Timeline for Definition of Function
Current License: CC BY-SA 2.5
24 events
| when toggle format | what | by | license | comment | |
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| Jul 24, 2010 at 9:32 | comment | added | Vag | @Joel David Hamkins: For example, every structure contains a (possibly empty) set of functional symbols so I guess there is notion of function that is completely theory agnostic. | |
| Jul 24, 2010 at 6:37 | comment | added | Vag | @Joel David Hamkins: In ambigious context (when it is unclear which known theory or which exact definition author uses) notion of function is indeed defined precisely (as 'isafun_1 or isafun_2 or logically equivalent'). Under 'contextless function' I mean not 'function in ambigious context' but precise meaning of word 'function' in heads of these who do not know neither isafun_1, nor isafun_2, nor their underlying theories but understands this notion perfectly. | |
| Jul 16, 2010 at 20:00 | comment | added | Joel David Hamkins | But the function concept has been made absolutely precise. In fact, it has been made fully precise twice, in two different ways. Each group prefers to use their own precise definition, for sound reasons, and I don't see how it would help anyone for you to give a third, incompatible definition, that was different than either of these. Rather than giving a "contextless" definition, you would instead merely be providing a third, unneeded, context. | |
| Jul 6, 2010 at 22:06 | comment | added | Vag | In programming I've found that giving absolutely precise meaning for ANY terms is very rewarding. For instance, more rigour means more human activities to automatize or made computer assisted. And while studying mathematics I've found that almost every time I formalize some important notion I discover various mistakes, misunderstandings, nontotalities, discrepancies, etc. Human intuition is good at abstracting and searching similarities but bad at case analysis. | |
| Jul 6, 2010 at 20:00 | comment | added | Joel David Hamkins | I think Carl was referring to the feature, as we've mentioned, that there is a common core part of the various definitions, the crucial fact that we can evaluate a function to produce a single value at any point in the domain, on which all the definitions agree. So it would seem that the word function would carry at least this part of the meaning even when there is very little or no context. But I don't agree that it is required or even sensible for us to provide precise absolute context-free meanings for mathematical terms in the way you seem to desire. | |
| Jul 6, 2010 at 19:39 | comment | added | Vag | With meaning in contexts we are done. But Carl Mummert said: "The term "function" without mentioning context is not meaningless". And it causes my asking: WHAT exactly is meaning of this word when NO context established? If this word without context is meaningless, there are no questions left. All clear. | |
| Jul 6, 2010 at 16:54 | comment | added | Joel David Hamkins | Many words lack meaning out of context, while becoming precise in a context. Why should you expect that there is a meaning for this word outside of any context? Our various mathematical contexts are set up specifically to give words precise meanings, and there seems to be little reason why these contexts must agree. In the case of function, I took as my answer task to explain that indeed there is a difference in meaning for different subject areas. But we can all translate easily from one context to the other, and any amibguity is therefore quickly resolved. Is there any issue left? | |
| Jul 6, 2010 at 6:16 | vote | accept | Vag | ||
| Jul 5, 2010 at 19:50 | vote | accept | Vag | ||
| Jul 6, 2010 at 6:16 | |||||
| Jul 5, 2010 at 19:47 | comment | added | Vag | So, for me notion of contextless function remains meaningless. | |
| Jul 5, 2010 at 19:45 | comment | added | Vag | No. Joel stated that isafun_1 is function in context of set theory and said nothing not vague about definition of contextless function. | |
| Jul 5, 2010 at 14:02 | comment | added | Andrea Ferretti | It seems that Joel answer actually gives an example (the one with the ordinals) where one would need to use contextless functions. So your claim is a bit unfair, unless you mean something else for contextless function (you mean isafun_1, right?). Anyway, psychologically I feel much more at ease using the Bourbaki definition of a function, unless I'm forced not to do so by set theoretical issues. | |
| Jul 5, 2010 at 12:57 | history | edited | Anton Petrunin | CC BY-SA 2.5 |
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| Jul 3, 2010 at 17:26 | comment | added | Vag | Thanks all for replies, almost all questions answered perfectly. But for the sake of honesty I must admit that all of you 1) do not know what contextless function is and 2) do not want to know. | |
| Jul 3, 2010 at 17:22 | vote | accept | Vag | ||
| Jul 5, 2010 at 19:50 | |||||
| Jul 3, 2010 at 15:02 | comment | added | Mariano Suárez-Álvarez | Vag, you are asking how we manage to teach the concept of function... just go into any class where it is taught, and you'll see that these problems you seem to be talking about simply do not exist. | |
| Jul 3, 2010 at 14:55 | comment | added | Vag | I'm not talking about precision in communication or daily operation. I'm talking about precision and uniformity in understanding and teaching. | |
| Jul 3, 2010 at 14:41 | comment | added | Joel David Hamkins | Vag, there are of course degrees of precision in mathematics, each serving a different kind of purpose. At the bottom underlying level, there is the hyper-precise accounts arising in the various formal systems, which would be more precise even than the definitions you and I mentioned, but few would argue that we should all operate only in that formal language just because we believe that it is possible in principle to do so. | |
| Jul 3, 2010 at 14:01 | comment | added | Vag | How do you teach somebody contextless notion of function without resorting to handwaving? | |
| Jul 3, 2010 at 13:34 | comment | added | Vag | But I can't understand how we may use a word successfully if everybody will understand something slightly different from others while using it. Can be that abstract contextless function defined in pure HOL as $$isafun(f)\leftrightarrow \forall x\forall y(x=y\rightarrow f(x)=f(y))$$? | |
| Jul 3, 2010 at 13:24 | comment | added | Carl Mummert | The term "function" is not meaningless, but the precise definition has to be established by context. The same is true for many other words: "ring" (does it have a unity?), "poset" (weak, strong, or the same as a preorder?), "graph" (is it a simple graph?), etc. We lie to students by telling them that each term in mathematics has an authoritative definition, which doesn't stand up to close scrutiny. But it has an inner truth: the precise definition of "graph" may change, but it's not going to change into the definition of a ring. | |
| Jul 3, 2010 at 13:24 | comment | added | Joel David Hamkins | For their most important uses, the two concepts are not so different---they share their most important feature, the functional relation between argument and value---and so outside of a definite context, one would still understand that part of the meaning. But I wouldn't take the union of a set of functions unless it was clear that I intended the set-of-ordered-pairs meaning. | |
| Jul 3, 2010 at 12:15 | comment | added | Vag | Thanks for answer. So, word function is meaningless if no underlying theory mentioned in context? If not, what meaning of this word is? | |
| Jul 3, 2010 at 11:26 | history | answered | Joel David Hamkins | CC BY-SA 2.5 |