05 February 2012

Lobachevsky and Kant

I attend Shimer College, and a few of my school-related friends are St. John's grads, so I've grown accustomed to friends opining on how one section of the great Books canon bears upon another section.  One popular such connexion is between Nikolai Ivanovich Lobachevsky and Immanuel Kant, and it goes thusly:

When Kant wrote the Transcendental Aesthetic, he was unfortunately constrained by an insufficient understanding of geometry: he suffered under the delusion that Euclidean geometry was the only possible variety, whereas Lobachevsky has demonstrated that we can just as easily have geometries that do not satisfy the Euclidean postulates.  Inasmuch, consequently, as the thesis that the One True Geometry must be Euclidean is fairly crucial to the Aesthetic, and the Aesthetic crucial to the rest of Kant's transcendental philosophy, the falsity of the first item renders the final pretty suspicious.

I wonder, however, how much Lobachevsky's discovery really speaks against Kant's doctrine in the Aesthetic.  My chief difficulties with the argument are two.

Firstly, Kant regards the chief consequence of the Aesthetic for the theorems of Euclidean geometry to be that the former establishes a synthetic basis for the validity of the latter.  Lobachevsky's insight, conversely, is more or less (unless I am profoundly misunderstanding him, which is completely possible) that certain Euclidean theorems (and the parallel postulate) can be denied without landing us in contradiction.  Requiring a contradiction, however, is the hallmark only of the denial of analytic truths.  Why, then, can't Kant simply say to Lobachevsky, "Yes, you're right: Euclidean geometry is not analytic.  But I never said that it was!  The reason I wrote the Aesthetic in the first place was to account for their being true necessarily without being true analytically.  All you have done is to rigourously prove my contention that geometry does not present us with an analytic doctrine."?

Secondly, I sometimes hear it objected that Kant overlooks the possibility of non-Euclidean forms of intuition (a possibility supported, it is claimed, by the non-contradictoriness of Lobachevskian geometry), which is supposed to somehow threaten the Transcendental Aesthetic.  But I don't see that the purported consequence follows.  Kant never claims that he is presenting the forms of intuition for anything other than humans, and Kant's treatment of space will be true of humans if of anything.  I seem to remember, in fact, him somewhere saying that a radically different being could be possessed of forms of intuition other than space and time, and he is certainly emphatic that there might well exist an intelligence possessed of intellectual intuition, which would thus not be affected with appearances and a fortiori not possess a means of organising them (i.e. a form of intuition like space or time).  So, I don't see how on either account Kant's doctrine can be impugned by appeal to the authority of Lobachevsky.

Any comments?


  1. I found this post by searching the term "Lobachevsky and Kant", since I caught myself wondering the very same thing while reading Kant's Prolegomena.

    I think the answer might lie in Kant's use of the term "anschauung", typically translated "intuition", but which I read is the German word used to translate the ancient Greek "phantazein" meaning "to present to the mind". As such, the term might have a visual connotation; it is related to our modern word "phantasm". In short, one is certain via anschauung because he has can "see" the truth in his mind.

    Therefore, I wonder if Lobachevsky, while he does produce a geometry that seems logically sound, doesn't fit Kant's terms because his geometry can't be drawn. True, Euclid's can't either, or at least not precisely, but it can be represented in a way that Lobachevsky's simply can't.

    It's true that Lobachevsky's geometry creates no contradictions of its own necessity (or at least it doesn't as far as I can recall it), but it also produces a geometry that can't really be visualized, or imagined; one is asked to believe that a straight line does not cross another straight line at an angle even if drawn out as far as you wish. Consequently, I think Kant would probably dismiss his geometry as an interesting exercise in logic but not a serious threat to Euclid.

  2. Hi. I am very interested in this topic and I (unlike most other philosophy students) agree with you. I am planning to write a dissertation on the connection between Lebachevsky and Kant and I am looking for sources that would aid me in this effort.
    Thank you

  3. The point that Leo seems to be missing is that Kant set himself up to be shot down when he showed what he meant by the existence of a priori synthetic truths by stating that an example of his principle was that the sum of the angles of a triangle must equal 180 degrees (half a circle) in all worlds (to use today's terminology). Lobachevsky showed a "world" in which this is not true. (By the way, "Anonymous", this "world" can easily be drawn on the surface of a sphere.) Hence Kant's assertion is false. The falsity of his assertion has been deepened by further results in mathematics since then, but that is another question. Ahmad: alas, there is no good popular source that gets very deeply into this, and there is the problem that there are good sources on this, known to mathematicians as results in the field of Model Theory (which essentially had its birth with Lobachevsky), but most philosophy students do not have the background to delve into this. Thus I would suggest getting some mathematician who specializes in Model Theory (because not all mathematicians are well versed in this field) to help you with your doctoral thesis.

  4. Postscript: I wrote too quickly in stating that you could draw Lobachevsky's geometry on a sphere; what I meant was that you can draw a geometry on the surface of a sphere which will also serve as a counter-example to Kant's idea, so the issue as to whether a geometry can be easily drawn does not have any bearing.