Here is a basic, though very important, example: Hilbert takes as primary the notion of “congruence” (or “equal”) between segments. His first axiom of congruence “requires the possibility of constructing a segment congruent to an assigned segment”. His second axiom reads: "if two segments are congruent to a third one they are congruent to each other." Then he goes to prove the standard properties of equivalence relations follow from the axioms: Since congruence or equality is introduced in geometry only through these axioms, it is by no means obvious that every segment is congruent to itself. However, this fact follows from the first two axioms on congruence if the segment AB is constructed on a ray so that it is congruent, say, to A'B' and Axiom III, 2 is applied to the congruences AB ≡ A'B', AB ≡ A'B'. On the basis of this the symmetry and the transitivity of segment congruence can be established by an application of Axiom III, 2.

Now it is clear how we may define the general notion of equivalence relation. This "Hilbertian" defintion has at least two advantages: first, It avoids (at the start) the somehow non-intutive property of reflexivity; second, and more importantly, it is in more harmony with the standard way of defining an equivalence class consisting of everything equivalent to a focal element.

Here are my sub-questions: Was the standard definition of equivalence relation just more lucky? Or, was there a rational choice involved? Do you know any other definition that has been preferred over the other? If yes, what was the reason? Do you have a personal example of such "choice"? If yes, what is the reason of your preference? Do you have an "instructional" defintion of a certain concept that you prefer it over the "standard" definition of the same concept when teaching? if yes,...

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    $\begingroup$ I like to think mathematicians work with ideas, not definitions or axioms, and a formal definition is just one way communicate the idea, but the "real" mathematical object is the union of all the different ways to define it and all the proofs of equivalence of the various definitions. Which one you (or Hilbert) use first depends on convenience and from what background you come to it. $\endgroup$ – Paul Apr 18 '13 at 22:17
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    $\begingroup$ Also, different ideas generalize better! For example, a commutative ring with unit is really the same thing as an affine scheme. There is no obvious way to generalize what a commutative ring is. However, generalizing a certain type of ringed space? No problem! $\endgroup$ – Robert Garbary Apr 19 '13 at 3:58
  • $\begingroup$ /paul I do certainly agree that mathematicians work with ideas when they are at their own desk. But, iamgine what would happen in a few years if they (as a community) do not organize their knowledge: no two mathematicians could speak with each other since that "union of all the different ways" of defining an object/idea/concept would be too big that makes it hard (if not impossible) even for the people working in the same area to communicate with each other. Thus, by and large, it seems there should be a "mechanism" to choose agreed upon definitions. – Amir Asghari $\endgroup$ – Amir Asghari Apr 19 '13 at 10:53
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    $\begingroup$ Post category theory it has become clear that it is often cleaner to define an object (up to canonical iso) by a universal property rather than explicit construction. This helps to clarify where it stands in relation to other objects. $\endgroup$ – Benjamin Steinberg Apr 19 '13 at 11:35
  • $\begingroup$ @ Amir: That's why we write articles ;-). And (in my opinion and experience) I think you have it backwards: the more one studies the different ways of defining a mathematical object and the proofs of the equivalences of the different definitions, and their implications and generalizations, the simpler the idea becomes (and therefore easier to communicate), rather than becoming too big or too hard. Definitions are awkward because language is not always perfectly suited to express a mathematical idea concisely. $\endgroup$ – Paul Apr 19 '13 at 13:31

To give one answer to the question in the title: A reason to prefer one way of defining the same 'idea' over another is that it generalizes better. (Where of course what better means can depend.)

I have notthing to say about equivalence relations but since also other examples are asked for:

The notion 'prime number' can perhaps serve as an example for what I mean.

For the natural numbers one can define this in various ways equivalent ways. In particular (we exclude $1$, and $0$ if one considers it as natural number):

  1. A number $p$ is a prime number if $p=ab$ implies $p=a$ or $p=b$. (Or, put differently, $p$ is only divisible by $1$ and itself)

  2. A number $p$ is a prime number if $p|ab$ implies $p|a$ or $p|b$.

These are equivalent for the natural numbers. But, I am convinced that the latter is the better definition of prime number. (Though, to answer side questions, when teaching introductory things I might use the former, since students might be already familiar with this being the definition and anyway I will not have the time to convey why the other is better. Also, I think this conviction is not universal (now) and certainly was not earlier.)

So, why am I convinced about this. Because, if you part from the natural number/integers to more general things, say rings of algebraic integers, Dedekind domains, domains in general, it is this definition that generalizes better to yield a notion of prime element.

The property given by the former is also interesting in more general situations, but I/one(?) would not call such an element prime but rather irreducible, for example.

Thus, in this case I would not say there are two definition for the 'idea' prime number that are equivalent.

But, rather there are really two 'ideas', the one of an irreducible element and the other of a prime element, and for the natural numbers (and the integers) one can show that they coincide.

The former definition yields the former, and the latter the latter. And, the latter being somehow more pertinent, as also documented by naming, this is the definition of prime number (in the naturals) that I prefer, since it generalizes better.

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