Talk:Mathematics/Archive 6

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Recent revert wars

I want to say that, even though in some sense I reverted it, I found Gububbu's version not particularly worse than the current one--while it was hardly complete, at least it avoided the sin of excessive ambition in "defining" math. But then Rick stepped in with a clearly unacceptable version, which was immediately vandalized. So rather than try to split the difference I reverted back before the latest outbreak of hostilities, hoping things would then be discussed according to procedure.

Let me explain my objections to Rick's version, in the hope that we can start to converge a little. I do not agree that mathematics is "the most certain [...] knowledge that we possess". For example, I am more certain of my own existence than I am of any mathematical proposition. Now, it could be said that that's my knowledge rather than ours, but nevertheless Rick's phrasing comes far too close to an assertion of apodeictic certainty. That topic could be discussed further down in the article together with objections to the concept; it should not occur unchallenged in the introduction.

The other objection is the bit about "using the method of deductive reasoning from definitions and axioms". Of course that is an important part of mathematics, but the phrasing suggests that it's the whole of mathematics, thereby excluding empiricism from mathematical epistemology. --Trovatore 19:01, 19 October 2005 (UTC)

Let me just add that this version, by The Anome, is really pretty good. Includes pretty much everything, and describes common methodologies without prescribing them. --Trovatore 22:14, 19 October 2005 (UTC)

Part of the revert wars was me reverting twice Gubbubu's version. Sorry for that. I don't have much against that text itself, I just hoped for some more discussion here rather than pushing one's version by brute force. By the way, isn't it overblown to put those "original research" and "neutrality disputed" templates just because we fail to use a dictionary definition and instead the text says that math is the study of space, change, etc? How about removing the templates? Oleg Alexandrov (talk) 23:54, 19 October 2005 (UTC)

I would be for removing the templates. I think Gubbubu just got a little hotheaded. Hopefully he'll cool off and remove them soon; otherwise we can see if anyone else wants to defend them here. --Trovatore 00:02, 20 October 2005 (UTC)

Gubbubu's remarks

1. My 100× reverted version contained mostly all disputed "definitons" of maths. What you have done was deleting a scientific definition from the article, and I think that is uneacceptable. You can debate that Webster's dictionary says truth or foolness, but you can't debate that Webster says that what it says. So editors reverted my version outraged our NPOV policy. Please send me the necessary apologetical letters with the donations.
2. Unlike you, I haven't deleted your work, your thougts from the article. I only moved down the recent "definition" with some row. But your reverts deleted important tings. I think your method was a bit impolite. Please in the future 1). READ, 2). THINK, 3). ASK/TALK, then 4). ACT before starting revert wars, in this order, especially when you reverting a logged-in, unanonim user's acts.
3. I said I haven't dleted anything. But I could do that without any talks and debates! Namely, the recent "definition" of maths ("study of structure, change" and what else) has no scientific resources (see Wikipedia:Cite sources) and then 1). it must be a serious pov, or/and 2) it must be Wikipedia:Original research! So I think anyone who debate this definition, cpuld delete it, only with a remark in the summary (no resource pov, original research deleted).
4. So please Trovatore, don1t delete pov/orig. res. templates, on no accounts. I think three of our policies shouldn't be injured. That is too much. Gubbubu 11:20, 20 October 2005 (UTC)

Rick Norwood's remarks

Please note that I was willing to let sleeping dogs lie until somebody, I forget who, broke the truce. Essentially, the conflict here over the past few months has been over whether mathematics is the study of certain subjects, such as shape, motion, and sparklie things, or a method, the method of deductive reasoning. Since mathematicians have been arguing over this for at least two thousand years, we may not be able to settle the question here. Thus the compromise, which wasn't a bad one.

I proposed using the definition from the OED, but I'm not sure if that is allowed.

On the subject of certain knowledge, Trovatore, you say that you think therefore you exist, but a Buddhist might say that there is no "you", no "thinking", and no "existance". It's all an illusion. But even Buddhists seem to agree that three sevens is twenty-one.

If my existence is an illusion, then whose illusion, exactly? This is essentially Descartes' argument, and he was completely right, of course. --Trovatore 03:54, 20 October 2005 (UTC)

Seriously, what I want in the introduction to this article is the following:

1) Mathematical knowledge is arrived at by the method of deductive reasoning. If you want to add to that something in favor of the people who believe that checking answers on a computer or making inspired guesses or recognizing patterns is just as much mathematics as theorems and proofs, I'll kick and scream, but I won't revert it, just so long as the bit about reasoning gets in there somewhere.

It's there in The Anome's version that I linked to above.

It's simply false that all mathematical knowledge is arrived at by deductive reasoning. Where then do the axioms come from? They are not assumed arbitrarily; they are assumed because we believe them to be true. In the case of simple axioms they are typically asserted to be "obvious", but this doesn't hold up forever--many large cardinal axioms are not remotely obvious, but we believe them to be true on the basis of the accumulated weight of evidence. Much like in any other empirical science. --Trovatore 03:54, 20 October 2005 (UTC)

2) That mathematical knowledge is universal. That is, that the Chinese had the same binomial theorem that Pascal had, that (contra the postmodernists) mathematical knowledge is not just a social construct.

This is a controversial point. I was listening to David Fremlin very recently musing about how most mathematics has been discovered only once (obviously, this doesn't mean people don't rediscover things), so we really can't know if other cultures would do it the same way. Would Tau Cetian mathematicians discover Woodin cardinals, sooner or later? I think so, but we currently have no way to run the test. Anyway it doesn't matter if I agree with you or not; it shouldn't go in the article without discussing opposing views, and therefore shouldn't go in the intro at all. --Trovatore 03:54, 20 October 2005 (UTC)

3) That mathematical knowledge is useful. That with no math, we would have no IPods.

4) That new mathematical knowledge is still being discovered today.

Most people do not know these things. They should. The Greeks turned their back on rational thought and look what happened to them. Rick Norwood 01:27, 20 October 2005 (UTC)

Points: (A) we settle nothing on Wikipedia; (B) people who edit with an agenda are menaces, and 'getting the word out' about anything counts as an agenda; (C) leave pomo out of it, please; (D) What about the Greeks? The Hellenic world gave us various things, mathematics and religion being obvious ones. This seems to be severely off-topic; (E) truce - what truce? This page is going to be edited, no question, and what ought to happen is not helped by reactiveness, verbosity, and people who can't or won't read the style guides. Charles Matthews 12:56, 20 October 2005 (UTC)

Is Charles saying that Rick has been pushing this a bit too hard too fast? If so, I would agree with that. Oleg Alexandrov (talk) 13:56, 20 October 2005 (UTC)

This is a good article.

intros summarize articles

Apparently we should rethink our article and add a section on philosophical views of mathematics and applications to say the least and _then_ try at the intro again. --MarSch 12:35, 20 October 2005 (UTC)

Whatever. :) Just let us not rush to edit the article without much consultation. That only leads to frustration on all sides. Oleg Alexandrov (talk) 13:58, 20 October 2005 (UTC)

the current intro

The current introduction seems fine to me. Sorry if I got a little hot under the collar yesterday. I just got back from a conference in which I learned that most high school geometry courses will no longer cover proof because it isn't on the high stakes "no child" test. Sigh. Rick Norwood 14:04, 20 October 2005 (UTC)

what's a "no child" test??? --MarSch 11:23, 21 October 2005 (UTC)

No Child Left Behind, President George W. Bush's education initiative. I don't know how good it is, but it made a lot of conservatives mad, and that is always a good thing. Oleg Alexandrov (talk) 14:09, 21 October 2005 (UTC)

Why templates been removed???

I haven't been given a clear answer who says that "maths is the study of structure, change and patterns"? Where this definition came from? Please answer, I esp. beg MarSch to answer me. Gubbubu 16:30, 20 October 2005 (UTC)

Well, you haven't said why you think that the article is NPOV, or original research. The only thing I could find is a complaint that no sources are mentioned. That is true, you do have a point here, and perhaps the best way to get a definition is to distill something from the definitions given in a couple of recent dictionaries. However, we don't go around and slap {{POV}} and {{OriginalResearch}} on every article which does not cite its sources. -- Jitse Niesen (talk) 17:12, 20 October 2005 (UTC)

I said it thousand times, you've just only slept it over. But I will sum my remarks soon again, for our EVERY new readers. Gubbubu 10:25, 21 October 2005 (UTC)

The "structure, change, and patterns" is a revision of "number, shape, movement, and patterns". There are a number of such formulations, generally by people on the math ed side of the math wars. In general, these people reject the importance of proof, and also reject the idea that education research should follow established statistical methods. According to some, it "empowers teachers" to allow anecdotal research to be published in refereed math ed journals.

The math wars have been going on for -- what -- ten years now? Neither side is going to budge, and so an introduction that incorporated both points of view was the best that could be worked out, to avoid endless reversion wars. Rick Norwood 22:59, 20 October 2005 (UTC)

I for one am a pure mathematician, not a math ed specialist (which doesn't mean I don't take my teaching seriously, just that it's not my area of research), and I certainly don't "reject the importance of proof", though I may not give it the unique epistemological status that you do. What I reject is overly prescriptive and restrictive epistemology that excludes attempts such as Woodin's to answer questions like the real truth value of the continuum hypothesis. --Trovatore 23:15, 20 October 2005 (UTC)

We agree. I also am a pure mathematician, but I have fallen among bad companions. It seems to me that "real truth", to whatever extent it differs from just plane "truth" is asking too much from an imperfect world. I haven't read Woodin (I'll follow your link) but I assume he wants to add a new axiom to ZF+C. Rick Norwood 23:29, 20 October 2005 (UTC)

Well, you can put it that way if you want to, but I don't know that the development of the theory was particularly axiomatic. I think he does think of it as an axiom now, rather after the fact. --Trovatore 23:32, 20 October 2005 (UTC)

The current construction of "quantity, structure, space, and change" in the lead section essentially comes (with the addition of "quantity") from the first two sentences of the section "Overview of fields of mathematics":

The major disciplines within mathematics first arose out of the need to do calculations in commerce, to measure land, and to predict astronomical events. These three needs can be roughly related to the broad subdivision of mathematics into the study of structure, space, and change (i.e. algebra, geometry and analysis).

I don't see these ideas as particularly POV or OR.

And by the way the lead should summarize the rest of the article. So, making substantive changes in the lead without changing the rest of the article is backwards. Paul August ☎ 00:19, 21 October 2005 (UTC)

It seems I don't need to explain myself anymore, since you all've done so for me :) Thanks. Gubbubu, you are welcome to constructive criticism or even non-constructive, but those templates seem not to have any support besides you, so I've removed them. They bear no relation to reality. --MarSch 11:30, 21 October 2005 (UTC)

This page

It was at 173K so most is now in /Archive4. I'm fairly happy with it now. It reads well. I think the 'common misconceptions' section might merit a page of its own. Charles Matthews 13:04, 21 October 2005 (UTC)

The "quantity, structure, change, and space" definition appears in several places, in the introduction, further down the article, and on the portal page (which doesn't have a link to this page -- should it?). Nobody seems to remember where it came from -- I remember somebody being pretty passionate about it, but I don't remember who. The OED says "space, number, quantity, and arrangement". I suppose we could do worse, but I can think of things that are clearly mathematics that don't fit this list, and things that clearly fit this list that aren't mathematics.

Rick Norwood 13:49, 21 October 2005 (UTC)

It's easy enough to see where it comes from: quantity = analysis, structure = algebra, space = geometry, plus change = differential calculus. Or for the OED arrangement = combinatorics, which is in a sense better because less redundant and more in tune with compsci. So this is all just using lay words for things that undeniably are mathematics. Charles Matthews 07:26, 22 October 2005 (UTC)

NPOV and original research?

I will reply for you now. Gubbubu 15:03, 21 October 2005 (UTC)

Answer to Rick Norwood

The "quantity, structure, change, and space" definition appears in several places,

Tell me two Wikipedia-independent source!! Tell me, please! Then I cuold accept it's not original research (but not that it is pov). Gubbubu 15:03, 21 October 2005 (UTC)

in the introduction, further down the article, and on the portal page (which doesn't have a link to this page -- should it?).

(Well, it doesn't matter in how much replication this pov appears in wikipedia. I asked that, please, where this original meme came from)? Gubbubu 15:23, 21 October 2005 (UTC)

Nobody seems to remember where it came from.

Then I just must say it is original research, sorry. Gubbubu 15:23, 21 October 2005 (UTC)

The OED says "space, number, quantity, and arrangement".

Well, what does it mean OED? Is it a source, indepentdent from us? Is it a dillettantocratic wiki like englishpedia, or a peer-reviewed source? Gubbubu

OED = Oxford English Dictionary. -- Jitse Niesen (talk) 15:52, 21 October 2005 (UTC)

Thanx. I delete the Original Research template, and if you think, you can delete the pov template (but I think the article is pov now, please if You (not only you, but any contributors) have time, answer my questions and remarks. Gubbubu 15:56, 21 October 2005 (UTC)

I suppose we could do worse, but I can think of things that are clearly mathematics that don't fit this list, and things that clearly fit this list that aren't mathematics.

Yes, I think. I will give You (not only you, but to all contributor) some questions below related to this remark given by you. Gubbubu

The "structure, change, and patterns" is a revision of "number, shape, movement, and patterns". There are a number of such formulations, generally by people on the math ed side of the math wars. In general, these people reject the importance of proof, and also reject the idea that education research should follow established statistical methods. According to some, it "empowers teachers" to allow anecdotal research to be published in refereed math ed journals.

I dont't think only formalist's views ("maths are deductive reasoning about theories of set theory") are npov. But they HAVE views and deleting them I think tha is not only so impolite contribution but it is the faking of the human beings avarage views about definition of maths. I think there are a lot of formalists. Gubbubu 15:23, 21 October 2005 (UTC)

The math wars have been going on for -- what -- ten years now? Neither side is going to budge, and so an introduction that incorporated both points of view was the best that could be worked out, to avoid endless reversion wars. Rick Norwood 22:59, 20 October 2005 (UTC)

I don't think recent definition incorporates anything. I think it is a convergence to - sorry, it cost me to tell xou, but - the lower limit of sillyness. I will write about this some rows under. Gubbubu 15:23, 21 October 2005 (UTC)

The current construction of "quantity, structure, space, and change" in the lead section essentially comes (with the addition of "quantity") from the first two sentences of the section "Overview of fields of mathematics":

Yes? But I don't see any wikipedia-independent resources at all. Original research? Gubbubu

The major disciplines within mathematics first arose out of the need to do calculations in commerce, to measure land, and to predict astronomical events. These three needs can be roughly related to the broad subdivision of mathematics into the study of structure, space, and change (i.e. algebra, geometry and analysis).

Then rather say that mathematics is the study of algebra, geometry and analysis. Gubbubu 15:23, 21 October 2005 (UTC)

I don't see these ideas as particularly POV or OR.

Yes, they are pov. If you "don't see", that is your business, open your mind (but I will open it, too with some rows under :-). Gubbubu

And by the way the lead should summarize the rest of the article.

That means all the article is original research. Oh my God! We must change it soon! Gubbubu

So, making substantive changes in the lead without changing the rest of the article is backwards. Paul August ☎ 00:19, 21 October 2005 (UTC)

Leaving the article in its recent and unscientific form is backwards from all the developments of philosophy of maths and of science theory.

Ask for Marsch

It seems I don't need to explain myself anymore, since you all've done so for me :) Thanks. Gubbubu, you are welcome to constructive criticism or even non-constructive, but those templates seem not to have any support besides you, so I've removed them. They bear no relation to reality. --MarSch 11:30, 21 October 2005 (UTC)

Yes it seems to be you need to explain your deletion. Sorry, I put back templates. Gubbubu 15:23, 21 October 2005 (UTC)

Gubbubu's critics on recent "definition"

Mathematics is often defined as the study of certain topics, such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions.

1. First of all, you mustn't delete alternative "definitions", especially from peer-reviewed sources. this is awful! We will censoring ALL scientific writings of definition of Maths????? Gubbubu 15:38, 21 October 2005 (UTC)
2. I think the expression "certain topics as ..." is a bit lexikon-unlike. I could say "physics is ther study of certain topics, such as canoeing, shooting and space". And this would be true, too.
3. Then, what are the (peer-revieved and wikipedia-independent) sources of this definition? You couldn't give me any yet.
4. This "definition" throws out scientific and traditional classification of disciplines of mathematics (foundations-algebra-analysis-topology-operation research and numeric methods - ) etc. Only because of this it is pov.
5. The definition is philosophically weak. What does it mean "the study of quantity"? Arithmetics? And arithmetics is why not "the study of structure"? Arithmetics examines a special ring, isn't it? And what does it mean "the study of patterns"? What does it mean "the study of space"? Geometry? And algebraic topology not? Or yes? But algebraic topology is why not "the study of structure"? ... etc.
6. So I think this definition is minimum pov, but if you can prove it is not original research, I think templates would be deletable.
7. Thanks for lavish time to hear me: Gubbubu 15:38, 21 October 2005 (UTC) (maths teacher, wrote thesis on foundations of maths)

Could you calm down on this issue? You've made your point. Often defined: this doesn't say rightly defined. It just means we start with one common view, before moving onto another. The rest of the article does plenty to support that first sentence; it does not say that the definition is correct. In fact I don't suppose the article now does define mathematics. Giving any narrow definition would be wrong, I think. Charles Matthews 15:48, 21 October 2005 (UTC)

I've taken down the NPOV tag. We get it that you hate the first sentence. I really don't think you can make that a POV issue. Tags like that shouldn't be used for leverage on content issues that are so tiny. Charles Matthews 16:01, 21 October 2005 (UTC)

Popper reference

I've reverted the following edit, added to "Is mathematics a science?"

On the contrary, if one accepts the definition of a scientific statement as "any statement that can be refuted" (Karl Popper) then mathematics is a science. Mathematics is different than other sciences like physics in the sense that its statements, like "if the axioms are true, then this theorem is true", are intrisecally true. However, if a mathematical proof is wrong, it can always be refuted.

While I do think Popper is relevant to the question, the passage above strikes me as a bit of a non sequitur. The content that's supposedly being refuted is not "Proposition A is true", but rather "Profferred proof P is a valid proof of proposition A". Thus the passage offers Popper in support of mathematics as a science only for those who believe that the content of mathematics is proofs.

A much more telling use of Popperian falsificationism relates to large cardinals, whose existence is potentially falsifiable. --Trovatore 20:02, 24 October 2005 (UTC)

In any case, Popper's definition of science is non-standard. Those of us who consider mathematical proof the essence of mathematics, and the reason why mathematics has been so successful, would define science as knowledge discovered by a combination of mathematics and physical evidence, including careful observation, measurement, and experiment. Rick Norwood 00:58, 25 October 2005 (UTC)

Popper is the most important philosopher of science of the 20th century. Doesn't mean he's right, but he can't be ignored.

Now as to your definition, I think it begs the question in favor of physicalism. Why should evidence be restricted to the physical? Mathematical reasoning is experiment. --Trovatore 01:26, 25 October 2005 (UTC)

I agree. Mathematical reasoning is experiment. --MarSch 09:55, 25 October 2005 (UTC)

Popper more important than, say Bertrand Russell? Not self-evident. In any case, the importance of definitions is not what they say but that they are common meanings. A rose by any other name would smell as sweet. Thus, the place to go for definitions is a dictionary, not a philosopher. Originality in the area of definitions is misplaced.

I also strongly disagree that reasoning is experiment. That conflates deductive reasoning and inductive reasoning. Mathematical proof is deductive, experiment inductive. Rick Norwood 14:17, 25 October 2005 (UTC)

Let me be more explicit: Mathematical reasoning is experiment in the sense that it is capable of refuting its premises, by finding a contradiction therein. The platonistic existence of mathematical objects has observable consequences, for example (though not necessarily limited to) the nonexistence of such contradictions. --Trovatore 21:23, 25 October 2005 (UTC)

I think it's very bad that you removed my input. You can always find an objection to every argument about wether mathematics is a science, and remove it; but I stand that my point was absolutely relevant and correct. 99.9% of maths is proofs based on Zermelo-Fraenkel. Not only very few mathematicians discuss which axioms should we use (they are more logicians than mathematicians I would say - or philosophers) but although this exists, that does not mean the essence of maths is not proofs. Mathematicians consider several systems of axioms, and they use one or another, and they have no problem with that. The questions in maths IMhO is to find the consequences of each set of axioms, the question "which axiomatic system 'should' be used" is not necessarily meaningful. As for Popper I think he has a very good and interesting definition of science, much better than the other definitions I've read. Anyway this is the kind of edits that will probably make me stop contributing to wikipedia, since there are always people to revert whatever you write. Herve661 18:52, 29 October 2005 (UTC)

I agree with you. But that doesn't mean that we can impose our will on common usage. I personally think, for example, that the =O and =o notation is dreadful, and ought to be purged. But that isn't going to happen, a sad fact that I have to learn to live with. Rick Norwood 00:21, 30 October 2005 (UTC)

I think that wikipedia should be changed into something more pyramidal with maintainers accepting or rejecting contributions and a hierarchy system based on meritocracy instead of the current laissez-faire, which has its own merit, but in the long run just gives predominance to the ones who spend most time enforcing their views. And when you spend a lot of time enforcing your view, this is a sign that you are likely wrong. Herve661 11:38, 30 October 2005 (UTC)

What we have works, if imperfectly. There is little actual support for changing to rationed editing. Goose, golden eggs. Charles Matthews 11:53, 30 October 2005 (UTC)

Arabic Numberals

Does anyone know what page I should go to in order to learn about the history of Arabic numberals and how they came to be the main symbols of mathematics? -- Thearticulator 21:18, 25 October 2005 (UTC)

Nothing could be simpler. The title of the article is "Arabic numerals" (note spelling). For more info, try "Numerals" and "Base ten". Rick Norwood 01:01, 26 October 2005 (UTC)

"try using the talk page more"

My revert of the edit that combines theorems and conjectures was reverted in turn with the comment that I should try using the talk page more. If you will look at the comments above, you will see that I use the talk page a lot. I did not see any need to use the talk page in this particular case, because I could make my point in a single sentence: nothing is gained by combining theorems and conjectures, which I appended to my edit. The other edits that were reverted seemed equally minor and pointless. But this is not that big a deal, and so I'll let someone else reseparate the theorems and conjectures if that is worth doing, and save my energy for more important battles. Rick Norwood 17:01, 30 October 2005 (UTC)

Something clearly has to be done about these two sections--right now "Famous theorems and conjectures" is followed by "Important theorems and conjectures" (it it's famous, it's not important?). And I really think that CH deserves independent mention, and mention of its rather unusual status as a conjecture that's still driving research forty years after those of a certain philosophical persuasion declared it "resolved". I'm confident this can be done in an NPOV way without taking up undue space. --Trovatore 17:56, 30 October 2005 (UTC)

It's a perfectly good distinction. You have to be a pro mathematician to have heard of Riemann-Roch, but it's of major importance. As for CH, until I have some assurance this page is not being used for pushing a POV on it, then I think I'm going to oppose it as just one of a number of Hilbert problems. (Yeah, yeah, someone's going to take the Riemann Hypothesis off, and we all suffer. But what was there before was entirely unacceptable.) Charles Matthews 18:11, 30 October 2005 (UTC)

It's logically defensible, but it just sounds bad. Surely we can do better, just from a stylistic point of view. --Trovatore 18:13, 30 October 2005 (UTC)

On CH: I think it's established that it's not POV to report genuine controversy among experts. I urge you to at least glance at the refs I pointed you to (you might also look at Woodin's papers in JAMS, also referenced in Continuum hypothesis; they're not what you'd call easy reads, but you should be able to get some sort of outline). Also you might look at Harvey Friedman's account of the panel discussion on CH at the AMS meeting in Atlanta in January, 2005 (Friedman himself is rather hostile to the idea, but just the fact that he reports on it confirms its status as a live issue). --Trovatore 18:31, 30 October 2005 (UTC)

Sorry, not JAMS but Notices of the AMS. My bad. Available for free download here (part I) and here (part II) (registration required). --Trovatore 19:47, 30 October 2005 (UTC)

Look, I'm not saying that the continuum hypothesis page shouldn't have all this and more. I will say that it is NP-completeness that is the only logic topic that needs a mention on this page. It would be scandalous to omit that. Charles Matthews 19:49, 30 October 2005 (UTC)

Certainly agree we can't omit P=?NP, but I think CH is just as important. It has run through the entire history of set theory, from Cantor to the present day. --Trovatore 19:58, 30 October 2005 (UTC)

Yes, but your opinion wouldn't be generally shared across mathematics, would it? Say we choose ten 'important' topics. I think logic gets one, and it has to be P and NP. Charles Matthews 20:10, 30 October 2005 (UTC)

If logic gets just one, I think more mathematicians would vote for Gödel's incompleteness theorems than P=?NP. If we're talking specifically about open questions it's another matter of course, but that's not the current division on the page. Besides, set theory should get one, not math just logic as a whole, and for set theory it has to be CH. --Trovatore 20:14, 30 October 2005 (UTC)

This discussion is certainly more interesting than most. But we all have preferences. I would put Gödel at 2, myself. Mathematical logic = recursion theory + proof theory + model theory, model theory = set theory plus the other stuff like Shelah. These are fairly conventional divisions, but for example topology = algebraic topology + geometric topology + differential topology, so the topologists will want Brouwer fixed point theorem + Jones polynomial + Stokes theorem, and rightly argue a case for each. Charles Matthews 22:03, 30 October 2005 (UTC)

I certainly don't consider set theory part of model theory! Set theory has chronological precedence over model theory, if nothing else. It's true that set theory is probably the biggest single consumer of model theory's product, but that doesn't make it a subtopic.

Set theory is "logic" only by historical convention. By my lights, it's an investigation into a particular mathematical structure (the von Neumann universe) that has the very nice property of being able to unify the rest of mathematics into a common framework. I choose that phrasing deliberately, to avoid foundationalism; I explicitly do not claim that set theory is the basis for the rest of mathematics. --Trovatore 22:17, 30 October 2005 (UTC)

May I point out, gentlemen, that the list should be of some interest to non-mathematicians. Mathematicians already have their favorite lists of famous theorems and of unsolved problems. The Jones polynomial is wonderful, but few people outside of algebraic topology have ever heard of it, while there have been articles on both CH and P=?NP in the New York Times. It is also entirely artificial to limit each area of mathematics to just one result. Somebody should separate the two articles again, obviously, and CH deserves a special mention of its own, since its status is both important and interesting. Rick Norwood 00:02, 31 October 2005 (UTC)

We could all lobby for our research interests. No mention of elliptic curves which go back to Fermat and now are used in strong crypto and so in dealing with online credit card transactions. Why is that not equally interesting? Charles Matthews 08:36, 31 October 2005 (UTC)

From the RFC

I read some of the article, the introduction and the history, and quite frankly I did not like it very much. It seems like a lot of original research. Can't you say that mathematics was invented (my view is that it was invented)? At least mention prominent mathematicians in the history sectoin, or the first person one might call a mathematician? The introduction seems very dense and doesn't sit well with me. Both definitions seem very forced. Check out this article and maybe it will help give some perspective on your dispute Math Invented or Discovered?. The history part seems sort of rambling about "abstraction" and things like that, and to me looks like it has suffered a long battle. I think at the heart of your problems are very deep philosophical issues, some of which philosophers have probably already sorted that out to your satisfaction, you just don't know yet, some of which you might agree or disagree on, some of which are just "who knows?". Personally, I would just say "Mathematics is the study of numbers" and leave it at that for this article. Leave it up to the reader whether they feel like reading into it or not. You should also check out Philosophy of mathematics. If you ask both people who disagree you might find they simply have a different outlook, and can maybe agree to disagree. If you look on the Internet, people have all sorts of opinions when you ask "What is math?," but everyone would agree that it is a study of numbers. I don't know if this is sufficient for this article (I found one article by a math professor that said saying math was the study of numbers was like saying zoology is the study of giraffes--he went on to say nobody really knows.), but try a different approach. There are many many different views. The last one I got was from here and it has lots of other views.--Ben 09:38, 2 November 2005 (UTC)

Mathematics is the study of numbers; I don't think any competent judge could accept that. Charles Matthews 09:47, 2 November 2005 (UTC)

Oh sorry, I didn't realize that you guys are having a contest :P Never mind then. (and ya missed my point too). --Ben 11:53, 2 November 2005 (UTC)

I don't think anyone missed your point, Ben, but just to be sure -- your point seems to have been that the introduction should be simpler. I tend to agree. But the question of whether mathematics is "numbers" (where does that leave geometry?) or "truth discovered by deductive reasoning" (the Sherlock Holmes approach) has not been settled by the philosophers, who also do not agree on whether mathematics is discovered or invented. An honest introduction to the subject has to present both points of view. Rick Norwood 13:04, 2 November 2005 (UTC)

"Numbers and shapes" then. That was just an example. I'd say why weigh in on whether it is "truth discovered by deductive reasoning" or not. Leave it up for the reader to decide, or put it on the Philosophy of mathematics page. Anyway, the first part of the intro, while I think it's awkward, doesn't negate the second, it doesn't say whether or not what is being studied is invention or discovery or whatever, but the way it is written it seems like that's what is being said. Anyway I was only responding to the RFC and trying to help and let you know what I thought of the article. Good luck with your dispute. --Ben 13:22, 2 November 2005 (UTC)

Reuben Hersh, What is Mathematics, Really?, gives it 250 pages and 65 pages of notes. I actually think it would be good to cover all the bases, with respect to things he brings up, somewhere here. But not on this page. Charles Matthews 13:12, 2 November 2005 (UTC)

I appreciate your comments, Ben. Rick Norwood 14:22, 2 November 2005 (UTC)

people?

I like the suggestion of adding a few names to the article. Archemedes, Newton, and Gauss? Aristotle? Any thoughts on the subject?

On another topic, I'm going to go back in and separate the theorems from the conjectures. Frankly, having "Important theorems and conjectures" and "Famous theorems and conjuectures" looks dumb. Rick Norwood 13:10, 2 November 2005 (UTC)

Probably does - to people who are confused as to whether celebrities are important people. Rick, there must be other edits you can make. Charles Matthews 13:12, 2 November 2005 (UTC)

There are indeed other edits I can make, and I make them. But this article needs work, and I am trying to help. Rick Norwood 14:24, 2 November 2005 (UTC)

Continuum Hypothesis

I object in the strongest possible terms to the repeated attempts to push CH on this page. You could poll all the Fields Medallists and all the Wolf Prize winners in mathematics about its importance. I think you'd only get one vote; and Paul Cohen presumably would reckon the matter was settled. This page needs perspective. The matter has been discussed above. Charles Matthews 14:20, 2 November 2005 (UTC)

I have no interest in "pushing" CH. If you want to delete that sentence, I have no objection. I agree with you. Since my goal was just to separate theorems and conjectures, I wanted to be inclusive. Rick Norwood 14:26, 2 November 2005 (UTC)

Charles Matthews edit

version 1) Further steps need writing or some other system for recording numbers such as tallies or the knotted strings used by the Inca empire, which as far as we know lacked any other system of writing. Numeral systems have been many and diverse.

version 2) + Further steps need writing or at least tallies. The Inca empire, which had no other writing system, represented and stored numerical data using a complex system of knots and rope called khipu. Numeral systems have been many and diverse.

Since the Inca system did not use writing or tallies, the second sentence of version 2 contradicts the first. Also, we do not KNOW the Incas lacked a written language. All we KNOW is that no written Inca language has been discovered to date. In any case, this is not relevent to the discussion at hand, and should be removed. The reference to khipu is valuable, and should be kept.

new version) Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse.

version 1) Historically, the major disciplines within mathematics then arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to measure land and to predict astronomical events. These three needs can be roughly related to the broad subdivision of mathematics into the studies of numbers, space and change.

version 2) + Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to measure land and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of structure, space and change.

The use of "then" to connect this paragraph with the previous paragraph is minor. On the other hand, clearly "calculations on taxation and commerce" have much more to do with number than with structure. I am changing structure back to number, but leaving the deletion of the word "then".

version 1) Mathematical discoveries have been made throughout history and continue to be made today. The number of mathematical discoveries in the last hundred years is greater than the number made in all the years that went before.

version 2) + Mathematics since been been much extended, for example to solve problems on light since the seventeenth century, and on heat, since the nineteenth century. In order for this to happen, mathematical physics has had to provide a bridge to fundamental mathematical areas. Mathematical facts have been discovered throughout history. Mathematical innovations in the last hundred years have been major, of the same importance to the subject as all that had been developed in the years that went before

To jump from taxation and land measurement to "light" and "heat" is too big a jump. If you want examples here, they should be chosen more judiciously. To suddenly introduce the influence of mathematical physics (following a paragraph on taxation and land measurement) is also too abrupt a transition. Finally, the claim that more mathematics has been written in the last century than in all the years that have gone before is easy to substantiate. That Twentieth Century mathematics is of the same importance as, for example, all of the mathematics done by Euclid, Archemedes, Euler, Newton, Cauchy, Gauss, Able and all the other pre-Twentieth Century mathematicians is quite a claim. It may be true, but it is certainly opinion. For these reasons, I am eliminating the references to mathematical physics, moderating the claim about the importance of Twentieth Century mathematics, and correcting a typo in version 1.

I would appreciate comments from a third party. Rick Norwood 16:18, 3 November 2005 (UTC)

I agree with most of your points. I don't agree with your (proposed) deletion of light, heat, math. physics. If you think of a way to make it better, then you can replace them, but till then it is better to let them stand. --MarSch 17:11, 3 November 2005 (UTC)

I've tried a sentence that is more general about the relationship about math and science. This seems in keeping with the rest of the article, which is very general and is, after all, a brief summary of the math history article. Rick Norwood 21:53, 3 November 2005 (UTC)

The thing about the Incas is too defensive: we'd say that the Greeks didn't have zero, not that as far as we know, they didn't have zero. Cutting out the physics because you saw a jump, rather than filling it - why do that? The thing is a summary, so anyone can find some gap. And you don't even mention your reversion to:

Mathematical discoveries have been made throughout history and continue to be made today. The number of mathematical discoveries in the last hundred years is greater than the number made in all the years that went before.

This is monotonous, flat English. And what is the point of implying 'discoveries' can be 'counted'? They can't. What's a 'unit discovery'? How many discoveries was Euclidean geometry? Come on, we need numbers. Charles Matthews 22:13, 3 November 2005 (UTC)

I don't like the "number of discoveries" thing either. Oleg Alexandrov (talk) 23:01, 3 November 2005 (UTC)

The comment about Inca writing didn't belong here in the first place. There is a nice wikipedia article on the subject of the Incas.

I had previously added this sentence to provide a less specific mention of science: "Mathematics since has been been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both."

How about "The number of mathematical papers published in the last hundred years greatly exceeds the number of papers published in all the years that went before."

Any writing you find flat, please add spice to. Rick Norwood 23:50, 3 November 2005 (UTC)

Why we get so many rewrites.

The "quantity, structure, space, and change" definition is like waving a red flag in the face of a pure mathematician. I googled that quartad, and came up with two other sources that have that definition of mathematics, but both have picked it up from wikipedia. As best I can tell, it was made up by someone early in the history of the math article here, and stuck.

One reason for keeping it -- it ties in nicely with the pretty pictures below. One reason not to keep it, and to replace it with something less original, is the lack of a reference and the constant rewrites it inspires. Of course, the rewrites may still occur, but that particular sentence seems to beg for a rewrite.

Since it is a definition, and since the gold standard for English language definitions is the OED, I suggest a paraphrase of the OED defintion.

Certainly, the article is not where it should be, and I don't think just reverting any attempt to change it is the answer, though reverting ill considered chances is certainly appropriate. The main problem, as I see it, is a lack of organization and too much repetition. Rick Norwood 14:07, 6 November 2005 (UTC)

Rick, you add large amounts of stuff that few like. You should take that on board. I get the feeling that I have to go through reasons that I don't like it, line-by-line; and then you still don't get it. Frankly it feels like my time is being wasted. For example, simply counting published papers in mathematics is just rubbish. No one would accept that as a measure. The costs have gone down, it is easier to get things into print. I don't see how anyone could take that as a useful measure. Charles Matthews 15:54, 6 November 2005 (UTC)

Surely Rick has a point, though, that the amount of mathematics being done, by whatever measure, has exploded in the last century. We ought to be able to come up with some way to take note of this, and I don't think it's unreasonable to do so in the history section, once a reasonable wording is found. --Trovatore 16:21, 6 November 2005 (UTC)

Charles Matthews -- I hope you can put aside what seem to be some strong personal feelings about me ("You still don't get it." "My time is being wasted.") I quote sources and think carefully before I write. The sentence you dislike is paraphrased from a history of mathematics -- I would quote the original except it is at my office and I'm at home. If you would like the exact quote, I'll post it here tomorrow.

The point of the quote, as Trovatore understands, is that people need to be informed that discovering new mathematics is an ongoing activity. Most people I talk to are under the impression that no new mathematics has been discovered for thousands of years -- that it is all stuff that was important to the Greeks and the Romans, but hardly worth bothering about today. One thing encyclopedias do is correct false ideas that have wide public currency. I'm reverting your revert, but at least I've taken the time to explain why.

By the way, your point on lower costs has much more to do with publishing in the 21st century than in the 20th. More mathematics was published in the 20th century because more people were supported, either by corporations or by universities, so that they could do research in mathematics. That has a lot more to do with the quantity of new mathematics published than the cost of publishing -- are you old enough to remember when authors were charged page costs for the mathematical articles they published? Rick Norwood 16:42, 6 November 2005 (UTC)

I'm not going to get into what people "need to be informed of". I just think it's an accurate point that fits naturally in that section; no need to bring social engineering into the discussion. --Trovatore 16:48, 6 November 2005 (UTC)

Part of the reason more math is published is also that there are so many more people doing math than ever before. Some very serious conjectures have been solved recently, and it does seem that the pace of math discovery is accelerating. But it is all subjective and I would not think it is a good idea to attempt to quantify it or spend too much time on that in a serious math article.

And I will reinstate my view that Rick spends disproportionately large amount of his time when editing math articles on Wikipedia on this proeminent mathematics article. That is the real reasons this article gets so much rewritten, people just would not get busy with other things, and there surely are plenty of other things to deal with. Oleg Alexandrov (talk) 17:56, 6 November 2005 (UTC)

Look, the number of novels published in the twentieth century will exceed the number for history up to then. But no one thinks this is fodder for the opening parts of the Literature article. Just because some quantitative facts are mentioned in some book - I'm not for a moment doubting there are such books - does not mean this has to be pushed at the reader in that way. You casn't weigh Shakespeare against any number of Tom Clancy books; history of mathematics is the history of a certain culture or cultures. The numerical aspect is part of the sociology of the subject; just like knowing about the top journals, awards, and so on. Certainly no one who hated mathematics at school will be wowed by knowing there is more and more of it. Rick has cut out things that should not have been, and inserted his own phraseology of little merit. Can we have some common sense and basic insight here? Charles Matthews 19:55, 6 November 2005 (UTC)

The OED definition of mathematics.

"Originally the collective name of geometry, arithmetic, and certain physical sciences (as astronomy and optics) involving geometric reasoning. In modern use applied (a) in a strict sense to the abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations, and which includes as its main divisions geometry, arithmetic, and algebra; and (b) in a wider sense to include those branches of physical and other research which consist in the application of this abstract science to concrete data."

Given this, our introductory paragraph doesn't seem so bad. In my personal opinion, mathematics is knowledge obtained by deductive reasoning from first principles and science is knowledge gained induction from careful observation and measurement. But that is neither here nor there. We are bound by what the primary sources say. Rick Norwood 16:56, 6 November 2005 (UTC)

"The number of papers thing sucks."

"The number of mathematical papers published in the last hundred years greatly exceeds the number published in all the years that went before."

Above is the sentence that is causing so much heat currently. I wrote it to replace a sentence that said that the mathematics in the last hundred years equaled in importance all of the mathematics done in all of the centuries that went before. I thought it was a relatively minor and harmless rewrite of a sentence that was unnecessarily controversial. It is, as I mentioned, a quasiquote from a book on the history of math. The response has been...well, you be the judge:

(All of this is from just two wikipedians.)

"Rick, you add large amounts of stuff that few like."

"You still don't get it. Frankly it feels like my time is being wasted."

"Rick spends disproportionately large amount of his time when editing math articles on Wikipedia on this proeminent mathematics article."

"The number of papers thing sucks."

"Rick has cut out things that should not have been, and inserted his own phraseology of little merit."

Is it just me, or is this response disproportionate and unnecessarily personal? I contribute to a lot of articles -- more than 100 to date -- and I keep coming back to this one because it obviously needs work. Does anyone deny that? Rick Norwood 22:36, 6 November 2005 (UTC)

You don't get it, right? Charles is an excellent editor who has been here for at least three years if not all four. If anybody can be called the founder of the math on Wikipedia it is him. He has added new significant content to hundreds of articles (the upper hundreds I mean) and worked on and copyedited thousands.

I have on my watchlist more than 1300 of the around 10,000 math articles, and your name pops up only here and at manifold — proeminent articles where you want to change the world. And your contributions history reveals just a little more than this.

Everybody is equal on the Wikipedia, right? No! Get down to business, spend at least six months doing dedicated work in here then one may pay more attention to your views.

This article is in good enough shape, and some of your edits make it worse not better. You are becoming a timesink rather than a helper. Again, there are plenty other places where a competent math editor can make a big difference. Oleg Alexandrov (talk) 22:49, 6 November 2005 (UTC)

The only reasonable question to ask about any contribution is, is it well thought out and backed up by research. There is no six month waiting period. However, your statement that my contribution history reveals just a little more than my contributions to Mathematics and Manifold is flatly wrong. Here is a list of the first twenty math articles I wrote or contributed to: manifold, classification problem, calculus, field, trancendence, transcendental, eigenvalue (etc.), magic cube, tetramagic cube tetramagic square, anti-racist mathematics, table of opposites, Pythagoras, Fibonacci number, Tribonacci numbers, Tribonacci constant, tetrakis hexahedron, theta function, tetrastigm, and tetraview. That brings us up to last September. You'll note that all of those contributions preceded my first contribution to the mathematics article.

All I ask is that you judge my contributions objectively, not personally. Rick Norwood 23:53, 6 November 2005 (UTC)

I agree that this article is in a bad state and that is the reason that it attracts big edits. People should edit where they want, there is no need to first edit so-and-so-many other math articles first. --MarSch 10:32, 7 November 2005 (UTC)

No, you can't forbit people from editing unless they contributed to such and such number of articles, as this is a wiki afterall. However, it is sad to watch that some people work their ass off trying to keep this project going and some other people have nothing to do all day than arguing about this paticular article and would not let go for months.

The biggest strength of Wikipedia — open access — is also one of its biggest problems. One's got to watch after and argue with each and every person who's got an itch to scrath. Articles get edited into oblivition, with subsequent versions often worse than the originals. I heard there is a proposal to promote articles to a stable status after a while, after which articles would be edited very conservatively and/or people who know what they are doing. Oleg Alexandrov (talk) 18:10, 7 November 2005 (UTC)

There is an issue, but it's certainly not personal. Extensive discussion on the mailing list shows that (a) some major people from Jimbo downwards (it's a flat structure, of course) think that writing quality is now more of a real issue than getting across the NPOV/NOR/sources basic definition of WP style and (b) many people want to stabilise versions. I don't want stable versions; it throws out the basic wiki advantage. Tomorrow someone announces a proof of the Riemann hypothesis, it can go into the page in 30 seconds. Literally; a friend of mine was amazed that the Topalov page was updated in seconds; he's not a Wikipedian, he asked me 'who are these people who do that'? So I want Wp as open as ever. But that means fighting 'drift': edits that go against the drive to make the articles ever better, tighter, better written. Talk pages are the place to make points. Edit wars settle nothing; so it all has to be done via talk, even tough talk. Charles Matthews 19:26, 7 November 2005 (UTC)

Our interests are the same. Thinking back on my very busy schedule, I have to wonder about Oleg Alexandrov's comment about people who have nothing to do all day. But, I won't take it personally. I do not make changes to any article unless it seems essential, and essential generally means either an error of fact or bad spelling and grammar. What brought me to the manifold article, for example, was a request for improvement on the Community Portal.

The most important thing that needs to be done in the Mathematics article is to eliminate or at least reduce repetition. I plan to help with that, but I will discuss any changes here before I make changes in the article. Rick Norwood 23:41, 7 November 2005 (UTC)

Request to Trovatore

You and I agree that something should be said about the amount of mathematical activity over the last hundred years. Oleg Alexandrov said, "I would concider accepting something in the spirit but not letter of this sentence." Would you attempt a sentence on this subject. Rick Norwood 23:57, 6 November 2005 (UTC)

To be honest I sort of thought Charles had a point with the comparison with Literature; I hadn't looked at it that way before. --Trovatore 00:08, 7 November 2005 (UTC)

Let's attempt a section on 'Mathematics as a profession' to accommodate this. Qua profession, mathematician has seen enormous growth, and no question. (By the way, despite what Oleg wrote above, User:Axel Boldt and User:Michael Hardy were in my view the important pioneers here in the mathematics sections.) Charles Matthews 10:52, 7 November 2005 (UTC)

overall organization of the article.

"quantity, structure, space, and change" "number, space, and change" "quantity, spatial relations, structure, and change"

These are all more or less the same, but I think the article would benefit if the same words were used for the same concepts in the different sections.

Beteeen "number" and "quantity", I prefer "number", since some numbers, i for example, do not represent quantity.

Between "space" and "spatial relations" -- I really don't like either. "Space" suggests "the final frontier". Spatial relations sounds like n-dimansional sex. How about "geometry"?

That would give us "number, geometry, structure, and change" in all three places, and elsewhere as needed (except no "structure" in the history section).

I welcome comments on whether or not it is a good idea to use the same words to express the same concepts throughout the article. Rick Norwood 00:00, 8 November 2005 (UTC)

Well, my (mild) defense of the "quantity, etc" definition has been based largely on the fact that it makes it fairly obvious that we're punting on the (doomed) idea of trying to define mathematics. I wouldn't be very happy about trying to take the non-definition seriously in the rest of the article. My real preference would be to punt explicitly, with something like "Mathematics is a discipline with no single agreed definition, but that includes..." but I don't feel like arguing about it now. --Trovatore 00:09, 8 November 2005 (UTC)

Me neither. I really was not trying to return to the question of definition. I think what we've got is as good as we're going to get, and our definition agrees substantially with the OED, except the OED just has number and geometry and omits structure and change -- which makes our version better, imho. I know Tolkien worked on the OED, I doubt Hardy did. What I'm aiming toward is consistant usage so that the article reads smoothly instead of the reading saying, "Wait a minute, didn't they just say..."

In any case, I'll wait a few days before changing anything, and will not make any change unless there is substantial agreement on this point. Rick Norwood 00:18, 8 November 2005 (UTC)

One may not think of i is a quantity, but I don't see that a good argument for replacing everywhere in the article "quantity" with number. I am not sure about the change either way.

I have no problem with "space" showing up in one place and "spatial relationship" in another place.

I would suggest we first talk about the inconsistencies. In one place on has math as "quantity, structure, space, and change", in another place as "number, space and change" and yet another place as "structure, space, and change".

I would not agree for the moment on replacing "space" with "geometry", let us see how it goes after the requests for more consistency/less repetition are dealt with.

One thing Rick. Yes, you are busy, so is everybody else. Waiting for even more than a couple of days would be good before implementing changes. And please, no big edits and no reverts. And right, all discuseed here beforehand. Oleg Alexandrov (talk) 01:16, 8 November 2005 (UTC)

We agree on the goal of this particular suggestion, to stimulate a discussion of inconsistencies. I'm not entirely happy with "number, geometry" either, since one is a class of mathematical objects and the other an area of study. I hope someone will come up with something better. No big edits. Every day, I revert two or three cases of obvious vandalism, typically a case of someone typing "Your queer" into one article or another, but aside from that, no reverts without discussion. Rick Norwood 17:07, 8 November 2005 (UTC)

After mulling it over for a while, I've come up with "number, shape, structure, and change" for the "subject matter" definition, in all three places. Does anyone have a better suggestion? Rick Norwood 22:35, 9 November 2005 (UTC)

"Another view"

I see people have been warring over the intro since time immemorial, but I still want to drop in two cents:

Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions.

This is vague. Which axioms and definitions? Different sets of them are the bread and butter of studies in formal mathematics. In this light it's peculiar to talk of the body of knowledge. Most people will read "knowledge" as "facts", which is inappropriate here—mathematics deals with truths supported by reasoning, but those are not the same as facts.

What is probably meant here is that mathematics is deductive, no more, no less. I don't know how you would write that as a definition of mathematics (if you would want to), but the above seems unsatisfactory. It suggests mathematics can be adequately described as a big bag of facts you get from applying (some) rules to (some) axioms. Is that really the view "many mathematicians" hold? 82.92.119.11 21:38, 9 November 2005 (UTC)

Unfortunately, I think it is. But then the vast majority of mathematicians (correctly) don't spend much time worrying about it; they're more interested in doing math than in finding a narrative that says what they're doing. Thus they latch on to a narrative, the formalist one, that allows them to dispense with the question and go on their merry way. Nothing really wrong with that, for their purposes. But when that narrative is seriously advanced as foundations, IMHO it doesn't hold up well at all. --Trovatore 22:16, 9 November 2005 (UTC)

Most mathematicians (i.e. academic pure mathematicians) would say this approach is a reasonable criterion: for what is and is not mathematics. Amid theoretical physics, for example, there is plenty of echt mathematics, and also plenty of stuff of high heuristic value that is not axiom-based reasoning. In that sense, this is a sensible description: not subject matter - some things about infinite-dimensional groups would pass the test 'how do you know that?', based on axiomatisation, and some would not. Charles Matthews 22:21, 9 November 2005 (UTC)

So in some sense what the dispute comes down to is whether you think the question of what the axioms should be is a mathematical one or not. In my view, of course, it absolutely is a mathematical question, and moreover, once one has admitted that there are (obviously non-axiomatic!) reasons for preferring one collection of axioms over another, the distinction between "axiom" and "mathematical statement in general" starts to look artificial. Thus things like the extremely convincing probabilistic arguments in favor of Goldbach's conjecture are also, in my opinion, mathematics. --Trovatore 22:35, 9 November 2005 (UTC)

Well, you won't get many in number theory to agree with you. Of course the value attributed to proof has oscillated over time. But the distinction between the proved and the unproven is reasonably stable, even if there are some interesting case studies (e.g. Italian school of algebraic geometry). Charles Matthews 23:04, 9 November 2005 (UTC)

I didn't claim the probabilistic arguments are proof--just that they're mathematics --Trovatore 23:22, 9 November 2005 (UTC)

The answer to "which axioms" is, usually, the Zermalo-Frankel axioms of set theory, plus the Axiom of Choice. For logic axioms, see Logic for Mathematicians by Hamilton. For geometry axioms, Hilbert's Axioms. I certainly agree that the choice of axioms determines the resulting mathematics, but attempts to produce something interesting or useful from abstract axioms (Every boffle is a biffle with toff.) have failed. My own view is that the reason deduction from axiom systems has been so successful is that axiom systems are abstractions from nature. For example, the Axiom of Union is an abstraction of pouring two pails into the same bucket, and the axiom "Two points determine a unique line" is an abstraction of actually drawing a line between two physical dots.

There are at least three reasons to think that the current body of mathematical knowledge deduced from axioms really does constitute "facts".

First, widely different cultures discover the same "facts". Negative numbers were discovered (or if you prefer, invented) three times in widely separated places, making transmission of knowledge unlikely. In Chinea, in India, and in Europe the "fact" that the product of two negative numbers is positive was also discovered. I know of no examples of different cultures discovering different mathematical "facts".

Second, the "facts" produced by axiom systems have lasted longer than any other "facts" known to man. The Pythagorean theorem is now at least 2300 years old, some would say 10,000 years old, based on megalithic structures that seem to incorporate it. What other "facts" have lasted so long and produced such universal agreement. Listen to historians argue about whether Pythagoras actually knew about the theorem that bears his name.

Third, you come up with the same set of "facts" no matter which of the major axiom systems you use. If you start with ZFC, define a function as a set of ordered pairs, a point as a single ordered pair, and define a line as ordered pairs that satisfy y = mx + b, you get the Euclidean axiom, "Two points determine a unique line." as a theorem in ZFC.

Finally, from a pragmatic point of view, mathematical statements are "facts" because when you use them in applications, the applications actually work.

Would probabalistic "proofs" work as well? We don't know. Rick Norwood 22:56, 9 November 2005 (UTC)

ZFC is a nice reference point; well-known, powerful enough for most mathematicians' purposes, and it's sociologically convenient to be adopt the convention that, when I make a claim without further qualifications, it can be interpreted as saying there's a proof that should (in principle) be formalizable in ZFC.

But it's a wholly unjustified leap from there to the notion that ZFC has some special epistemological status. Suppose we claim, for example, that theorems of ZFC are reliable, but things not theorems of ZFC are not reliable. Then what about the claim "ZFC is consistent"? It's not a theorem of ZFC, so it's not reliable. But if it's not reliable, then neither is ZFC, because if ZFC is inconsistent, then it proves falsehoods.

As to your third point, that's true for lots of things, but it's not true in general. For example, ZFC does not prove that every projection of an analytic set is Lebesgue measurable, whereas ZFC+"there exists a measurable cardinal" does prove that.

Oops, I goofed here -- make it "every projection of a co-analytic set". --Trovatore 00:06, 10 November 2005 (UTC)

And on your final point, well, how do you know the applications actually work? By observation, right? The probabilistic proofs are also observed to work. You have some subtle argumentation ahead of you if you want to draw an epistemological line here. --Trovatore 23:07, 9 November 2005 (UTC)

This is an entertaining discussion, but doesn't really have anything to do with the article, since the article that has all this stuff in it is Philosophy of mathematics

I have no desire to have the mathematics article discuss these issues. I just want to make sure the naive formalist view is not presented as the only or preferred one. --Trovatore 00:11, 10 November 2005 (UTC)

However, just for fun -- I would say that ZFC has earned a special epistemological status, first by allowing mathematicians to prove the same theorems that could be proven in older axiom systems, second by not (so far) allowing anyone to prove both A and NOT A. But, of course, you are absolutely correct in that the statement "ZFC is a very fine axiom system indeed" is a philosophical statement, not a mathematical statement.

ZFC+"there exists a measurable cardinal" also satisfies your criteria. And I agree that ZFC is a very fine axiom system; I just don't think we should stop there--it isn't complete, and we can still discuss mathematically the truth values of the statements it doesn't decide. --Trovatore 00:11, 10 November 2005 (UTC)

We agree. Rick Norwood 00:15, 10 November 2005 (UTC)

Deductive reasoning has worked for a very long time, now. Arguements of the form -- "We have tested this statement in a very large number of special cases and have never found a counterexample." have really only been offered in mathematics in the last fifty years or so. And, as I'm sure you know, for any natural number n it is possible to generate a statement which will past the statistical test for the first n natural numbers and fail for n+1. The statement "x is less or equal to n" is such a statement. If you let n = 10^10^10, every statistical test you can devise will support the assertion, "All natural numbers are less than n." But from this statement, bad consequences follow.

This sort of thing is unavoidable in science (yes, I think math is a science). Yes, you have to give up the notion that math is apodeictically certain, or that (as Petry quotes Gauss as claiming) there is no genuine controversy in mathematics. Good riddance, I say. --Trovatore 00:11, 10 November 2005 (UTC)

I donno. Gauss was a pretty smart fella. Rick Norwood 00:15, 10 November 2005 (UTC)

Back to the discussion of the article. What do you think of "number, shape, structure, and change". I know it's bad, but is it less bad than what we've got now? Rick Norwood 23:38, 9 November 2005 (UTC)

No opinion on "number", etc.? Rick Norwood 00:15, 10 November 2005 (UTC)

Not really, no. Probably could come up with one if I tried, but I don't really care enough to bother, as long as it doesn't slide back towards POV. --Trovatore 00:18, 10 November 2005 (UTC)

number, shape, structure, and change

I had resolved to let things settle down, and a consensus be reached, before inserting "number, shape, structure, and change" everywhere a similar list appears as a definition of mathematics. But somebody has changed the first sentence unilaterally, which got me to thinking (always dangerous).

Most authorities define science according to method rather than subject matter. For example, from the Oxford American Dictionary "a branch of study requiring systematic study... ." On the other hand, most authorities define mathematics according to subject matter. For example, "The science of number, quantity, and space".

But every mathematician can think of ways of studying number, quantity, and space that are not mathematics, numerology for example, and can also think of areas of mathematics, logic and set theory for example, that are not about number or quantity or space.

So, how about this:

Mathematics is usually defined as the study of such subjects as number, shape, structure, and change. Many professional mathematicians, however, use mathematics to describe any study in which theorems are deduced from axioms and definitions, including such areas as mathematical logic and set theory.

Then we can use the "number, shape, structure, and change" ruberic throughout the rest of the article.

Comments? Rick Norwood 21:19, 13 November 2005 (UTC)

I think the introduction the way it is now sounds better while saying the same thing. Oh, and I reverted the anon change you mention above. I like the "often" part which the anon removed. Oleg Alexandrov (talk) 23:02, 13 November 2005 (UTC)

I'm ok with that. So, should it be "quantity, structure, space, and change" everywhere, or one thing in one place and other things in other places? Or was your comment just about the shorter version vs. the longer, rather than about the list of the four "things" that mathematics is about? Rick Norwood 23:33, 13 November 2005 (UTC)

I would think we should indeed put "quantity, structure, space, and change" everywhere for consistency, but one should look carefully around the place those occur, to see if they match with what is in there. But let us see if there are other opinions too. Oleg Alexandrov (talk) 23:46, 13 November 2005 (UTC)

Well, I've waited more than a week, and you are the only person to offer an opinion on "quantity, structure, space, and change" v. "number, shape, structure, and change", so I will defer to you, and to tradition, and go with the former. I only know of a couple of places that need changing, so I'll do that now. Rick Norwood 20:26, 17 November 2005 (UTC)

That was rather fun. It is surprising how easily the various topics fall into place under the headings "quantity, structure, space, and change". Now all of the definitions of mathematics on the page agree, and the topics are covered in the same order in each subsection. Form follows function. Rick Norwood 21:25, 17 November 2005 (UTC)

Recent changes

Rick, from what I remember and from what is written right above, I thought the talk was about doing a replacement here and there. There was no mentioning of rewriting entire paragraphs.

I would like to reiterate what was said before. It is not good to do too many changes too fast with too little prelimiary discussion.

Now, let us leave this article alone for a couple of weeks until we digest what changed, and if necessary, what to move back. Oleg Alexandrov (talk) 02:30, 18 November 2005 (UTC)

I agree -- for one week at a minimum. Actually, I think the article is in pretty good shape now. I've read it through from beginning to end several times, and it seems to flow.

Most of what I did was to put things in the order: quantity, structure, shape, and change, when they were in a different order. Also, where it said something like "number and shaping" I changed that to "quantity and shape". Sometimes it was necessary to write a sentence or two, as I did on Pythagorean triples, when the article had examples of quantity, shape, and change but no example of structure. The only other thing I did was to eliminate repetition -- there were several places where the article repeated itself almost word for word. Rick Norwood 14:22, 18 November 2005 (UTC)

Aesthetic value

Automated theorem proving has some value as evinced by the existence and value of formal verification. The assertion that they will generate nothing of value therefore has to be qualified. "Widely recognized esthetic value" is a reasonable formulation of such a qualification.--CSTAR 23:01, 20 November 2005 (UTC)

I understand your point, and it is not worth getting in a fight over, but to say that automated theorem proving has not generated anything of esthetic value is to suggest that it has generated things of practical value. I'm a big believer in the esthetic value of mathematics, but I thought in this context the adjective sounded a bit snobby to me. Rick Norwood 01:13, 21 November 2005 (UTC)

Automated theorem proving has roiled the theory of hypergeometric series, where new identities can be found at will. Aesthetically, the new identities are similar to the old, hand-derived ones. Are the newer identities "less beautiful"? The analogy is that of using a computer to multiply numbers: it has allowed the level of abstraction to jump up a notch. In at least this area, automated theorem proving actually has resulted in new and aesthetically pleasing identities (and I know this contradicts my earlier statements). linas 15:33, 5 December 2005 (UTC)

publication

would you like to publish this article? -- Zondor 22:13, 27 November 2005 (UTC)

Prehistoric math

I think the following could be more accurate with the changes I indicate--I am writing as an anthropologist who did field research in New Guinea among technically neolithic people, whose number system I saw first hand. Another anthropologist studied it in detail later. The Oksapmin people I lived with didn't count seasons and didn't recognize years because it was the equatorial tropics. They did count objects, however; but not abstractions. You could ask how many times were you angry today, and they would be able to give numerical values to it. But I doubt that it happened every day. (This is just an impression.)

The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years.

I would change it this way:

The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have quantity in common, for example, that they both correspond to "two" fingers, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples probably also recognized how to count abstract quantities, like time -- days, seasons, years. Addition was also likely in prehistory, where, e.g., three men coming up a hill could be called "three" men or "one-two" men. --samivel 22:33, 27 November 2005 (UTC)

I don't like your suggested change, but I appreciate your thoughtfullness in discussing it instead of just making it. First, it is dangerous to extrapolate from neolithic peoples today to prehistoric peoples. Second, this is too technical for a general article. An article on "neolithic mathematics" might be nice. There are some references you could use in the article Pythagorean theorem. Rick Norwood 23:15, 27 November 2005 (UTC)

Agree with Rick. Let us keep it simple. An article on neolithic mathematics would be nice. Oleg Alexandrov (talk) 23:50, 27 November 2005 (UTC)

Inspiration, aesthetics, and pure and applied mathematics

Things seem to have quieted down, so I'm going to work on this section. I've spotted one grammatical error, and a couple of infelicities. Rick Norwood 14:10, 5 December 2005 (UTC)

If you disagree with my removing string theory, then change that, instead of reverting everything. You comment that string theorists work with number theorists. Of course they do. Fuld Hall has room for many disciplines. But both are pure mathematics. The claim in this paragraph was that string theory was driven by a combination of pure reasoning and physical insight. I see the beauty of the pure reasoning, but it seems to me that, so far at least, the physical insight is notoriously missing. Rick Norwood 14:42, 5 December 2005 (UTC)

I reverted the edit for a variety of reasons:

• It removed mention of commerce, astronomy as place where math originated.
• It removed statment that physics is the source of much new math
• Last I looked, string theory was considered to be a branch of physics, not math. As to the nature of "physical insight", this is easily detected by the screams of pain that emanate from mathematicians when the string guys ride roughshod. During talks, I've heard the following exchange more than once: question from the audience: "can you prove that x?" to which the speaker (a physicist) replies "I've never proven anything in my life". linas 15:22, 5 December 2005 (UTC)

Overall, the recent edit seems to be a watered-down milque-toast version of the punchier, more direct and entertaining earlier version. linas 15:26, 5 December 2005 (UTC)

I actually quite like Rick's edit, the previous version was convoluted. However, I do have reservations:

1. I'd certainly classify string theory as physicists; they do use a certain insight and they claim themselves it is physical insight (continuing Linas' exchange: a similar but rather old-fashioned reply is "Gentlemen don't ask for proofs"; this was related to me in the context of the tradition of "British applied mathematics").
2. I don't like the first sentence ("Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change") though the previous version ("Mathematics arises wherever there are difficult problems that merit careful mental investigation") was not any better.
3. I prefer commerce to economics, etc.
4. The end of the first paragraph hints at the split pure/applied, then there is a sentence about specialization, and then the pure/applied distinction is mentioned.
5. I'm not sure one could say that stats is split of.
6. I disagree with "especially those that replace a long proof with a much shorter proof": firstly, this is implied by simplicity; secondly, shorter proofs are not always more elegant than long proofs.

I addressed points 1, 3 and 6 in my edit. The rest needs some more thought. -- Jitse Niesen (talk) 16:03, 5 December 2005 (UTC)

The result of today's edits isn't bad. Some comments:

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change.

This is now almost identical to the very first sentance of the article. Surely we can start in a different direction.

At first these were found in commerce, land measurement and later astronomy;

This sentance highlights the weakness of the previous section, on the history of math. That section should have at least telegraphed Egyptian land-surveying (geometry) and the Sumerian "futures contracts", the clay seals, the ones containing the embedded tokens for bushels of grain and heads of cattle that lead directly and immediately to the written Sumerian numeral system. The Inca knotted string reference has no such direct connection, I don't beleive.

nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself.

Maybe true, it is a dis-service not to highlight the importance of physics in spurring the advancement of math over the centuries. I'm also not entirely convinced that, as of the late 20th century, "all sciences suggest problems studied by mathematicians". In fact, I can't even think of examples for most sciences. What problems have medicine and ecology suggested, that mathemticians study? Predator-prey relationships? Chaotic arhythmias of ventral fibrilations? By contrast, all of the early DNA/evolutionary phase transition and scaling papers from the 1980's seemed to come from Soviet particle physicists, and these flowed in the reverse direction: some known results from the theory of infinite-dimensional spaces were applied to problems from biology. linas 23:56, 5 December 2005 (UTC)

Mathematical biology is a big and growing field. In my department of mathematics, we have for instance people working on cancer tumours [1] and the modelling of ecological systems [2]. Of course, one could argue that they are not mathematicians, but given that they are in a maths department that does not seem a strong argument. So, I remain of the opinion that "all sciences suggest problems studied by mathematicians".

However, I agree that physics has had more influence than the other sciences, and if you look at the work awarded with a Fields medal, I can't find anything inspired directly by any science other than physics. I even have a vague recollection of Michael Atiyah saying something similar as Linas, but a quick websearch only turns up this interview. -- Jitse Niesen (talk) 15:18, 7 December 2005 (UTC)

And responding to Rick's remark From what I have read by Witten...: although "pure" string theory as envisioned by Witten is not testable, there already have been accelerator-based searches for extra dimensions and branes, see e.g.[3] at CDF. See also the proposed gravitation experiment STEP (satellite): in short, physicists find this stuff beleivable enough to be funnelling (tens of) millions of dollars (and more) to build equipment to search for and measure stuff that's essentially one-off from Wittens's pure string theory.

The other remarkable thing about string theory, as opposed to the more laboratory-freindly chaos theory, is that string theory has made remarkable contributions to "pure math" such as number theory and geometry e.g Donaldson theory or Gromov-Witten invariants, that the number theorists and geometers had been unable to make progress on thier own. (This may be a statement about funding, and not brains per-se). By contrast, I'm not aware of contributions of equal depth from chaos theory, except maybe from David Ruelle?? linas 00:15, 6 December 2005 (UTC)

My (former) reversion

This comment has been edited a bit, you are welcome to see its old version though.

Rick, if you think I am overprotective of this article, then the answer is yes, I am. This article is one of the most visited articles on Wikipedia, #5 to be exact, see Wikipedia:Most visited articles. And it was not so long ago that you were attempting to edit this article by sheer pushing (thankfully you learned a lesson or two from that period).

That is to say, I really don't like to wake up in the morning, see big changes to this article as "accomplished fact", and a comment on the talk page saying "If you disagree with my removing string theory, then change that, instead of reverting everything.". And yes, everything started with you "spotting a grammatical error".

So, please, from now on, if you do anything else than just very minor edits, please make your case first on the talk page for what you want to edit, point by point. In this way you save yourself the embarassment of being reverted, and maybe save some time and frustration to other people involved. This is a proeminent article. Let's be conservative. Oleg Alexandrov (talk) 17:01, 5 December 2005 (UTC)

Thank you for reading my edit. It was cautious, discussed here, referenced in the Edit Summary line, and the only edit to this article that I intend today.

There seem to be only three points of contention between my edit and the latest version (as I write this). The first is the line about "commerce, land measurement, and astronomy". The only problem with that line is that it is a repetition, almost word for word, of a line in the previous section. The second is the question of whether or not mathematicians consider clever or short proofs beautiful. We do. Shorter is not always more beautiful. How about just "clever proofs"? The third question is whether or not string theory is inspired by physical insight. Please note that it is not a question of whether physicists like proofs or not. It is specific to string theory. From what I have read by Witten and the other major string theorists, their claim is that string theory must be true because it is beautiful mathematics, but they admit (brag?) that it cannot be tested physically, at least, not with any of the equipment available today. Therefore it seems to me a poor example of physicists using a combination of mathematics and physical insight to come up with new mathematics. If you want a modern example of that, choas theory is a better one. Rick Norwood 21:03, 5 December 2005 (UTC)

Rick, I may be very wrong, and I have been wrong before. But discussing things on the talk page does not mean writing on the talk page "I am going to rewrite this and that right now". It means that you specify what you think is wrong with the article, wait for a couple of days, and see if people have comments in any way. If they don't have any comments, or if they tell you "looks good", let us see how it turns out, then you go ahead. If they tell you "wait a second, I don't think you are quite right" on this, then you clarify yourself.

This kind of caution is not necessary most of the times, but this is an important enough article that I think it is. I hope you understand what my point is. If you don't, let us discuss this. Thanks. Oleg Alexandrov (talk) 21:49, 5 December 2005 (UTC)

Yes, I understand your point, and will take it into consideration next time I work on the article. I assume that errors of fact, grammar, and spelling can be corrected out of hand, however. Rick Norwood 15:07, 6 December 2005 (UTC)

Minor changes of course can be done without consultation. :) I meant big things and things which cause lots of discussion on the talk page later. Oleg Alexandrov (talk) 18:48, 6 December 2005 (UTC)

Rick's new suggestions

With that in mind, as the article currently reads, this sentence appears:

"Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events."

In the next section, but only seven lines below, we have this sentence.:

"At first these were found in commerce, land measurement and later astronomy; nowadays..."

1. Does anyone object to eliminating this repetition?

2. Next, I believe that many mathematicians find clever proofs beautiful. Does anyone object to this being included along with generalization, simplification, and clever algorithms?

3. Finally, I think chaos theory is a better example than string theory, of modern mathematics that is driven by both reason and physical intuition . (See discussion above). Are there any objections to that change?

I will wait one week before making any changes. Rick Norwood 15:19, 6 December 2005 (UTC)

I replied above about the "talk page consultation issue". The issue with eliminating repetition is I guess not a big one, as long as you don't use this as an excuse to rewrite entire paragraphs without asking first (happened a while ago :). The other two are more complicicated, I guess they would need some thought. Oleg Alexandrov (talk) 18:48, 6 December 2005 (UTC)

Rick, I numbered your questions to make it easier to reply to them; I hope you don't mind.

Ad 1. Hmm, the repetition does not bother me, but on reflection, I don't care greatly about having "commerce, land measurement and later astronomy" mentioned in the "Inspiration etc." section.

Ad 2. I agree with Rick, adding clever proofs is a good idea.

Ad 3. The problem that I have with chaos theory is that I'm not sure what it is. I only know it as a part of dynamical systems. I think it has also been taken up by New Age adherents, like the term nonlinear, but I guess that's not what we are concerned with. I don't see much physical insight being applied here. Furthermore, Linas raised a valid point above:

"The other remarkable thing about string theory, as opposed to the more laboratory-freindly chaos theory, is that string theory has made remarkable contributions to "pure math" such as number theory and geometry e.g Donaldson theory or Gromov-Witten invariants, that the number theorists and geometers had been unable to make progress on thier own. (This may be a statement about funding, and not brains per-se). By contrast, I'm not aware of contributions of equal depth from chaos theory, except maybe from David Ruelle??"

Finally, I raised some more points before:

4. The end of the first paragraph hints at the split pure/applied, then there is a sentence about specialization, and then the pure/applied distinction is mentioned.

5. I'm not sure one could say that stats is split of.

Especially the last one, I'd like to see discussed. It is my impression, without doing any research, that in many universities computer science and mathematics are seperate departments, but statistics is within the mathematics department. -- Jitse Niesen (talk) 15:18, 7 December 2005 (UTC)

The point is not that string theory isn't good science and good math, though I seem to remember Roger Penrose writing against it. The question is -- did physical intuition help drive string theory, as the article claims. Chaos theory is clearly driven by both pure reason and by physical intiution. The main result is this: even in a deterministic system computational complexity can prevent accurate prediction (in a sense that can be made precise with epsilons and deltas). Not our fault that New Agers have picked up on it, they're big on string theory, too.

In saying that stat is split off, I am not expressing my own opinion, but the universal opinion of all my statistical colleagues, who keep telling me that, even though they are stuck in the math department, statistics is not math. I just assume they know what they are talking about. Ask a statistician. There probably are not many statisticians on wiki -- they're too busy consulting. Rick Norwood 22:15, 7 December 2005 (UTC)

A week has gone by. How time flies when you're having fun! I'm going to eliminate the repetition, add "clever proofs", but, since there is no consensus, leave string theory alone. I'm also going to talke a look at Jitse Niesen's point 4 above. Rick Norwood 14:47, 13 December 2005 (UTC)

On consideration, I only made one change -- I added "clever proofs" and gave Euclid's proof that there are infinitely many primes as an example. Rick Norwood 14:52, 13 December 2005 (UTC)

Church's Thesis

I removed Church's Thesis from the list of theorems, since it is in fact not a theorem. It's not even provable.

CraigDesjardins

moving right along

I am now looking at the next section, Notation, Language, and Rigor. I fixed a comma error. I do not think it necessary to ask permission to correct errors in punctuation, grammar, or spelling.

This section seems very good, but there are a few changes I propose. No change in paragraph one. In paragraph two I think that something needs to be said about the origin of mathematical notation, and the transition from notation to language is not clearcut. Here is my suggestion for paragraph two in this section (now split into two paragraphs, one on notation, the other on language). Note: modified per comments below on Dec 14 and again on Dec 15.

Most of the mathematical notation we use today was not invented until the 16th Century. Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict grammar (under the influence of computer science, more often now called syntax) and encodes information that would be difficult to write in any other way.

Mathematical language also is hard for beginners. Even common words, such as "or" and "only", have more precise meanings than in everyday speech. Mathematicians, like lawyers, strive to be as unambiguous as possible. Also confusing to beginners, words such as "open" and "field" have been given specialized mathematical meanings, and mathematical jargon includes technical terms such as "homeomorphism" and "integrable". It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. But there is a reason for special notation and technical jargon: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".

In the next paragraph, I disagree with the statement that "Mathematicians want their theorems to follow mechanically ... " I don't think "mechanically" is quite the right word here. A mathematician would say "rigorously", but of course that doesn't work in this context because it uses the word we are trying to explain. I suggest dropping that word, and also dropping the repetition: "axiom" ... "axiomatic". Also, I think "many" works better than "plenty of". These changes would make this paragraph read:

Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions, of which many instances have occurred in the history of the subject (for example, in mathematical analysis).

I have no changes to suggest in the final paragraph of this section. Comments on my suggestions, pro or con, are welcome. Rick Norwood 15:50, 13 December 2005 (UTC)

A few comments ...

1. Your rewrite looses the point that mathematical language and mathematical notation have been developed in order to support precision and rigor in mathematical arguments. I think this is an important point that should not be lost.
2. I wouldn't say mouthfulls - instead I would perhaps say specialised terms or technical terms.
3. You could link to the article on mathematical jargon, which contains more examples of mathematical language.

Gandalf61 09:38, 14 December 2005 (UTC)

Thanks for taking the time to comment. I have modified the suggested rewrite to include the three points you raise. Rick Norwood 15:54, 14 December 2005 (UTC)

I have doubts about the statement "Mathematical language also is hard for beginners." The concept of proof (rigour) and the level of abstraction required often cause problems, but language? I wonder what others think about this.

About "mechanically": I agree it's not the right word, but I'm not happy with a plain "follows". Unfortunately, I can't find a good replacement. "follows without any doubt"? "follows undoubtedly"? "indisputably"? I should probably leave this to the native speakers.

By the way, thanks Rick for going through the article. It is improving. -- Jitse Niesen (talk) 18:18, 14 December 2005 (UTC)

I think the need for abnormally precise language is part of the rigour required for proof, and does seem to cause problems for some people. Mathematicians needs to be even more unambiguous than lawyers.

Doesn't the "by formal reasoning" at the end of the sentence cover what was meant by mechanically? If not, and an extra word is needed, I think it should be here - "mechanical formal reasoning", "unquestionable formal reasoning" or something like that. JPD (talk) 19:28, 14 December 2005 (UTC)

You have a good point about the "by formal reasoning" bit; that indeed covers the "mechanically" bit. However, it uses the word "formal" which I think might be another piece of jargon. How about "systematic reasoning" or "methodical reasoning"? Or do you think "formal" will be clear? -- Jitse Niesen (talk) 12:16, 15 December 2005 (UTC)

I agree, "systematic" is a better word than "formal". I've made the change above.

As for students having trouble with the langauge of mathematics, I have often taught courses of the "math for poets" variety and, yes, the langauge of mathematics gives non-mathematicians conniption fits. For example, they have a great deal of difficulty seeing that "All crows are black." and "All non-black objects are non-crows." say the same thing. To their mind, one statement says something about crows and the other statement doesn't. The class is usually divided about fifty fifty between the students who don't understand but are willing to take my word for it and the students who think I'm wrong. Rick Norwood 15:20, 15 December 2005 (UTC)

I've never taught maths to arts students, but I'd say that they're having trouble with logic, not language. But I can see your point. Okay, then what about

Also confusing to beginners, words such as "open" and "field" have been given specialized mathematical meanings, and mathematical jargon includes technical terms such as "homeomorphism" and "integrable".

Especially the last point I'm not sure about. Is this a big problem? Is mathematics worse in using jargon than other academic disciplines? -- Jitse Niesen (talk) 22:53, 15 December 2005 (UTC)

Hmm. Obviously jargon makes people have trouble with the language, but as you say, this isn't particular to mathematics - I'm not sure whether it's worth mentioning or not. JPD (talk) 09:05, 16 December 2005 (UTC)

A week having gone by, and hearing no objection, I'm going to move the rewrite above, as modified by reader comments, into the article. Rick Norwood 15:11, 20 December 2005 (UTC)

Rick the rewrite of the "Notation, Language, and Rigor" section has made it much better. Thanks to you and the other editors who helped — good work. Paul August ☎ 18:15, 20 December 2005 (UTC)

And in conclusion...

Oleg Alexandrov may be relieved to hear that, after reading through the article from start to finish, I have no further changes to suggest. In fact, it looks pretty good to me. Rick Norwood 15:30, 20 December 2005 (UTC)

Yes, the ever nagging, change averse, never appreciating, always reverting, you-know-who will finally get some (well-deserved) peace. :)

Good (team) work. Oleg Alexandrov (talk) 22:53, 20 December 2005 (UTC)

Vote for new external link

My website has tons of example problems and solutions from all areas of graduate level mathematics. Please add my site to the external links section of the main article if you think it is helpful.

http://www.exampleproblems.com

-Thanks - Tbsmith

It was a nice factoid, though.

Sometimes it's the factoids that make articles interesting instead of dull.Rick Norwood 01:04, 31 December 2005 (UTC)

• "They didn't want 'em good -- they wanted 'em Tuesday!"

---- Ronald Reagan

Thanks, and a Tip o'the Hat to: Jon Awbrey 01:16, 31 December 2005 (UTC)

I was blocked from the Mathematics article by a spam filter!

I tried to add the following category entry:

[[Category:Top 10| Mathematics]]

But I got a spam filter message when I tried to save the page. I went to the spam blacklist, and comcast.net (my ISP) wasn't listed. Does anyone know what is going on?

If someone could add this to the article for me, I'd appreciate it. Please note the blank space right before the word Mathematics. Thank you. Go for it! 13:40, 4 January 2006 (UTC)

I don't see a need to add [[Category:Top 10| Mathematics]] to the mathematics article. Category:Mathematics is already in Category:Top 10, so that should be enough. There is a rule somewhere about not overcategorizing things; an article should not show both in a category and in a subcategory of that cateogory. Oleg Alexandrov (talk) 20:05, 4 January 2006 (UTC)

Request for link to mathematical computing

I looked through Mathematics looking for a link to a page about computer applications and libraries for mathematics (hoping to get from there to a page about the PARI math library).

I couldn't see a relevant link, but maybe no article or category exists to link to quite as I've defined it; I did at least find Category:Computer arithmetic, a link to which would be a good start. Not sure though whether it should go under 'Change', 'Applied mathematics' or 'Mathematics and other fields'. Hv 17:49, 11 January 2006 (UTC)

Update: finally found the page I was looking for under PARI-GP computer algebra system, categorised under Category:Computer algebra systems. Hv 18:30, 11 January 2006 (UTC)

Proposed changes to "Major themes in mathematics" section

I've created a new version of the "Major themes in mathematics" section: here.

Basically the changes are as follows:

1. I've created an image box for "Foundations and methods".
2. In the list of links below the image boxes, I've removed any links that appear in the image boxes (e.g. I removed Natural numbers from the the list of links below the "Quantity" image box.)
3. I've rearranged the "Structure" image box to be more compact and better match the others.
4. I've made the images all the same size.
5. Finally, I've reorganized things slightly. There is probably much to quibble with about which topics belong in which section — there are things I would quibble with myself.

Comments? If nobody has any complaints I will just go ahead and make the changes. Paul August ☎ 21:25, 24 January 2006 (UTC)

Looks good to me, but if I were you I'd wait until two or three others weigh in on the subject.

While you're at, there are a few changes I've wanted to make but didn't know how. First, you can't measure numbers! How about, for the caption under quantity, just "Quantity starts with measurement." And there should be a ..., before the -1 under Integers. Rick Norwood 22:01, 24 January 2006 (UTC)

Well if you place a ruler along the diagonal of a unit square, aren't you measuring the square root of 2? Any way I would have no problem with changing the caption to read something like what you suggest, say: "Quantity starts with counting and measurement". As for there needing to be "..." before -1, I think the idea is that each of the sets of number is supposed to be a list (not necessarily in order nor complete!) of some of the given type of number. So the list of integers starts with -1, thus no ellipsis is needed. If we were to place an elipsis before -1, then what about the lists of rationals and reals? Do they also need and ellipsis in front? Alternatively we could do away with the ellipses altogether. What do you think? Paul August ☎ 05:14, 25 January 2006 (UTC)

Looks good to me Paul. -- Fropuff 17:20, 25 January 2006 (UTC)

I agree with your point about the elipsis. I hadn't looked at it that way. On the other hand, I like "Quantity starts with counting and measurement." I would say that when we measure the diagonal of the unit square, we are measuring the diagonal. The square root of two is the measurement. Rick Norwood 18:57, 25 January 2006 (UTC)

Ok I've made the above changes. Paul August ☎ 20:12, 27 January 2006 (UTC)

could someone please clarify....

What exactly does this sentence mean:

attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias

I can't quite figure it out. Does it mean that the person who is attempting the circumvention is assuming that the journal's editors are biased? Or something else? Could it be made clearer please? Dmharvey 21:45, 24 January 2006 (UTC)

I think it means what you suggest it means. Pseudomathematicians may claim that they don't publish in standard journals because standard journals are biased against them. -lethe ^talk 22:16, 24 January 2006 (UTC)

It would be a big help if you told us where the quoted phrase appears. Rick Norwood 22:02, 24 January 2006 (UTC)

It's there, in the article. -lethe ^talk 22:16, 24 January 2006 (UTC)

OK thanks guys. Hopefully it's clearer now. Dmharvey 23:03, 24 January 2006 (UTC)

I assumed it was somewhere in the article. I was hoping to learn which section it was in. Rick Norwood 18:58, 25 January 2006 (UTC)

Recent edits to this article

What do people think of Rick's recent edit to this article. I can't make up my mind if this info is worth including or not? Oleg Alexandrov (talk) 01:11, 10 February 2006 (UTC)

Keep, but as a statment with a citation, rather than as a quote (I hsoudl earn how to do footnotes...) Tompw 23:26, 10 February 2006 (UTC)

Recent revert/mystery of spammy (graphic) images

I just wanted to explain my last revert a little better. I was using my firefox browser, and when I arrived at the math page there were three big graphic images at the top of the page. I tried to edit the page, but could not see code for the images!! The images were there for the last 5-6 edits, which I found really quite impossible to believe. Now I am using Internet explorer and I also see no sign of the images anywhere! I don't know if this is just a problem for the firefox browser, I will try look again when I am at the same computer which has that firefox browser. I also left a message about this at wikipedia title=vadalism talk page so hopefully it will get resolved and it can be prevented from happening again. Sorry for the confusion about this, I am relatively new to the workings of wikipedia. --Masparasol 8:55 a.m., 10 February 2006 (UTC)

Good

I was surprised but pleased to discover that Mathematics is now a good article. Somehow, even though I visit this article every day, I missed that. In any case, now that the article is proposed for the article improvement drive I've been reading over it to see what remains to be done.

The first thing, which I think would kill any chance of a move to excellence, is that fact that the new picture, while very nice, covers up some of the words. This is something I don't know how to fix.

Second, there has been a gradual drift in the first two paragraphs, which I think now need polish. Here is how they currently reads:

"Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions.

"Mathematics, in nearly every society, is used in fields such as the natural sciences, engineering, measuring land, predicting astronomical events, medicine, accounting, and economics. Some of these fields often give rise to and make use of new mathematical discoveries. Mathematical study often involves discovering and cataloging patterns, without regard for application.

"Mathematics is often defined as..." No. I am fairly sure that mathematics is defined that way only in Wikipedia, and nowhere else. How about "Mathematics can be defined as ..."

"Mathematics, in nearly every society...such as...some of these...often give rise...often involves..."

I understand the use of qualifiers to avoid dogmatic statements, but one "nearly", one "some" and two "often"s is too much weaselwording. How about "Mathematics is used throughout the world in fields such as science, engineering, surveying, astronomy, medicine, and economics. These fields both inspire and make use of new discoveries in applied mathematics. Pure mathematics is also widely studied for its own sake, without any particular application in view."

Comments? Rick Norwood 21:25, 15 March 2006 (UTC)

That would definitely be an improvement, in my opinion. I'd remove astronomy from the list, because it is part of science. Furthermore, I don't quite like how you sneak in the applied vs pure distinction at the end; it smacks of using a term before it is defined (but perhaps it will be clear to the reader anyway). Unfortunately, if one removes "applied" and "pure", the word "mathematics" appears twice next to eachother. -- Jitse Niesen (talk) 23:56, 15 March 2006 (UTC)

Thanks, Jitse. Hearing no objection, I'm going to make those two changes. Rick Norwood 21:40, 16 March 2006 (UTC)

I object! (no, just kidding :) Oleg Alexandrov (talk) 02:44, 17 March 2006 (UTC)

Can someone help with the picture?

The picture still covers up some of the words. Does anyone here know how to fix this? Rick Norwood 22:03, 17 March 2006 (UTC)

What are you refering to?—Daelin @ 2006–03–18 10:08Z

When viewed on my browser, the picture covers up the last two letters of the word "medicine". Rick Norwood 02:19, 19 March 2006 (UTC)

I think you must be using a horribly broken browser like Netscape 4 or IE 4, or you've scaled your font up enormously. What you seem to be describing is not something that can be caused by the page's markup, and I cannot reproduce it in the three browsers I use. Upgrade or downgrade to something that works, or change your default theme in "my preferences".—Daelin @ 2006–03–21 06:21Z

I'm using Internet Explorer. Is that a bad thing? I tried to find the version number but it didn't seem to want to tell me. Does no one else see part of the word "medicine" covered up by the picture? Rick Norwood 13:52, 21 March 2006 (UTC)

I think you'll find that the obstructed word will change if you change the width of your browser window. You'll find the version number under Help->About Internet Explorer in tiny light grey lettering. IE has a number of serious problems, but only IE4 is actually technically broken, and unable to properly render floats according to any interpretation of the standard. The page will render properly in IE2, IE3, IE5, and IE6. —Daelin @ 2006–03–21 19:17Z

Thanks. It says version 6.0.2900 and then some more stuff. Rick Norwood 21:43, 21 March 2006 (UTC)

Definition as study of pattern added

I've added 'the study and investigation of pattern' to the two existing definitions. This is widely used:

http://www.google.co.uk/search?q=%22mathematics+is+the+study+of+pattern%22

158-152-12-77 16:48, 22 March 2006 (UTC)

I've no wish to get involved in revert wars here, but even if it is held that pattern 'is' form and structure - which is debatable - surely the word pattern should go in one of the definitions, since it is referenced widely in serious definitions of mathematics.

Also what is the problem with using bullet points in the first paragraph? They would make it easier to read IMO, for those many who come here to find out what serious definitions of mathematics are currently in use.

As it stands, the first paragraph is lopsided, saying that mathematics "can be defined as [X]", and "another view, held by many mathematicians" is that mathematics is [Y]. Many mathematicians would define the subject as the study of pattern.

Meanwhile...the article on pattern needs a lot of improvement. Sounds very much as if it were written by a computer programmer/Chomskyist linguist. Enough said!

158-152-12-77 00:24, 23 March 2006 (UTC)

The first paragraph as it now stands presents two different views of what mathematics is. Roughly speaking, the first view is the view of the applied mathematician, who concentrates on what mathematics does. The second view is the view of the pure mathematician, who concentrates on what mathematicians do.

The emphasis on patterns is very new, almost postmodern. While mathematicians have always looked for patterns, patterns were until recently secondary to the use of logic. I am tempted to say that, because the modern world has turned its back on logic, patterns are all we have left. Rick Norwood 14:53, 23 March 2006 (UTC)

Mathematics doesn't do anything, any more than Zeus does. It is something that people do. As for the emphasis on pattern, I don't think the application of logic was the basis of the construction of Stonehenge or the pyramids at Giza, nor of Mesopotamian calendrical achievements either. Moreover, the divergence of viewpoint is not really about emphasis, it's about essence, foreign though the notion might subjectively be felt to be, from the viewpoint of the modern computer-programming-influenced thinker. I can't speak for others, but there is nothing postmodern about my approach, since it does not relativise truth, and it does not rest on a notion of what's left after junking logic. The culmination of the line of thought in Euclid's 'Elements' was...the construction of the Platonic solids! He was not just justifying statements by deducing them from axioms and definitions. He was doing that, of course, but to think that that was the essential of his field of study would be to 'think' like a robot. (Or functionary, bureaucrat, etc.) A Baconian version of religion, basically, as dry as dust. Aka 'science'. Never mind the Dawkinsian phony references to inspiration etc. with which this religion can and does cloak itself, in its (did I mention military-funded?) obscurantist march 'forward'. It's the belief in that - the taking of expediency for profundity, which should clearly never be judged in its own terms, which are what it basks in - that's very new on our planet... Of course, the very idea of 'NPOV' is wrapped up with the same mental drunkenness...

Aka is an abbreviation for "also known as", not short for "also". Just thought you would like to know. Rick Norwood 01:28, 25 March 2006 (UTC)

No need to be sarcastic, Rick. I used 'aka' to mean 'also known as'. If you thought I was using it to mean 'also', you were in error. If you feel the need to snap sarcastically when you've misunderstood something, maybe it's time to consider attitude readjustment? I also noticed your out-of-place use of paraleipsis above, when you typed 'I am tempted to say'. Maybe you use paraleipsis too much, perhaps as an alternative for deepening your understanding? Did it occur to you that I might have known what 'aka' was an abbreviation for, when I typed it? Did it occur to you to try to understand what I wrote, taking 'aka' in its usual meaning? I doubt it. Your use of short and snappy sarcasm in your reply suggests otherwise. It gives me an image of someone standing in front of a miror. OK, so what are you going to give us now? More paraleipsis, some sarcasm, or maybe an arrogant bit of litotes? Here's a concept for you: mental armour. 158-152-12-77 11:08, 25 March 2006 (UTC)

I was trying to provide information, not be sarcastic. I did not understand the reading, "A Baconian version of religion, basically, as dry as dust. Also known as 'science'." Now, reading more carefully, I can see that you were saying that science was Bacon's religion. The first time through, I thought you were saying that Bacon's science, like Bacon's religion, was dry as dust. I apologize for the misreading on my part. Rick Norwood 21:43, 25 March 2006 (UTC)

"References" and "Further reading" sections

I have renamed the "References" section to "Further reading". After looking at the works listed there, and reviewing who added each work when, I doubt whether any of these were actually used in the writing of this article. The section was originally (Nov 25 2001) called "Further reading", was renamed "Bibliography" (Dec 13 2003) and only recently renamed "References" (Dec 16 2005). Four of the entries have annotations (e.g. "Boyer … A concise history of mathematics from the Concept of Number to contemporary Mathematics), which also supports my guess that they were added as further reading rather than used as references.

Thus, as the article now stands it has no references section. That is a problem which we should fix. If any of the current works actually were used as references (or can be cited as references for any of the assertions in the article), they should be moved to the new section. Also any contributors to this article should list any other such references actually used. And of course any editor can try to find appropriate references for various parts of the article.

In addition we should also review the "Further reading" section (i've copied it below for easy reference). There are a lot of books on mathematics, is the current list the "right" one? Are there works we want to add? (For example, the original list contained: "Mathematical Society of Japan: Encyclopedic Dictionary of Mathematics, 2nd ed., MIT Press, Cambridge, Mass., 1993. Definitions, theorems and references." which got deleted somewhere along the way.) — or delete? (we don't want everybody just adding all their favorite math books).

Paul August ☎ 20:28, 23 March 2006 (UTC)

JA: After a little more experience with the usages that folks seem comfortable with around WikioPolis, it seems that they're mostly Ok with Further reading — Bibliography seems too intimidating somehow, maybe it suggests exhaustiveness. When there are a lot of Works cited though, they need to be in a separate References section, before the See also section, otherwise it's way too much eyestrain trying to find the work cited in the text. Jon Awbrey 20:58, 23 March 2006 (UTC)

JA: On a related note, I've been thinking that it might be good to form Cumulative Bibliography pages for various areas, like the ones we have for glossaries and other lists. I'm mostly getting tired of typing in the same references multiple times. What do other folks think? Jon Awbrey 20:58, 23 March 2006 (UTC)

This is probably a good idea. What I've done myself is create my own bibliography subpage. We could create a common one for mathematics as a subpage of the project page. We can discuss this further at the project talk page. Paul August ☎ 21:26, 23 March 2006 (UTC)

In general I think it's unnecessary to distinguish cited from uncited works in a refs section. I don't ever recall seeing this done in a journal article. Occasionally WP articles have references that are not only uncited, but not even directly on-topic to the article; these might reasonably go in a separate section, assuming they're to be kept at all. --Trovatore 21:03, 23 March 2006 (UTC)

Common WP practice regarding this is not yet very mature. But many would say that it is intellectually dishonest to list a work as a reference if it has not actually been used as such. Paul August ☎ 21:26, 23 March 2006 (UTC)

This has me a bit concerned. My understanding of references is "this is where you can find such and such", not necessarily "this is where I found it". Have I really misunderstood this? Where would I put, say, Hallett's book referenced at limitation of size, a book I don't actually own, but is clearly the correct place to look? It seems really pedantic to me to give the section a different name for this reason. --Trovatore 22:56, 23 March 2006 (UTC)

How do you "know" this is where you can find such and such unless you've looked there yourself? For me a reference is "this is where I know it can be found because I've seen it there". Paul August ☎ 00:20, 24 March 2006 (UTC)

Well, I have browsed the book, but that was a long time ago. In any case it's mentioned in the abstract of another paper disputing what the author says is Hallett's thesis that Cantorian set theory was "a theory of limitation of size from the beginning". --Trovatore 00:40, 24 March 2006 (UTC)

JA: Journal articles don't allow non-cited works, unless they are "survey articles" in which case they may have humongous annotated or analytic bibs. But encyclopedia articles are often more like survey articles than research articles. Of course, it only really matters much in practice if there are, say, more than 4 or 5 of each. Jon Awbrey 21:18, 23 March 2006 (UTC)

Um, I think that's just not true about what journals allow. Nobody ever complained about that in my papers. I typically use the BibTeX \nocite command once or twice, to mention a work that the reader may need to understand the background of the question, but which I don't find necessary to cite in any particular line. --Trovatore 21:29, 23 March 2006 (UTC)

JA: Good, then all us WikiPedants understand the distinction, along with the rule, the contingency, and the rarity of its exception. Jon Awbrey 23:02, 23 March 2006 (UTC)

References (current)

• Kline, Morris, Mathematical Thought from Ancient to Modern Times, Oxford University Press, USA; Paperback edition (March 1, 1990). ISBN 0195061357.

What works should be listed as References?

In my contributions to this article, I used information from the book by Kline, so it is a genuine "Reference". Other works cited in the text inclued Newton's Principia, Wigner's Unreasonable effectiveness..., and Hardy's Apology. Whether they were actually used or not I do not know.

Ok, I've started a "References" section, and moved Klein there. Paul August ☎ 00:20, 24 March 2006 (UTC)

Further reading (current)

• Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies, Oxford University Press, USA; New Ed edition (December 14, 2000). ISBN 0195139194.
• Boyer, Carl B., A History of Mathematics, Wiley; 2 edition (March 6, 1991). ISBN 0471543977. — A concise history of mathematics from the Concept of Number to contemporary Mathematics.
• Courant, R. and H. Robbins, What Is Mathematics? : An Elementary Approach to Ideas and Methods, Oxford University Press, USA; 2 edition (July 18, 1996). ISBN 0195105192.
• Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Mariner Books; Reprint edition (January 14, 1999). ISBN 0395929687.— A gentle introduction to the world of mathematics.
• Gullberg, Jan, Mathematics--From the Birth of Numbers.W. W. Norton & Company; 1st edition (October 1997). ISBN 039304002X. — An encyclopedic overview of mathematics presented in clear, simple language.
• Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. — A translated and expanded version of a Soviet mathematics encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
• Pappas, Theoni, The Joy Of Mathematics, Wide World Publishing; Revised edition (June 1989). ISBN 0933174659.

What works should be listed as "Further reading"?

As for "Further reading", that section should point only to major works about mathematics in general. For example, much as I love Kelly and Herstein, they should be listed in Topology and Modern Algebra, not here. Essential works of general interest that come to mind are Men of Mathematics by Bell, The Encyclopedic Dictionary of Mathematics, The World of Mathematics by Newman, The Mathematical Tourist by Peterson, Journey through Genius by Dunham, Godel, Escher, Bach by Hofstadter, Elements by Euclid, and Writing Mathematics Well by Gillman. Anything on this list that does not belong? What should be added? Rick Norwood 22:50, 23 March 2006 (UTC)

Yes I agree about Kelly and Herstein, only general works should be listed here. Rick, you've given your opinion on some works to add (all of which seem ok to me), do you have an opinion about the works already listed? Paul August ☎ 00:20, 24 March 2006 (UTC)

What about the annotations? Do we want to keep them? Should each work have them? Paul August ☎ 00:20, 24 March 2006 (UTC)

I have not read any of the books currently listed as "Further reading" so I really don't know if they are outstanding or not. Another book I used in this article, though I used it more heavily in the History of Mathematics article, is An Introduction to the History of Mathematics by Howard Eves. I don't know how it compares to the History of Mathematics listed above, but I like it. Rick Norwood 13:37, 24 March 2006 (UTC)

Include contributions men have made to mathematics?

We should append a new section to the article that discusses the contributions great mathematical thinkers like Euler, Gauss, Archimedes, and Newton have made and how those contributions further advanced mathematics thought. - Christopher 20:55, 29 March 2006 (UTC)

I do not think that is import on the mathematics page, because we have a history of mathematical ideas but maybe on the mathematician page it might work. Besides how did those guys advance mathematical thought? No one was able to pass Archimedes. Timothy Clemans 20:59, 29 March 2006 (UTC)

Math as science again

I reverted an anon addition claiming that mathematics fails the Popper criterion of falsifiability; it has since been re-added ad the first edit of a new editor User:Meznaric, presumably the same person.

The current wording is not acceptable; mathematical axiom systems can be falsfied, for example by discovering that they are inconsistent. (That may not be the only way, but it is the clearest way.) I invite the new editor to propose wording that takes this into account. --Trovatore 17:14, 10 April 2006 (UTC)

Popper remark 2

Dear community

I think it is very mean to simply delete views you do not believe in. I added the note that Karl Popper defined science as a experimentally refutable theory and therefore mathematics cannot be science. I am sure there are many people who do not believe Karl Popper was right. There are many people who do not agree with large portions of Wikipedia. Does that mean they have the right to simply delete anything they do not believe in? You may continue doing that but perhaps then we should ask whether there is any point to continue contributing to this encyclopedia. I would appreciate if such short comments (like Popper's views) were left on the page as they only add to the content.

Best regards —Preceding unsigned comment added by Meznaric (talk • contribs) 17:17, April 10, 2006

Mathematics (or at least parts of it) is experimentally refutable, and therefore in fact meets Popper's criterion. However it is quite possible that Popper (who was a great philosopher but no mathematician) didn't realize this. In that case it's more complicated than asking whether "Popper was right", because there are two questions about which he could be right or wrong, the first being whether his criterion is right, and the second being whether mathematics meets it. If you can provide a source that Popper held that mathematics is not a science, then yes, that information can be provided, but the article should not uncritically state that mathematics fails Popper's criterion (a distinct question from whether Popper thought it failed his criterion). --Trovatore 17:28, 10 April 2006 (UTC)

The method employed by mathematics is to reduce propositions to tautologies, thereby proving their truth. Therefore once a proposition has been proven there cannot exist a counter-example to that proposition. Therefore the proposition (now a theorem) cannot be falsified by any means, let alone experimentally. Perhaps it should not be worded so as to suggest Popper held the view (I am not completely sure whether he did or not) but the statement itself is logical and that is important. —The preceding unsigned comment was added by Meznaric (talk • contribs) 17:37, 10 April 2006 (UTC)

OK, I'm not sure how much you understand of mathematical logic; maybe you could provide a little background about yourself. Your claim that mathematics proceeds by reducing propositions to tautologies is simply not true. This sounds like an old and mostly discredited view called logicism.

To get non-trivial mathematical results, you have to assume non-trivial axioms. It could be that your axioms are inconsistent and you simply don't know. If they are inconsistent, then obviously they are not true, or at least not all of them at once. Thus the theory—treated as a whole—would be falsfied. --Trovatore 17:42, 10 April 2006 (UTC)

I do not want this to become a debate in style of "Oh I am that and that therefore I am more credited to say something". Let us just stay with the arguments rather than personal backgrounds. What Popper meant was not to test the theory logically but rather experimentally. This means you need to set up a real device in the real world that will test your theory. If the results of the experiment disagree with the prediction of the theory, then the theory is refuted. Now obviously that cannot be done with mathematics. I checked his views in "The Logic of Scientific Discovery". -- Meznaric

A pencil and paper, or a computer, is a "real device in the real world". The claim that a set of axioms is consistent implies that certain things will never be observed, namely a proof of a statement and its negation from those axioms.

Again, I agree that it is reasonable to report that Popper believed that mathematics failed his criterion, if it's true that he said that, which I think he probably did. What I oppose is presenting Popper's criterion and then claiming uncritically that mathematics fails it, and therefore Popper thought it was not a science. The truth is that mathematics passes Popper's criterion, whether Popper thought so or not. --Trovatore 18:45, 10 April 2006 (UTC)

The way I presented it when my contribution was deleted was that Popper believed mathematics wasn't science and not vice that mathematics wasn't science and therefore Popper believed it. Besides as a pencil and paper is concerned - when using pencil and paper you are not an inactive observer but an active participant in the process of obtaining the result. Therefore this cannot be an experiment (an observation). In any case I would be very grateful if my contribution was put back on the article. If you want to reword it so as to make it clear that Popper believed it you have my blessing. Best regards. --Meznaric

The way you put it was that, if the definition of science is that which is experimentally falsifiable, then mathematics is not science. That's wrong; mathematics is experimentally falsifiable. --Trovatore 13:53, 11 April 2006 (UTC)

As I said, the experimenter cannot be an active participant of an experiment. Experiment must be independent of an experimenter. Since mathematics is done by people, it cannot be an experiment. Therefore if mathematics is not an experiment by itself, it does not contain experiments. And if the whole of mathematics is an experiment, what is the theory? So you see it just does not make sense to say that mathematics is experimentally falsifiable. I have put the thing back worded in a way that makes clear Popper believed what was said. --Meznaric

"The experimenter can't be an active part of an experiment"? Why not, exactly? Anyway, the questions about consistency can be formulated in terms of what you will observe in the output from a computer, which the experimenter doesn't have to be part of. The current wording is not acceptable because it claims that Popper's formulation supports the claim that math is not a science, which it doesn't. It appeared thus to Popper only because he didn't really understand mathematics that well. --Trovatore 15:52, 11 April 2006 (UTC)

Even if one believes Popper to be wrong, his view still does support the claim that maths isn't science simply because of Popper's significance and authority as a philosopher of science. But I believe him to be right. The experimenter cannot be an active part of an experiment because an experiment is something that must be independent of human activity in all non-negligible ways other than the set-up of the experiment. If mathematics was a theory predicting an output from a computer then I would say: "Ok, then mathematics is science". However, mathematics rather than predicting the output IS the output. So a computer simulation cannot be considered an experiment. Meznaric 16:14, 11 April 2006 (UTC)

This is easily settled by a quote (or the absence of one) from Popper. The debate in the absence of such a quote is pointless. Rick Norwood 20:16, 10 April 2006 (UTC)

Well, no, it's not really settled by a quote from Popper. I think Popper did believe that mathematics failed his criterion, but I think he was wrong about that. While (with such a quote) it is reasonable to say that Popper thought mathematics was unfalsifiable and therefore not science, it is not reasonable to say mathematics is unfalsifiable, and therefore Popper thought it was not science. --Trovatore 20:30, 10 April 2006 (UTC)

Yes, of course. The quote only settles the question of what Popper thought. In any case, this discussion belongs in Philosophy of mathematics rather than in this article. Rick Norwood 20:47, 10 April 2006 (UTC)

I agree. I think that if the comment "Karl Popper believed..." is to remain, a citation must be provided. But even given such a citation, is this comment necessary here? Yes, Karl Popper was important and influential, but was he so much so that he should be quoted by name in this section? I don't really associate his name with work on math. per se, and I don't see the need for bringing him in. In the Phil. of Math. entry, sure. Incidentally, speaking for myself, I think math. is not faslifiable in the sense of Popper, and is not therefore a science. Not being a science isn't necessarily a slam against something. Perhaps the Demarcation Problem entry needs a bit on math. too. JJL 16:23, 11 April 2006 (UTC)

Yes, I think the Popper remark should be removed. From what I've read Popper was interested in the sciences (i.e. those things which describe the world we live in) rather than the abstract/formal systems. Mathematics was not really what concerned him. To include the remark leeds to confusion, and perhaphs extends Popers falsifiability beyond the scope where he intended it. Without a quote we are on the verge of commiting that sin of sins WP:OR. --Salix alba (talk) 14:10, 15 April 2006 (UTC)

Sorry to reopen this debate, but another page Talk:Anthroposophy cited this in an argument so we do need to establish the case and not propagate a speculation. For now I've added a {{fact}} tag. We could reword this in an objective way by saying something like Popper used a test of experimental falsifiability to distinguish whetehr a field followed the scientific method. However, it is debatable as to whether Poppers work should be applied to mathematics. --Salix alba (talk) 15:27, 2 May 2006 (UTC)

Well, if we're going to open the question up that far, then we also need to acknowledge that, even if Popper's work is applicable, it's debatable which direction it cuts. My position is that mathematics is an experimentally falsifiable natural science; for example, the proposition that inaccessible cardinals (Platonistically) exist would be refuted by a proof of a contradiction from their existence. According to User:Kenosis, this is the position of Imre Lakatos (though the latter doesn't sound like someone with whom I'd agree with on much else). --Trovatore 15:58, 2 May 2006 (UTC)

Not sure where to put this

This seemed like the best place to get people's attention about the article PDE Surfaces, written by Zer0 cache. I suspect that it's promoting research, but I can't be sure. It would be appreciated if other editors can check this out. I've also left a small query at PDE surfaces talk page. MP (talk) 11:28, 11 April 2006 (UTC)

I've copied this to Wikipedia talk:WikiProject Mathematics which is really the propper place. --Salix alba (talk) 14:19, 15 April 2006 (UTC)

Good article nominations

There are a few maths articles looking for reviewers in Wikipedia:Good articles/Nominations. Some of them are quite specialised and I suspect that it would be a while before anyone got around to reviewing them so I thought I would point it out here in case anyone wanted to look. --Richard Clegg 16:18, 24 April 2006 (UTC)

Call for others to revert

It seems me and another user are at loggerheads about the validity of the 1+1=1 statement. I posted a proof that I find valid. I'm asking people who agree to step up and revert it back. I won't revert for the third time since it has been reverted three times already.

I will not revert again today either. Please see User talk:128.84.178.82 for our conversation about this issue. -lethe ^{talk +} 07:21, 26 April 2006 (UTC)

Also, this is the link to the proof that user:128.84.178.82 wants to add to the article. -lethe ^{talk +} 07:28, 26 April 2006 (UTC)

Please visit Division by zero#Fallacies based on division by zero for some relevant comments. -lethe ^{talk +} 07:32, 26 April 2006 (UTC)

The link cited as reference doesnt fall under Verifiability policy. It is an anonymous proof on a free-hosting website. Sorry. (april 1 was three weeks ago?) --Quiddity 08:56, 26 April 2006 (UTC)

Come on guys. The author is cited as Brilliant Mathematician. What's this anonymous talk? LOL -—The preceding unsigned comment was added by Veritas4U (talk • contribs) .

The 1+1=1 comment is just silly. It should be reverted when it appears. It's true that (apparent?) contradictions can drive new mathematical advances, but this is no example of that. JJL 15:04, 26 April 2006 (UTC)

By the way, division by zero issues aside, I'd like to address your other point. You say "the simple fact is that contradictions are not fatal to mathematics". I don't believe that's true. In classical first order logic, if the theory contains P as well as ¬P for any proposition P, then it contains all propositions in the language, basically because the implication "false implies anything" is true. Thus a theory with a contradiction is completely useless. Contradictions are fatal. This is why resolving Russell's paradox with axiomatic set theory was so important to the foundations of mathematics. Gödel's work implies that we can't know for sure that our mathematics doesn't contain contradictions, but it is widely believed that ZFC does not contain any contradictions, and there are some "meta" proofs of this fact which assume the existence of large cardinals. So take this point: if you think you have a proof that 0=1, all you've really shown is that your proof is wrong. -lethe ^{talk +} 11:13, 26 April 2006 (UTC)

Going off on a tangent here: There are viewpoints from which a sufficiently obscure contradiction might not be "fatal", even using first-order logic. What if one of our foundationally relevant theories (ZFC, say, or even PA) proves a contradiction, but the shortest proof has more steps than can be written down in the physical universe, and has no underlying idea? Then, while the theory would in some Platonistic sense prove every sentence of its language, maybe no sentence and its negation would both have proofs accessible to us, and the theory would still appear to us to describe a coherent reality.

Hmmm, interesting. Sounds like something a constructivist or finitist might believe. I can see how someone with this viewpoint would see mathematics as an experimental science, or at least more similar to one than is usually imagined. Personally I find it distasteful, but I guess I'm in the Platonistic school. -lethe ^{talk +} 20:32, 26 April 2006 (UTC)

That strikes me as a bit of a non sequitur. If mathematical objects are things with real existence, then questions about them are questions of fact, "synthetic" rather than "analytic"; that should make empirical methods more, rather than less, applicable. Arguably it takes at least a mild Platonist even to make sense of the hypothetical I posed above; a strict finitist like Edward Nelson might well regard it as a null assertion: Where is the proof of a contradiction supposed to exist, if not in the physical world? --Trovatore 21:04, 26 April 2006 (UTC)

What bit is the non sequitur? My connection between finitism and the experimentalness of mathematics? -lethe ^{talk +} 21:20, 26 April 2006 (UTC)

Yeah, or rather between mathematical realism and non-empiricism. --Trovatore

I guess I'm just thinking out loud in response to your interesting observation, and I'm outside of my area of expertise, so forgive me if I'm conflating unrelated topics. But here's how I'm seeing it at the moment: some futuristic descendent Gödel, using a quantum computer, comes up with a metamathematical proof that there is a contradiction in ZFC whose proof has a length that excedes the number of particles in our Hubble volume. The empiricist thinks "no problem. That's not a mathematical proof, because it doesn't have a physical existence", while the Platonist thinks "the proof of that length may not have a physical existence, but as long as Gödel's proof is correct, I know it exists, therefore ZFC contains a contradiction". Seems to me that the Platonist has to believe in more things than just those which are empirical or constructible or whatever. Just like the geometric circle, which no compass is perfect enough to make a physical reconstruction of, so this very long proof. -lethe ^{talk +} 22:08, 26 April 2006 (UTC)

Obviously that doesn't apply to trivial proofs with trivial errors, such as the one posted by the individual in question. --Trovatore 15:13, 26 April 2006 (UTC)

More importantly, if we forget about the glaring error in the proof for a moment, this argument doesn't apply to the proof anyway, because the proof is quite short, essentially one line. If someone found a paradox with an accessible proof, it would be bad. But actually, I guess "fatal" is still too strong a word. I mean, Russell did it once already, right? And it's not like all of mathematics died that day. Even Frege himself was allowed to continue living, and probably lots of people continued doing math without noticing the lack of a foundation. A couple decades went by, and Zermelo et al. fixed things by choosing some more careful axioms. If today, a contradiction were found in ZFC, it would require at least some kind of response, either choosing some better axioms or at least making some yucky philosophical justifications like the above why the contradiction is OK. -lethe ^{talk +} 20:32, 26 April 2006 (UTC)

More on Trovatore's tangent. Consider the following logical system: propositional calculus with a hyperfinite (as in Nonstandard analysis) number of atomic propositions. We can consider this as the free (internal) algebra with a hyperfinite set P of generators. To say the logical system is inconsistent means that using hyperfinite (and internal) derivations we can derive an inconsistency. Algebraically this means that the free internal boolean algebra generated by P is trivial. It is possible for P to be internally inconsistent but externally consistent.--CSTAR 00:57, 27 April 2006 (UTC)

Here, trivial means that the Lindenbaum construction yields the Boolean algebra with one element? Yes, that must be it. And what does "externally consistent" mean? There is a model of this algebra in ZFC? And why hyperfinite? Is that essential to this construction? Would it not suffice to just take a proposition calculus with an uncountable number of atoms? -lethe ^{talk +} 01:24, 27 April 2006 (UTC)

Re: Would it not suffice to just take a proposition calculus with an uncountable number of atoms? . I don't think that approach will work. The point of the construction is to regard a formal theory as a mathematical object. We can consider this in two different models of ZFC (although using the Zakon superstructure construction we don't have to be so high-brow). The point is that a finite derivation means quite different things in each one of these models.--CSTAR 01:53, 27 April 2006 (UTC)