noahfor

Here is my essay on the Kant, if anyone cares. It's not good either, but I had a hard time understanding the relationship between non-Euclidean geometries and Kant that would make the geometries existence undermine Kant. I don't really even think they do.

Propositions can be either analytic or synthetic. An analytic proposition is one whose truth is contained within the meaning of its parts, and states nothing more than its parts alone. For instance, “If A then A” is true because states nothing more than “If A,” and so it is analytic. An analytic proposition is necessarily true. A synthetic proposition is one that states more than its parts alone. For instance, “If A then B” cannot be known by reference to its parts because nothing of A’s relationship with B is contained within A, and so it is synthetic. As well, proposition can be either a priori or a posteriori. A proposition is a priori if its truth does not depend on experience. A proposition is a posteriori if its truth can only be concluded by reference to experience.
There are four possible combinations of the types of propositions: analytic a priori, analytic a posteriori, synthetic a priori, and synthetic a posteriori. Analytic a priori propositions are possible because, by definition of an analytic proposition, their truth is necessitated by the meaning of their parts and not by any connection to experience. For the same reason analytic a posteriori propositions are not possible. From common experience we know that synthetic a posteriori propositions are possible. That leaves one last possible combination, synthetic a priori. Kant claims that not only are synthetic a priori propositions possible, but that they are also the basis for all of mathematics and natural science.
Kant believes that all perceptions must take place within space and time, but that the sense data which supplies the material for perception does not contain any information about its existing in space and time. Instead, Kant believes that space and time are structures that are added to perception in order to facilitate our understanding of sense data, and that space and time are understood a priori. Kant calls space and time intuitions. He argues that synthetic propositions can be made whose truth can be verified by reference only to space and time, and since space and time are understood a priori, these propositions will also be a priori. For instance, “Space has three dimensions.” can be verified by our understanding of space, and is synthetic because there is nothing of three-dimensionality inherent in the concept of space. It is Kant’s belief that all mathematical and geometrical propositions are synthetic a priori propositions generated and verified in this manner.
Of the synthetic a priori propositions based on the spatial intuition, were Euclid’s postulates, on which all of Euclidean geometry is based. However, in the 1800’s geometers began experimenting with geometries in which Euclid’s fifth postulate – “If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough” – is excluded. They found that by proposing a space shaped like a sphere, defining a line as great circle around the surface of the sphere, and a point as two opposite points on the sphere, they could create consistent geometries in which there are no parallel lines.
In Kant’s view it is our understanding of space that provides the basis for geometrical synthetic a priori propositions, such as Euclid’s fifth postulate, and it is the a priori nature of the spatial intuition that makes propositions based on it a priori, and a proposition derived from a non-Euclidean geometry, such as “no parallel lines are possible” can be seen as being of the same sort to propositions that are Euclidean in nature, except for the space that they describe. However, in the case of non-Euclidean propositions, the positions are not based on our intuitive space, but are also not based on any experienced space, so they must be analytically true. Thus, due to their equivalency to non-Euclidean propositions, Euclidean propositions must also be analytic in nature.
Kant also proposes that synthetic a priori propositions are the basis for the natural sciences. These, however, are not based on the intuitions, but on something he calls Categories. The Categories are to propositions what the intuitions are to perceptions, and that is the extent to which I understand them. From the categories, Kant believes propositions such as “Every happening has a cause.” can be known. It is on statements like this one that Newtonian physics is based. However, with the development of quantum mechanics we have learned that some happenings do not have a cause. For instance, the decay of an atomic nucleus happens completely by chance, and even if all the physical circumstances surrounding the nucleus of an atomic known, one could still not predict with certainty whether, at a given time, it would decay or not. Thus, Kant’s proposition “Every happening has a cause.” is falsified. Hence, it was never known, and whatever method used in obtaining the supposed knowledge failed to obtain any knowledge at all.

Thanks to everyone that helped me out.