Talk:Mathematics

Lead paragraph (sorry)

Given the warning in the article pseudo-code about changing the opening, I decided to bring my proposal here, since the way it currently reads is awkward, in my opinion. I propose:

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of abstract objects and the logical relationships among such objects. Mathematics encompasses topics including quantity, structure, space,and change, although it has no generally accepted definition.

...for the following reasons: 1) Mathematical entities, as represented by symbolic notation, are abstract. They are well defined for the purposes of an axiomatic system, and the rules and operations between mathematical objects are logical in character. I'm sure there is no objection here. 2) The topics listed are not well defined, and while math is used to study things like structure and change, it is NOT the study of naturally occurring structure or change, but the study of abstract representations of such. That is to say, the quantum zeno effect negates the direct correspondence with reality of infinitesimal calculus. Topology deals with abstract surfaces ect. 3) The "and more" is amateurish and doesn't do anything to inform a reader about what mathematics actually is. 4) I don't even know what is meant by "the abstract study of subjects" - the verb study is surely only undertaken by a physical human or physical computer, an abstract object cannot "study" anything as far as I know. The word "subjects" is too ambiguous and probably incorrect. In colloquial parlance "subjects" can mean topics of learning in school or whatnot, but "fields" or "disciplines" works better if I understand the connotation correctly. In any event, Math is the the discipline that makes use of well-defined abstract objects and manipulates them by logical rules and operations. One might even include "rigorous" before "study" in my proposed intro, but it's not particularly important. What I see as important is to do away with "abstract study" because nobody even knows what abstract study is. Either everything that could conceivably studied is abstract or nothing is. My point is that I could see dog feces on my shoe and look at it carefully and in my brain associate dog feces with my previous understanding of dog feces, and the structure of the smear on my shoe could be "structure", and by the current lead paragraph I would be doing math. To me, a definition that excludes nothing is a poor definition. -Fcb981(talk:contribs) 00:56, 29 August 2012 (UTC)

This is, as you might guess, a discussion that has been held many times before. The conflict is between what non-mathematical sources say mathematics is and what mathematicians say mathematics is. While I agree with you, non-mathematical sources such as dictionaries and encyclopedias say mathematics is the study of numbers and shapes. Wikipedia reflects sources, not truth. Sigh! Rick Norwood (talk) 17:21, 29 August 2012 (UTC)

Fcb981, I sympathize with your complaints about the opening quasi-definition: "abstract study" is weird, the mathematical study of dynamics is awfully static and certainly different from how change is studied in the natural sciences, etc. I'd like to see a better definition. However, here are the three big reasons favoring the current definition:

(1) The breakdown of math into major topics corresponds to the structure of the article, at least the subheadings under "Pure mathematics". A good lead should summarize the body of the article; see WP:LEAD.

(2) The definition of mathematics is highly controversial. Please see Definitions of mathematics for a sampling of leading definitions and the unresolved conflict between them. There is not even a consensus among mathematicians on how to define mathematics; see Mura, Robert (1993). "Images of Mathematics Held by University Teachers of Mathematical Sciences". Educational Studies in Mathematics. 25 (4): 375–385. {{cite journal}}: Unknown parameter |month= ignored (help). For this reason, it's not appropriate for us to give a proper definition that draws a clear boundary between what is and is not mathematics; we settle for a rough distinction that leaves the boundary indeterminate.

(3) As Rick Norwood said, Wikipedia is a summary of sources, not truth for which there is no consensus in previously published, reliable sources. Lacking a consensus among mathematicians, we've opted for a definition similar to that of other general reference works, but tuned a bit to the body of the article. The current definition has its flaws, but it's certainly well-sourced (except for "change").

These reasons don't mean that the current opening can't or shouldn't be greatly improved. They're obstacles that need to be overcome in order to reach that improvement. Here are three things that you can do. First, edit the body of the article to the point where it calls for rewriting the definition; see WP:LEADFOLLOWSBODY. The body of the article is begging for all kinds of improvement, and those would likely do as much or more to get across the nature of mathematics than rewriting the definition. Second, find a way to word the new definition so it doesn't take one side in the conflict over definitions. Your current proposal takes one side, or at least opposes some of the leading definitions. And third, find good sources to support the new definition. The reasons you gave above refer to the nature of mathematics, not to sources.

The reasons you gave regarding clarity are well taken, though. I just changed "subjects" to "topics". Of course, this is only a small improvement. It might also be possible to improve the "abstract" part without first overcoming the above obstacles. —Ben Kovitz (talk) 21:03, 10 September 2012 (UTC)

Well...does seem to be written by mathematicians and not writers.  :-) In addition to starting with a single awkward sentence (instead of a well formed paragraph), it then goes on to provide a laundry list of quotes about math from famous people. I feel that the field is so broad and there's no "Joe Math" who started it, so a good lead on the topic should probably avoid naming any specific mathematicians as best possible. I'm also about the "holy trinity of paragraphs" so I'd say paragraph one would go approximately like this (solve for X)

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the abstract study of topics including quantity,[2] structure,[3] space,[2] and change.[4][5] While the term has been around since X[?], there is no generally accepted definition of everything it encompasses [7][8], and practicing Mathematicians analyze many other properties [6]. Though often pursued for its own sake, many of today's scientists rely heavily on applied mathematics for their work—and practical uses for what began as pure mathematics are frequently discovered.[17]

For what would be the third paragraph, start with a variant of Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.[11] and finish with a couple more sentences about applications. The second paragraph is whatever's left when the quotes are cut—basically expanding on the definitions of the field and its sub-branches. Well, there's my "vision statement". Metaeducation (talk) 08:09, 6 January 2013 (UTC)

Group theory picture

As a graduate student in group theory, I do find it a bit alarming that this picture of a rubix cube has become the cornerstone of Wikipedia's imagery on groups. Yes, the rubix cube does form a group under composition of turns, but is it a key, or even interesting example of hows a group can function? I motion that it is changed to something more relevant to the field itself (e.g. an illustration of a dihedron group might be nice as it is the lodestone of much of most introductory texts). If I am alone on this issue, I will simply retract my argument, but I do find it a tad annoying.

Benjamin Peirce a logicist?

A C.S. Peirce scholar brought this line in the article to my attention:

Two examples of logicist definitions are "Mathematics is the science that draws necessary conclusions" (Benjamin Peirce)[24] and "All Mathematics is Symbolic Logic" (Bertrand Russell).[25]

Russell's quote is quite logicist, but Benjamin Peirce's quote doesn't seem so, since logicism usually means the idea that much or all of mathematics is reducible to logic, not merely the idea that mathematical conclusions are logical or deductive. Have anti-logicists generally held that mathematical conclusions are not generally deductive?
Now, the B. Peirce quote is from "Linear Associative Algebra" in which he goes on to say, beginning lower on the same page (the article's first page),

Mathematics, under this definition, belongs to every enquiry, moral as well as physical. Even the rules of logic, by which it is rigidly bound, could not be deduced without its aid. The laws of argument admit of simple statement, but they must be curiously transposed before they can be applied to the living speech and verified by observation. In its pure and simple form, the syllogism cannot be directly compared with all experience, or it would not have required an Aristotle to discover it. It must be transmuted into the all the possible shapes in which reason loves to clothe itself. The transmmutation is the mathematical process in the establishment of the law. [....]

That really doesn't sound like logicism at all. It's more to say the same as his son C.S. Peirce said, that the mathematician aids the logician, not vice versa. (I'm not arguing about whether they were right, just that that was their view.) Anyway, I suggest that Benjamin Peirce's definition of maths not be characterized as "logicist." The Tetrast (talk) 01:00, 7 September 2012 (UTC).

Nobody has commented, so I'm not sure whether anybody cares. I propose to revise the line in question to

An example of a logicist definition is "All Mathematics is Symbolic Logic" (Bertrand Russell).[25].

If nobody comments during the coming five days or so, I'll go ahead and make the change. The Tetrast (talk) 02:56, 8 September 2012 (UTC)

I care. I agree that we should not call Benjamin Peirce a logicist. I'm no expert, but I believe the literature says that he led the way toward logicism but predated it. Strictly speaking, the quotation doesn't say that he was a logicist; the definition is meant to give the spirit of logicism. Still, the juxtaposition certainly suggests that Pierce was a logicist. Can you think of a graceful way to word that paragraph so that Peirce's definition is still included? (A nuanced but still brief exposition of Peirce's and Russell's definitions is here. Could you double-check that, too?) Many people come up with the "necessary conclusions" definition on their own, and they think it's the only reasonable definition. It would be nice if the main Mathematics article mentioned that definition and its originator, alongside the main competing definitions. Also, I don't think Russell's definition is clear to someone who doesn't know much about the field; the Peirce definition is much clearer. —Ben Kovitz (talk) 21:37, 10 September 2012 (UTC)

I just had an idea: distinguish between logicism and defining mathematics in terms of logic—since they're not the same, anyway. This provided a nice way to put Peirce's definition first, just by going in chronological order. It's in this version. Please take a look, see what you think, and improve. —Ben Kovitz (talk) 22:59, 10 September 2012 (UTC)

I would not even agree that Benjamin Peirce offers a "definition of mathematics in terms of logic". "The science that draws necessary conclusions" could only be said to be drafted "in terms of logic" if logic is the only science where necessary conclusions are found. But one might equally argue that mathematics and logic both deal with necessary conclusions but not in the same way. Just like philosophy and psychology both deal with the mind but not in the same way. I would rather amend the whole paragraph to read something like this:

"In Principia Mathematica, Bertrand Russell and Alfred North Whitehead advanced the philosophical program known as logicism, and attempted to prove that all mathematical concepts, statements, and principles can be defined and proven entirely in terms of symbolic logic. A logicist definition of mathematics is Russell's "All Mathematics is Symbolic Logic."[25] Related but more subtle views were put forward by Benjamin Peirce, who wrote that mathematics is "the science that draws necessary conclusions" [24], and his son Charles, who wrote that logic is "the science of drawing necessary conclusions".

I can reference the latter quote if this change is liked. (Greatcathy (talk) 09:31, 14 September 2012 (UTC))

Thanks, Greatcathy. I understand Benjamin Peirce's definition to be in terms of logic in that "drawing conclusions" and "necessary conclusions" are entirely concepts from logic. That is, it distinguishes mathematics from other topics by the character of its propositions and reasoning rather than by the content of its propositions. Maybe you can find a way to improve the wording in the article so this point is made clearly.

Here are a couple quotations that might be helpful for figuring out what to say in this paragraph:

1. From Boyer & Merzbach's History of Mathematics: "[Benjamin Peirce's] son was in wholehearted agreement with this view, as a result of Boole's influence, but he stressed that mathematics and logic are not the same. 'Mathematics is purely hypothetical: it produces nothing but conditional propositions. Logic, on the contrary, is categorical in its assertions.' This distinction was to be argued further throughout the mathematical world in the first half of the twentieth century." I take the "but" to mean that Benjamin Peirce's definition is so clearly related to logic, it suggests that the two are easily confused, and C.S. Peirce found it necessary to explain the distinction.

2. Here's an excerpt from C.S. Peirce's writing: "…[T]he mathematician does not conceive it to be any part of his duty to verify the facts stated. … All features that have no bearing on the relation of the premises to the conclusion are effaced and obliterated. … [T]he mathematician frames a pure hypothesis stripped of all features which do not concern the drawing of consequences from it, and this he does without inquiring or caring whether it agrees with the actual facts or not; and secondly, he proceeds to draw necessary consequences from that hypothesis."

It would be best, of course, if we can refer to more-recent commentaries that reflect scholarly consensus on the meaning and significance of the B. Peirce definition.

I'd prefer not to include both "[mathematics is] the science that draws necessary conclusions" and "[logic is] the science of drawing necessary conclusions". I think it's inviting confusion. If there is a strong reason to include them both, let's make that explicit in the text. If they're both important but require more exposition, then maybe the two juxtaposed definitions should go into Definitions of mathematics. BTW, I invite you to have a look at that article, too; it also needs many improvements, and you might have a lot to contribute. —Ben Kovitz (talk) 01:53, 19 September 2012 (UTC)

Why mathematics is more important than physics and chemistry?

We have mathematics subject since we are in elementary school, but not physics or chemistry until high school180.194.246.163 (talk) 10:14, 12 November 2012 (UTC)

Which is more important is a matter of opinion. You need math to understand physics and chemistry. That is one reason math comes first. Also, the public schools are paid for by taxpayers. Physics and chemistry labs are expensive. Mathematics only requires pencil and paper. Rick Norwood (talk) 12:58, 12 November 2012 (UTC)

Pure mathematics on top section

It is just a minor change but i cant touch it.

On top section last paragraph,it said, "Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind."

It is ambiguous. A mathematician can engage in "pure mathematics" without mathematics and any application in mind. The sentence isn't exactly false but ambiguous. If i can edit i would just delete ", or mathematics for its own sake, ". If you think we must mention something like "pure mathematics often has mathematics in mind", then try split the sentence in better shape.

Also,this wiki article is trivial and important for all, extra care on wordings/semantics must be given, so it doesn't spread any misleading information.14.198.221.131 (talk) 16:34, 23 December 2012 (UTC)