Talk:Mathematics

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This is the talk page for discussing improvements to the Mathematics article. This is not a forum for general discussion of the article's subject.

Put new text under old text. Click here to start a new topic. New to Wikipedia? Welcome! Learn to edit; get help. Assume good faith Be polite and avoid personal attacks Be welcoming to newcomers Seek dispute resolution if needed Article policies Neutral point of view No original research Verifiability

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I don't know of a better place to discuss this than the wiki page for Mathematics, although the question pertains to any page under the section of Mathematics.

Can we implement mathematical symbols that are ALSO hyperlinks to wiki pages for each symbol?

I do not think all mathematical symbols have wiki pages, but I don't see why not.

At least they could link to a relevant page in which the symbol is heavily used.

I believe this would make it significantly easier to learn mathematics from wikipedia. — Preceding unsigned comment added by 140.247.59.253 (talk) 23:45, 27 July 2011 (UTC)[reply]

List of mathematical symbols gives a extensive list of symbols.--Salix (talk): 08:03, 28 July 2011 (UTC)[reply]

The idea is not just to have a list of symbols, but to use hyperlink versions of the symbols on any or all wiki pages within Mathematics. Like many special terms, the first usage of a symbol on any wiki page could be a hyperlink version of the symbol. Much in the same way that unique terms can be clicked on to bring the wiki-reader to the definition of that term, so too could she more quickly learn about the mathematical symbols that crop up in whatever section of mathematics she is currently browsing. — Preceding unsigned comment added by 140.247.59.84 (talk) 15:29, 9 August 2011 (UTC)[reply]

Sounds great: do you have an implementation available?Knwlgc (talk) 06:12, 18 September 2011 (UTC)[reply]

I don't know HTML really, but if you look at List of mathematical symbols many are hyperlinks, i.e. =. We could change the first usage of any symbol on any wikipage to such a hyperlink version to the symbol's page, and make a new page for it if it does not already exist. Are there wikipedia guidelines regarding such a change? Personally I don't see any reason not to. — Preceding unsigned comment added by 140.247.59.29 (talk • contribs)

Seems like a good way to build the web. No need for HTML— wiki markup is pretty easy to use. For example, [[π]] makes a link to π. Also, on talk pages, four tildes ~~~~ "signs" your comment, adding a date/time stamp.

@others, is there a better place to promote this idea? Seems like the kind of thing that would go well if multiple people start picking at it. __ Just plain Bill (talk) 01:21, 30 November 2011 (UTC) see below Just plain Bill (talk) 02:51, 30 November 2011 (UTC)[reply]

I'm sure we've had this discussion before as I remember thinking it a very bad idea at the time, for various reasons.

• symbols are often very small, so links are difficult to see.
• it doesn't work with PNG formulae generated by LaTeX rendering, which is used very often as either editor preference or because the formulae are too difficult to render in HTML
• Some symbols like ≤, ≥, ±, ⊆, ⊻ are already 'underlined', while adding links to others will change their meaning, to make them look like those or other underlined symbols (links always underlined is a user option).

If a symbol really needs explaining then add a sentence, e.g. from Euler's formula:

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x,

${\displaystyle e^{ix}=\cos x+i\sin x\ }$

where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians.

This is far clearer than linking any symbol.--JohnBlackburne^words_deeds 02:34, 30 November 2011 (UTC)[reply]

Even better. Was unaware of previous discussion, and those points seem valid. __ Just plain Bill (talk) 02:51, 30 November 2011 (UTC)[reply]

The work in the Centre for Experimental and Constructive Mathematics is more than twenty years ahead of Mathematics as an international discipline. This has been inside information which might now be public.

It seems as though there could have been some confusion about what Applied Mathematics was. Generally, Applied Mathematics has been a variety of subsets of ad hoc amalgamations of Theoretical Physics, Statistics, Computing Science, Mathematics, Engineering Science, and almost anything else. The common denominator clarifying what Applied Mathematics has been was tricky to find. If/when ad hoc disciplines (an oxymoron) try existing primarily as theoretical constructions built for the purpose of trying to get money any old way, the result is internal organizational inefficiency. It would be unfair and dreamy of Mathematics, as the international discipline this is, to ask other organizations to have internal cohesion due to our recent update of the definition of if, without first demonstrating what we mean by internal cohesion.

(Instantiations and examples differ slightly: instantiations are generalizable and examples may have generalizable properties and features. However, part of this work includes teaching mathematics to seven billion people, thus for now I preferentially use the word example.)

An example of confusion arising from lack of internal organizational cohesion due to presence of ad hoc discipline, is the 50% vote of support Jonathan and Peter Borwein received from participating voters (abstention rate unknown) for establishing the Centre for Experimental and Constructive Mathematics at Simon Fraser University, a tie which was broken in preference of establishing Experimental and Constructive Mathematics by someone in senior administration circa 1992, and the upshot of which includes the Organic Mathematics Project which singularly redefined Mathematics online education and collaboration; correction of Aristotle; redefinition of the word if; free demonstrative and instructional tutoring services for the world's central banks with respect to the additive and multiplicative identities; open questions including where Mathematics proofs come from, ownership of intellectual property in collaborative processes, how Mathematics and Mathematically informed disciplines develop communities; and the intellectual property ownership question: who owns Mayer Amschel Rothschild's intellectual property conceived circa 1794, still in circulation, and which I previously cited in my work as osmosis.

Generally and obviously, a discipline is not yet qualified to offer its services to customers until after the discipline has demonstrated the same expertise internally. Having organizations' internal and external services match works. The Centre for Experimental and Constructive Mathematics and our network is perfect for driving optimization by osmosis and naturally occurring, real selection processes. Therefore handling the question << what is Mathematics >> is part of this constructive instruction. This exemplifies what Applied Mathematics really is, both in this self-referencing demonstration and explanatory definition update, and in real world ubiquitous application across all disciplines everywhere; therefore we acknowledge Mathematics was previously domesticated partly under Philosophy and partly under Science, and might be correctly understood as a profession under the Institute for Electric and Electronic Engineers, who as an organization has the highest standards in ethics and professional conduct. This Applied Mathematics includes Information Theory and Computer Architecture. Having the discipline Mathematics perfectly located under the IEEE solves all problems related to franchising Mathematics other than my unique personal problem if The Rothschild Family prefers to take me to court for accidental intellectual property theft.

References:

http://www.cecm.sfu.ca/

http://www.cecm.sfu.ca/organics/project/

http://www.ieee.org/index.html

Founder by Amos Elon, ISBN 0 670 86857 4

JenniferProkhorov (talk) 19:23, 19 October 2011 (UTC)[reply]

I don't understand. In any event, this talk page is specifically for discussing changes to the Wikipedia Mathematics article. It is not a general discussion forum about mathematics or its philosophy. So please describe what concrete changes you wish to make to this article. Mgnbar (talk) 22:04, 19 October 2011 (UTC)[reply]

Does mathematics really belong to the mathematical sciences? Mathematical sciences says

• "Mathematical sciences is a broad term that refers to those academic disciplines that are primarily mathematical in nature but may not be universally considered subfields of mathematics proper"
• "Computer science, computational science, operations research, cryptology, econometrics, theoretical physics, and actuarial science are other fields that may be considered mathematical sciences."Brad7777 (talk) 15:30, 6 November 2011 (UTC)[reply]

The question is, what is mathematics? I think mathematics is that body of knowledge arrived at by deduction from axioms. But the dictionary disagrees with me, and says mathematics is the study of numbers and geometry. Wikipedia says mathematics is the study of quantity, space, structure, and change. The Wikipedia definition was a compromise, not perfect, but nobody wants to open that particular can of worms again.

One moderately authoritative list of what mathematics is is the AMS subject classification, which lists 00A06 Mathematics for nonmathematicians (engineering, social sciences,etc.); 03A10 Logic in the philosophy of science; 03B70 Logic in computer science; 08A70 Applications of universal algebra in computer science; 35Q68 PDEs in connection with computer science; 35Q91 PDEs in connection with game theory, economics, social and behavioral sciences; 35Q92 PDEs in connection with biology and other natural sciences; 46B85 Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science; 46N60 Applications in biology and other sciences; 47N50 Applications in the physical sciences; 47N60 Applications in chemistry and life sciences; 62P05 Applications to actuarial sciences and financial mathematics; 62P10 Applications to biology and medical sciences; 62P25 Applications to social sciences; and whole sections on 68-XX COMPUTER SCIENCE; 90Bxx Operations research and management science; 91-XX GAME THEORY, ECONOMICS, SOCIAL AND BEHAVIORAL SCIENCES; and 92-XX BIOLOGY AND OTHER NATURAL SCIENCES. And that's not even getting into the final section on math ed. What does that tell us?

It seems to say that we should differentiate between mathematics, and applications of mathematics. It also seems to suggest that computer science, game theory, economics, social and behavioral science, and biology and other natural science are close enough to mathematics to get sections of their own. And where the hell do you put statistics?

Once you reject my definition, and try to define mathematics by the subjects it investigates, I think the task is hopeless. Try to tell me a subject that is not investigated using mathematics.

Rick Norwood (talk) 18:43, 6 November 2011 (UTC)[reply]

Mathematics isn't primarily mathematical in nature. It is COMPLETELY mathematical in nature.

Why do we have long paragraph of quotations about mathematics in the lead? Shouldn't that go in Wikiquote? Kaldari (talk) 22:18, 4 December 2011 (UTC)[reply]

How about adding "Mathematosis" to the "See Also" section? 164.107.189.191 (talk) 14:37, 6 December 2011 (UTC)[reply]

Because there is no such article. There once was but it was deleted: Wikipedia:Articles for deletion/Mathematosis.--JohnBlackburne^words_deeds 15:23, 6 December 2011 (UTC)[reply]

It's nice to see this right in the first paragraph of a significant article: "Galileo Galilei (1564-1942) said" I never knew the man lived to be almost 400. Good job, Wikipedia. And the article is locked so I can't even fix this boneheaded error. Ugh. — Preceding unsigned comment added by 131.193.127.17 (talk) 16:03, 6 December 2011 (UTC)[reply]

Fixed, thanks !--JohnBlackburne^words_deeds 16:10, 6 December 2011 (UTC)[reply]

That's the downside of "protecting pages". Protecting pages can "protect" vandalism, yet it can also "protect" from perfectly harmless and beneficial contributions. Creating an account is one way to circumvent this issue, but if you're going to edit one freakin' little typo rather than being a lifelong editor, there is no incentive to register, much less contribute. 164.107.189.191 (talk) 16:58, 6 December 2011 (UTC)[reply]

hi.

I would replace

"However, mathematical proofs are less formal and painstaking than proofs in mathematical logic"

with

"Mathematical proofs are written in a formal language provided/analysed by mathematical logic".

The main reason for this exchange is that mathematical logic is itself a part of math! Therefore, the above statement means something like "trains are faster than TGVs". It is just nonsense. Another reason is that proofs in e.g. algebra are just as formal and painstaking as proofs in mathematical logic...

best regards a phd-student in math — Preceding unsigned comment added by 138.246.2.177 (talk) 17:48, 20 December 2011 (UTC)[reply]

agreed. as a computer programmer i would also like to note that the statement is nonsensical in that they both use a completely strict and formal grammar, and it is the grammar and rules thereof that determines the formalness, not what is said with it, so to say one is more or less formal than the other is absurd. one may have more letters or steps in one proof vs another, but in either case each step of the proof is no less a faithful application of the grammar rules than any other. anyways, if the change hasn't been made yet, i'm going to make it. Kevin Baas^talk 17:59, 20 December 2011 (UTC)[reply]

Hold on a minute here. The text as given is certainly bad, but the proposed "correction" is if anything even worse. Proofs are not to be identified with formal proofs. Proofs are arguments directed at human mathematicians (including oneself); they are not ordinarily in any formally specified language.

The reason the existing text is bad is that mathematical logic is a branch of mathematics, and proofs in mathematical logic need not be any more or less formal than in any other branch. In that sense the IP contributor is right. But the proposed correction is wrong. --Trovatore (talk) 18:12, 20 December 2011 (UTC)[reply]

alright, suggestions? Kevin Baas^talk 18:33, 20 December 2011 (UTC)[reply]

I'd just remove the sentence outright. I think the idea it was trying to convey is precisely the opposite of what it says now, namely that mathematical proofs are not ordinarily completely formal. But it did a bad job of that, and we don't need that idea at that point in the text, because there's nothing there it's contrasting with. --Trovatore (talk) 18:59, 20 December 2011 (UTC)[reply]

Mathematical proofs really are less formal and painstaking than proofs in mathematical logic. Open any math book to a proof. I'll pick one at random off the shelf behind me, and open it to a random page. "Proof: Recall that a subspace Y of L is said to be convex if for every pair of points a, b of Y with a < b, the entire interval [a, b] of points of L lies in Y." I think this is fairly typical of how a mathematical proof is written. Now, compare with a proof in mathematical logic:

• (1) (~B->(~A->~B)) (L1)
• (2) (~A->~B)->(B->A) (L3)
• (3) (~B->(B->A)) (1),(2) HS.

More formal. More painstaking.

Mathematicians usually assume that the kinds of proofs we do in our work could, if necessary, be reduced to mathematical logic, but we never, in practice, do that.

Rick Norwood (talk) 19:13, 20 December 2011 (UTC)[reply]

The problem is the phrase "proof in mathematical logic". The term mathematical logic doesn't normally mean things like your lines (1), (2), (3); it normally means set theory, model theory, recursion theory, and proof theory. Those are branches of mathematics, and proofs in those branches of math are not really different in character from proofs in, say, differential topology. --Trovatore (talk) 19:19, 20 December 2011 (UTC)[reply]

Mathematical logic usually means mathematical logic, as in Hamilton's Logic for Mathematicians or Manin's A Course in Mathematical Logic. The other topics you mention are in Foundations, rather than in Mathematical Logic. I agree that the proofs in essentially all areas of mathematics except formal mathematical logic are in the metalanguage rather than in the object language. Rick Norwood (talk) 19:53, 20 December 2011 (UTC)[reply]

I tend to agree with trovatore that the difference between logic and other fields of mathematics is not the level of painstaking care and/or formality (the idea of a logician as some kind of an OCD is an erroneous one), but rather in the field of investigation. Tkuvho (talk) 20:02, 20 December 2011 (UTC)[reply]

The main point here is that both the grad student and the computer programmer above are confused about the relationship between formal mathematical logic and ordinary mathematical proofs. Mathematical proofs are almost never written in formal language, rather they are written in the metalanguage. And the are almost never analyzed using mathematical logic, they are analyzed by a mathematician trusting his ability to think logically. Rick Norwood (talk) 20:06, 20 December 2011 (UTC)[reply]

The term mathematical logic will not work here. I'm sorry, but it means what I said and does not mean what you said. Other than that I basically agree with you. --Trovatore (talk) 20:23, 20 December 2011 (UTC)[reply]

Sorry, I'll amend that. It can mean what you said. It just doesn't usually. The main use of the term is as in The Handbook of Mathematical Logic. --Trovatore (talk) 20:27, 20 December 2011 (UTC)[reply]

there is no confusion here. i know what is a mathematical proof and what is not. and i understand perfectly the relationship between formal mathematical logic and ordinary mathematical proofs. and i couldn't care less what a few nondescript sloppy pompous self-described "mathematicians" have to say about there oxymoronicly non-rigourous "mathematical proofs". Kevin Baas^talk 21:33, 20 December 2011 (UTC)[reply]

when ever i hear "proof" in mathematics i think "formal proof". after all, the whole point of mathematics is to formally prove or disprove propositions. (well, there's applications outside of that, obviously, but you get the point.) if you want to talk about some loose logic in an informal grammar and call it a "proof", well that's your perogative, but from the first time i learned about mathematical proofs (in middle school, mind you - i was an advanced student), we never did it that way. (always w/formal grammar. we wrote it in informal grammar in the left margin, yes, but the formal grammar, or at least exactly stating the official name of the rule applied, was mandatory.) and i will never trust a mathematician whose "proofs" cannot be directly translated into a formal grammar and shown to be valid and consistent. and if i were a math teacher, they'd get an F.

having said that, i agree with what someone said earlier that it might be best to just remove the sentence altogether. Kevin Baas^talk 21:19, 20 December 2011 (UTC)[reply]

03-XX   Mathematical logic and foundations 
 03-00   General reference works (handbooks, dictionaries, bibliographies, etc.) 
 03-01   Instructional exposition (textbooks, tutorial papers, etc.) 
 03-02   Research exposition (monographs, survey articles) 
 03-03   Historical (must also be assigned at least one classification number from Section 01) 
 03-04   Explicit machine computation and programs (not the theory of computation or programming) 
 03-06   Proceedings, conferences, collections, etc. 
03Axx  Philosophical aspects of logic and foundations 
03Bxx  General logic 
03Cxx  Model theory 
03Dxx  Computability and recursion theory 
03Exx  Set theory 
03Fxx  Proof theory and constructive mathematics 
03Gxx  Algebraic logic 
03Hxx  Nonstandard models [See also 03C62]

previous unsigned comment by User:Rick Norwood 21:25, 20 December 2011 (UTC)[reply]

thanks for that, rick. see? "Proof theory and constructive mathematics" one section. notice the absence of a separate, independant section on "Proof theory and constructive mathematics for mathematical logic, specifically, which is for some reason different". Kevin Baas^talk 21:27, 20 December 2011 (UTC)[reply]

Kevin_Baas: while Travatore and I disagree on some things, we both understand the difference between a "formal proof", which can be checked by a computer program, and most mathematical proofs, which cannot be checked by a computer program. Yes, most writers of mathematics use formal grammer, in the sense that we use the grammar of authors, not of twitter. But in mathematics "formal proof" has a more technical meaning. It doesn't mean the same thing as "rigorous". It has to do with the distinction between an object language, in which the formal proof is written, and a metalanguage, in which this paragraph is written. The point on which we disagree is over which of the topics on the AMS list fall under "mathematical logic" and which under "foundations". Rick Norwood (talk) 21:29, 20 December 2011 (UTC)[reply]

I am not saying that a mathematical proof must be written in a formal grammar. i am saying that it must be capable of being faithfully and unambiguously represented in one. Kevin Baas^talk 21:35, 20 December 2011 (UTC)[reply]

I reverted your change because, though you say one thing above, the change you made says another. We need to either omit this entirely or find a way of saying it that is both intelligible to the layperson and mathematically accurate. Rick Norwood (talk) 14:14, 21 December 2011 (UTC)[reply]

So, Kevin Baas, you agree that a typical mathematical proof is not "written in a formal language provided/analysed by mathematical logic", but merely could be written in such a language? So the proposed change, as written, is incorrect? Mgnbar (talk) 14:42, 21 December 2011 (UTC)[reply]

Ah, and therein lies the danger of insufficient rigor. Where I to take off my proverbial mathematics hat for a while, I would say to the first sentence, "close enough". In the second sentence, however, you put forth a causal proposition so i have to put it back on, and say, which proposal, and how does that follow? Also i think the sentence should just be removed altogether. It doesn't do anything there anyways, besides add confusion, that is. Kevin Baas^talk 15:41, 21 December 2011 (UTC)[reply]

I see, the one offered by anon. Well, you got a lot of subjective words there. For instance, if he would have said "formal grammar" it would'be been stricter. "formal language" however, could just be normal everyday english, but where care is taken in one's communication to unambiguously present spatial relations. In that case, i'd say no definition of "mathematical proof", however broad, would include a situation where the proof is less rigourous and sound than any other. And that is where I take issue with the original sentence. TYhis is math here. It's either 100% proven, or 0%. There is no middle ground. Kevin Baas^talk 15:49, 21 December 2011 (UTC)[reply]

And you see now the dangers in using informal language (<-langugage here, not grammar) when describing formal concepts. Case in point of something ambigiuous which by my criteria would ipso facto not qualify as part of a valid mathematical proof. Kevin Baas^talk 15:56, 21 December 2011 (UTC)[reply]

In the mathematical sense of the phrase "formal language", it is not possible to be "more formal" or "less formal". A "formal language" is one where the proofs depend only on the form the symbols take, and not on the meaning of the symbols. If you prefer "formal grammar", that is also used in the same sense. I'm primarily following Hamilton's Logic for Mathematicians.

But, back to the point of this discussion. I agree the disputed sentence should either be improved or, if nobody can come up with a good way to improve it, removed.

Rick Norwood (talk) 16:29, 21 December 2011 (UTC)[reply]

A new suggestion:

"For convenience, most proofs are written in a metalanguage and, therefore, have to deal with the insufficiencies of each metalanguage. Nonetheless, mathematical proofs should (!) be written in such a way that a mathematician could translate them into a more formal language with an unambiguous grammar. This more formal proof could then even be checked by an computer. Usually, one of these more formal languages is taught at the beginning of mathematical logic"

btw.: less attacking and more suggestions and we could have closed this secition yesterday....!!! — Preceding unsigned comment added by 138.246.2.177 (talk) 17:14, 21 December 2011 (UTC)[reply]

I actuallyLOVE the lead, and appreciate the approach of quoting a few important folks as they described mathematics. However, I found the context lacking, especially for Galileo's quote. It is too often that the poor metaphors of natural law and language are used to describe mathematics. Such conceptions are invaluable to its history and this article, but there is a responsibility to more properly contextualize this paragraph I'm question. I believe the final sentence, a quote by Einsteiniis intended to achieve this effect, but j would rather see a punchline less punched if it meant clarity that could prevent further propagation of naive interpretations, despite also being valuable for other reasons. — Preceding unsigned comment added by 67.161.64.224 (talk) 07:43, 6 January 2012 (UTC)[reply]

The choice of entries in the See also section looks rather arbitrary to me and is, in my opinion, quite uneven. Iatromathematicians is an extreme obscure topic. Why is Self-similarity on this list? Any suggestions for a criterion to decide what should be on this list?  --Lambiam 12:42, 12 January 2012 (UTC)[reply]

I agree and have been bold and reduced it to what I think makes sense. There's certainly no need to mention individual topics: the article mentions many, including all major ones; they are all available in the template below. The same goes for particular aspects of mathematics education/ability, particular lists and particular groups. That's left the definition article and two relating mathematics to other topics. I would suggest only similar articles are added, i.e. not too narrow/specialist and not covered by one of the existing links (so no topics, no lists).--JohnBlackburne^words_deeds 13:04, 12 January 2012 (UTC)[reply]

This edit request has been answered. Set the |answered= or |ans= parameter to no to reactivate your request.

Please transpose the words space and quantity. It was recently found that the "all pages lead to philosophy" loop was broken by an edit to the mathematics page and the problem would be easily fixed by transposing these two nouns. menaing of the page would not be changed and many meme-ers would be made very happy. Thank you.

66.99.120.222 (talk) 15:24, 20 January 2012 (UTC)[reply]

Sorry, to edit articles to cause them to lead to philosophy violates the idea that all articles naturally lead to philosophy. Also, the current word order is used throughout the article and in several other articles. Rick Norwood (talk) 16:02, 20 January 2012 (UTC)[reply]

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