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3 ideas
17715 | The truth of the axioms doesn't matter for pure mathematics, but it does for applied [Mares] |
Full Idea: The epistemological burden of showing that the axioms are true is removed if we are only studying pure mathematics. If, however, we want to look at applied mathematics, then this burden returns. | |
From: Edwin D. Mares (A Priori [2011], 11.4) | |
A reaction: One of those really simple ideas that hits the spot. Nice. The most advanced applied mathematics must rest on counting and measuring. |
10216 | We master arithmetic by knowing all the numbers in our soul [Plato] |
Full Idea: It must surely be true that a man who has completely mastered arithmetic knows all numbers? Because there are pieces of knowledge covering all numbers in his soul. | |
From: Plato (Theaetetus [c.368 BCE], 198b) | |
A reaction: This clearly views numbers as objects. Expectation of knowing them all is a bit startling! They also appear to be innate in us, and hence they appear to be Forms. See Aristotle's comment in Idea 645. |
17716 | Mathematics is relations between properties we abstract from experience [Mares] |
Full Idea: Aristotelians treat mathematical facts as relations between properties. These properties, moreover, are abstracted from our experience of things. ...This view finds a natural companion in structuralism. | |
From: Edwin D. Mares (A Priori [2011], 11.7) | |
A reaction: This is the view of mathematics that I personally favour. The view that we abstract 'five' from a group of five pebbles is too simplistic, but this is the right general approach. |