149
$\begingroup$

I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in practice. Are there examples of equivalent definitions where one is more natural or intuitive? (I'm meaning so greatly more intuitive so as to not be subjective.)

Alternatively, what common examples are there in standard lecture courses where a particular symbolic definition obscures the concept being conveyed.

Improve this question
$\endgroup$
14
  • 9
    $\begingroup$ This definition of the product topology is really not so bad when you correct the typos and translate it to words: it says every point X x Y should have an open neighborhood that's a product of open sets in X and in Y. What's wrong with that? $\endgroup$
    – Jonathan Wise
    Dec 2, 2009 at 16:52
  • 14
    $\begingroup$ Well, the natural generalization of that definition is the box topology, whereas the natural generalization of Daniel's definition is the (categorical) product topology. $\endgroup$
    – Qiaochu Yuan
    Dec 2, 2009 at 17:07
  • 16
    $\begingroup$ My second comment: (2) The definition in terms of open sets is spiritually a construction, not a definition. It may be described as "a construction in terms of open sets that works only for finite products". The definition in terms of coarsest topology is a genuine definition, and is generally accepted as the correct definition, but it doesn't give you a construction. The genuine definition gives you much more intuition about the product, but sometimes you need a construction. Some of my fellow category theorists regard that bit about needing a construction as a heresy. $\endgroup$
    – SixWingedSeraph
    Dec 2, 2009 at 17:24
  • 12
    $\begingroup$ The definition in terms of a coarsest topology gives you a perfectly valid construction: take the inverse image of every open set. $\endgroup$
    – Qiaochu Yuan
    Dec 2, 2009 at 17:28
  • 13
    $\begingroup$ This general point about definitions needs to be made: The definition is intended to give a (more or less) minimal technical description of the concept that implies all true theorems about the concept and nothing else. It doesn't matter if the definition emphasizes technical aspects and doesn't mention some big intuitive ideas about it. That's not what definitions are for. A teacher should provide many ways to think about the concept, some of which might constitute definitions. $\endgroup$
    – SixWingedSeraph
    Dec 5, 2009 at 1:35

31 Answers 31

Reset to default
1
2
-6
$\begingroup$

The first sentence in a probability talk is likely to be, Let $X$ be a random variable. As a non-probabilist who dabbles occasionally in probability, I find the notion of rv difficult to absorb; and then they have the expectation operator (integration), characteristic function (with a different meaning---indicator function means what characteristic function means in measure theory), and in general, it is a distinct language.

Improve this answer
$\endgroup$
1
2

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