*cognitive psychology*. Then, how physics views the question whether space is Euclidean and 3-dimensional. Finally, I argue that modern psychology and physics together imply a conclusion that

*contradicts*the central assumption of Kant's Transcendental Idealism -- namely, Kant's claim that space is the form of external intuition.

Space, the regions that physical things like rocks and chairs move around in, the arena of spacetime regions and spacetime points, to which may be assigned co-ordinates, upon which physical fields, tensor fields, etc., are defined, is

*mentally represented*in the human cognitive system, and presumably this is quite similar to how it works in the primates, and

*all*creatures with anything like a simple visual system (even bees and flies, say). Now we

*mentally represent*(in perception, I mean) space as 3-dimensional and Euclidean. Only three (and no more) co-ordinate axes can be placed

*perpendicularly to each other*. Presumably, the perceptual system in humans, primates, bees, etc., is

*innate*. And presumably this is why we tend to consider its output as

*a priori*. Kant uses the phrase "form of external intuition" to refer to this mental representation of space (or, if you prefer, to its organizing pattern). So, using Kant's terminology, the form of external intuition is 3-dimensional Euclidean geometry, $\mathbb{E}^3$.

It is very unclear how this works, of course. But let us take this also as something outputed by modern cognitive psychology:

(1) The form of external intuition is $\mathbb{E}^3$. (Psychology)Next, turning to space -- i.e., the regions that physical things like rocks and chairs move around in, the arena of spacetime regions and spacetime points, to which may be assigned co-ordinates, upon which physical fields, tensor fields, etc., are defined, and so on. This is

*not a definition*. On the contrary! For space (or spacetime points, events, etc.) is assumed as a

*primitive*in modern physics (classical mechanics, electromagnetism, special relativity, general relativity, quantum theory; various quantum gravity programmes such as superstring theory, canonical quantification, supergravity, loop gravity, causal set theory, etc.).

In a physical theory, we have $M$ (i.e., space) assumed as primitive. Then there are various functions and whatnot on $M$ (word-lines, fields, fibres, and all sorts of weird stuff, etc). In fact, space and time, have, since Einstein and Minkowski, been fused together, and usually, $M$ has the structure of a Riemannian manifold, with a special metric $g_{\mu \nu}$ which tells you how "far apart" neighbouring points are; but it needn't have this structure. Physicists are very unsure what properties space has. For example, space might be a manifold of some kind -- perhaps a compactified 10-dimensional manifold, an idea that goes back to Kaluza-Klein theories. Space might be something finite and/or discrete (such as in causal set theory). Or perhaps something quite different.

However, physicists agree (this is called The Correspondence Principle, and is why, e.g., Einstein aimed to get Newton's Laws as approximations from his field equations) that any theory of space will recover the 3-dimensional Euclidean space $\mathbb{E}^3$ as an

*approximation*. But this approximation is, to repeat, an

*approximation*. So, space, whatever it is, is approximately $\mathbb{E}^3$ "at a certain scale". That's what we "observe" at the medium-scale. But this does not imply that space

*is*$\mathbb{E}^3$. In fact, it

*isn't*, on any modern theory.

Let us therefore take this as something outputted by modern physics:

(2) Space isTaking together, these two statements (1) and (2) imply:not$\mathbb{E}^3$. (Physics)

(3) Space is not the form of external intuition. (Physics, Psychology)Move now to Kant and his argument for Transcendental Idealism (TI), which is Kant's claim that,

Time and space, and all objects of a possible experience, cannot exist out of and apart from the mind.(See Kant,

*Critique of Pure Reason*, Transcendental Logic, Second Division: Transcendental Dialectic, BOOK II: The Dialectical Inferences Of Pure Reason, Chapter II: THE ANTINOMY OF PURE REASON, SECTION VI. Transcendental Idealism as the Key to the Solution of Pure Cosmological Dialectic)

Kant's argument for (TI) it is based on his assumption that space

*is*the form of external intuition. More exactly, his argument for (TI) uses the assumptions:

(A1) Space is the form of external intuition.So, in particular, Kant's argument for (TI) is based on (A1).

(A2) Space (and time) is necessary for the representation of objects (of a possible experience).

(A3) External intuition is a property of the mind.

But the problem is that Kant's assumption (A1)

*contradicts*(3).