Talk:Mathematics

One-line definition

I have been musing on the definition of mathenatics as "the study of quantity, structure, space, and change". It strikes me that I could take my camera, get out there and contrast a single human being with a crowd, compare the structure of a leaf with a network of roads, picture the clever use of a tiny volume in a yacht and the humbling vastness of the mountains, and document the changes of seasons. This would provide me material for a show "the study of quantity, structure, space, and change", and we would not recognise any of it as mathematics.

Rather than characterise mathematics by WHAT it studies (even though the list is very compact and comprehensive), I would attempt to characterise, in one sentence, HOW it does it. So my two cents: "Mathematics is the art of rigorous abstract thinking"

• art: as in artificial and artisan: human made, and a skill
• rigorous: the greek started it, then came the study of axioms, the widespread use of manipulation of symbols etc...
• abstract: the moment someone realised that sharing 5 roots or 5 fruits between 3 people is the same challenge, (taste does not matter) we were started...
• thinking: as in "thinking before acting", mathematics as a tool to conquer the world. And seen like that, mathematical concept do not preexist their invention, even though some of them are so uncommonly useful - contentious obviously!

Does it make sense?

Obviously the proposed definition touches on many elements already well discussed in the article.

Philippe Maincon (talk) 17:24, 15 July 2011 (UTC)

This has been discussed heavily. You raise good points, but your points seem like original research to me. Unfortunately, so does the current one-line definition in the article. It lacks citation. At the risk of beating this to death yet again, let me give two definitions of mathematics with citations:

• "Mathematics is the science of quantity and space" (p. 6).
• "The study of mental objects with reproducible properties is called mathematics" (p. 399).

The source is The Mathematical Experience, winner of the American Book Award and a classic of popular mathematics. Mgnbar (talk) 17:38, 15 July 2011 (UTC)

The battle over the lead occupied a great deal of time and effort, and many sources were cited, which appear in the article. Generally, the lead reflects the article, and the footnotes appear in the body of the article. The "quantity, structure, space, and change" formulation was a compromise, and is reflected in several other articles, and in the subsections of this article.

For what it is worth, I was on the opposing side of the battle, favoring, "Mathematics is the body of knowledge discovered by pure reason, as contrasted with science, the body of knowledge discovered by experiment." This formulation lost, because most major reference works define mathematics by the subjects it studies, rather than by its method.

It may be that someday the lead may be changed, but anyone attempting such a change needs to prepare for strong resistance. Rick Norwood (talk) 12:53, 19 July 2011 (UTC)

I'm a fan of both "Mathematics is the science of quantity and space" and of well-sourced defns., but all things considered the current version is a pretty stable compromise. The proposed version is much too broad to define math. Philosophers, for example, would surely claim they do the same thing, and no mathematician would describe what he did for a living as "Think rigorously abstractly." JJL (talk) 17:00, 19 July 2011 (UTC)

Rick Norwood: I know that this has been battled; I said so twice in my post. Your comment that the lead reflects the article is spot-on. I have always viewed the overarching article structure of "quantity, structure, space, and change" as a gross violation of NOR. (How many mathematicians would recognize "change" as a "field of mathematics"?!) In the past, I have proposed that the article instead categorize fields of math based on the American Mathematical Society's (or other similar organization's) categories for new Ph.D.s: logic, algebra/combinatorics/number theory, geometry/topology, analysis, applied, etc. This idea gained no traction. Mgnbar (talk) 17:55, 19 July 2011 (UTC)

There are many competing definitions of mathematics. Some, like most definitions of topics, define mathematics by the things it studies; others by how it studies them. It's not Wikipedia's place to settle the matter. —Ben Kovitz (talk) 03:38, 23 August 2011 (UTC)

Peirce: "The science that draws necessary conclusions" (1870)

Charles Sanders Peirce's New Elements of Mathematics has an thorough and stimulating discussion of previous definitions of mathematics.  Kiefer.Wolfowitz 10:49, 29 July 2011 (UTC)

What's your point? The last time I checked, Peirce's view was mentioned. What else do you want? Peirce is a fascinating figure, but his outlook is very far from universally shared among experts. --Trovatore (talk) 21:00, 29 July 2011 (UTC)

It seems that his point was to directly respond to the issue of defining mathematics, by pointing to a discussion of definitions of mathematics. See page 25 of the linked material. Kiefer Wolfowitz's comment is exactly relevant. Mgnbar (talk) 14:02, 30 July 2011 (UTC)

It's a book by Peirce. Peirce is not a neutral observer; he's a proponent of a particular POV. That POV should be mentioned, and indeed, it is mentioned. --Trovatore (talk) 21:15, 30 July 2011 (UTC)

Peirce's discussion of definitions, on page 25 and even earlier beginning on page 3, discusses the history of older definitions better than this article, and therefore it is a resource for somebody wishing to improve this article to 1900 standards.  Kiefer.Wolfowitz 22:46, 30 July 2011 (UTC) Peirce criticizes the Kantian definition of mathematics, which devolved via De Morgan & Hamilton and others to this articles's shopping list and political compromise.

C.S. Peirce's definition is "mathematics is ... the study of hypotheses, or mental creations, with a view towards drawing necessary conclusions" (p.4) He demolishes the shopping-cart definition (parroted still on Wikipedia) on pages 5-7, as (in conclusion) "probably ... the very worst". K.W. 03:20, 31 July 2011 (UTC)

Anything other than a "shopping-cart definition" is necessarily going to be POV. The ideal solution would be simply to dispense with "defining" mathematics in the lead, and leave the various definitions people have proposed, all of which carry philosophical baggage, to later in the article. Everyone knows, roughly speaking, what mathematics is, and nothing we can put in the lead is going to be better than roughly speaking in any case. So it's a useless exercise to try to define it at all in that location.

Unfortunately, purely pro forma, we have to put something there; it's expected. I think the current definition does about as little damage as anything that has been proposed. --Trovatore (talk) 04:08, 31 July 2011 (UTC)

Peirce's definition has never had widespread acceptance. Here is Florian Cajori mentioning dissatisfaction with it in 1919. Any attempt to characterize the essence of mathematics takes a side in a long-running philosophic controversy; see Definitions of mathematics for a small sample. I think we are wise to follow the other standard reference works, which have also opted for a "shopping-cart definition". Whatever its flaws, our definition lays out the topics covered in the body of the article, which is the most you can realistically hope for. —Ben Kovitz (talk) 03:26, 23 August 2011 (UTC)

Definition as description vs definition as demarcation

Looking over the above exchanges, I find that I have left some things unexplained that I now believe I know how to say more clearly, and on a related note, that I also do not like the first paragraph as it stands. (Well, I never did really, I just thought it was "least bad", but I now find that it has evolved in a way that I think is suboptimal, and I have a candidate point in the past where I think the text was better, that we should consider a starting point.)

What is a definition? In mathematics itself, our definitions, say of a "ring" just for example, provide precise demarcations. They divide things into rings and non-rings, with nothing in between (ignoring quibbles about whether you require a unit). It's not unnatural that mathematicians would like to be able to provide that sort of a definition for mathematics itself.

Perhaps such a project is possible; I am skeptical, but for the sake of argument suppose that it is possible to define "mathematics" in such a way that it precisely demarcates that which is mathematics from that which is not mathematics. We are left with the problem that no such definition is agreed among mathematicians or philosophers of mathematics. It is not Wikipedia's function to pick one from among them. We simply may not do that; it is a blatant violation of WP:NPOV.

To preserve neutrality, we could futz around with competing definitions and say who uses them. But I hope everyone agrees that the lead is not the place for that. The article is about mathematics, not about how to define the term mathematics, and more than two or three sentences is too much to spend on the definition in the lead section.

But luckily, we don't have to. All we have to do is recognize that the sort of "definition" required in the lead paragraph is not a demarcation at all. The first sentence of dog does not give you the information required to divide all objects into dog and non-dog, and it can't be expected to. Rather, it gives you enough information to identify what the article is talking about, and possibly tells you something you might not have known about what is included. (Are dingoes dogs? Does mathematics deal with things other than numbers?)

That's why an NPOV definition in the lead will necessarily have a "shopping-cart" aspect to it. It can't demarcate what is mathematics from what is not, but it can say some things that we all agree are mathematics, including some that some readers may not have realized are mathematics. --Trovatore (talk) 07:50, 1 August 2011 (UTC)

The better historical lead

Here are the first three sentences as they stand: (I'll leave the refs but won't put a reflist):

Mathematics (from Greek μάθημα (máthēma) — knowledge, study, learning) is the study of quantity, structure, space, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity.

Here is the version I like better, from 1 February 2009:

Mathematics is the academic discipline, and its supporting body of knowledge, that involves the study of such concepts as quantity, structure, space and change. Some practitioners of mathematics maintain that mathematics is the science of pattern, and that mathematicians seek out patterns whether found in numbers, space, science, computers, imaginary abstractions, or elsewhere. Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions. The mathematician Benjamin Peirce called mathematics "the science that draws necessary conclusions".

Here's the main reason I like it better: The current version pretends to be a demarcation. It lists four (vague) things; those are mathematics, nothing else is. As I argue above, that cannot possibly meet NPOV. The Feb 2009 version does not; rather, it lists some of the things that mathematics studies, without claiming to exhaust the subject.

That's not to say it can't be improved. I would change "the academic discipline" to "an academic discipline", making the non-demarcative nature more explicit. Also I would probably incorporate some of the language of the current version as well; I have no detailed proposal at this time for how to do that. But I hope my main point is clear. --Trovatore (talk) 07:50, 1 August 2011 (UTC)

I'm only going to address one of your points: "academic discipline" was discussed, and it was pointed out that while many mathematicians work in academia, many others work in industry, and historically there were many "amateur" mathematicians, and therefore "study" is better than the more limited "academic discipline". Rick Norwood (talk) 12:36, 1 August 2011 (UTC)

Overall I agree with Trovatore and I prefer the Feb 2009 version. Precise definitions about the practice of math are not necessary or even possible. (Topology as a math concept is precisely defined, but topology as a discipline of math is only vaguely defined.) The Feb 2009 version offers several views, as concisely as possible, which to me is ideal. I agree with Rick Norwood that the "academic" is undesirable. Mgnbar (talk) 13:13, 1 August 2011 (UTC)

Yes I prefer the 2009 version, which seems to capture most views on what mathematics is.--Salix (talk): 13:48, 1 August 2011 (UTC)

I applaud Trovatore's clear and stimulating discussion today. However, the 2009 definition is problematic because of several issues

1. the academization error, as noted by Mgnbar and by Rick Norwood (who referenced archived discussions here).
2. "Rigorous deduction": It is clear that mathematical proofs aim at convincing other mathematicians and at meeting the contemporary standards of rigor, but non-logical mathematical proofs fall far short of the rigor displayed in proofs in philosophical logic, especially in mathematical logic (or logic programming with relevant intuitionist logic or other paradises ...).
3. "Some practitioners of mathematics": should be "some research mathematicians and philosophers of mathematics". ("mathematical practitioners" include "junior high-school teachers of mathematics and mathematical educationalists", which are unreliable sources here).
4. The intentional definition from Benjamin Peirce should be upgraded to the definition by C.S. Peirce, noted above.
5. There should be the mention of mathematics as "the science of quantity" (sic.), which was popular in 19th century German and has been embalmed in many "social sciences". This is the definition of mathematics most familiar to the general public (who remember their time with arithmetic and plane geometry (and perhaps $\mathbb {R} [x,y]$ or univariate calculus). This definition's inadequacy became apparent with the rise of projective geometry and topology.
6. It should mention that mathematics is dynamic: It develops by adding new fields, like mathematical statistics (Laplace, Gauss, and Karl Pearson, and for the needs of physics, biology, and insurance), or functional analysis and operator theory (for the needs of physics and classical mathematics, e.g. differential & integral equations). The fields of set theory and general topology were developed to unify and better understand the foundations of other mathematical disciplines, illustrating the importance of abstraction/abduction/retroduction/hypothesis formation. It also drops fields like projective geometry and summability theory, which were central to mathematics c. 1870, but now are studied as examples of more general theories; invariant theory has made a comeback.
7. The definition focuses on pure mathematics, which has been practiced by a rather narrow group of mathematicians beginning in the 20th century. The best mathematicians, like Kolmogorov and von Neumann, contributed to the development of science and modern technology. I believe that the NSF noted c. 1996 that most mathematics covered by mathematical reviews were in non-mathematical academic departments. Some discussion of vital applied-mathematics, e.g. by Kolmogorov or Arno'ld or Atiyah or Lax etc., should appear.

Those caveats aside, the 2009 definition is better because it clearly notes the traditional major fields of mathematics (analysis, algebra/number theory, geometry & topology) in popular versions "change, structure & quantity, space", as conventional.

Despite my criticisms of the 2009 definition, it has several virtues:

• It signals the incompleteness of this shopping-cart definition of mathematics with the phrase "such as". That incompleteness then motivates perhaps the most prominent, accessible definition of mathematics, Benjamin Peirce's (which should be replaced by C.S. Peirce's).

Thanks,  Kiefer.Wolfowitz 17:41, 1 August 2011 (UTC)

P.S. popularizers of mathematics like Steen and Devlin (who notoriously discussed "Bayesian mathematics" (sic.) on Science Friday would seem to be not the highest quality sources. What about definitions from Saunders MacLane or von Neumann or Kolmogorov (e.g. in his leadership of the 3 volume survey of mathematics) or Peter Lax or Garrett Birkhoff or Michael Atiyah, etc.? (The younger Fields Medalists Tim Gowers or Terrence Tao are smart and have written popular treatments of mathematics, but they are not yet as broad or as profound as the fellows I mention, who have led the development of whole fields of mathematics.)  Kiefer.Wolfowitz 17:35, 1 August 2011 (UTC)

I agree with your criticisms of the Feb 2009 lead (except #4). Here are a couple more. (1) Quoting Peirce's definition in the lead makes it sound more widely accepted than it is. (2) "Some practitioners" introduces a philosophical argument even before we've said much of anything about math. I do like that the Feb 2009 definition is worded to suggest that the "shopping cart" does not exhaust the subject. Maybe pursuing that idea further could lead to really good first para, simultaneously (loosely) demarcating the subject and introducing it. —Ben Kovitz (talk) 04:02, 23 August 2011 (UTC)

I'm really against demarcating it whatsoever in the opening paragraph. --Trovatore (talk) 04:22, 23 August 2011 (UTC)

Oops, I thought we were in agreement. Maybe I confused matters by misusing the word "demarcating". I'll try explaining what I think is both our concern again, a little differently. Normally an article leads with a definition of the topic, telling some kind of essential characteristic that sets it apart. With mathematics, though, all attempts to define it are controversial. But, there is near-universal agreement about what subjects mathematics includes. So, we can serve the same purpose as an ordinary definition by listing some subjects within mathematics. But, no list could cover all of mathematics, so it would be wrong to present it as if it were exhaustive. Is that your concern regarding demarcation? I'm proposing to provide a list but word it so it's clearly just some examples of subtopics to give the flavor, with no pretense at exhaustion. Maybe something in the spirit of, "Mathematics is the study of stuff like numbers, shapes, spatial relationships, correspondences between sets, probability, symmetry, paths through networks, approximation, infinity, ways to shuffle groups of things, and, well, you get the idea, eh?" OK, not so long and not so informal, but you get the idea, eh? —Ben Kovitz (talk) 15:17, 23 August 2011 (UTC)

As unsatisfactory as the "quantity, structure, space, and change" list is, it is now embedded not only in this article but in several other Wikipedia articles. If we want to avoid giving different "definitions" in different articles, we need to either change it everywhere, or keep what we've got. I don't think "qss&c" is so bad. How about just adding "such as" in front of the list? Rick Norwood (talk) 14:03, 24 August 2011 (UTC)

Instead of saying "which are arguments sufficient to convince other mathematicians of their validity", it should something along the lines of "which are arguments sufficient to logically imply some mathematical statement. Of course, interpretation of these arguments are subject to human error, as in any other field." The former quote almost implies that mathematicians only convince, and don't prove. Consider this page http://en.wikipedia.org/wiki/Physician. It has the line "...which is concerned with promoting, maintaining or restoring human health through the study, diagnosis, and treatment of disease, injury and other physical and mental impairments" however, it doesn't add "Any method of restoring health is not certain, but only has convinced other doctors that it will restore health." — Preceding unsigned comment added by CBribiescas (talk • contribs) 16:06, 27 September 2011 (UTC)

Mathematical Symbols: Clickability,Hyperlinking, Own Pages

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)

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

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)

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

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)

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,

$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)

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

Edit request from Knwlgc, 13 September 2011

{{edit semi-protected}} After " Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although practical applications for what began as pure mathematics are often discovered.[10]" add "At times it is difficult to know where pure mathematics ends and applied mathematics begins." Knwlgc (talk) 05:12, 13 September 2011 (UTC)

Well, it's a true statement; I doubt anyone here really wants to argue that the line between pure and applied mathematics is sharp. But what's your rationale for adding it at that point in the text? What problem are we trying to solve? --Trovatore (talk) 23:28, 14 September 2011 (UTC)

The preceding sentences in the article introduce the concepts of 'pure' and 'applied' mathematics. Suppose you are someone unfamiliar with either the concept of 'pure' or 'applied' mathematics, then having read a brief statement about both you might be curious as to their relation with one another. In some way mathematics as a whole is characterized by the relation between its parts, and, having come to the article to read about mathematics as a whole, it would be reasonable to assume you might want to know a bit about the relation between the specific parts 'pure' and 'applied' mathematics. Since it would seem relevant to make a brief comment on the nature of their relation with one another, I suggested a statement which is seemingly unobtrusive, undoubtable, vague, but which points towards the complex relation between pure and applied mathematics. My goal is this: paint a picture of mathematics that is enticing and suggestive enough that someone reading a wikipedia article might want to know more about mathematics as a whole or about the relation between its parts. Knwlgc (talk) 17:54, 16 September 2011 (UTC)

I think that it's reasonable to add. A non-mathematician may not know that the line between pure and applied is fuzzy, because they may not realize that disciplines of math are fuzzy at all. I'd like a citation though. Mgnbar (talk) 19:53, 16 September 2011 (UTC)

In the preface to Keener's Principles of Applied mathematics he states "Much of applied mathematical analysis can be summarized by the observation that we continually attempt to reduce our problems to ones that we already know how to solve." While this is not the same as saying "At times it is difficult to know where pure mathematics ends and applied mathematics begins," it has some significant baring on this statement. Many of the problems which "we already know how to solve" are both 'pure' and 'applied' and it may not be clear where pure mathematics ends and applied mathematics begins in any such reduction. I should also say that I am not an expert in either 'pure' or 'applied' mathematics and thus do not have access to the knowledge needed to prove the statement "The line between pure and applied mathematics is vague." either true or false (or if it is anything more than nonsense).Knwlgc (talk) 23:35, 16 September 2011 (UTC)

I now believe that my statement may fall into the category of "original research" as I can not find a direct source which states clearly or precisely "The boundary between pure mathematics and applied mathematics is vague". Since I do not have experience identifying what is or is not O.R. I am willing to discard my edit request.Knwlgc (talk) 23:35, 16 September 2011 (UTC)

Procedural note: I'm removing this 'edit semi-prot request' for now, pending consensus  Chzz  ►  01:37, 17 September 2011 (UTC)

My request still exists, it has not gone away. It was assumed when I sent in my edit request that there was not consensus (otherwise this article would not need semi-protection), how has the state of my request changed since its conception? I should add that an admission of willingness to discard my edit request, is not a request to discard my edit request (had it been then I would have discarded it).Knwlgc (talk) 03:08, 17 September 2011 (UTC)

Since everyone seems to agree that your statement is true, I've shortened it slightly and added it as a clause in the final sentence. Rick Norwood (talk) 11:52, 17 September 2011 (UTC)

The sentence "Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, but there is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered." is a run on sentence. Perhaps one could change it to "Additionally, mathematicians engage in pure mathematics without having any application in mind. Though, there is no clear distinction between pure and applied mathematics as pure mathematics sometimes has practical applications."Knwlgc (talk) 16:52, 17 September 2011 (UTC)

I disagree that it is a run-on sentence; I have no trouble parsing it. But the sentence is quite long. So I have broken it into two pieces in the article. Mgnbar (talk) 22:11, 17 September 2011 (UTC)

I am not a grammarian: my claim that the sentence was a run-on has no baring on whether the sentence was or was not a run-on. I apologize for any confusion my comment may have caused.Knwlgc (talk) 06:05, 18 September 2011 (UTC)

There is no clear line separating run-on sentences from non-run-on sentences, and sentences that began as short sentences often turn into run-on sentences by subsequent edits, but occasionally editors disagree as to at which point this actually happened. Marc van Leeuwen (talk) 09:46, 21 September 2011 (UTC)

Marc, I think we need to add that to the page on run-on sentences. I hope it's not semi-protected. Knwlgc (talk) 00:22, 27 September 2011 (UTC)

Unintended article misinformation? This perspective from field leadership.

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)

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)

mathematical science

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)

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)

Quotations paragraph

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)

Mathematosis?

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

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)

Galileo's Death Year

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)

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

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)

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

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)

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)

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)

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

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

Galileo in lead third paragraph, quote

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.