Quora Answer: Do mathematical scholars agree with Kant’s view that math is synthetic a priori?

Oct 18 2014

It is not mathematics in general that is the Synthetic a priori as far as I know but Geometry which synthesizes space. Time is an Analytical a priori and related to Arithmetic for Kant. The the categories which are also synthetic a priori are schematized and thus connected to the series in time. The series in time is discontinuous so that the connection via the schematization and the categories gives a connection between the continuous and the discontinuous. The status of time is up in the air in Kant as we can see by Heidegger’s interoperation of Kant. Mathematical Categories like groups, rings, topologies are obviously synthetic, but it is unclear if they are a priori or a posteriori, i.e. platonic source forms or something we construct after the fact. What seems to like time and number is the series. But moments of time are not continuous in the same sense that geometrical objects in space are continuous. There is one school of thought on time that only the Now moment actually has Being which we ascribe to Zeno and Parmenides. There is another school of thought that time is continuous in its flowing like a river which is attributed to Heraclitus. But unlike the continuity of time we do not have access to the prior moment once it is gone in time the same way we have access to a point on a surface that we have moved away from and then can move back to. Some therefore say that each moment of time is itself discontinuous from the other moments of time like Heidegger which attributes different existentials to the different temporal ecstasies. When we consider a group as a table then it appears Parmenidian. But if we consider it as rotations that take the same thing back to the same position again at the end of each group operation then it appears discontinuous oscillating between the sameness of the outcome and the operations that give the group structure. But which group operation we execute seems arbitrary even though the group itself is well-ordered, so that makes it appear as if the group operations are discontinuous with each other. So the question of time in mathematics is problematic and up in the air and it is unclear whether time is a synthetic a priori or analytic a priori or synthetic a posteriori. When given a mathematical structure we analyze it then that is analytic a posteriori. But the givenness of math categories themselves is where the problem arises and there are different philosophical positions on that which it is difficult to decide between.

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