Kant's Mathematical World: Mathematics, Cognition, and Experience

by Daniel Sutherland

Kant's Mathematical World aims to transform our understanding of Kant's philosophy of mathematics and his account of the mathematical character of the world. Daniel Sutherland reconstructs Kant's project of explaining both mathematical cognition and our cognition of the world in terms of our most basic cognitive capacities. He situates Kant in a long mathematical tradition with roots in Euclid's Elements, and thereby recovers the very different way of thinking about mathematics which existed prior to its 'arithmetization' in the nineteenth century. He shows that Kant thought of mathematics as a science of magnitudes and their measurement, and all objects of experience as extensive magnitudes whose real properties have intensive magnitudes, thus tying mathematics directly to the world. His book will appeal to anyone interested in Kant's critical philosophy -- either his account of the world of experience, or his philosophy of mathematics, or how the two inform each other.

Genres: Philosophy

300 pages, Hardcover

Published October 28, 2021

Reviews

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The author postulates that Kant postulated that the conditions of possibility for experience are the same as those of mathematics: it is not a question of applicability. The process of intuition itself is non-spatial, though it generates the necessary representation of space which underlies the application of the pure concepts of reason. This representation of space is that of indeterminate quanta, the abstract reign of pure quantity and mereological austerity, which tries to explain things like placements of objects without recourse to conceptuality. This is a tricky part, because "number" is the schema of the category of quantity in Kant's philosophy, but the indeterminate quanta of intuition have precedence to that: they constitute a homogenous manifold deprived of difference that is required for the representation of the quantitas-sense of magnitude, schematized by number. This all is in response to Leibniz's principles of the identity of indiscernibles and the principle of sufficient reason: the doctrine of intuition, and what is sometimes called a "discursivity thesis" is the antithesis of that. A good illustration that goes against Leibniz could be something like imagining different quantities in different dimensions, in a area or a line, and noting that a similarity can be shown between the changes in quantities despite this radical difference, requiring a pre-conceptual representation of indeterminate quanta which are not contingent on the conceptual differences that PSR holds to exist with every object. It is also important to note that these are even required for the representation of a single line, rather than just closed representations of manifolds that already presuppose lines.

The book is no light reading: as typical of Kant scholarship, it rummages throughout the man's bibliography for references. This book also concerns itself with the most forgotten parts of CPR like the Axioms of Intuition, a tendency which may have you grasping for your copy. Some of it was way too deep into the game for me, but I think the basic ideas outlined above are interesting. The book regrettably does not contain much speculation on the status of the thing-itself: it seems to work with the "problematical concept" theory by default. There are ways to defend a more realistic view even about the thing-itself, considering that the concepts and intuitions alike cannot exhaust themselves, as seen in the Dialectic, which is something that Schopenhauer probably hit on with his identification of the thing-itself with the Will: what's left over really seems to be something like mind (NOT the idealist, Berkeleyan Mind - but perhaps more like reverse-engineered Neoplatonic ascent to Soul from reason, to atman from the self - or else simply some other mind that is communicated with)

Another, very different angle from which you could approach the problem of discursivity is by recognizing the vagueness of our representations of shape, how there's always at least a couple interpretations of even the simplest geometrical shapes. Three-dimensional objects can have their angles turned inside-out, while plane objects can be imagined as floating in three-dimensional space, exemplifying either a front-or backside of the object. With lines you have the non-euclidean idea of straight lines whose parts exist at different depth levels. The role of the concepts of the understanding would be to fix this indeterminacy into one particular representation, like how we usually automatically see cubes and figures with depth. The possible divergence between animals and humans, and plane-beings and three-dimensional beings, would be located on the level of the concepts of the understanding, while the indeterminate data of intuition would be something similar for everyone. This gives rise to the possibility of the many different kinds of proportions between conceptual thought and acceptance of the truth of the intuitive faculty on the other hand. Strangely, it would seem that the plane beings, meaning here only creatures with a space conceptualization that does not allow them to perceive a singular object while, say, a cube is rotating, are probably more reliant on the concepts: they have broken up the original intuition into further segments, while the three-dimensional man can appreciate the concept of the plane, but at the same time he can simply "see" something more. The ultimate result of the application of the concepts of reason is necessarily a limitation of the whole and at worst, downright ignorance of it in favour of the limited object. Imagine the current reductionist-oriented, conceptually analytic era and the possibility that in locating these divisions the tendency grows similar to where an animal sees a bunch of squares one after another, twisting and changing shapes before simply disappearing as if by magic, when, on a higher level of unity, it is really a rotating cube. It sounds dire and vitalism and a sort of pseudo-holism seems ever-so tempting at this point, but on the other hand the animal is at least noticing something strange. Which is precisely what humans are noticing in modern physics, as a result of this regression and reduction. So, the situation is tense with possibilities: there's an increase of limit knowledge and a potential to "see" higher unity but also the force of analytic reason, the method responsible for the current success, driving mankind lower, to a more limited conceptuality of isolated objects.

So, admittedly I read this very selectively seeking mostly stimulus and reflection to my own ideas, but it's still a recommended, if difficult, look into obscure areas of Kant's philosophy.