be assumed ... It is therefore allowable to use the system of the world of sense as the type of a supersensible system of things, provided I do not transfer to the latter the intuitions, and what depends on them, but merely apply to it the form of law in general."

Meanwhile, Kant also (elsewhere, here and here) says of space and time:

But apart from this [transcendental] relation, a priori synthetical propositions are absolutely impossible, because they have no third term, that is, no pure object, in which the synthetical unity can exhibit the objective reality of its conceptions. ... Phenomena as objects of perception are not pure, that is, merely formal intuitions, like space and time, for they cannot be perceived in themselves.

So I will say that either Kant is not using his terminology in the clearest possible way, or at least the translations I'm using murk the scene gravely enough for me to also say that it seems possible to represent Kant as claiming that pure mathematical knowledge depends on a relativization of the concept of noumena such that we do have noumenal intuition of space and time in themselves, but that:

Although, then, respecting space, or the forms which productive imagination describes therein, we do cognize much a priori in synthetical judgements, and are really in no need of experience for this purpose, such knowledge would nevertheless amount to nothing but a busy trifling with a mere chimera, were not space to be considered as the condition of the phenomena which constitute the material of external experience.

Whether Kant could've arrived at a more Frege-aligned stance on his own general terms is not necessarily clear, either. That is, his bifurcation of the faculties of understanding and sensibility might not be so strict, even per his own thesis. In the Transcendental Dialectic, he says:

The different phenomenal manifestations of the same substance appear at first view to be so very dissimilar that we are inclined to assume the existence of just as many different powers as there are different effects—as, in the case of the human mind, we have feeling, consciousness, imagination, memory, wit, analysis, pleasure, desire and so on. Now we are required by a logical maxim to reduce these differences to as small a number as possible, by comparing them and discovering the hidden identity which exists. We must inquire, for example, whether or not imagination (connected with consciousness), memory, wit, and analysis are not merely different forms of understanding and reason. The idea of a fundamental power, the existence of which no effort of logic can assure us of, is the problem to be solved, for the systematic representation of the existing variety of powers. The logical principle of reason requires us to produce as great a unity as is possible in the system of our cognitions; and the more the phenomena of this and the other power are found to be identical, the more probable does it become, that they are nothing but different manifestations of one and the same power, which may be called, relatively speaking, a fundamental power.

But then:

The logical principle of genera, which demands identity in phenomena, is balanced by another principle—that of species, which requires variety and diversity in things, notwithstanding their accordance in the same genus, and directs the understanding to attend to the one no less than to the other. This principle (of the faculty of distinction) acts as a check upon the reason and reason exhibits in this respect a double and conflicting interest—on the one hand, the interest in the extent (the interest of generality) in relation to genera; on the other, that of the content (the interest of individuality) in relation to the variety of species. In the former case, the understanding cogitates more under its conceptions, in the latter it cogitates more in them. This distinction manifests itself likewise in the habits of thought peculiar to natural philosophers, some of whom—the remarkably speculative heads—may be said to be hostile to heterogeneity in phenomena, and have their eyes always fixed on the unity of genera, while others—with a strong empirical tendency—aim unceasingly at the analysis of phenomena, and almost destroy in us the hope of ever being able to estimate the character of these according to general principles.

And his earlier tangent about levels of representation offers definitions of roughly eight terms, which definitions the understanding/sensibility distinction is itself embedded in or interpolable with. I wonder, then, if Kant could have allowed for a finer-grained compartmentalization of our intuition than he appears, on the surface, to have; or then does he already have possible room for a nonspatial/nontemporal but also nondivine intuition? (For one might object that knowing space and time in themselves, by intuition, is not yet knowing Fregean logical objects, since those are abstract and "not in space and time." However, here we face the puzzle of space and time in themselves, relative to the abstract/concrete distinction; and Kant says of a cosmological space that it is not within itself,F after all, so that space as a whole is not inside of itself, i.e. is abstract by the usual criterion, despite enveloping concreta by the by.)


F"... all places are in the universe, and the universe itself is, therefore, in no place."


Condensed:

  1. The closest thing to the OP question's target, in Kant's own writings, would seem to be Kant's talk of intellectual intuition and intuitive understanding. The former seems ruled out as a human faculty; the matter is less clear with respect to the latter. Or perhaps Kant thought of moral objects like "we" think of mathematical objects nowadays, and so would have had all abstract objects as moral objects, and so still with intellectual intuition for them (yet not for us).

  2. Alternatively, due to obscure terminological factors, the formal intuition of space and time is an intuition of pure mathematical objects, however subjective they might be in a deeper sense.

  3. Or maybe Kant's differentiation-of-faculties principle could be applied so as to devise a Kant-inspired network of types of intuitions that has something between formal intuitions of space and time on the one hand, and intellectual intuition/intuitive understanding on the other, something more accessible to the human mind than the latter and more Fregean than the former.

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  • Sorry, but I don't understand what you're trying to say. You seem to quote random passages which have relatively little to do with the topic. Commented May 23 at 0:53
  • @abracadabra mostly I am citing direct and indirect references to Kant's talk of intellectual intuition and intuitive understanding, which have everything to do with the topic, after all. The other main part is explaining that in Kant, space and time are pure objects and the basis of mathematical knowledge, by intuition, so that there is a sense in which we have an intuition of space and time "in themselves." Since Kant is not clearly terminologically consistent, I think I am being sufficiently fair to him to allow that he is not inconsistent if different passages are mutually interpreted. Commented May 23 at 1:04
  • Sure, some of the passages are directly connected with the question, but, e.g. the passage that you cite from "On regulative use..." appendix to the dialectic, is mostly unrelated to the topic (although it is in a way that I cannot explain in this brief reply related to issues of completeness/incompleteness in phil. of math, but your comments don't touch on this question at all). My main objection is that I think that Kant's theory of moral knowledge etc. are mostly grounded in his theoretical philosophy and not vice-versa. Therefore it makes no sense to mention them... [1/2] Commented May 23 at 1:59
  • ...when discussing philosophy of mathematics. I cannot agree with Foerster that Metaphysical Foundations are somehow an afterthought to Kant's project. Most of the Critique doesn't make sense without MFNS. Most of the changes made to the B-Edition of the Critique don't make much sense without the MFNS either. But the 2nd and the 3rd Critiques were an afterthought. So basically, although there is, perhaps, broadly speaking, a link between the Third Crtitique and Kant's philosophy of mathematics, I just wouldn't look there before settling exegetical issues with the First Critique. [2/2] Commented May 23 at 2:01
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Kant doesn't think that mathematics is concerned solely with concepts. He in fact thinks that mathematics is concerned with constructing objects in intuition (which is a view that people like Brouwer later adopted under the name of "intuitionism", sometimes "constructivism", also cf. Doctrine of Method of the first Critique). This is indeed related to the question of the thing in itself, but not in the way you think it is.

It shouldn't be seen as anachronistic to ascribe to Kant the view that mathematical theories have their models only in appearances, since Kant's discussion of these issues in the Transcendental Dialectic links directly to modern metamathematics through Leibniz's proto proof theory. There is no actual infinity (nowadays we use the term "transfinite" to name this concept) in the world of appearances - this is something that Kant stresses in relation to our empirical knowledge in the Antinomies (cf. Suppes' great article) section of the Critique. But it also has implications for mathematics, since, if our mathematical theories are realised by the appearances, this implies a finitistic, proto-intuitionistic view of mathematics.

Kant's view has some further implications for his general epistemology, since it implies that there is no clear disambiguation between a mathematical picture and a sensible picture in his writings (e.g. his Metaphysical Foundations of Natural Science are concerned with schematizing the pure empirical concepts of matter and motion onto space, which is directly related with giving mathematical meaning to the dynamical categories, especially causality, which becomes force; for Kant these goals are basically identical, since space and time are the sources of all mathematical relations).

Kant's philosophy of mathematics of course doesn't end here. There is some contention regarding his motivations for the view that mathematics is related to intuition in this way. This view was not at all uncommon among people that Kant read, like Euler and Newton, however, so his views weren't very controversial at the time. Some however suggest that Kant was mainly persuaded by limitations of the logic of his time (e.g. what you mention). There are also some questions regarding the differences between mathematical (pure) and empirical intuition, since the latter is transcendentally constrained in a way in which the former isn't and cannot be (we can't demand mathematical figures to be composed solely of material points, for example, that would be nonsensical). David Hyder in his great book The Determinate World shows that Kant's disagreement with Newton regarding the nature of space-time must extend to the realm of philosophy of mathematics, if Kant is to be consistent. Kant also elaborates a theory of how algebra and analysis work under his model which is crucial for understanding some parts of the Critique, but mostly unrelated to his broadly constructivistic views.

What is then the link between the concept of thing in itself and philosophy of mathematics, especially the dispute between logicism and intuitionism? Well, if we allow mathematics to have models not just in the appearances, but in things in themselves (which are on Kant's view basically "bare particulars"), then we adopt basically a Platonistic, Fregean-Gödelian, infinitistic, extensionalistic view of mathematics. Kant is quite explicit about the relation between the idea that things in themselves are bare particulars (the relevant section of the Critique is Amphiboly of the Concepts of Reflection) and the redundancy of (what we would call) singular terms (intuitions) in place of (what we would now call) definite descriptions. This means in practice, as we know and Kant probably didn't, that various features of classical logic, its "non-perspectivality", hold, including the LEM.

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According to Kant

if, however, I suppose that there be things that are merely objects of the understanding and that, nevertheless, can be given to an intuition, although not to sensible intuition (as coram intuiti intellectuali), then such things would be called noumena (intelligibilia). (A249)

Paraphrasing

If, however, I suppose that there be things that are merely objects of the understanding and that, nevertheless, can be [given?] to an immediate understanding without evident conscious thought or rational inference, although not to sensible intuition (as before looked at known), then such things would be called noumena.

Now Kant says the understanding has no concepts available to it other than the 12 categories. The number one, would thus be an object of the category Unity. Thus, the number one would be a noumenon. Understanding of the number one is obtained in utero, before the mind is flooded with sensory data, by the fetal mind knowing the self is one thing. The understanding of god as Kant imagined, is obtained according to Kant by intellectual intuition, an ability foreign to mortals. Intellectual intuition is Kant's term for non-sensible intuition. Thus Kant is wrong that only god has intellectual intuition, since any fetus understands the number one.

Now to Frege.

From the SEP

Frege's abstract/concrete division is made through use of the statement

A thing is abstract if and only if it's non-mental and non-sensible.

The problem is, electrons, protons, etc. are abstract, according to Frege's definition of abstract objects.

So the answer to your question is at an impasse.

Now to the OP.

Kant considers mathematical objects to be objects, but he considers them to be cognized at least in part through sensibility. He doesn't seem to make Frege's ontological distinction between objects connected to the senses in some way (such as physical objects and geometrical objects) and objects that are not, such as numbers. Frege wants to prove that numbers can be known through pure logic (that is, without an intuition), but this is problematic, and an alternative view is that numbers are known through an entirely non-sensible intuition. Kant doesn't have a form of intuition that would work for this--at least not a form of intuition available to mortals, but it seems like a fairly straightforward extension to add a form of intuition connected to the understanding instead of to sensibility. And in this case, it seems to me that those abstract objects would be noumena.

  • I don't see how Kant sees numbers as cognized at least in part by sensible intuition. The natural numbers are built up from the number one, and the operation of addition. Thus they, even by Kantian confusion, are noumena.

  • Kant might not explicitly distinguish between objects connected to the senses in some way, and those that are not e.g. numbers, but surely he was aware of the division.

  • Frege wanted to prove that numbers can be known through pure logic, because of his set theoretic definition of the number one. He wanted to bypass intuition, and this is indeed problematic since a fetus understands the number one by intuition. Kant does have a form of intuition that would work for this, intellectual intuition, but sadly Kant saw this as a non-mortal ability.

Now OP clearly states their position.

it seems like a fairly straightforward extension to add a form of intuition connected to the understanding instead of to sensibility. And in this case, it seems to me that those abstract objects would be noumena.

Indeed, had Kant viewed his intellectual intuition as a mortal ability, connected to the understanding of numbers, it appears to me also that numbers as noumena would be abstract objects. And generalizing this, we could say Kantian noumena are Fregian abstract objects.

Nous is the ancient Greek word for mind, so if Kant meant by the term noumena mind-things, your conclusion is further supported.

The main problem in equating the two seems to be, Frege's definition of abstract objects is unclear.

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