Non-Euclidean geometry does follow Euclid's assumptions at the scales that are meaningful to humans. So there is no conflict.A quote from the essay that summarises the problem:
"....Geometry that fails to follow Euclid's assumptions is, according to Kant, literally inconceivable."
Frank Wilczek — Perplexed
I expect Kant would have been entirely comfortable with the notion that our in-built mechanism for arranging information is an approximation to a paradigm whose differences are only visible at scales that are beyond ordinary human experience. — andrewk
we process raw inputs within a framework of three space dimensions and one time dimension. — andrewk
For me they seem to. When I visualise many-dimensional branes colliding in even higher dimensional space, I visualise hanging, wobbly two-dimensional sheets banging into one another in 3D space. The calculations will be different from the 3D case but for me the visualisation has to remain 3D (or at most 4D - I sometimes use time as proxy for a fourth spatial dimension) as I am not capable of visualising anything higher.I wonder if those concepts could fit into the transcendental aesthetic since they're so far beyond the normal framework of space/time.
Did he do that? I don't know. I never read the original, being a secondary-source kind of chap. I think we'd need to dig out a quote, both in the original German and a diversity of English translations, and analyse it to see whether we can reach that conclusion from it. My loose observation about Kant scholarship is similar to that commonly made about economics: If we put n Kant scholars in a room there will be n+1 opinions about what Kant meant by any particular passage he wrote.Isn't the issue that Kant elevated the postulates of Euclidean geometry to the level of a metaphysical certitude
The curvature that gives us our fairly modest Earth gravity is a curvature of spacetime, not a curvature of space. The difference between the two concepts is crucial. IIRC, it is possible to have a spacetime that is curved but for which all spatial slices are flat. The curvature is only in the relation between space and time, not in the space itself.I'm not sure we can rely on geometry remaining Euclidean at human scales because if one takes general relativity into account then masses are warping spaces and causing gravitational geodesics that support our entire existence. — Perplexed
Space is not an empirical concept which has been derived from outer experiences. For in order that certain sensations be referred to something outside me (that is, to something in another region of space from that in which I find myself), and similarly in order that I may be able to represent them as outside and alongside one another, and accordingly as not only different but as in different places, the representation of space must already underlie them. Therefore, the representation of space cannot be obtained through experience from the relations of outer appearance; this outer experience is itself possible at all only through that representation — Kant
Space is not something objective and real, nor a substance, nor an accident, nor a relation; instead, it is subjective and ideal, and originates from the mind’s nature in accord with a stable law as a scheme, as it were, for coordinating everything sensed externally — Kant
Space is a necessary a priori representation that underlies all outer intuitions. One can never forge a representation of the absence of space, though one can quite well think that no things are to be met within it. It must therefore be regarded as the condition of the possibility of appearances, and not as a determination dependent upon them, and it is an a priori representation that necessarily underlies outer appearances. — Kant
Space is not a discursive, or as one says, general concept of relations of things in general, but a pure intuition. For, firstly, one can represent only one space, and if one speaks of many spaces, one thereby understands only parts of one and the same unique space. These parts cannot precede the one all-embracing space as being, as it were, constituents out of which it can be composed, but can only be thought as in it. It is essentially one; the manifold in it, and therefore also the general concept of spaces, depends solely on limitations. It follows from this that an a priori intuition (which is not empirical) underlies all concepts of space. Similarly, geometrical propositions, that, for instance, in a triangle two sides together are greater than the third, can never be derived from the general concepts of line and triangle, but only from intuition and indeed a priori with apodeictic certainty — Kant
Perception, partly pure a priori, as establishing mathematics, partly empirical a posteriori as establishing all the other sciences [...] We demand the reduction of every logical proof to one of perception. Mathematics, on the contrary, is at great pains deliberately to reject the evidence of perception peculiar to it and everywhere at hand, in order to substitute for it logical evidence. We must look upon this as being like a man who cuts off his legs in order to walk on crutches [...] Whoever denies the necessity, intuitively presented, of the space-relations expressed in any proposition, can with equal right deny the axioms, the following of the conclusion from the premises, or even the principle of contradiction itself, for all these relations are equally indemonstrable, immediately evident, and knowable a priori — Schopenhauer WWR Vol I §14-15
In fact, it seems to me that the logical method is in this way reduced to an absurdity. But it is precisely through the controversies over this, together with the futile attempts to demonstrate the directly certain as merely indirectly certain, that the independence and clearness of intuitive evidence appear in contrast with the uselessness and difficulty of logical proof, a contrast as instructive as it is amusing. The direct certainty will not be admitted here, just because it is no merely logical certainty following from the concept, and thus resting solely on the relation of predicate to subject, according to the principle of contradiction. But that eleventh axiom [11th axiom is equivalent in the context of Euclidean geometry with Euclid's Fifth Postulate] regarding parallel lines is a synthetic proposition a priori, and as such has the guarantee of pure, not empirical, perception; this perception is just as immediate and certain as is the principle of contradiction itself, from which all proofs originally derive their certainty. At bottom this holds good of every geometrical theorem — Schopenhauer WWR Vol II §8
As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality
I mean what makes geometry - if anything - certain? Before, it was taken to be certain since it was merely the reflection, in the understanding, of the a priori spatial form of the sensibility, and as such, there was nothing "external" involved in it, but rather everything external had to conform to it. — Agustino
And where do the axioms and postulates get their certainty from?From axioms and postulates. — Response
In Euclid's Elements Book I, Euclid gives definitions, postulates, common notions and propositions (the latter of which are derived from the definitions, postulates and common notions or each other).
There are 5 postulates given:
1. To draw a straight line from any point to any point. {this is to be read in the sense of "it is always possible to draw a straight line from any point to any other point"}
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any centre and distance.
4. That all right angles are equal to one another.
5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which there are angles less than two right angles.
My question is "which of these 5 postulates are synthetic a prioris, which are synthetic a posteriori, and which are analytic"? — Agustino
That essay is a pile of manure.There’s an essay on it here. — Wayfarer
>:O And Kant wasted his time saying that the propositions of geometry are synthetic in order to tell us they can be denied without contradiction, because that certainly proves the reliability of mathematics when applied to physics (which is what Kant was trying to do - just open a page of the Prolegomena, instead of reading 20,000 secondary sources who don't know what they're talking about). Of course, this isn't what Kant did. Kant aimed to say that the propositions of geometry don't derive their CERTAINTY (because he took their certainty for granted) due to the law of non-contradiction (hence the synthetic part). Rather they derive their certainty from their a priority, rooted as they are in the pure form of sensation, space.The Euclidean nature of our imagination led Kant to say that although the denial of the axioms of Euclid could be conceived without contradiction, our intuition is limited by the form of space imposed by our own minds on the world. While it is not uncommon to find claims that the very existence of non-Euclidean geometry refutes Kant's theory, such a view fails to take into account the meaning of the term "synthetic," which is that a synthetic proposition can be denied without contradiction.
Nope. This is a mistaken view. Kant was trying to show why mathematics is so effective at describing physical space - if mathematics is just a human construct, its effectiveness cannot be accounted. So Kant resolved the problem by saying that it is not just a human construct. We have this form of pure intuition, space, from which we derive the axioms of geometry. So they are synthetic, since we can imagine the opposite to be the case, but they are a priori, because they stand as being true prior to experience. His point was that I don't need to run a physical experiment to see that Euclid's Fifth Postulate holds (which of course turned out to be false). It is simply impossible, according to Kant, for Euclid's Fifth Postulate not to hold - not due to its synthetic nature, but due to its a priority, and the role the pure form of sensation (space) has in constituting all (spatial) experiences, and hence any possible experimental result.Non-Euclidean geometries are just as intuitive (synthetic a priori) within their contexts as Euclidean geometry is within the context of everyday experience, so I don't see the problem. — Janus
That is correct.Kant considered all of mathematics and geometry to be synthetic a priori knowledge, unless I am mistaken. — Janus
Space is not something objective and real, nor a substance, nor an accident, nor a relation; instead, it is subjective and ideal, and originates from the mind’s nature in accord with a stable law as a scheme, as it were, for coordinating everything sensed externally — Kant
Sorry, but I will disagree with you on this.Meh, you miss the point. — Hanover
Reality is a hazy word. Why is the noumenon reality, and the phenomenon not? Don't forget that it is the phenomenon that is the empirically real according to Kant, not the noumenal. Kant's notion of a noumenon, at any rate, is confused. He talks of the noumenon causing the phenomenon, which is nonsense, since causality is a category of the understanding, and hence can only apply to the phenomenon. It takes Schopenhauer to clarify this aspect of Kant.Kant says nothing of reality (noumena) — Hanover
Precondition of the sensibility, not of the understanding. Kant talks of space (and time) in the Transcendental Aesthetic, and labels them as forms of the sensibility (as opposed to the matter or content of the sensibility, the sensations themselves), which comes before he goes into the categories of the Understanding.Space is a precondition of understanding — Hanover
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