Something about definitions in mathematics has always interested – confused? - me, I call it “arbitrariness in Mathematics” - it's a bad name, but I don't know a better one. Let me explain:

1st - Have you ever asked yourself why matrix multiplication is so strange?

When I discovered that matrix multiplication is “just” a definition (not a law of nature or something similar), and mathematicians could have defined it in other ways, I asked myself why they defined it like that. The answer is that by defining matrix multiplication in the “usual” way it matches with a lot of other concepts in mathematics, for example with composition of linear transformations. That is a reasonable answer. (In Graph Theory we define a different “product” to match “incidence” properties of the graph)

2nd - Have you ever asked yourself why (-1)(-1) = 1 ?

This one has a simpler answer, it is defined that way because it's the only way it could be defined for the properties of arithmetic operations to work. That is a definite answer.

When I find a new definition I always ask myself what is the reason (first or second?) for that definition to be so.

But that recently raised another question when I use the first explanation:

How do we know – do we? - that defining something in another way won't lead to useful properties and new discoveries? How do we know we don't “miss” anything not investigating other possible definitions?

Why it's not interesting to define and study other operations and functions for the real numbers?

I tired to explain it to myself by using concepts like isomorphisms and similar, but I'm not satisfied. This question seems a little bit vague but that is the best I could explain.

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    $\begingroup$ Most times things are defined after somebody began playing with them and found them useful/fun. One can try to define things out of the blue and see if they're useful somehow, but unless things were used and proved before, odds are that new definitions without any cause will lead nowhere. $\endgroup$ – Timbuc Nov 23 '14 at 4:38
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    $\begingroup$ Not to mention, the definitions that survive the test of time are the ones that have the nice properties you describe; the lame definitions aren't remembered. $\endgroup$ – Greg Martin Nov 23 '14 at 4:49
  • $\begingroup$ One good example is the concept of integration. There are different definitions (Riemann, Cauchy-Riemann, Lebesgue and Henstock-Kurzweil (maybe written differently)). Of these, the historically early ones lack a lot of nice properties (no satisfactory theory for interchanging limits and integrals). The Lebesgue definition is considered the "correct" definition for most purposes nowadays. $\endgroup$ – PhoemueX Nov 23 '14 at 4:59
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    $\begingroup$ An example to follow with Greg's statement and your initial question is that of matrix multiplication. One of those "other ways" to define matrix multiplication is the Hadamard product (entrywise product) which is occasionally used, but few people know the correct symbol and name to use for it. $\endgroup$ – JMoravitz Nov 23 '14 at 5:02

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