EFRAÍN LAZOS | Disonancias de la Crítica. Variaciones sobre cuatro temas kantianos | Instituto de Investigaciones Filosóficas UNAM 2014


By Edgar Valdez

Efraín Lazos’s Disonancias de la Crítica is a collection of four essays that seek to resolve four of the most prevalent polemics in contemporary interpretations of Kantian epistemology. Each essay provides new insights into debates that have held the attention of contemporary Kant scholarship for decades. In the first, Lazos deals with the relationship between concepts and intuitions, making a contribution to the debate between conceptualism and nonconceptualism. In the second and third essays, he seeks to distinguish Kant’s idealism from the idealism of Berkeley and Descartes respectively, by overcoming their dogmatism and scepticism. Lazos in turn shows why Kant is not susceptible to the same criticisms as the others. In the final essay, Lazos considers transcendental apperception and in distinguishing it from Descartes’s cogito considers what is unique to Kant’s theory of self-consciousness. The essays do not comprise a single argument but they certainly comprise an integrated view of Kant’s epistemology.

I shall limit my remarks here to the first essay and its contribution to the conceptualist debate in part because in my view it is the most consequential of the issues brought up in the book but also because Lazos poses some challenges for all sides of the debate concerning Kant’s claim that concepts without intuitions are empty and intuitions without concepts are blind. The famous dictum from Kant affirms the epistemic complementarity or interdependence of concepts and intuitions. Lazos’s point is that such interdependence requires a psychological independence. The latter is given in Kant via a difference in quality—not degree—between the two representations. I doubt there is any way around Lazos’s reading of the Elementarlehre as requiring interdependence and qualitative distinction. Problematic for both sides of the conceptualism debate is that the conjunction of interdependence and qualitative distinction requires a balance that is easily disrupted in choosing sides. Emphasising concepts or intuitions can lead to a priority of one that forms a one-sided dependence. While a strong interdependence that puts concepts and intuitions on a par with each other can erase the qualitative distinction. And so Lazos, though he has chosen a side, has tethered both sides of the debate to the centre with a Kantian truism.

I first want to acknowledge that overall I think conceptualist readings of the Elementarlehre do a disservice to the great lengths Kant goes to insist on a distinction between the faculties of sensibility and understanding. While there is value in investigating what conditions and guarantees the application of concepts as general representations to intuitions as particular representations, a solution that simply subsumes one under the other—regardless of which is subsumed—seems contrary to what Kant lays out in the Transcendental Aesthetic and the Prolegomena. Likewise, I agree with Lazos that the significance of Kant’s assertion that concepts and intuitions are both required for cognition is undercut if we do not first acknowledge the difference between concepts and intuitions. Only if there is a difference of kind between the two representations can it be revolutionary to highlight their working in consort to yield cognition. Identifying the difference between them should also provide insight into the cognitional mechanism Kant outlines.

One way in which I resist Lazos’s characterisation of the Elementarlehre is in what he calls the psychological independence of concepts and intuitions:

[P]sychological independence is derived from the Kantian conviction that there is a difference in nature, not in degree, between intuitions and concepts; such conviction entails that, in general, and for human beings, each of our capacities of representation, the receptive or the spontaneous one, may be exercised without its counterpart. (p. 19)

This psychological-independence thesis is often what he calls on to refute elements of conceptualism, particularly McDowellian conceptualism. For Lazos, psychological independence can be affirmed in the light of Kant’s claim that the differences between concepts and intuitions cannot be reduced to differences of degree and that there is between them a difference in kind. Intuitions are particular representations that are directly related to an object while concepts are general representations that relate to objects only via marks that can be shared by many objects. These two kinds of representation correspond to two faculties involved in cognition, sensibility and understanding. The former is receptive to intuitions while the latter is active in its employment of concepts. For Lazos, the psychological independence of the two means we can exert one faculty while the other remains inert. This principle is then employed when he wants to reject elements of conceptualism as it would seem to require concepts to always be employed when we have intuitive representations.

It certainly follows that in order to salvage something genuinely Kantian we must be able to find sufficient qualitative difference between concepts and intuitions. Our cognition requires two different kinds of representation and the error of the metaphysicians that predate Kant is, in his view, their inability to recognise this distinction. When we reduce all cognition to a single kind of representation we either deflate our cognitional content or allow our metaphysical ambition to run unchecked. We do not, however, want to push the psychological independence too far or demand that it play too large a role in the exegesis of Kant’s epistemology.

The first reason for this is that I think something like divisibility will suffice to make Lazos’s claim concerning transcendental idealism. If we can distinguish the two kinds of representation and identify what each contributes to cognition we need not separate them in order to affirm Kant’s opening remark from the Elementarlehre. More importantly, the psychological intermingling of our representations suggests that a genuine independence may be out of reach. Lazos’s standard for independence is that we can exercise one without recourse to its counterpart, but a disconnect between a given representation and its corresponding counterpart is not the same as a separation of the faculties more generally. I can be intuitively receptive to redness without recourse to the concept of red but I am not in that moment conceptually inert. Kant is not capriciously emphasising the need to recognise the twofold nature of the representations involved in cognition. Rather he is making such an appeal because of our constantly being immersed in the duality.

The German rationalists who deduced God’s existence failed to connect (or attempt to connect) their concepts to a corresponding intuitive representation but they were not without intuitive representation altogether. Their employment of concepts necessitated spatiotemporal receptivity. Their affirmation of the existence of God was for Kant accompanied by intuitive representation, their failure was in not letting intuition serve as guide for their deduction as well instead of simply providing content to the conclusions. My worry in this case is that Kant may not be equipped to bridge the divide between concepts and intuitions if it is too deep. As it stands, the schematism (more on that later) is supposed to provide an explanation for how concepts apply to intuition. Kant thinks that he may fall short of explaining the applicability of concepts to intuitions if the schematism is merely a higher order conceptualism. In that case, intuitions may not derive their form from concepts but schemata would.

Lazos takes the oft-quoted dictum that “[t]houghts without content are empty, intuitions without concepts are blind” (A51/B75) to emphasise the independence that he finds conceptualists to be ignoring. But to really find the dictum and the conceptualist view at odds with one another we need a strong reading of both. Lazos’s reading must insist that there are concepts employed without any reference to intuitions and intuitions without any connection to concepts and that we employ them in some form of proto-cognition. He then must also read conceptualism as the view that ascribing any conceptual markers or traits to intuition is in conflict with affirming that there are blind intuitions. But there is an implied epistemic normativity that calls on us to unite concepts and intuitions. We may fall short and fail to unite an intuition with the correct corresponding concept or a concept with a corresponding intuition. Immediately following the dictum Kant says:

It is thus just as necessary to make the mind’s concepts sensible (i.e., to add an object to them in intuition) as it is to make its intuitions understandable (i.e., to bring them under concepts). (A51/B75)

This suggests that the exercise of either faculty at the very least anticipates or calls for, if not requires, the exercise of the other. This in no way compromises the distinction of the faculties and their corresponding contributions. I also think that many ways of resisting conceptualism would still involve intuition having certain properties that predispose it to be conceptualised. Lazos’s definition for nonconceptualism is that “taken generally, nonconceptualism states that human beings are in possession of certain cognitive capacities that are pre-discursive” (p. 19), but this language of thinking of intuitions as pre-discursive implies intuitions may not be discursive but call for discursivity about them. But it is certainly not a conceptualist reading of Kant to suggest that intuitions anticipate conceptual application. That is, conceptualism must involve something stronger than anticipating concepts to really connote a different way of reading the Elementarlehre.

I think the best way to distinguish between conceptualist and nonconceptualist readings of Kant is by how one reads the footnote at B160:

Space, represented as object (as is really required in geometry), contains more than the mere form of intuition, namely the comprehension of the manifold given in accordance with the form of sensibility in an intuitive representation, so that the form of intuition merely gives the manifold, but the formal intuition gives unity of the representation. In the Aesthetic I ascribed this unity merely to sensibility, only in order to note that it precedes all concepts, though to be sure it presupposes a synthesis, which does not belong to the senses but through which all concepts of space and time first become possible. For since through it (as the understanding determines the sensibility) space or time are first given as intuitions, the unity of this a priori intuition belongs to space and time, and not to the concept of the understanding (§24). (B160–1n.)

This footnote gives rise to several interpretative problems but most fundamentally brings into focus Lazos’s concern about the relationship between sensibility and understanding. On the one hand, the footnote suggests that space admits of a unity that is prior to all concepts. On the other hand, it claims that all unity must be attributed to the understanding. The clause that takes priority in a given interpretation operates as determining factor in the conceptualist/nonconceptualist divide. Onof & Schulting (2015) call Hermann Cohen, one of the conceptualist targets of Lazos’s criticism, a strict conceptualist in that he holds that all the unities in question rely on the understanding by virtue of conceptual unity. Onof and Schulting identify Éric Dufour (2003) as following in Cohen’s footsteps in holding conceptual unity to be the mode of affection of the understanding on sensibility although he limits the unities in question to geometrical ones. Such strict conceptualism that seems to entirely subsume intuitive representation under conceptual representation seems especially in conflict with Kant’s dictum if only because Kant’s reproach is reserved especially for the rationalists who have overlooked intuition. But the psychological-independence thesis, though it stands opposed to a strict conceptualism does not provide an explanation for spatial or even geometrical unities.

Béatrice Longuenesse (1998) suggests that after reading the footnote we must reread the Transcendental Aesthetic taking the unity of the intuitive representations of space and time to result from the understanding. Onof and Schulting, however, consider Longuenesse and Michael Friedman (1992, 2000, 2012) to represent a broad conceptualism in that they hold the unity of space to be owed in part to the understanding but is a pre-discursive application “in the sense of being prior to the application of any particular concepts of space in a judgment about space, but not prior to the application of the categories” (2015:9). And so the understanding affects sensibility in a way that makes our receptivity to objects amenable to cognition but does not necessarily reduce intuitive content to conceptual activity. This way of reading the footnote imposes a kind of categorial property on intuitive representation that affords concepts greater significance than they are afforded in the Transcendental Aesthetic. As such, it suggests that the understanding is responsible for the structure of our receptivity. It is possible that the Elementarlehre betrays him but Kant certainly does not want such a priority: “Neither of these properties is to be preferred to the other” (A51/B75) If anything, Kant’s insistence that that intuition deals with objects directly and in concreto and his claim that rationalist metaphysicians have ignored the intuitive element of cognition might leave open a possibility of a priority of intuition over concepts.

While ultimately Longuenesse’s position is untenable for Lazos, his psychological-independence thesis makes impossible any such compromises—in terms of reading the footnote in a noncontradictory manner—that would leave room for an effect of the understanding on sensibility that does not impose a conceptualist priority on the faculties. In my view, Longuenesse’s tempered approach to the effect of the understanding on sensibility still prioritises concepts but interpretations like it that attempt to situate the unity attributed to space while preserving the contribution of sensibility would seem to be ruled out by the idea of being able to exercise one faculty without the other.

The uniqueness of Kant’s epistemological contribution comes from affirming not just complementarity but a unity between heterogeneous faculties, one that is best understood by looking at pure mathematics. Kant’s Copernican Revolution in philosophy calls for a critical step in philosophy that is a matter of reason examining its own capacities in order to comprehend what it can and cannot know. As a result of this autocritique, Kant holds that we must abandon traditional metaphysics in favour of rigorous epistemology.

The shift from metaphysics to epistemology involves both a negative and positive step. There is, on the one hand, the negative element that denies the possibility of noumenal knowledge of the traditional objects of metaphysics. On the other hand, there is the positive step that makes an epistemological affirmation about how we come to synthetic a priori knowledge. The former occurs when we recognise that dogmatic rationalism does not know the limits of reason and ventures into subreption. The latter occurs when we turn to mathematics to reveal the epistemological truths about how synthetic a priori judgements are possible. The negative step provides us with a criterion for the kind of cognition we want to admit while turning to mathematics is the first affirmative of how such a criterion is met. One element that is often missed in considering such a revolution is Kant’s view on the significance of mathematics. That is, part of this revolution involves considering pure mathematics in a way that Kant also sees as distinct from his predecessors. The negative and positive elements correspond to a claim that distinguishes the faculties and the step that maintains their unity. Lazos’s psychological-independence thesis could leave us affirming the negative step without recourse to the positive step.

The possibility of pure mathematics points us to the unity in cognition between the particulars of experience and universal concepts. This unity is the one between intuitions and concepts for Kant; it is the unity between sensibility and understanding. Once we have admitted that our knowing involves heterogeneous faculties and that knowing cannot be attributed to either of those faculties in isolation, we must account for the applicability of those faculties to one another. Our cognitions must always be a unity of intuitive particulars and conceptual universals. This unity of universals and particulars is at the core of Kant’s notion of schema and it is central to the epistemic status of categories as applicable to objects of intuition. For Kant, mathematics is in a unique position to inform our epistemology because we are assured of both its necessary truth and its applicability to possible experience.

For Kant, the possibility of synthetic a priori principles is the general problem of pure reason and he sees a Copernican character to his shift in attention away from objects and towards the human epistemic condition to understand the limits of reason. Just as Copernicus sought to turn away from the nature of the objects in the sky to the epistemic conditions for such appearances, so Kant wants to turn away from the nature of the objects of metaphysics and towards an understanding of the epistemic conditions of the representations of those objects. Instead of asking how our cognition can correspond to objects, for Kant we must ask how objects conform to our cognition. But the conformity is one Kant takes as a given. Kant’s reversal is a reversal in identifying the source of the conformity, not one of verification.

The failure of the metaphysicians before him to ask about how objects can conform to cognition has allowed them to affirm and deny the existence of objects that reason cannot cognise. What is dogmatic about them is the inherent assumption of the unity of objects and our cognitions of them. Such a position assumes that our concepts conform to objects and that as such the proper focus on the source and nature of the objects will yield accurate cognition. In so doing, both rationalists and empiricists alike mischaracterise aspects of our knowing in subsuming them under their methods. The rationalist characterises the receptive elements of knowing as active or innate while the empiricist characterises the active elements as reproductive and contingent. For Kant, rationalists reject the particulars of experience as having anything to do with a priori knowledge and empiricists reject relations of ideas as having any content for cognition. Kant’s transcendental idealism ultimately holds that they each are involved in the process of us coming to synthetic a priori knowledge and each has a different contribution. A reading of Kant that prioritises conceptualism or nonconceptualism would seem to overlook his commitment to unity.

Kant makes it clear that both necessary universals and contingent particulars contribute to our knowledge but not in the way that rationalists and empiricists have previously thought. In this, he is uniting sensibility and understanding in their ability to yield knowledge. Neither can yield knowledge alone. Rather, they always yield knowledge together. Without both, we do not have cognition. While we may sometimes inquire about universals or particulars alone they both contribute to the knowledge that we have. Of course, Lazos does not deny the need for both in cognition but merely that it might be possible to employ one without the other. But it seems that though we can focus on one as our object of investigation, our investigation is necessarily employing both. Kant is affirming the importance of both for us as knowers. In the discipline of pure reason, Kant further discusses this unity between universals and particulars:

Philosophical cognition thus considers the particular only in the universal, but mathematical cognition considers the universal in the particular, indeed even in the individual, yet nonetheless a priori and by means of reason, so that just as this individual is determined under certain general conditions of construction, the object of the concept, to which this individual corresponds only as its schema, must likewise be thought as universally determined. (A714/B742)

When philosophy considers the particular it looks at the universals present in it. It understands those particulars by understanding the universals that condition our cognition of them. We access universals through concepts belonging to the faculty of the understanding and particulars through intuitions on the part of the faculty of sensibility. And so the question of objects conforming to cognition concerns how our intuitions can conform to concepts when they have their origin in faculties that are not just numerically distinct but qualitatively distinct. The discursivity of philosophy brings us to understand particulars by turning to discursive concepts (e.g. does this object have property x? Does that mean it cannot have property y?). But if this discursive activity were not about intuitive representations it would not reach the level of cognition. It would be mere play.

We process phenomenal experience by turning to concepts. Without both, we do not have human experience. In mathematics, we come to understand universals by turning to particulars that instantiate them. Because mathematics constructs its concepts it can provide particulars of experience that instantiate a universal without compromising apriority. Thus in mathematics we come to understand a universal concept by looking at the particular intuition that exhibits it. The use of the unit circle is a perfect example of how we use a particular example to come to understand universal relationships. When we employ the unit circle to aid us in remembering trigonometric relationships, there is no need for the circle we consider to have certain a posteriori measurements. Rather because we stipulate a particular hypotenuse to be of unit length we can consider the universal trigonometric values of all triangles with certain interior angle measurements. But if this example does not link to universal concepts it does not yield knowledge.

In both cases, our concepts and intuitions have been integrated and the universal and particular are united. In philosophy, there can be no understanding of the particular intuitions without the concepts that condition our access to the particular. In mathematics there can be no understanding of universal mathematical concepts without particular intuitions that exhibit those universals that we have put into them.[1] It is this aspect of our epistemic condition to which Kant wants to draw our attention before proceeding in our metaphysical investigations. Kant wants metaphysics to also come to know what we have ourselves put into objects. He wants philosophy to recognise the necessary unity that underlies cognition. Rationalism and idealism seek universals or ideals with no tether to the particulars of experience and empiricism is concerned with the immediacy of particulars while sceptical of universals. Kant sees them as necessarily united in human knowing. He turns to mathematics as a way to see the necessary integration of concepts and intuitions and as the key to moving beyond Humean scepticism since the denial of pure a priori concepts of the understanding is

an assertion, destructive of all pure philosophy, on which he would never have fallen if he had had our problem in its generality before his eyes, since then he would have comprehended that according to his argument there could also be no pure mathematics, since this certainly contains synthetic a priori propositions, an assertion from which his sound understanding would surely have protected him. (B20)

For Kant, mathematics looks at particulars to understand what we have already put into them. When we construct mathematical concepts, we use the intuitions of space and time as our formal conditions and turn to our constructions to understand the universals that reveal themselves through the constructive process. When he considers whether or not synthetic a priori judgements are possible, mathematics is the first affirmative that gives us license to proceed in search of a system of such judgements. Kant’s phrasing of considering what we have ourselves put into objects means that our encounter with objects is already framed or influenced by sensibility.

This unity is central to Kant’s understanding of cognition. We need to turn away from seeking the traditional objects of metaphysics and turn to the structure of reason where we find the union of general principles and particular experiences. This integration of concepts and intuitions is made possible by the transcendental schema:

Now it is clear that there must be a third thing, which must stand in homogeneity with the category on the one hand and the appearance on the other, and makes possible the application of the former to the latter. This mediating representation must be pure (without anything empirical) and yet intellectual on the one hand and sensible on the other. Such a representation is the transcendental schema. (A138/B177)

The transcendental schema ensures the applicability of concepts of the understanding to objects of intuition and necessarily belongs to both understanding and sensibility. It is an act by the understanding on sensibility that generates the form of how we are to be receptively affected. It is the determination of the structure of the form of intuition. Whether we move from the particulars of intuition to concepts or from universal concepts to particulars, cognition is only possible given the unity between the two, a unity that goes unaccounted for in traditional attempts at metaphysics. While the discussion of schemata arises much later in the First Critique, Kant is sure that they provide the link between the two faculties, “[t]hus the schemata of the concepts of pure understanding are the true and sole conditions for providing them with a relation to objects” (A145–6/B185). The schema links immediately to a concept and immediately to an object. It relates immediately to the understanding because it is the ground for its determination and it relates immediately to an object because it is the structure through which the object is received. The heterogeneity between a concept and an object that falls under it keeps the concept and object from relating immediately. Without the transcendental schema we would have distinct elements to our knowing but there would be no link between them. We could have particular intuitions and we could discourse through universal concepts but our concepts would not be about our intuitions. The transcendental schema is what ensures that the objects we receive are objects we can cognise.

Kant sees the schema as “a way of establishing composition in the manifold in space and time” (Refl 5552, AA, 18:220). The transcendental schema is an act of the understanding that generates the spatiotemporal manifold and so ensures the application of non-spatiotemporal categories to spatiotemporal intuitions. If what makes Kant’s transcendental idealism critical is the fact that he has ventured to ask how it is possible that objects can conform to our cognitions rather than assumed it then we must recognise the transcendental schema as the crux of his answer to the general question of the possibility of synthetic a priori judgements. His characterisation of the faculties of understanding and sensibility falls short of answering the Critical question if there is no way to integrate their disparate traits. He would have described faculties but would not have accounted for the possibility of the relevant judgements. While he does not elaborate much on the genesis of the schema, its significance is not lost on him:

The difficulty [in accounting for the possibility and mediation of the application of the categories to objects of intuition] seems to arise because the transcendental time-determination itself is already a product of apperception in relation to the form of intuition and thus itself raises the question how the application of the categories to the form of intuition is possible, since the categories and the form of intuition are heterogeneous. In general, the schematism is one of the most difficult points.—Even Herr Beck cannot find his way about in it.—I hold this chapter to be one of the most important. Refl. 6359, 18:686)

The schema is a necessary bridge between our active understanding and receptive sensibility and not one that is dormant when psychologising cognition. The schematism serves as the integrative act in which the understanding is active and sensibility is accordingly receptive but neither is independent of the other:

Without schemata, therefore, the categories are only functions of the understanding for concepts, but do not represent any object. This significance comes to them from sensibility, which realizes the understanding at the same time as it restricts it. (A147/B187)

Schemata determine the form of receptivity of sensibility and make possible the conformity between objects and cognition. Lazos cites Kant’s discussion of other animals having sensibility but lacking the faculty of the understanding as an example of the independence he aspires to, but human sensibility is different because it has been affected by human understanding.

Pure mathematics is far from an arbitrary choice of science that manages to draw our attention to the central problem of a critical philosophy. The concepts of pure mathematics are at the same time universal and in concreto. As such, only pure mathematics can inform us of how a representation can be at the same time homogeneous with the intellectual and the sensible, as representations must be when it comes to the pure concepts of the understanding if we are to have genuine cognition. Pure mathematics is comprised exclusively of synthetic a priori judgements and so is the starting point for understanding the kind of knowledge that is possible under such circumstances. Other realms of inquiry, even other sciences, include increasingly more empirical and abstracted judgements. Kant holds that in other realms of inquiry these elements are more closely linked and so there is no difficulty in admitting their applicability to particular experiences. The genesis of certain concepts is empirical and we arrive at that through mere abstraction. The concepts may be empty but their generation did not forego intuitive representation. In these cases the concepts are still linked to empirical representations through schemata but there is homogeneity between concepts and their representations and so a transcendental account of their link is not necessary.

For example in a social science like economics we would view the judgements as descriptive and not apodeictic so we would not consider the concepts of economics to be drastically unlike the objects of economics, a data set and a conceptual abstraction from said data set would both be empirical. The study of economics does not admit of the kind of concepts and cognition that Kant seeks for metaphysics. Of course, when there is such homogeneity we are not confronted with the problem or fundamental question of reason critiquing itself, namely how concepts can conform to our cognition because such homogeneity keeps us from recognising the distinct character of the aspects of our knowing and thus the need for their unity. It does not bring us to examine our epistemic condition. It is precisely this kind of homogeneity that is assumed by dogmatic metaphysics and that is at the crux of Lazos’s nonconceptualism. In mathematics and philosophy there is a stark contrast between the concepts that exhibit universals and those that deal with empirical particulars.

So while other investigations might be able to proceed without accounting for the unity of the heterogeneity of sensibility and understanding, philosophy can only do so only upon an autocritique. It is only when turning to mathematics that this autocritique can uncover the schemata that condition this necessary unity. Kant realises that before his Critical turn he himself had assumed this unity rather than aimed to account for it. When he decides to inquire into this unity and its possibility for all cognition, he begins the turn towards a critical philosophy.

In contemporary debates about Kant’s characterisation of mathematical reasoning, one of the polarising issues is the role that intuition plays in mathematical construction with the divide seeming to mirror the sides of the conceptualist schism. Scholars like Friedman and Jaakko Hintikka (1969) have argued that intuition allows Kant to provide singular terms that logic and elementary construction cannot generate. On the other side of the issue thinkers like Emily Carson (1997) and Charles Parsons (1964, 1983, 1990, 2008) have argued that Kant’s inclusion of intuition in mathematical reasoning stems from a phenomenological acquaintance with certain spatiotemporal facts upon which we base mathematical axioms. These views place the emphasis and significance of the appeal to intuition in different aspects of spatiotemporal receptivity on the part of sensibility. For the logical view the significance rests on the singularity of the objects generated or presented and for the phenomenological view the significance rests on the immediate relation between the knower and spatiotemporality. Both the logical and phenomenological positions must view mathematical reasoning as essential for revealing the schemata that ensure that objects conform to cognition. That is, with regard to mathematics, both sides of the debate uphold the unity of concepts and intuitions, of sensibility and understanding. Regardless of the basis for the turn to intuition, it must ultimately be united with concepts in order to meet the conditions of cognition and mathematics reveals to us how that unity with concepts is made possible. What is at stake in the debate is not whether or not mathematics can tell us about cognition but rather what it tells us about cognition.

Recognising the unity of particulars and universals is the first authentic moment of reason’s critique of itself. A system of metaphysics that dispenses entirely with either is a system that has fallen short in its epistemology and does not understand the nature of human cognition as always surrounded by particulars but always seeking universals. As Kant relays in his now famous letter to Marcus Herz, this recognition is a driving force of his turn to a critical philosophy:

I noticed that I still lacked something essential, something that in my long metaphysical studies, I as well as others, had failed to consider and which in fact constitutes the key to the whole secret of metaphysics, hitherto still hidden from itself. I asked myself this question: What is the ground of the relation of that in us which we call ‘representation’ to the object? (Corr, AA 10:130)

As I mentioned earlier, I am fundamentally in agreement with Lazos’s attempt to stave off conceptualism and the idea that the value of Kant’s claims that cognition requiring both sensibility and understanding hinges on being able to understand that heterogeneity between the faculties. So perhaps this serves more as an invitation to explore what is at stake for Lazos in psychological independence instead of mere divisibility or isolability. Though he does not expound on this explicitly, I take it that his view of Kant’s nonconceptualism—and in turn of his affirmation of the psychological independence of sensibility and understanding—also serves to bolster his view of rejecting idealism, Descartes and considerations of transcendental apperception and perhaps those areas of significance that would be impacted by compromising on psychological independence can also be identified.


[1] As has often been noted, for Kant the geometer may well be able to manœuvre geometrical concepts without drawing them or even consider their relationships to other concepts but without the turn to intuition this manœuvring would not attain the level of cognition and would remain mere play.



Carson, E. (1997), ‘Kant on Intuition in Geometry’, Canadian Journal of Philosophy 27(4): 489–512.

Dufour, E. (2003), ‘Remarques sur la note du paragraphe 26 de l’Analytique transcendantale: Les interprétations de Cohen et de Heidegger’, Kant-Studien 94(1): 69–79.

Friedman, M. (1992), Kant and the Exact Sciences (Cambridge, MA: Harvard University Press).

——— (2000), ‘Geometry, Construction, and Intuition in Kant and His Successors’, in G. Sher & R. Tieszen (eds), Between Logic and Intuition (Cambridge: Cambridge University Press), pp. 186–218.

——— (2012), ‘Kant on Geometry and Spatial Intuition’, Synthese 186: 231–55.

Hintikka, J. (1969), ‘On Kant’s Notion of Intuition’, in T. Penelhum & J. Macintosh (eds), The First Critique: Reflections on Kant’s Critique of Pure Reason (Belmont: Wadsworth), pp. 38–53.

Longuenesse, B. (1998), Kant and the Capacity to Judge (Princeton: Princeton University Press).

Onof, C. & D. Schulting (2015), ‘Space as Form of Intuition and as Formal Intuition: On the Note to B160 in Kant’s Critique of Pure Reason‘, the Philosophical Review 124 (1): 1–58.

Parsons, C. (1964), ‘Infinity and Kant’s Conception of the “Possibility of Experience”’, the Philosophical Review 73(2): 182–97.

——— (1983), ‘Kant’s Philosophy of Arithmetic’, in C. Parsons, Mathematics in Philosophy. Selected Essays (Ithaca, NY: Cornell University Press), pp. 110–41.

——— (1990), ‘The Structuralist View of Mathematical Objects’, Synthese 84: 303–46.

——— (2008), Mathematical Thought and Its Objects (Cambridge: Cambridge University Press).

© Edgar Valdez, 2018.

Edgar Valdez earned his Ph.D. in Philosophy from Binghamton University (SUNY), USA. His research focuses on Kant’s theoretical philosophy and how it relates to mathematics and natural science. His book Kant, Space, and the Foundations of Geometry is forthcoming from Lexington Books.