Timeline for Why do mathematicians prefer one definition over the other when they both define the same concept?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 7, 2021 at 4:10 | answer | added | Wlod AA | timeline score: 1 | |
| Sep 28, 2021 at 5:34 | answer | added | Wlod AA | timeline score: 6 | |
| Sep 27, 2021 at 16:32 | comment | added | Jon Bannon | @LSpice ROFL. This made my day... | |
| Sep 26, 2021 at 16:21 | answer | added | Timothy Chow | timeline score: 10 | |
| Aug 2, 2020 at 14:44 | comment | added | LSpice | @RobertGarbary, there is no obvious way to generalise what a commutative ring is? | |
| Apr 19, 2013 at 13:31 | comment | added | Paul | @ 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. | |
| Apr 19, 2013 at 11:35 | comment | added | Benjamin Steinberg | 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. | |
| Apr 19, 2013 at 10:53 | comment | added | Amir Asghari | /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 | |
| Apr 19, 2013 at 6:50 | vote | accept | Amir Asghari | ||
| Apr 19, 2013 at 3:58 | comment | added | Robert Garbary | 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! | |
| Apr 19, 2013 at 0:37 | answer | added | user9072 | timeline score: 19 | |
| Apr 18, 2013 at 22:17 | comment | added | Paul | 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. | |
| Apr 18, 2013 at 21:18 | history | asked | Amir Asghari | CC BY-SA 3.0 |