Everyone once in a while I run across the opinion that non-Euclidean presents a serious problem for Kantian epistemology. While I've rebutted this notion before, it's common enough that I thought I'd have another go at explaining why it's a misconception.
For Kant we can't know the universe to be spatial "in itself" (as in "things-in-themselves"), Euclidean or Non-Euclidean or otherwise. Time and space are something supplied by our cognitive faculties. To say what really is the case here means we're making objective judgments about the world of object-appearances (which is to say we're correctly analyzing what we've already synthesized).
If we discover that the universe is actually, objectively (in the Kantian sense) non-Euclidean when our spatial intuition suggests it is Euclidean, then there is a conflict here between the faculties of understanding and intuition. If you've studied non-Euclidean geometry you'll readily see what this means: the denial of the parallel postulate violates our intuition (unless we model the new geometry within Euclidean geometry but as occurring on a hyperbolic surface); but it does not produce any logical inconsistency. And in fact this is the whole point of Kant calling our perception of Euclidean space "intuition": I have no other basis for the parallel postulate -- I cannot argue for it as following from a principle of logic or arithmetic; nor can I argue about it from some a posteriori discovery in physics about the nature of the world.
Nothing says that our spatial intuition has to be "right," either in the metaphysical thing-in-itself sense, or in the Kantian sense of always being confirmed by higher level theoretical judgments of the understanding, as in physics. And the following remains a fact: from the standpoint of cognitive science, we perceive the world in Euclidean terms. No discovery outside of cognitive science could change the fact that this is how we intuit the world. All of this is a way of saying that the space of physics is not the same thing as the space of ordinary, every-day experience (the former is conceptual and a posteriori, the latter is intuitive and a priori). If non-Euclidean geometry is useful for physics and is better at modeling "space," this means not that Kant must revise his concept of intuition as being of Euclidean space; rather, he must revise his conception of the relationship between understanding and intuition (accounting for the possibility of conflict and a posteriori, non-intuitive conceptions of space).
In the end, this conflict between understanding and intuition is not something Kant foresaw; and I doubt he would have liked it. But ultimately, his system doesn't depend on the absence of such a conflict any more than our discovery of the non-Euclidean nature of space requires that we intuit our living room in non-Euclidean terms.
-- Wes Alwan
Michael Burgess says
To put it in more broad idealist terms, a transcendental structure (physics-space) does not have to equate an immanent structure (consciousness-space) even if one is a condition for the possibility of the other.
Alex Perez says
Very nicely done–and succinctly, too! It’s always driven me crazy when people (like my old philosophy of science prof) insist that non-euclidean geometry generally and Relativity theory specifically somehow demonstrate that Kant was “wrong” (the implication being that we can toss his silly old books into the fire, where they and the rest of history’s outdated, if quaint, missteps belong). A pretty straightforward category mistake, it seems to me, but it’s amazing how stubbornly people will cling to their misapprehensions when they feel their particular branch of philosophy is being threatened. I hope this blog post goes some way toward preventing undergrads (and grads!) from being steered away from Kant by narrow-minded anti-Kantians.
It’s easy to obtain a reason for believing the parallel postulate, just draw two parallel straight lines on a piece of paper and think of extending them, it is obvious that no way of extending them will make them cross paths.
Now if you believe the parallel postulate to be something one intuits it’s all fine and well but then the act of intuition is quite useless, it’s just saying, well human beings are inclined to believe something of this sort, could be right, could be wrong, and they could arrive at the same ideas just as well using other means than intuition.
On the other hand if you are interested, as for example Henri Poincaré was, in figuring out the truths of mathematics you wan’t to remove untrue intuitions and keep true intuitions, infact you want intuitions to be true by definition:
“What is the nature of geometrical axioms ? Are they synthetic à priori intuitions, as Kant affirmed ?
They would then be imposed upon us with such a force that we could not conceive of the contrary proposition, nor could we build upon it a theoretical edifice. There would be no non-Euclidean geometry. To convince ourselves of this, let us take a true synthetic à priori intuition — the following, for instance, which played an important part in the first chapter: — If a theorem is true for the number 1, and if it has been proved that it is true of n + 1, provided it is true of n, it will be true for all positive integers. Let us next try to get rid of this, and while rejecting this proposition let us construct a false arithmetic analogous to non-Euclidean geometry. We shall not be able to do it.”
This whole “Kant is wrong, because of non-Euclidean geometry” thing is stupid. For one it rests on this argument that “we intuit a Euclidean space”. From my own experience as a human being…that just seems completely wrong; I don’t in truth “intuit” any damn thing of “what space truly is”. I think people are imposing abstract models of geometry onto our experience and saying “we literally experience space as it is through Euclidean eyes”. Bullshit. As I sit here now, I will be the first to admit that my direct experience as a human being does not entail anything “Euclidean” about space or even “of space itself” (and this is what Kant also said: that we don’t “experience space” as an object, our experience itself is shaped by space/our intuition). Simply put: people who use this argument are lying to themselves about their “experience” through confusing a conceptual abstract model (geometry) with “experience” …these are truly different “things”…i.e. they are making a common simple category mistake and basing their “critique of Kant” on a category mistake.
Also, Kant never (as far as I understand it) makes the claim “space is Euclidean”; as far as I understand it he would reject any statement making such a claim as “pointless metaphysics trying to get at the thing in itself which we can’t do”. So essentially Kant says this: we intuit space, but even though we intuit space, we still cannot answer the question without any shred of doubt as to “what our intuition (i.e. space) really is”. This is the “divine paradox” if you will. Our intuition is simultaneously the “closest to us, yet the farthest away”.
People make this mistake and attribute Kant to making a claim that “space/our intuition is Euclidean”, when he actually just says “we intuit space, but our intuition of space paradoxically does not grant us access to what space really is”; i.e. we might discover a new type of geometry of space tomorrow called “uber ultra Euclidean” and find that it “agrees with our perceptions of experiments” (i.e. agrees with our experience, BUT IS NOT OUR EXPERIENCE; again emphasizing that the conceptual mathematical model IS NOT the experience itself) better than both Euclidean and non-Euclidean models, but even this would not guarantee with absolute certainty that we now truly “know what space (i.e. our intuition) really is” (since our “perceptions of experiments” again is pre-conditioned/intuited by the very thing we are trying to understand/model…space). Simply put: our intuition (space) conditions our experience (observations, measurements, etc.) which in turn conditions our models of our experience (geometry).
One can of course question whether the models of our experience somehow “loops back” and conditions our experience as well; this is possible and is a more conservative argument against Kant which is highly debatable, but acceptable to me. So then one says “I do acknowledge that experience and models of experience are different categories, but I argue that they interact at a fundamental level”; i.e. you don’t make the stupid category mistake most non-Euclidean critics of Kant make, but you present an argument whereby our models of our experience are not completely separate from our experience itself, thus criticizing Kant for not acknowledging it. The other option is to say that our models of experience (i.e. any conceivable geometry) somehow does not just influence our experience, but actually reflects “how space really is” (i.e. that our very intuition is shaped by our models) and this is a far more liberal critique of Kant and one I presume he would just say: No, you are wrong, you are now emphasizing complete idealism and this is just not the case: we never “get at the thing itself” nor do our models somehow “change it” and even if it did, we could never know it. You can of course then criticise Kant for “building into his system a claim that cannot by the nature of the Kantian system be refuted/falsified”, but this is a completely different line of argumentation and one with its own sets of problems and assumptions.
Just for interest sake, if someone can point me to the exact sentence in Kant’s work where he makes the claim or postulate “space as we intuit it is Euclidean” I would be much obliged and would agree then that he made a mistake and if his theory as I understand it rests on such a claim then indeed his theory might be more suspect, but as I understand it Kant never makes such a claim and in fact warns against exactly such. I ask for this since I (as many other Kant fans I assume) have read works about his work, but have never had the guts to work through the Critique itself…but one day I will, I feel I owe it to Kant; he was indeed a brilliant philosopher.
Oh, and by the way: great post Wes, thanks for it.
Wayne Schroeder says
From Kant and Mathematical Knowledge-McFarlane
“Kant’s views on where the distinction [synthetic/analytic] is to be drawn merely reflects the knowledge of his day, and its limitations. But if the a priori synthetic truths condition the possibility of experience, and if the synthetic/analytic line can be shifted, this implies that experience is, in fact, malleable. Like learning to visualize non-Euclidean space, once we recognize that what was previously taken for granted is actually an arbitrary structure, we are free to change that conditioning. In this manner, the very definition of experience could be altered, for we have changed the rules which constitute it. As Kant said, “we can know a priori of things only what we ourselves put into them.” And who’s to say what we put into things can not be changed? While it is one thing to recognize the possibility of altering our preconditions of experience, it is quite another matter to actually transform those conditions and experience a different world.”
Wayne Schroeder says
From J.M Bernstein’s lecture on the Third Analogy (http://www.bernsteintapes.com/):
In his Refutation of Idealism, Kant holds that “actuality is in all cases either direct perceptual awareness or something that is connectable to direct perceptual awareness.”
“So Kant thinks anything is real or actual that either is or can be conceived or is connectable to what is or can be perceived.”
“Therefore it is perfectly compatible with Kantian idealism and realism that there be in principle theoretical particles that are invisible—just as long as they are causally connected—that is, we can get the effects of them perceptually.”
“And this is what we think today. There is no mystery about quarks and the like, about sub-atomic particles, their existence is not directly perceptible, but their effects can be perceived—a particle going through a cloud chamber leaves a line. We don’t see the thing but we see its effect on its medium that allows us to record its movement in a cloud chamber.”
“That’s all Kant means by connectability. So everything must be causally connected. “:
Leland Gregory says
I don’t understand the premise you’re refuting. It seems obvious to me that no amount of empirical understanding of the world is going to invalidate transcendental idealism. Perhaps you destroyed the argument so well that I can’t even pick up the pieces to reconstruct it.
Anyway, for the example of intuition vs “reality” I would love to hear your thoughts on how the checkerboard illusion is similar or dissimilar to the idea of euclidean space. It seems to me like this example matches up 1:1. Our intuition/perception tells us that square A is a very different color than square B, yet it is completely wrong. The thing I love about this illusion in particular is that there’s no way to see around it, as with some visual illusions. Even if you know how it works, and you prepare yourself, you’re always going to see the 2 blocks as different colors.
My mind has come preloaded with image processing that sees those colors inaccurately. And this inaccuracy happens before the moment when I as the subject apprehend it, but after light has entered my eyeballs. Amazing.
Amazing post. Thanks for voicing my opinion.
Orwin O'Dowd says
A timeous post: this issue is simmering, even among journalists, as the Enlightenment stirrs in the East: http://www.journalism-now.co.uk/seminar-blog-kant/
Phil, Kant’s pointed remarks are tucked away in his notebook Reflections, for which one had to go to the German Akademie Gessamelte Werke. Reflections 5-11 were analyzed by Michael Friedman in Logical Form and the Order of Nature (Princeton 2007) and he missed the point! Kant dispiutes Euclid’s Parallel postulate in favour of a constructive assumption, where parallel lines have a common orthogonal. So do longitudes at the Equator as any navigator knows!
The point was well made by Helholtz in the 1870s: one could operate normally in non-Euclidean spaces, as in a distorting mirror. But he didn’t go back to Kant, and all the academic commentary after that is just INCOMPETENT where it really matters. Friedman’s lapse is quite unfogivable, but Helmholtz was wrong about conservartion: force is not conserved – but his rant is reprinted in Harvard Classics, from the home of legal pragmatism which now justifies the cynical realism of the world’s tyrants.
Without independent bloggers and open comment we’d be chocking off around now, smothered by the sheer weight of battered dogma and careerist complacency.
I think what irks most mathematicians about the debate is the whole edifice of ideology itself. In fact many of the above comments make reference to “which” ideological edifice we should “support”. It is then odd to see many of those same comments then apparently attacking the “institution” of ideology.
Frankly speaking, any ensconced ideologue (philosophy, humanities, “social-science”) is reduced to defending their Ivory Castle, and it is sad to see that much of modern philosophy revolves around defending castles, even amongst its insiders (The David Foster Wallace episode comes to mind). On the other hand, the discipline of mathematics is actually in motion and actively explores the consequences of ambiguity rather then trying to dome-over foggy-gray with the philosophers’ beloved “Categorical Blanket”.
Most of the debate seems to revolve around what “category” the 5th “postulate” is in and such politicking doesn’t let things (like the oddity of the 5th “postulate”) stand for themselves. I wish “philosophy” as a practicing discipline would be less ideological and just accept that it doesn’t hold consequence over the the vast no-man’s-land of intellectual territory it often claims.
Mathematics as “an institution” is a long history of mavericks, and most of the advances made in the field are counter-intuitive to the prevailing ideology (mathematical / philosophical or otherwise). Yes there is an edifice of mathematics but it is an active one and not one that so resistant to new ideas that it must shut-down creative ways of application and relevance – in short it is willing to entertain and explore with rigor the notion that it “could be wrong”.
I know it is sad to say but I don’t think it matters that much if Kant really is “Correct”. The ambiguity of Euclid’s 5th “postulate” will remain no matter what category we decide to assign it to “axiom” or not or otherwise. Mathematics as as discipline is at least willing to encourage and entertain the notion that there may be “holes” or “cracks” in its edifice. The even more amazing part is that it then rigorously studies those “cracks” and moves knowledge forward as a whole in so doing, and as we can see it also moves (into) other standing edifices.
So really the subtext in this whole debate is this: Why is so much of modern philosophy simply about defending the walls of the old castle and not about moving knowledge forward? Was this motion of knowledge not the original concern of the discipline as it was first practices all those years ago?
You may need to actually read some Philosophy of mathematics or geometry – your description of how mathematics ‘works’ is nowhere near real life. Euclidean vs. Non-Euclidean vs. Projective Geometries hardly engender ‘cracks’ to be filled – they ask unanswered questions about our view of the universe.
And the Kantian argument/project – still ongoing – formed at least part of Russell’s and Poincare’s mathematical reasoning.
So – no.
Bill Kelleher says
I just discovered this essay. Astoundingly clear. It is as if you summarized in common sense language (the only language available for describing physics) the works of Tim Maudlin. Great!
One would have to indulge in delusions of denial, to insist empirical evidence did not destroy Kant’s a priori. The empirical overthrow of Euclidian geometry is, in fact, a huge slap in the face to pure reason. Apparently, some rationalist have not awakened. Kant was the same guy who couldn’t construct a scenario where would be the right thing to do. Not exactly a master of free invention, but his philosophy has been, nonetheless, been glorified by the confused masses.
You are not making any arguments – just throwing around generalized and proby incorrect ideas.
michael adams says
I was just outside looking at the railroad tracks. do you know that they meet way off in the distance?