Talk:Axiom schema

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I think this entry needs to reference the inference rule of Uniform Substitution (which doesn't yet have an entry). Uniform subsitution, in sketch: given wff X, in which variable y is free, and another wff Y, we can create a new wff X\[y/Y\] in which every occurrence of y is replaced with Y. 'Uniform' indicates that every y has been replaced with the same wff. The inference rule of Uniform Substitution is: if X is a theorem, and Y a wff, then X\[y/y\] is also a theorem. If Uniform Substitution exists as a given or derivable inference rule in a deductive system, then a great many axiom schemata can be dispensed with (though, given how general the definition in this entry is, not necessarily all). Thoughts?

I have just rewritten this entry, unintentionally doing so without implementing your suggestion. I have no quarrel with what you say, but I do find it very hard to be very precise about substitution rules. I wrote the substitution rules in Laws of Form. I have read that even Alonzo Church himself had trouble getting substitution right in his early formulations of the predicate calculus.132.181.160.42 05:33, 8 August 2006 (UTC)

ELEGANCE of MATHS

Beauty of mathematics, elegance this is all very very nice. And I am a person who is alive, believe or not, medium or slightly subnormally intelligent than the normal 150 IQ mathematician. And I do like maths, why? Elegance, beauty, God where are we getting? Aesthetics or art pour art or something even higher high in the sky and heavens above us outside the scope of our prying telescopes, where these Deities and Go... dwelllll - no this is not a misprint: this gives the special higher emphasis on the highnesss.. NO!!!

Maths is a very useful and SO nice and lovely tool in your mental arsenal.
Sharp, swift if necessary, slow if we can give us the privilege of having enough time.
This is no ranting, this is a cry of a humble wikipedian to experts who write highly theoretical articles thinking the elegance might get spoiled if we used
PLAIN language, less special symbols, more daily life examples: let us make the mathematical experience come to life through getting down to earth and always when boldly abstracting on things like countability of infinity reconsider whether we
UNDERSTAND OURSELVES and

can make use of our lofty thoughts ONE second later.

Sorry for talking too long and thank you for the delight your artices convey even to those who are not as clever, smart, elegant as you are - there is not irony in these words not at all - caution here please.

--Capekm (talk) 02:52, 1 May 2012 (UTC)

I took the liberty to get rid of the old "elegance" stuff. I agree that mathematicians and physicists talking about "beauty" as if it weren't subjective is cringe, and it's easier to get rid of than the universally used sketchy Platonistic idioms by which things "exist" which don't really, which is an applicable consideration in the light of "infinity" and projects to avoid it in axiomatizations. I wish I could have elaborated more, in a factual way, but I'm not the right person for that. — Preceding unsigned comment added by 24.56.247.67 (talk) 20:03, 2 July 2023 (UTC)

"Loss of 'elegance'"?

collapse ugly rant from 2007, it's not worth starting an archive page for a talkpage this short

"(...) can be finitely axiomatized, but only with some loss of elegance." What the diddle is that supposed to mean? Who gives a diddle about elegance, as long as it makes sense? That's not diddly relevant. Isn't anybody else out there tired of the use of "elegance" as though it was a relevant mathematical attribute? I really think assessing the matter of "elegance" of certain mathematics shouldn't have place in Wikipedia articles. I am a little extra pissed off because I am just coming from complex number, which full of this "elegant" this and "elegant" that shit. The fact that "elegant" is slang for the mathematical community doesn't make any less slang of it. Talking about about "elegance" of mathematics isn't any different from talking about it's "coolness". The fact that you are so angry for no reason and appear to be not only a person of severely limited understanding but also one who is unwilling to understand, detracts from your credibility. Michael Hardy 21:07, 16 March 2007 (UTC) I don't need credibility, I am not trying to get a loan or running for a public office. It's just a matter of common fucking sense, and it's out there for everyone to see, independent of my "credibility". "Loss of elegance"? Are you fucking kidding me? That's outright ridiculous. I don't give a shit if you think mathematics is some form of art in which aesthetic concepts are intrinsic and important, it isn't. It simply isn't. Mathematics is a formal science, and that's it. Beauty has nothing to do with it's investigation (bear in mind I am not talking about mathematicians' personal motives and inspirations in their study, that's a whole other story), and aesthetic concepts shouldn't contaminate the articles, slightly POVing into this aesthetic view of mathematics, as if the "loss of elegance" was going to cripple mathematics from it's "beauty" (as if this was a relevant criterion for it's study). —The preceding unsigned comment was added by 200.164.220.194 (talk) 23:42, 16 March 2007 (UTC).

countably infinite

Article says

Given that the number of possible subformulas or terms that can be inserted in place of a schematic variable is countably infinite, ...

I think that is incorrect or needs clarification. I believe it's common in model theory to have languages with uncountably many constant symbols, say one symbol for each real number x. So you could have uncountably many terms to put in place of a free variable. Any restriction against that should be stated explicitly. 69.228.170.24 (talk) 05:31, 24 May 2010 (UTC)