## Thursday, 27 June 2013

### Is Space the Form of External Intuition?

First, a bit of 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 is not $\mathbb{E}^3$. (Physics)
Taking together, these two statements (1) and (2) imply:
(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.
(A2) Space (and time) is necessary for the representation of objects (of a possible experience).
(A3) External intuition is a property of the mind.
So, in particular, Kant's argument for (TI) is based on (A1).

But the problem is that Kant's assumption (A1) contradicts (3).

1. Isn't Kant free to insist that, if the physics tells us that space has n dimensions, then the form of external intuition is not as it appears to introspection? It's not implausible to me that I might be able to represent things in more than 3 spatial dimensions, without being able to identify more than 3 directions in introspection. (It's already unclear to me whether I can identify more than two dimensions in introspection. Or whether I can tell if I represent parallel lines so that they do not meet.)

On the physics side, ignoring the possibility that space is discrete for now (though that wouldn't be disastrous for Kant, since maybe the form of external intuition is discrete, just not in a way accessible to introspection - the retina and the network of nerve endings in the skin are both discrete, after all), if there are more than 3(+1) dimensions, but 3 of them clearly correspond directly to observable space, wouldn't that be a good reason to call only the subspace spanned by those 3 'physical space'. After all, we know reality can have dimensions that are non-spatial, because time is one.

Less seriously, to try to prove Kant's psychologism about mathematics instead of his idealism, you could adapt A1 and A2 to say 'E^3' instead of 'space' ("E^3 is the form ...", "E^3 is necessary for...").

2. Thanks, Anon,

Isn't Kant free to insist that, if the physics tells us that space has n dimensions, then the form of external intuition is not as it appears to introspection

This might be ok; but it looks now, on this interpretation, that

"the form of external intuition"

means

"the referent of external intuition",

in something like the modern sense of theories of mental representation.

But I'm pretty sure Kant didn't mean that. For if it did mean that, the argument for (TI) would not go through. That is, we'd lose the link between the form of external intuition and being mind-dependent. Or, to put it another way, if it did go through, then all referents of mental representations would be mind-dependent! A very easy way to get a kind of idealism, and to be honest, I think it's a bit like McDowell. And though it's ages since I read Russell's criticisms of Idealism, I think Russell attributes to them precisely the view that all referents of mental ideas are themselves mind-dependent.

On the other point, I think Kant already gave that argument! And Brouwer did too, as I understand Brouwer.

Cheers,

Jeff

3. Shouldn't (3) be
(3*) Space, as conceived by physics, is not the form of external intuition. (Physics, Psychology)

4. Antonio, thanks.

I think (3)'s ok, because it has the bracketing "(Physics, Psych)"; so it already states the source, as it were, and says

"Space is not ..., according to physics and psych".

Whereas (3*) would be a double relativization, saying

"Space, as conceived by physics, is not ..., according to physics and psych"

(There's a puzzle about what "..., as conceived by X" means: in all likelihood, it's an epistemic operator, "X says that: ....".)

Cheers,

Jeff

5. Oops - I should have added also that,

(a) Physics and psych imply that space is not the form of external intuition
(b) Kant says that space is the form of external intuition.

are not contradictory, of course! Though the semantic contents are.
So, the crucial issue I'm raising can focus on whether (a) is true.

Cheers,

Jeff

6. Thanks Jeffrey. Yes, (a) and (b) are not contradictory of course. And yes, the crucial issue is (a). And yes, I see that 3* is redundant, but it seemed to me it could help making my point. That is, that one can distinguish between space as mentally represented and space as conceived by physics. Let me explain a bit better what I'm thinking. To show that the most important difference between the positions of a mathematician and of a physicist is that the mathematician is in much more direct contact with reality, Hardy wrote: "a very little reflection is enough to show that the physicist’s reality, whatever it may be, has few or none of the attributes which common sense ascribes instinctively to reality. A chair may be a collection of whirling electrons, or an idea in the mind of God: each of these accounts of it may have its merits, but neither conforms at all closely to the suggestions of common sense". In this perspective, if one takes the view on space within such common sense and refine it into a transcendentally idealized form (what Kant seems to do), then "the form of external intuition" needn't be the "the referent of external intuition", where the latter has those properties that physics tells us (and yes, this is the meaning I'm giving to "as conceived by physics" :) ) I'm not saying this is Kant's view (arguably, the opposite - since he wanted to offer a theoretical justification of Newton's laws), but still it seems to me that one can agree with Kant's account of our intuition and modern physics at the same time.

7. Antonio,
Yes, I think the whole problem then resurfaces with the phrase "X, as conceived by A, ...". (I think Alan Musgrave once referred to this as talk of Kantian-hyphenated-entities!) But one can then just reject all talk of Kantian hyphenated entities, as it were; or make the epistemic relativization explicit. Or to put it a bit differently, "X, as conceived by A, has property P" and "X, as conceived by B, lacks property P" are semantically hard to make sense of, unless they just mean "A implies (says, claims, ...) that X has property P" and "B implies (says, claims, ...) that X lacks property P".

So, I think it's clearer to simply try and get the contents clear, along the lines of,

The mind's a priori system thinks that space is 3-D & Euclidean
But physics implies that space is not 3-D & Euclidean.

Then the problem is that Kant's "space is the form of external intuition" is the claim that space *is* what the mind's a priori system thinks it is; whereas physics says space isn't what the mind's a priori system thinks it is (because the mind's a priori system generates a misrepresentation of space).

Cheers,

Jeff

8. Thanks Jeffrey.

"Then the problem is that Kant's "space is the form of external intuition" is the claim that space *is* what the mind's a priori system thinks it is; whereas physics says space isn't what the mind's a priori system thinks it is (because the mind's a priori system generates a misrepresentation of space)."

The mind's a priori system *represents* space in 3-D and as Euclidean
Physics implies that space is not 3-D & Euclidean
Then, "the mind's a priori system generates a misrepresentation of space"

This is not problematic in its own, right? So I would say that
(i) if Kant says that our representation of space *is* the space, then he is wrong - based on modern cognitive psychology and physics
(ii) if he says that "space is the form of external intuition" meaning "the form of external intuition is the Euclidean 3-d space", then he is right or wrong independently from what Physics… says :) And he is probably right, based on modern cognitive psychology.

You seem to hold the antecedent of (i). I'm not sure of this and can't recall the details of the Critique.

What do you think?

Cheers,
Antonio

9. ... where "our representation of space *is* the space" stands for "the space is how we represent it", of course

10. Antonio, that's all pretty much right. The antecedent of (i) is "Kant says our representation of space is space", so in the end everything comes down to whether Kant thinks:

the form of external intuition = our representation of space

But I think that is Kant's view: "the form of external intuition" is more or less a synonym for "the form of the representation of space".

Cheers,

Jeff

11. All of this seems to be a consequence of a tacit assumption that substantivalism about space (or, if you prefer, spacetime) is both coherent and true. A central part of Kant's argument for (A1) is that substantivalism is both false and unable to explain our knowledge of space. So aren't you just arguing that hard-core realism about contemporary science (which, I take it, is what you need for your (2) to mean the right thing) is incompatible with transcendental idealism?

12. Thanks, Brian,

You've hit the nail on head there, for I think the Leibniz vs Newton issue is what lies behind Kant's view of space, and how Kant arrives at (TI). But on the underlying question, no - I'm an anti-substantivalist, and what's formulated here is certainly compatible with that. And this rejection of substantivalismt is actually my main interest here too, as I'm interested in the relationism/substantivalism debate as it reappears in modern physics, and not Kant's (TI) view per se. (I've put up various posts here on "Leibniz Equivalence".)

Before CPR, Kant was a Leibnizian relationist as against Newton's substantivalist:

(L) S = the abstract form of the spatial relations of material bodies.
(N) S = an enclosing absolute container in which material bodies were "located".

But then, under Hume's empiricist attack, Kant concluded that relationism (L) made space unknowable: i.e., not deducible from sensory impressions. So, for epistemological reasons, in order for space to be knowable (indeed, a priori knowable), Kant replaced (L) by his own Idealist account,

(K) S = the form of external intuition

(And similarly for time.)

From this (TI) then follows, assuming external intuition is mind-dependent, etc.
So, that's how I see all this.

On the question though, I think ordinary physics and psychology (rather than meta-theories about physics and psychology) are incompatible with "space is the form of external intuition", because together they imply that space isn't the form of external intuition.
We just need the psychological and physical premises,

(Psych) The form of external intuition is E^3.
(Phys) Space is not E^3.

Then these scientific premises imply that space is not the form of external intuition.

Cheers,

Jeff

13. Probably I should add that I'm not saying science is incompatible with Idealism.
Science is compatible with, e.g., Berkeleyan Idealism, and one could view Russell's Our Knowledge of the External World and Carnap's Aufbau like this. So, physical entities could be "constructions from sense-data", or something along those lines.

The point I'm arguing for is that science is incompatible with Kant's view of space, and therefore that Kant's argument for Transcendental Idealism doesn't go through.

Cheers,

Jeff

14. Jeffrey, I believe somehow that 'Space' as in (K) may be not the same as in (Phys), but I have to think about why it could be so. (Maybe the difference between intuitions and concepts in Kant is a sufficient reason.) Of course, this doesn't influence your conclusion from (Psych) and (Phys), but you do need those two to be the same for the contradiction to hold.

15. I'm pleased to hear you're an antisubstantivalist -- but then I'm left at a loss as to how to interpret your claim (Phys) in such a way as to render Kant's claim false. Because then it seems to me that Kant can escape refutation unless you can show that what is meant by 'space' in (Phys) is the same as what Kant means by it. If, for example, Poincaré has convinced us (as well he might) that we ought to be conventionalists about applied geometry and use whichever is most convenient/parsimonious, then I suppose we'd lost any claim to the thought that Physics shows us what space "really" is. Kant could nevertheless still claim (as Antonio says) that transcendental psychology allows us to understand the empirical (but of course not the transcendent) reality of space.

1. Hi Antonio and Brian,

So I think your proposal is this. Focusing on the inconsistent triad,

(Psych) The form of external intuition is E^3.
(Phys) Space is not E^3.
(KTS) Space is the form of external intuition

we could try and protect Kant's theory of space from inconsistency with physics and psychology by postulating that "space" is the incorrect translation of Kant's use of the word "Raum".

But the objection to this is that Kant's translators do translate it as "space"; and second, what should its translation be?

Cheers,

Jeff

2. Hi Jeffrey,
Just to offer an example, possible in principle, consider these claims:

(KTS) Space = the form of external intuition;
(Psych) The form of external intuition is E^3;
(Phys) Space = the abstract form of the spatial relations of material bodies (Leibniz) & Space is not E^3.

These imply that the abstract form of the spatial relations of material bodies is not the form of external intuition, on which Kant agrees.
Of course, it also implies the opposite, by immediate substitution. But in this line the disagreement is then shown to be about the very *definition* of Space, and it seems to me that your line of thought doesn't take in consideration the possibility of such disagreement: "The point I'm arguing for is that science is incompatible with Kant's view of space, and therefore that Kant's argument for Transcendental Idealism doesn't go through."

Any thought?

Cheers,
Antonio

3. Hi Antonio

Yes, this is a semantic reinterpretation. The prima facie problem is that extending three inconsistent claims with a definition can't make it consistent. (Kant might even have thought "space is the form of external intuition" is a definition!)
The only way to make the inconsistent set of three sentences consistent is to change the word "space" in Kant's theory of space to another word, say W.

But this implies that all of Kant's translators mistranslated his word "Raum". And what would this word W be?

Cheers

Jeff

4. Hi Jeffrey, thanks. :) Yes - the inconsistency is definitive, of course. Just thought that could help to visualize better where the divergence was.
But I must say I now see the whole point, even if it was all there already:

From
(1) The form of external intuition is E^3 and
(2) Space is not E^3

which are both *scientific* claims,

one gets that Space, whatever it is, is not the form of external intuition.
So either Kant or science is wrong. But science holds the true, then Kant is wrong.

Yes, one can hardly disagree.
And I should add that it's been a pleasure going maieutically through the details of this.

Cheers,
Antonio

5. Hi Antonio, thanks, I think the best resolution of the triad isn't to change "space" somehow, but rather to change how it's related to "external intuition", along the lines we were discussing before. Roughly,

(Psych) The form of external intuition is E^3.
(Phys) Space is not E^3.
(KTS*) Space is *represented* (approximately) by the form of external intuition.

Then, everything, including the modification of Kant's theory of space, is ok. The mind's external intuition has some form (i.e., a structural mental representation which is, or somehow generates, E^3, and thus grounds our a priori synthetic knowledge), and then this form represents, albeit not 100% accurately, space.

This is consistent, and (KTS*) is, so far as we can tell, true. But then it doesn't yield (TI), because although perhaps the representation, the external intuition, its form, etc., can't exist mind-independently, still the referent of a representation can, particularly if it's a misrepresentation. The mind's cognitive system generates (I'd guess largely innately) an a priori representation, and this is, or yields, E^3 (though no one quite knows how). This mental representation of space is a good approximation as a representation of what space is. But, crucially, it is still not 100% correct, because what space is isn't what our mental representation says it is. For (TI), Kant needed space to be our mental representation ("the form of external intuition"); but then physics eventually said: no, they're, very probably, different.

Cheers,

Jeff

6. I don't think it's a translational issue, Jeff. Rather, I think that the semantics of your (Phys) are not at all straight-forward in the way that your (KTS) can be. In short, if you're an antisubstantivalist, then I assume you aren't going to treat space as a simple, denoting noun -- whereas Kant (and a substantivalist) would. Hence, there's going to be more semantic unpacking to go on in your (Phys), hence we can't draw surface conclusions about its coherence with Kant.

7. Hi Brian.

"Hence, there's going to be more semantic unpacking to go on in your (Phys), hence we can't draw surface conclusions about its coherence with Kant.". As you already noticed, originally this was more or less my suggestion too; and it still attracts me somehow. The problem is as to how to express this suggestion so that it works.
After a little thinking on this, it turns out that, whatever is the meaning of the term 'space' in (Phys), the contradiction will still remain unresolved; and rightly so. Specifically, the contradiction will be there also whatever is the type of the term 'space'; that is, for any possible "semantic unpacking" - if I take your comment correctly. Indeed, assume the type of 'space' in (Phys) is different from the type of 'space' in (KTS). Then trivially, what the former 'space' means cannot be what the latter 'space' means. Hence, the former 'space' is not the form of external intuition. And this contradicts (KTS).

So I eventually concluded that it was a flawed suggestion after all.

Any thought?

Cheers,
Antonio

8. Brian, thanks - the modern debate, since Earman & Norton's paper on the Hole argument, focuses on the question of gauge freedom in GR and the right way to think of Leibniz equivalence/anti-haecceistism. I'm a Leibnizian on this, and have given maybe five or six talks on this topic over the last year or so. I agree with the formulations given by physicists: e.g., Wald's standard formulation in his 1984 textbook, General Relativity:

http://m-phi.blogspot.co.uk/2013/06/walds-formulation-of-leibniz.html

But modern Leibnizianism and "sophisticated substantlvalism" are extremely close, and it's quite hard to see what they differ on. Technically, it concerns whether isomorphic spacetime models $(M, g)$ and $(N, g^{\prime})$ represent the same worlds or not (physicists say they do) and what to make of this for the status of the spacetime points themselves. This is a hard conceptual problem.

However, on any such clarification, the premise

(Phys) Space isn't Euclidean 3-space.

comes out true. For example, each spacetime model $(M, g, T, \dots)$ solving Einstein's equations represents a world (and any two isomorphic ones represent the same world); and, usually, (the spatial part of) $M$ is not diffeomorphic to $\mathbb{E}^3$.

So, here, I don't think that challenging (Phys) will work, as substantivalists and Leibnizians say more or less the same thing.

Cheers,

Jeff

16. Here I think there is a intellectual domain problem with comparing the understanding of physics with the nature of physics itself, and the same with psychology. We can't assume that our present beliefs constitute the nature of matter or mind.

But what I really mean, is that E^3 is not always necessarily external to the psychologist; according to the view that every-thing has it's own mind, that such minds are embodied in matter might be a sheer coincidence. Consequently it is entirely open to interpret intuition as non-intuition, and space as non-space, etc. If there is such a radical contingency as this, there is no saying that there is an overlap between physics and psychology when they refer to the word 'space' or 'time'. These might be utterly separate majesteria, by virtue of the idea that both are interpretations.

I don't think that Kant would say that psychology is immaterial, but neither would he say that matter must be psychological. The notion of intuition might involve some degree of extreme independence from definitional certainty. And it is going still further to suggest that physical formulae as they are conceived by mathematicians have any direct psychological bearing, except in extreme degrees of complexity. External intuition, as simple as it seems, might be as far from quantum mechanics as quantum mechanics is from relativity. And, afterall, there has not been a grand unified theory that I know of. In some sense, physics is still obverving discrete areas, which are not identical, even within that discipline. And that is still further not to say that physics has 'discovered everything'. Psychology is one of the disciplines that says consistently that we do not fully understand ourselves, to say nothing of the nature of the external world.

What if, for example, the exterior is vastly contingent on the interior, such as in a case of being 'parsed by the universe'? Physical law would be determined to be a corollary to interior truths, not theorems in any permanent sense of the word. Then it is announced that theorems define mathematics permanently, and this rejects outright (assumes) that exterior intuition does not exist. It's a very typical problem in mathematics, that it makes assumptions. For example, the second zero is assumed to be decimal, but decimal is a small detail compared to things like pi and relavity. The decimal system is vastly more contingent and uncertain, unless we determine that mathematics is itself a form of 'external intuition'.

I don't place much cachet in the idea of external intuition. I'm a believer in vast complexities and unrealized truths. Saying the world is a product of the mind is a trick of words. For there is no way of saying that mental-physical-things-rendered-real are not themselves material entirely, and if so, there is no way of guaranteeing that that materiality does not itself have mastery over the mind. If that is the case, it is a semantic difference what is meant between some psychological theories of space and time, and similar or opposite theories put out by mathematicians. Both can be observed, or not. Neither is changed by nomenclature, unless in some subtle and specific way.