How is a system of axioms different from a system of beliefs?

Other ways to put it: Is there any faith required in the adoption of a system of axioms? How is a given system of axioms accepted or rejected if not based on blind faith?

Answer

To paraphrase Robert Mastragostino’s comment, a system of axioms doesn’t make any assertions that you can accept or reject as true or false; it only specifies the rules of a certain kind of game to play.

It’s worth clarifying that a modern mathematician’s attitude towards mathematical words is very different from that of a non-mathematician’s attitude towards ordinary words (and possibly also very different from a classical mathematician’s attitude towards mathematical words). A mathematical word with a precise definition means precisely what it was defined to mean. It’s not possible to claim that such a definition is wrong; at best, you can only claim that a definition doesn’t capture what it was intended to capture.

Thus a modern interpretation of, say, Euclid’s axioms is that they describe the rules of a certain kind of game. Some of the pieces that we play with are called points, some of the pieces are called lines, and so forth, and the pieces obey certain rules. Euclid’s axioms are not, from this point of view, asserting anything about the geometry of the world in which we actually live, so one can’t accept or reject them on that basis. One can, at best, claim that they don’t capture the geometry of the world in which we actually live. But people play unrealistic games all the time.

I think this is an important point which is not communicated well to non-mathematicians about how mathematics works. For a non-mathematician it is easy to say things like “but i can’t possibly be a number” or “but can’t possibly be a number,” and to a mathematician what those statements actually mean is that i and aren’t parts of the game Real Numbers, but there are all sorts of other wonderful games we can play using these new pieces, like Complex Numbers and Projective Geometry

I want to emphasize that I am not using the word “game” in support of a purely formalist viewpoint on mathematics, but I think some formalism is an appropriate answer to this question as a way of clarifying what exactly it is that a mathematical axiom is asserting. Some people use the word “game” in this context to emphasize that mathematics is “meaningless”. The word “meaningless” here has to be interpreted carefully; it is not meant in the colloquial sense (or at least I would not mean it this way). It means that the syntax of mathematics can be separated from its semantics, and that it is often less confusing to do so. But anyone who believes that games are meaningless in the colloquial sense has clearly never played a game…

Attribution
Source : Link , Question Author : Gabriel , Answer Author : Qiaochu Yuan

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