Timeline for Why is Set, and not Rel, so ubiquitous in mathematics?
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 23, 2013 at 13:37 | comment | added | Ronnie Brown | I am reminded that Paul Suppes has a book on logic which defined a function in terms of a set of ordered pairs, but also used the term "onto" or "surjective" for a function. He was amazed when I pointed out the difficulty of this, and discussed it at a conference, when others were also suitably amazed! | |
| May 20, 2013 at 5:20 | comment | added | Włodzimierz Holsztyński | It took me only a dozen editions to get $\LaTeX$ right above. It still doesn't look pretty but I am not going to beautify it :-) | |
| May 20, 2013 at 5:18 | comment | added | Włodzimierz Holsztyński | @mbsq: I'll guess and signal a direction toward an answer (I don't know your formalism): $\{f|f:\omega\rightarrow \omega_1\} =\bigcup_{\alpha < \omega_1}\{i_{\alpha\omega_1}\circ(f:\omega\rightarrow\alpha)\}$. (you and OM are killing me with $\LaTeX$. Yes, this is all just a matter of convention--said Alice in Wonderland. | |
| May 20, 2013 at 0:26 | comment | added | Monroe Eskew | @Wlodzimierz: Now here's an example for you. Consider the set-theoretical statement. $\{f|f: \omega\to\omega_1 \}= \bigcup_{\alpha<\omega_1} \{f|f: \omega\to\alpha \}$. This is true in ZFC, but in the category theory conventions it could not be. Now I'm sure there is an elegant way of handling this, as set theory would have a way of handling your homology example by noting the functors are not functions with inputs just these identity functions (qua graphs) but rather some n-tuple of relevant information including the codomains. I suppose this is just a matter of convention. | |
| May 19, 2013 at 23:53 | comment | added | Peter LeFanu Lumsdaine | @msbq: if a function is defined as simply a set of ordered pairs, then indeed your claimed absurdity, “there’s nothing to distinguish them” is one. But if a function is defined as including its domain and codomain, then it’s not an absurdity at all — they’re distinguished by their codomains! Which definition is the right one is clearly a debatable and somewhat subjective question — many people would argue each, and perhaps the right answer is that depending on the field, each can sometimes be more fruitful — but calling this issue an “absurdity” is rather unhelpful. | |
| May 19, 2013 at 21:03 | history | edited | Włodzimierz Holsztyński | CC BY-SA 3.0 |
typo
|
| May 19, 2013 at 20:29 | history | edited | Włodzimierz Holsztyński | CC BY-SA 3.0 |
2nd part (but the system is acting up)
|
| May 19, 2013 at 19:26 | comment | added | Włodzimierz Holsztyński | @Qfwfq, there was plenty of (objective) evidence during my years. I'll expand my "Answer" above to reflect it somewhat. -o=o- @mbsq, what you call so cheerfully "absurdity" is the mathematical reality at least since Eilenberg + (Mac Lane, Cartan, Steenrod, Grothendieck, ...), while I am old enough to remember the good set-theoretical, pre-category times. | |
| May 19, 2013 at 16:12 | comment | added | Monroe Eskew | @Qfwfq. Yes you can easily express surjectivity, you just have to specify with respect to what codomain, which in practice is often clear from context. I think my point trumps any formal concern about convenience of language-- The notion that codomain is a part of the identity of a function leads to the absurdity that there are two "distinct" functions that comprise the same set of ordered pairs, yet "have different codomains." But there's nothing to distinguish them! The correct thing to say is the are many ways of talking about the same function. | |
| May 19, 2013 at 14:57 | comment | added | Ronnie Brown | To add to the last comment: Ehresmann's interest in analysis led him to grouoids which he applied to foliations, and also developed the notion of differential groupoid, now called Lie groupoid, and also topological groupoid, and fibre bundles. He used in essence pseudo groups, and related them to ordered groupoids. See also my article `Three themes in the work of Charles Ehresmann: Local-to-global; Groupoids; Higher dimensions', [147] on my publication list. By contrast, the notion of groupoid has been notably absent or not much developed in most algebraic topology books. | |
| May 19, 2013 at 14:21 | comment | added | Ronnie Brown | By a partial function $f : X \to Y$ I mean a triple consisting of $X,Y, Gr(f)$ where $Gr(f)$ is the graph, a functional subset of $X \times Y$, so that $dom(f), range(f)$ are subsets of $X,Y$ resp. Historically, category arose from algebraic topology, where all functions are total. Ehresmann's approach to category theory arose from analysis, and local-to-global questions, hence his difference in style, and relation to geometry. My definition of "higher dimensional algebra" is the study of algebraic structures with partial operations whose domains are defined by geometric conditions. | |
| May 19, 2013 at 11:36 | comment | added | Qfwfq | @Wlodzimierz Holsztynski: is the historical reconstruction you hint at based on some objective documents, or is it just a conjecture? | |
| May 19, 2013 at 11:33 | comment | added | Qfwfq | @mbsq: (continued) Codomain is very important in the everyday treatement of functions, otherwise you wouldn't be able to conveniently express "surjectivity": we want to be able to neatly express -say- that an operator between Hilbert spaces has dense range (note that the range may not be a complete inner product space) but is not surjective. | |
| May 19, 2013 at 11:33 | comment | added | Qfwfq | @mbsq: I think the identification of functions with their graphs, which is prevalent in Logic I presume for practical reasons (you just say $f\subseteq g$ to express that "$g$ is an extension of $f$"), doesn't reflect the mathematical practice, in which a function is actually a triple $(f,\mathrm{dom}(f),\mathrm{cod}(f))$. | |
| May 19, 2013 at 0:28 | comment | added | Monroe Eskew | What does it mean for two functions f,g to have the same graph but different codomains? In what way are they distinct functions? Clearly they are not. Set theory says they are the same object. What does is mean for f to "have codomain B"? Doesn't that depend on the presentation or definition and not some intrinsic property of f? I think that the standard category theory treatment where cod(f) is a single well-defined object was an error, although not one of any real mathematical consequence. | |
| May 18, 2013 at 23:34 | history | answered | Włodzimierz Holsztyński | CC BY-SA 3.0 |