10
$\begingroup$

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,...

$\endgroup$
7
  • 24
    $\begingroup$ 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
  • 5
    $\begingroup$ 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! $\endgroup$ Apr 19 '13 at 3:58
  • 5
    $\begingroup$ 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. $\endgroup$ Apr 19 '13 at 11:35
  • 4
    $\begingroup$ @ 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
  • 2
    $\begingroup$ @RobertGarbary, there is no obvious way to generalise what a commutative ring is? $\endgroup$
    – LSpice
    Aug 2 '20 at 14:44
19
$\begingroup$

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 nothing 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 $p\mid ab$ implies $p\mid a$ or $p\mid b$.

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.

$\endgroup$
1
  • 5
    $\begingroup$ FWIW, this explanation does also apply to equivalence relations. Starting from the standard definition, one can (1) remove reflexivity to obtain the notion of "partial equivalence relation", (2) remove symmetry to obtain the notion of preorder, or (3) remove transitivity to obtain a notion of undirected simple graph. These generalizations are not as clear with Hilbert's definition. $\endgroup$ Sep 26 '21 at 19:42
10
$\begingroup$

I can think of several distinct reasons.

  1. Definition A may coincide with Definition B in a special case. Definition A may be more elementary, or easier to understand, while Definition B may generalize better. When we don't care about more general scenarios, we may prefer Definition A for its simplicity, but when we have the general case in mind then we may prefer Definition B. One example is the integral of a continuous function over the interval [0,1]; the Riemann integral is (arguably) simpler to understand, while the Lebesgue integral generalizes better.

  2. Definition A may be "axiomatic" while Definition B may be "constructive." For example, the real numbers may be defined axiomatically as a complete ordered field with the least upper bound property, or they may be defined in terms of Dedekind cuts or Cauchy sequences. (Also, as mentioned in a comment, other examples come from category theory, which allows us to define many things in terms of a universal property; but to prove that the definition is not vacuous, we typically need a way to define the object constructively.) The axiomatic definition is typically preferable for developing the theory, because it shows that the theorems depend only on the stated properties and not on the details of the explicit construction, but the explicit construction is needed at some point to prove that the object actually exists.

  3. Definitions A and B may be cryptomorphic, which roughly speaking means that they are logically interchangeable. In some ways this is the most interesting case, because the reasons for preferring one over the other can be subtle and subjective; e.g., the proofs of certain theorems may be easier or more natural using Definition A, while the proofs of other theorems may be easier or more natural using Definition B. Or, Definition A may be more convenient for explicit computations, while Definition B may be more suitable for proving theoretical results. For more discussion, see this MO question.

$\endgroup$
13
  • 1
    $\begingroup$ Instead of constructive I guess you mean "analytic" (the opposite to synthetic) $\endgroup$ Sep 26 '21 at 23:35
  • 1
    $\begingroup$ @DavidRoberts "Constructive" is not the best word, I agree, but I don't think I mean "analytic" either. I almost want to say that one definition focuses on uniqueness and the other focuses on existence. $\endgroup$ Sep 27 '21 at 0:28
  • $\begingroup$ I'm not sure how one proves existence except via building a model, or else something like the compactness theorem (?). Certainly the discussion and example you give seems analytic to me, but if you mean something else, then fair enough... $\endgroup$ Sep 27 '21 at 6:52
  • $\begingroup$ @DavidRoberts Is the example of the real numbers not clear enough? One proves the existence of an ordered field with the least upper bound property by using Dedekind cuts or Cauchy sequences. $\endgroup$ Sep 27 '21 at 11:32
  • $\begingroup$ Yes, it's clear, but that's what I would call an analytic definition: building a model of some axioms out of some other stuff, rather than postulating properties and working with them (=synthetic definition). Uniqueness is a separate issue, cf synthetic plane geometry (w/o parallel axiom) vs building models using coordinates etc. $\endgroup$ Sep 27 '21 at 12:28
6
$\begingroup$

Many mathematicians prefer to have (at least) two equivalent and elegant(!), but different definitions of the same notion/topic/... (such definitions either both exist or one formulates one of them or even both and proves the equivalence).

One of the reasons might be that one definition may look as weak as possible, while the other one may look as strong as possible.

Then when you want to prove that a construction/theory/... is an example/model/... of the given theory, use the seemingly weak definition.

But when you want to prove hard theorems that follow from the definition then it'd be much easier to apply the strong/advanced definition.

There is intellectual energy stored between the weak and the strong definitions.


First Example: there are easily a dozen and more elementary axiomatizations of the so-called Euclidean plane geometry. They often aim at elegance understood as making the axioms as weak as possible. Given an algebraic model as Cartesian $\ \mathbb R^2\ $ it's easy to prove such weak axioms.

However, an excellent definition of such plane geometry is as follows: the study of metric invariants of the complex plane $\ \mathbb C.\ $ Suddenly you can truly prove powerful theorems in an algebraic clean way, and often even without relying on ingenuity as it is invariably in the case of the classical approach. You also get a clear understanding of the border (degenerated) cases. Finally, "elementary" proofs consider several cases due to the different situations w.r. to the "in-between" relation. This, however, is not any problem in the case of algebraic proves based on the complex plane.

PS. While the Cartesian plane is adequate for the affine geometry of the plane (and for the classical mechanics -- classical mechanics is a natural and mild extension of the affine geometry), it is the complex plane $\ \mathbb C\ $ that handles also angles.


Second Example: Elementary definition of determinant is elementary but the axiomatic definition is easier to apply and provides understanding. For instance, it's natural to use the axiomatic definition to derive the formula for the derivative of a square matrix when the entries are differentiable functions.

PS. The said axioms of the determinant of a matrix are formulated as a linear function of its columns, alternative, and normalized -- the determinant of $\ \mathbb I_n:=1$.

$\endgroup$
2
  • $\begingroup$ Nice idea, but could you also provide one example please, two will do as well :) I understand that how unconsciously people might switch from one to the other, but consciously? $\endgroup$ Sep 28 '21 at 13:58
  • $\begingroup$ What I have described seems ubiquitous. I'll still try to provide examples (just give me some time, please). The issue is a bit obscured by the fact that instead of actual definitions, there are constructions, often not equivalent, and then someone comes as Samuel Eilenberg and his partners who give justice to the apriori not-so-clear ideas. $\endgroup$
    – Wlod AA
    Sep 28 '21 at 23:15
1
$\begingroup$

The Cauchy definition of continuity makes Lipschitz functions continuous, and it's easy and natural. In general, it's good to prove the continuity of functions -- especially, that the whole function is continuous (or at a whole region) rather than at just one point.

On the other hand, the Heine definition of continuity makes it easier to prove that a function is not continuous at a given point.

The two styles of the definition of continuity were extended over all metric spaces, and then for all topological spaces.

$\endgroup$
1
  • $\begingroup$ I am guessing that "Heine continuity" means preservation of convergence of sequences? I have never encountered this terminology before. $\endgroup$ Oct 8 '21 at 19:37

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.