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Naturals, Nonnegatives, Positives

JA: A lot of fuss can be avoided if it is simply acknowledged that the term "natural number" is ambiguous in practice, replacing it with the use of "nonnegative" and "positive" to say exactly what you mean. Jon Awbrey 17:26, 28 April 2006 (UTC)

RN: Ah, but the phrase "natural numbers" has a long and honorable history, and refuses to die. Rick Norwood 19:12, 28 April 2006 (UTC)

JA: Yes, indeed, in fact it has two long and honorable histories, one beginning on Day 0 and one beginning on Day 1. Jon Awbrey 04:20, 29 April 2006 (UTC)

And it's not as though the ambiguity of the term "natural number" is all that problematic. -lethe 19:14, 28 April 2006 (UTC)

To put it in nonnegative terms, natural numbers are not that ambiguous, I am positive about that. Oleg Alexandrov (talk) 04:09, 29 April 2006 (UTC)

I always learned that natural numbers started at 1, but many people disagree. I would change it either to positive or nonnegative. --Mets501 01:18, 1 May 2006 (UTC)

I'll chime in here. I believe the term "natural numbers" MUST stay in the article for three reasons. The first and most important is that the term is a well known part of the history of mathematics. Secondly, "natural numbers" when used in this article links to the article on Natural numbers, and the very first paragraph of that article discusses the two conventions regarding the inclusion of zero (number theory vs. set theory & computer science). Third, the use of the capital blackboard bold font "N" to denote the set of natural numbers is a standard convention, and the "N" stands for "natural" (although I suppose someone could try to rewrite history to claim that it stands for "non-negative"). capitalist 02:24, 1 May 2006 (UTC)

Everyone should know 0∊ℕ ;-) That is why we have ℤ⁺. We could always have a poll. Stephen B Streater 08:08, 24 May 2006 (UTC)

Reasons for removing new "triangle" section

An anon contributor added a new section headed Example of Physically "False" Mathematics, with various examples related to triangles and non-Euclidean geometry. I removed it because it intertwines the non-Euclidean 4D geometry of relativistic spacetime with the 2D spherical geometry of the Earth's surface. The implication that there is some sort of link between these two concepts because "our universe is not described by Euclidean space" is both confusing and misleading. Gandalf61 15:35, 22 May 2006 (UTC)

Please discuss reverting changes

Instead of simply reverting them and saying they make the page look "awful". What I think looks awful is the big white space next to the table of contents and the math portal link in the wrong section. VegaDark 00:38, 24 May 2006 (UTC)

Yes you are right, it is good to discuss things. But your changes filled in the white space to the right of the table of contents with a very narrow column of text, making that article section hard to read and not so pleasing. You also moved the nice picture from the top somewhere down under the narrow column of text I mentioned above, and putting the TOC to the right is not standard in the article namespace. Since it was only a minor edit, I took the liberty of reverting it. My edit summary was surely poorly thought out, sorry. Oleg Alexandrov (talk) 01:13, 24 May 2006 (UTC)

I've seen articles that have the TOC off to the side. I think it is a good way to get rid of the white space if there is an especially long TOC. It can go off to the right side if you want, that will look fine as well. I just don't think it looks good with all the white space as is. Also, as far as I know links to the portal an article is in should go in the see also section. At least that is how I have seen it linked in other articles. VegaDark 02:09, 24 May 2006 (UTC)

But again, it mangles the text. On my screen the TOC takes at least half of the width of the screen, and then the text has to fit on the other half. I think that space is too small to be filled usefully. Maybe if we have a picture or two we cut stick them into that space. But I agree with putting the portal down, I just did. Oleg Alexandrov (talk) 04:03, 24 May 2006 (UTC)

Hmmm...on my screen the TOC only takes up a third of the space, so it doesn't look bad at all with text next to it. I wouldn't object to a pic or 2 there instead. VegaDark 05:38, 24 May 2006 (UTC)

Handling uncertainty

I propose adding a new Major Theme: Handling uncertainty. This would cover Probability Theory, Measure Theory, Statistics, Brownian Motion, Quantum Theory and Information Theory. Stephen B Streater 08:15, 24 May 2006 (UTC)

I tend to agree, there is a tendancy to focus on pure mathematics reducing applied mathematics and statistics to a foot note. I've also though about moving the whole Major Themes section up in the article, maybe just after the introduction. --Salix alba (talk) 09:01, 24 May 2006 (UTC)

I agree with this idea of moving the themes up. It lets people know how varied and powerful Mathematics is. Most laymen identify Mathematics with Arithmetic, and it is important to remove this misconception as soon as possible. Stephen B Streater 09:20, 24 May 2006 (UTC)

I support the new theme. Uncertainty has occupied the minds of many great mathematicians and would be a worthy theme. -- Avenue 09:40, 24 May 2006 (UTC)

What we need is a reference naming the themes of mathematics. --MarSch 10:22, 24 May 2006 (UTC)

Take your pick - Dewey Decimal, AMS Mathematics Subject Classification, Library of Congress Subject Classification, another Mathematics Classification System, yet another MCS; no lack of choices, but they are all different. Gandalf61 12:23, 24 May 2006 (UTC)

I feel that the AMS classification system is the professional standard. I would name the theme simply Chance, myself. Measure Theory and Chaos wouldn't quite fit there, to my mind. Prediction is another possibility. JJL 14:44, 24 May 2006 (UTC)

Of course, Chaos is deterministic, unlike (for example) Brownian motion. I agree that this section should not include deterministic branches of Mathematics. Stephen B Streater 17:35, 24 May 2006 (UTC)

Chance is a catchy title. Perhaps Stochastic Processes would be a more general name. Probability Theory is a special case of Measure Theory, which is why I included it in my original concept. Stephen B Streater 17:35, 24 May 2006 (UTC)

The AMS list is pretty comprehensive! I think we need a shorter list of themes. Being Mathematics, perhaps some things should be in several themes. For example, Quantum Theory could come under Chance and Theories of the Universe, though I can't see a lot of obvious candidates yet. Stephen B Streater 17:35, 24 May 2006 (UTC)

Understanding and describing change

I suggest to put in bold characters the words "Understanding and describing change ", undoubtely it's a key concept Brian Wilson 13:28, 26 May 2006 (UTC)

Yes, it is a key concept, but why does it need to be in bold? JPD (talk) 13:40, 26 May 2006 (UTC)

Becouse readers can more easily find and remember it.Brian Wilson 13:54, 26 May 2006 (UTC)

It doesn't fit with the format of the rest of the article. No other key concepts are bolded. JPD (talk) 14:02, 26 May 2006 (UTC)

The question would be whether bolding under these conditions is "judicious," as used at WP:MOS#Legibility. I think bolding all "key concepts" in mathematics might not be a good idea for legibility. John (Jwy) 14:04, 26 May 2006 (UTC)

Ok, maybe you are right, but possibly ALL key concepts should be put in italics. Brian Wilson 14:09, 26 May 2006 (UTC)

My view is that lots of things in Maths are important. I wouldn't bold the change idea in the text (at least at the moment), as bolding everything makes text illegible. Change has its own theme, with some nice pictures, so people will realise it is important from this. Stephen B Streater 15:22, 26 May 2006 (UTC)

Opening paragraph

The opening paragraph introduces two views as if competing, whereas I think one elaborates on an aspect of the other. I think it is also important to bring in somehow that maths is a large edifice, in which methods and techniques are an important part. Here is the present text:

Mathematics can be defined as the logically rigorous 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.

and here is a suggested new text:

Mathematics is the study of topics such as quantity, structure, space, and change, using logically rigorous methods. This means that mathematical results can be justified by deductive reasoning, starting from axioms and definitions. Using this, a large body of knowledge has been created over the course of many centuries that is constantly expanded, covering many fields and comprising definitions, theorems, methods, and techniques, all of which build upon each other.

Reactions? --Lambiam 20:07, 27 May 2006 (UTC)

I would not be happy about the implication that what's not rigorous is not mathematics; I don't think that's true at all. Were Newton's and Leibniz's approach to calculus, not mathematics? Certainly the demand for rigor has increased since their time, but in many subfields (including some very pure ones) I suspect hardly anything that's published is truly rigorous, because it's just too difficult to make progress that way. --Trovatore 20:15, 27 May 2006 (UTC)

Agree. Ramanujan did mathematics. I believe he was often unable to provide justifications for his calculations.--CSTAR 20:24, 27 May 2006 (UTC)

The implication is from the original version; I did not bring it in. Actually I weakened it, precisely for the reasons mentioned by the two of you. The original has "knowledge justified by deductive reasoning". I inserted "can be" in front of "justified". We have to be careful though not to weasel the text down to something absolutely true and absolute useless (another definition of mathematics). Undoubtedly the version above is not perfect, but what do you think, is it an improvement over what we have now? I've inserted the present version for easy comparison. --Lambiam 21:11, 27 May 2006 (UTC)

Well, OK, that's not the way I was reading the original version. My point is that there's a clear philosophical dispute here, between the Euclidean axiomatic approach referred to in the second sentence, and the more "scientistic" one from the first sentence (note that "scientism" is usually pejorative, but I don't mean it to be). The original version, while not clearly explaining what the dispute is, at least acknowledges that there is one, whereas your version seems to pull everything into the Euclidean paradigm.

That said, your version does read much nicer, but I think it achieves that smooth flow at an unacceptable cost in terms of obscuring a genuine controversy. --Trovatore 22:22, 27 May 2006 (UTC)

If that genuine controversy is in the text, I missed it, and I'm afraid so will the reader. The link "logically rigorous" redirects to Rigor, which gives several meanings. I assume that not Medical rigor is meant here, but "Mathematical rigour". I quote: "Mathematical rigour is often cited as a kind of gold standard for mathematical proof. It has a history traced back to Greek mathematics, where it is said to have been invented. Complete rigour, it is often said, became available in mathematics at the start of the twentieth century. This relies on the axiomatic method, and the subsequent development of pure mathematics under the axiomatic umbrella." This seems to me very much like the anti-scientistic Euclidean police at its worst. If it is important to present the clear philosophical dispute, there must be a clearer way. --Lambiam 23:34, 27 May 2006 (UTC)

How about just deleting "logically rigorous" from the present version? -- Avenue 00:51, 28 May 2006 (UTC)

Try to imagine you are a reader who does not know what mathematics is, and who wants to find out. An older defintion here had: "Mathematics is the study of quantity, structure, space and change. It developed, through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the study of the shapes and motions of physical objects." Now that reads nicely but is awfully vague. You still won't know. You might think it is architecture and choreography. Leaving out "logically rigorous" from the present version brings this back. Also, the impression is now created that the mathematicians who "side" with the use of deductive reasoning deny that maths is involved with the study of QSS&C. I'm not too happy with "logically rigorous" either, but more because of a connotation I have with rigidity, which may be idiosyncratic. We need some wording to set mathematical thought apart from the kind of thought involved in science issues like whether birds are dinosaurs, or what happened at the Big Bang. It has something to do with absolute, unconditional and unqualified certainty. --Lambiam 01:23, 28 May 2006 (UTC)

To my mind, mathematical thought has much more to do with abstraction than with certainty. Requiring certainty is far too limiting. Mathematics includes conjectures as well as theorems. It includes proofs with mistakes or gaps in logic. If mathematics required robotic infallibility, it would progress at a snail's pace. So I think overemphasizing "absolute certainty" in mathematical thought would be a mistake. -- Avenue 03:27, 28 May 2006 (UTC)

OK, I missed that "logically rigorous" had been added to the paragraph at some point (the paragraph has been argued over so much that I thought I knew what was in it and didn't really bother to read it again). Yes, that should be removed.

I agree with Avenue on the issue of certainty, and in fact would go further: Math is not apodeictically certain, not only because the proofs might be wrong, but because the axioms might be wrong. In fact axioms are not a cleanly distinguished category of mathematical statement; mathematical systems considered as wholes may be evaluated for coherence, conformance with intuition, falsifiability (without actual falsification), and power to predict conclusions later independently verified. Then among the statements you believe are true, it's rather arbitrary which ones you choose to set aside as axioms (compare Replacement versus Collection, say). --Trovatore 06:34, 28 May 2006 (UTC)

On a related subject to the certainty idea (which I agree doesn't reflect the creative process which generates the theorems and proofs), I'm planning to add a whole section on chance. Stephen B Streater 09:24, 28 May 2006 (UTC)

Is there some way to make progress towards convergence? I see several things wrong with the present text that I have tried to address. An objection has been voiced to "logical rigor", which I did not introduce. However, I object to just removing it. If you do that, the definition becomes too vague. Also, the real or supposed controversy becomes really weird. It should not be removed but be replaced by something better. What? I don't know. I did not propose to state in the article: "mathematical thought gives absolute certainty". Nevertheless, "compelling" in "a compelling mathematical argument" has a qualitatively different meaning than in "a compelling legal argument". The difference is that outside of mathematics it is always possible that new evidence will overturn the verdict. It is the difference between induction and deduction, or between synthetic and analytic judgment. Of course I'm aware that mathematics is a human endeavour and as such fraught with lapses of judgment and all other effects of mental frailty. I also know that the foundational crisis of mathematics has never been satisfactorily resolved. But in spite of all that, there is something unique to mathematics, and I think we should try to convey that. --Lambiam 10:39, 28 May 2006 (UTC)

So I specifically deny that mathematics is analytic; I say it's synthetic. The claim that mathematics is analytic is the central thesis of logicism, which (at least in its original form) has been pretty thoroughly discredited.

Do you mean analytic as in analytic philosophy? I've never understood what that was supposed to mean. I meant it in the sense of Kant's analytic reasoning. To believe otherwise means you are a diehard Platonist who is not only amazed at the effectiveness of mathematics as a modelling tool in the natural sciences, but equally amazed that the theorems laid out for us to discover keep behaving as if they follow logically from certain premises. --Lambiam 18:13, 28 May 2006 (UTC)

I meant it the same way you did. My personal views (which, yes, do lean fairly strongly Platonist) aside, realism is an important strain of thought in philosophy of mathematics, and the article (a maggior ragione the first paragraph) should not be written in a way that excludes it.

Your last sentence above seems to be arguing whether or not Platonism is right, which is a bit OT here, but I'll address the point briefly: We don't have a fixed set of premises from which our theorems follow logically; we have to keep finding new ones. In the last fifty years this has mainly meant large cardinal axioms, but there are now new candidates that go beyond those, such as Woodin's Ω-logic. This discovery process, in my view, is definitely mathematics, and is definitely not analytic. --Trovatore 18:39, 28 May 2006 (UTC)

OT as in Operating Thetan? If we ever discover Platonist extraterrestrial intelligence out there, we'll find that they made different but equally consistent "discoveries", and no conceivable argument can show one set of axioms is more right than any other. As an intuitionist (not really covered in the article) I personally think all large cardinals are devoid of meaning anyway. Well, I give up here. --Lambiam 19:22, 28 May 2006 (UTC)

We still have to figure out what to do about it; the current version with "logically rigorous" is unacceptable. Just removing "logically rigorous" would be an improvement but not that much of one, as it makes the paragraph choppy and disjointed, and doesn't clearly convey that there are two distinct views being described (definiing mathematics by subject matter versus defining it by methodology). I'll think it over. --Trovatore 19:35, 28 May 2006 (UTC)

We should not try to give mathematics a less vague definition than it in fact has. I really think the opening paragraph should not attempt to "define" mathematics at all, but just describe it a bit. Any "definition" is going to come up too foundationally limiting. --Trovatore 15:41, 28 May 2006 (UTC)

Maybe we should just not have a first paragraph. --Lambiam 17:58, 28 May 2006 (UTC)

Etymology of "mathematics"

While Template:Polytonic can mean "fond of learning", which is cute, a more basic meaning is: "related to Template:Polytonic", which for one of the meanings of Template:Polytonic amounts to "mathematical". Hence (with the female form) Template:Polytonic "the mathematical art", which can be shortened to Template:Polytonic, and with the neuter plural form Template:Polytonic "things mathematical". I think that the plural form we have in English comes from the last. But I must admit my classical Greek is rusty, and I have no sources for any of this. Anyone with a good Greek dictionary? --Lambiam 20:35, 27 May 2006 (UTC)

Well the definitive Greek lexicon, Liddell and Scott, is online at Perseus project, but seems to be not responding at the moment, and I don't have my paper copy handy. But for the purposes of etymology it is useless to translate mathema as mathematics. The word originally and literally means learning, so the word mathematics means "things related to learning". I think that would be a better etymology. There is nothing about fondness in the word. But I don't think you should change the translation to "mathematics". That would be like saying that the etymology of "geometry" is that it is the Greek word for "geometry". While true, it is not a very useful etymology, etymology is supposed to give you some source meanings. -lethe 20:40, 27 May 2006 (UTC)

But the general meaning of "learning" already acquired during classical Greek times among its several meanings the more specialized meaning of "the mathematical sciences". What is wrong with saying that "landscape" comes from Dutch "landschap", meaning "landscape"? --Lambiam 20:59, 27 May 2006 (UTC)

It's incomplete. I wonder what the purpose is of including an etymology anyway? Etymologies are interesting when they come from simpler roots that allow you to recognize the word or tell you some history of the word. Both of these are accomplished by saying that "geometry" comes from the Greek "geos" = Earth, "metron" = measure. Nothing useful at all is conveyed by saying "geometry" is from the Greek word "geometria" meaning "geometry". I think the latter applies to the word "mathematics" as well. Don't know about the etymology of "landscape", though I'll bet there is a Dutch root "land" which has a relevant meaning, which should be mentioned in any useful etymology of the word. -lethe 21:22, 27 May 2006 (UTC)

What exactly is incomplete? My remarks addressed correctness. We should not substitute an interesting but incorrect etymology for a correct but boring one. By the way, this fondness thing is one of the meanings. You can say there is nothing about excellency in the word "cool" since it originally means "cold", but in current English "cool" can and does also mean "excellent". --Lambiam 23:44, 27 May 2006 (UTC)

If you give the shortest possible etymology which contains no information other than the source language then you are doing your reader a disservice. If you don't mention that "cool" derives from an english word meaning "cold", then you haven't given a good etymology. -lethe 01:55, 28 May 2006 (UTC)

And if you say that "cool" comes from "ghoul", you're wrong. And if you give the longest possible etymology, occupying many pages, the reader won't be happy either. I agree with all of this. But what does that have to do with the question whether the etymology currently given on the page is correct? --Lambiam 02:13, 28 May 2006 (UTC)

Liddell and Scott is back online responding now, and you can see that "fond of learning" is exactly the translation it gives for mathematikos. So the etymology in the article is a good one. It also lists "scientific, esp. mathematical", "mathematics" as definitions, so those might also provide good etymologies. For mathema (of which mathematikos is just an adjectival form), it gives "that which is learnt", "lesson", "learning", "knowledge", "esp. the mathematical sciences". and manthano, "learn", "perceive", etc. Now the question is only: which of these lexemes do we want to list in the article? Which will be most useful to the reader? Should we simply list the L&S first translation of the word mathematikos? This is what's currently done. I guess I'm OK with it. -lethe 02:34, 28 May 2006 (UTC)

I think etymology is more subtle than that. Just because mathematics is derived from mathematikos and one of the meanings of mathematikos is "fond of learning", does not mean that that is the meaning from which mathematics is derived. I'd be more comfortable if we used some kind of dictionary. For instance, the Oxford English Dictionary says that mathematics is derived (via French and Lati) from the Greek mathematikos meaning mathematical or mathematics; it does not mention the "fondness of learning" meaning of mathematikos, but it does say that derives from mathema = knowledge and the suffix -ikos. So, just from the OED, I'd say that the best is to say that it derives from mathema = knowledge, without the "fond" part. -- Jitse Niesen (talk) 02:53, 28 May 2006 (UTC)

I was actually thinking that as far as listing the root from which the obviously derived word comes, manthano is the better choice. As for the "fond of learning" in Liddell and Scott, I think they were just searching for an adjective to which to attach "learning", since there isn't a neutral adjectival combining form in English. Anyway, I would be in favor of a longer etymology than just mathematikos. That's what I was trying to persuade Lambian of, afterall. My only point was, nothing in the article is wrong at this point, so if we change it, it's only because we like something else better. -lethe 03:04, 28 May 2006 (UTC)

I was about to change it according to what you said, but I see that it already mentions mathema. I suppose there is nothing to be done, the article is fine. -lethe 03:05, 28 May 2006 (UTC)

Phew ;-) Stephen B Streater 09:18, 28 May 2006 (UTC)

The article states or suggests that "mathematics" comes from a word meaning "fond of learning". I think that is cute but misleading. When the ancient Greeks used "mathematikos", they meant: "relating to mathema". What did they mean by "mathema"? They could mean "learning", or more specifically "science", or quite specifically that which we now call mathematics. When they said: "This young man is mathematikos", they might well mean: "He is studious". When they said: "This problem is mathematikon", they might well mean: "It is a mathematical problem". At least that's my best guess until someone who has a classical dictionary shows otherwise. --Lambiam 09:37, 28 May 2006 (UTC)

I looked up mathematics in The Oxford Dictionary of English Etymology (1983 reprint). According to the ODEE mathematics has a dual pedigree. Middle English has matematik (attested 14th c), via Old French/Spanish/Italian from Latin (feminin singular) mathematica, namely mathematica ars or mathematica disciplina, from Greek mathematike, namely mathematike tekhne or mathematike theoria. The current plural form (attested 16th c.) is said to be probably modelled after French les mathematiques (also 16th c.), from Latin (neuter plural) mathematica (Cicero), from Greek ta mathematika (Aristotle). There is no explanation of mathematikos except that it is from mathema something learnt, science, from manthanein learn. Jasper Klein.

The verbal noun from Template:Polytonic, "learn", Template:Polytonic "that which is learnt, learning" and its adjective Template:Polytonic "related to learning" are used of science in general and the mathematical disciplines in particular by Archytas, by Plato and by Aristotle. One may assume that this use of the word originated in the school of the Pythagoreans. Archytas, a close friend of Plato's, was a Pythagorean philosopher and mathematician, and he wrote the book about mathematics with the title Template:Polytonic or Template:Polytonic, which is known only in fragments. The prominence of numbers and mathematics in the Pytheagorean tradition is wellknown, and it may have been the reason why mathematical studies would be labelled simply "learning". Enkyklios 11:48, 28 May 2006 (UTC)

Lambiam, will you be satisfied if we change "fond of learning" to "related to learning"? I'm cool with either one. -lethe 18:30, 28 May 2006 (UTC)

That is half the point I was trying to make; it is an improvement. I don't understand why it is so problematic that mathema already took on its meaning of maths in ancient times, but if that is considered a humiliating revelation, then let's leave that out. --Lambiam 19:10, 28 May 2006 (UTC)

Hmm well I guess that is where you and I disagree. I really think that the purpose of providing the etymology of a technical word is to see where the technical word took its meaning from nontechnical language. The incomplete etymologies do not do this, and providing the technical usage in Greek conveys exactly zero information about the origin of this word. I think the translations provided by our Ancient Greek literati friends above testify the interesting fact that the word has a nontechnical origin in Ancient Greek. -lethe 19:25, 28 May 2006 (UTC)

In German, they use the word der Blackout. It means "blackout", the same as in English. If you were writing a German dictionary, and simply sited that "the blackout" is the English word for der Blackout, it would be an entirely useless etymology. Much better to mention the words "black" and "out", to give some small inkling of the feel and history of the word. This is the purpose of etymology, I think. Your position on the word mathematikos ignores this purpose, it makes it rather just a useless translation. -lethe 19:50, 28 May 2006 (UTC)

I don't understand the problem. Can't we state that Greek mathema is the verbal noun of manthano "learn" and had acquired at the time of Plato and Aristotle, possibly even earlier, various increasingly specialized meanings: from "learning" to "study", "(study of) science", and "mathematical sciences"? And that "mathematikos" is the corresponding adjective? --Lambiam 00:34, 29 May 2006 (UTC)

Perhaps there is no problem. I have no problem mentioning that the technical meaning was the same for the Greeks as it was for us (and in fact I made an edit to mention that fact). I just had a problem with what I thought you were proposing: replacing the nontechnical definition with the technical. Now it seems you favor including both, something I can definitely get on board with. How do you like the edit I already made? -lethe 03:12, 29 May 2006 (UTC)

I've put the etymology in a section of its own and expanded it further with the information supplied or verified by Jasper Klein and Enkyklios. I hope I did not go overboard, but you won't be able to complain it's too terse :) --Lambiam 22:37, 29 May 2006 (UTC)

New try at first para

Here's another attempt:

Mathematics is a discipline that studies topics including quantity, structure, space, and change. Its methodology emphasizes deductions made in logically rigorous fashion from axioms and definitions. Some mathematicians of a formalist bent consider this methodology actually to define mathematics, whereas those of a more realist viewpoint tend to find such a definition overly restrictive.

This seems to me to present both sides concisely while emphasizing what they genuinely have in common (both schools would agree that these topics are studied and these methods are used; they may disagree about what other topics are studied and what other methods are used.) --Trovatore 19:56, 28 May 2006 (UTC)

The first paragraph is too early for dueling philosophies. I like the first two sentences well enough. It's an improvement over the current opening paragraph. JJL 21:13, 28 May 2006 (UTC)

Yes I would agree with JJL, I like the first two sentences better but I'm not sure why we even need this sentence: "Some mathematicians of a formalist bent consider this methodology actually to define mathematics, whereas those of a more realist viewpoint tend to find such a definition overly restrictive." in the intro at all, let alone the first paragraph. Paul August ☎ 21:32, 28 May 2006 (UTC)

While we are talking about the first paragraph, and keeping in mind that the intro should essentially be a summary of the rest of the article, I think it would be good to reinsert the sentence: "It developed, through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the study of the shapes and motions of physical objects." which used to be there. Combining this sentence with Trov's might give:

Mathematics is a discipline that studies topics including quantity, structure, space, and change. It developed, through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the study of the shapes and motions of physical objects. Its methodology emphasizes deductions made in logically rigorous fashion from axioms and definitions."

Paul August ☎ 21:32, 28 May 2006 (UTC)

That's not bad. A close eye will have to be kept on it to make sure "emphasizes" doesn't get changed to something more absolutist, but that's an unavoidable problem. I can live with that version if others can (it might also be copied to the portal). --Trovatore 21:45, 28 May 2006 (UTC)

I like this except for the last sentence, which I propose to replace by a modified version of my earlier last sentence, giving:

Mathematics is a discipline that studies topics including quantity, structure, space, and change. It developed, through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the study of the shapes and motions of physical objects. Using deductive reasoning, starting from axioms and definitions, a large body of knowledge has been created over the course of many centuries that is constantly expanded, covering many fields and comprising definitions, theorems, methods, and techniques, all of which build upon each other.

--Lambiam 21:48, 28 May 2006 (UTC)

I'm not entirely happy with the possible implication that only deductive reasoning is used; that's what I like about the "emphasize" language. I'm also not sure "comprising" is quite the right word; again, there's a possible inference that mathematics is limited to these things (say, if you take definition 2 of "comprise" at http://www.m-w.com/dictionary/comprise). --Trovatore 21:52, 28 May 2006 (UTC)

How do you get the implication only? But what else could be mentioned? "Using now deductive reasoning, then , ..." (I'd be happy to include divination or handwaving if you can find some notable theorems obtained that way.) --Lambiam 22:00, 28 May 2006 (UTC)

The sentence seems at least to suggest that deductive reasoning was sufficient to do all this. I don't think it was, at all. A great deal of mathematics is discovered inductively (obviously I'm not talking about mathematical induction here; I mean "via abstraction from examples"). And it's not limited to theorems; a big part of my point here is that mathematics is more than theorems proved from known collections of axioms.

But you never said what you didn't like about the last sentence from Paul's version. Is it just that it doesn't talk about the large body of knowledge? The part of your version starting with "a large body of knowledge" might possibly be incorporated into that sentence, or added as a fourth sentence, without changing the "emphasizing" language. --Trovatore 22:09, 28 May 2006 (UTC)

I don't like the part "Its methodology emphasizes deductions made in logically rigorous fashion". This may be true in theory, but it bears little relation to mathematics as she is practiced, especially the "rigor". Rigor is appealed to when a field turns into a mine field of paradoxes; after a rigorous examination definitions are made precise, conditions are tightened, some useful lemmas are formulated, and otherwise life goes on as before. I tried to bring something to the text of what mathematicians do when they don't quibble over the definition of what mathematics is. Deduction is of course not all of mathematics, in particular there is also what we select as interesting and why, but nevertheless the activity of examining a topic by giving definitions, proofs, theorems, etcetera, also for justifying and communicating results, is at the heart of mathematics. In a nutshell, I think the lead section should say something non-trivial about mathematics as an activity, not only what is studied but also how. And yes, I'd like to bring in the "large body of knowledge", including the part "all of which build upon each other". —The preceding unsigned comment was added by Lambiam (talk • contribs) 23:10, May 28, 2006 (UTC).

I like the "starts with counting" language. Apart from that, deduction/induction seems to be a matter of how much one emphasizes the proved results that make up the accepted corpus of math. or the inducted ideas that give one something to prove in the first place. Wiles moved FLT from inducted to deducted, I'd say. I agree with Philip J. Davis that proof is over-emphasized. But, as taught, math. emphasizes deduction; as done, it isn't nearly so simple. So, I think the suggested language is misleading about the practice of math., and favors the presentation of math. Still, I don't object to it. Math. is the subject that mathematicians study. It ain't gonna get better than that. JJL 01:09, 29 May 2006 (UTC)

Paul's suggestion looks good to me. Of course, maths is more than proofs, but proofs are a large part of what separates maths from other subjects. Abstraction is another part, which is mentioned in the second sentence. I don't quite understand why Lambiam objects to "Its methodology emphasizes deductions made in logically rigorous fashion" and favours "Using deductive reasoning"; to me, and apparently also to Trovatore, the latter formulation gives rigour an even bigger place in mathematics, which seems opposite to what Lambiam wants. -- Jitse Niesen (talk) 03:01, 29 May 2006 (UTC)

Basically every mathematical article, lecture, argument, whether using a blackbord or a napkin, uses deductive reasoning, like "but X is also Y, therefore it must be in Z, which means that...". Little of this is in any sense logically rigorous. This was the way long time before anybody started thinking about "methodology". --Lambiam 09:02, 29 May 2006 (UTC)

How about this:

Mathematics is a discipline that studies topics including quantity, structure, space, and change. It developed, through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the study of the shapes and motions of physical objects. While, guided by physical and mathematical intuition, its methodology emphasizes deductions made in logically rigorous fashion from axioms and definitions. It represents a large, constantly expanding body of knowledge, built up one idea upon another, over the course of many centuries."

I'm not completely happy, with the above, but I've tried to deal with Lambian's concerns by tempering the "methodology emphasizes deductions made in logically rigorous fashion" language, and adding some "large body of knowledge" language. Paul August ☎ 15:56, 29 May 2006 (UTC)

I appreciate the effort to deal with my concerns, but it doesn't scan. Actually, the second sentence sounds like it stops in the middle, like it begs being finished off thus: "While, guided by physical and mathematical intuition, its methodology emphasizes deductions made in logically rigorous fashion from axioms and definitions, working mathematicians could not care less and produce their results by divination and handwaving." I'd prefer the term "methodology" not to be used at all – as far as I'm concerned nowhere in the article, but most definitely not in the lead section. I'll try once more, then I'll give up:

Mathematics is a discipline that studies topics including quantity, structure, space, and change. It developed, through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the study of the shapes and motions of physical objects. Over the course of many centuries a large body of knowledge has been created, covering many fields and comprising definitions, theorems, methods, and techniques, all of which build upon each other. Mathematicians justify their results mainly by deductive reasoning, starting from axioms and definitions.

--Lambiam 16:49, 29 May 2006 (UTC)
Oh, and I think "such as" was better than "including", not because of content but because of language usage. --Lambiam 16:53, 29 May 2006 (UTC)

As a practical matter, anyone who can read surely has at least a basic idea what mathematics is, so arguing over how to define it is a bit obsessive. I like the "It developed..." line; does it need to be said that it's a large body of knowledge? JJL 17:04, 29 May 2006 (UTC)

Many people have a basic idea that is wrong. I've met people who could not understand that there was such a thing as mathematical research. They thought that you put in the problem, apply the formulas, and out pops the answer; what's there to research? They pitied me that I'd had to memorize all these formulas, like "b plus or minus the square root of 2a". They definitely had no idea that there is more there than a person can master in a lifetime. --Lambiam 21:58, 29 May 2006 (UTC)

I've met many people like that. Anyway, I like the "justify their results" version, although I agree that the "large body of knowledge" sentence isn't obviously necessary. JPD (talk) 09:42, 30 May 2006 (UTC)

I agree that sentence is not obviously necessary. But do you like it? What do you feel, in the balance of things, does it add sufficient value, or does it distract too much from the essence? --Lambiam 13:00, 30 May 2006 (UTC)

I compared Physics, Chemistry, and Biology. They have a relatively similar style--not perfectly uniform, of course--that the first two sentences here fit reasonably well but which the last does not. A third line that would fit the style of those entries might be something like: "Today mathematicians study a broad range of topics, from the most arcane results in logic to state-of-the-art high performance computing, and apply their skills to many other areas in the sciences, social sciences, engineering, and other fields." It seems to me that the other entries I mentioned tend to end on a "Today..." note and an indication of breadth/diversity. I think it'd be good to have the math. article do something similar. The current suggestion does give some indication of breadth and implicitly refers to the current state of things, but I think it can be made better. JJL 14:47, 30 May 2006 (UTC)

Another try:

Mathematics is a discipline that studies topics such as quantity, structure, space, and change. It developed, through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the study of the shapes and motions of physical objects. It involves choosing appropriate axioms and definitions, and using deductive reasoning to prove theorems from these. In this way, over the course of centuries, a large constantly expanding body of knowledge has developed, covering a broad range of topics.

Again I've tried to incorporate everyones ideas. As for JJL's last suggestion, I think it has merit, but the ideas contained in his "Today ..." sentence are to some extent, at least, already covered in the second paragraph. Paul August ☎ 16:37, 30 May 2006 (UTC)

I'm sufficiently happy with this. I hope that eventually the article will also cover, somehow, the phenomenon of cross-fertilization and inspiration between maths areas (such as Algebra+Topology = Algebraic topology and Analysis+Geometry = Analytic geometry, or Homology theory sparks Abstract nonsense theory is used in Topos theory, Formal semantics, ...), or in any case make clear this is not just a tree with many leaves but a deeply intertwined network. --Lambiam 17:51, 30 May 2006 (UTC)

I've been busy for a few days, and I like the new paragraph. However, I do think one important area is missing: the area of uncertainty (see my earlier section "Handling uncertainty"). I have listed various areas eg probability, statistics. One advantange of adding this in in the first paragraph is that many people have the preconception that Mathematics is all black and white, absolute truth, and nothing to do with the real world. By adding in "uncertainty" or "chance" in some way, it will surprise and inform many readers, and broaden their understanding of Mathematics. Stephen B Streater 08:24, 31 May 2006 (UTC)

Just my 2 cents, I like the first paragraph, but I do agree a mention of the study of uncertainty should be made. Is there any reason the word can't be added to the end of the first sentence? Also, should centuries not be millennia? --darkliight 12:19, 31 May 2006 (UTC)

OK - I've added "uncertainty". It has a much more relevant article than "Chance". Stephen B Streater 14:10, 31 May 2006 (UTC)

(edit conflict) I've reverted this change for now. I don't think we should add "uncertainty" to the first sentence without changing the rest of the article appropriately. The intro should essentially be a summary of the rest of the article. The theme that mathematics is broadly speaking the study of quantity, structure, space, and change is a theme that is used through out the article. In any case such a significant change should be discussed and agreed to here on the talk page first. Paul August ☎ 14:56, 31 May 2006 (UTC)

I disagree with adding the "uncertainty". Having the four "quantity, structure, space, and change" in there provides a unifying theme which goes throughout the article and I like it that way. Of course math is also about "uncertainty" and a lot of other things, but I don't think the more stuff you add the better the text becomes. I say we stick with the text the way it is now, the "four pillar thing" if you wish. Oleg Alexandrov (talk) 14:51, 31 May 2006 (UTC)

Uncertainty is definitely an important part of math. today, but is a relatively new addition to math. (let's say 1600s and not quibble about what came before). So, the "Big Four" emphasizes what historically laid the foundations, but now there is much more diversity. Personally, I wouldn't list uncertainty alongside the "Big Four," but then the first three of those read to me like the bases for analysis, algebra, and topology, resp., which to me are the real Big Three. I'd subsume change in the other three--the interest is not in change but in a changing quantity, or shape, etc. (I am not proposing changing this at this late point, though; just saying I already think the initial list is long enough.) I would save uncertainty for a later point in the article. Stylistically, I don't like the "over the course of centuries" language. A nod to applied math. might be nice too, with something about modeling alongside the deductive logic bit. The emphasis is still on how math. is taught and written over how it's done. A suggestion:

Mathematics is the discipline that studies topics such as quantity, structure, and space, and how they change. It developed, through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the study of the shapes and motions of physical objects. It involves choosing appropriate axioms and definitions, and applying deductive reasoning to establish conjectured results concerning them; it also involves the application of those results to the physical world via modeling. Today mathematics covers an impressivley broad range of topics.

This might require tweaking paragraph 2 too. All that having been said, apart from dropping uncertainty I do not object to the current suggested form of the first paragraph. JJL 15:11, 31 May 2006 (UTC)

I thought we were talking about Mathematics today, not in the 1600s. Today, particularly in the branch of Financial Mathematics, communications theory, statistics and Quantum Theory uncertainty is what Maths is used for. Stephen B Streater 15:41, 31 May 2006 (UTC)

Another thought, I would change It developed to something that doesn't imply completion. Maybe it's just me, but the sentence seems to imply that mathematics was developed, but does not continue to develop. As for this paragraph suggestion, the line to establish conjectured results irks me, we should emphasise that mathematics is focused on proof. Mentioning the application of mathematics is a good idea, but I think the Today.. line should be left to the next paragraph, allowing a little more discussion of it. As for leaving out uncertainty, fair point about it not being discussed in the article, so I agree with it being left out until something does appear, per the handeling uncertainty section above. Cheers --darkliight 15:44, 31 May 2006 (UTC)

What I hoped to accomplish with conjectured is to counter the impression, which I fear the current form gives, that math. proceeds by straight deductive logic without any creativity. What's missing is the creative nature; as it stands, it reads as though a mathematician simply sits down and produces classical geometry proofs all day, one after the other. There's too much emphasis on logic/reason/proof and no mention of conjecturing/modeling/estimating. I disagree that math. as done focuses on proof. It focuses on ideas, relations, etc. There's a lot of industrial math. being done with little to no proof. There's a lot of modeling and computing done. It can't all fit in one para., but it can be acknowledged.

As to uncertainty, Stephen B Streater is right that it's important in applications. I see it as a subset of analysis (quantity, change), and think for the first sentence it doesn't fit. It comes at the next hierarchical level, to my mind. Certainly, it needs to be in the article. JJL 16:01, 31 May 2006 (UTC)

(Edit conflicted too). I think the point about raising "uncertainty" in the article is a good one. But not as good as this one: We've already discussed giving uncertainty its own pillar in its own section, and it had a positive response (see talk page section "Handling uncertainty"). I think that adding it in is only a matter of how not when. Stephen B Streater 15:36, 31 May 2006 (UTC)

As to the word "developed" suggesting completion, perhaps some inspiration can be found in my first attempt, way back at #Opening paragraph, where "is constantly expanded" took away that suggestion. --Lambiam 16:39, 31 May 2006 (UTC)

Can we come to some agreement?

Ok can we come to some agreement on a new first paragraph? Unless someone objects I would like to replace the existing version with:

Mathematics is a discipline that studies topics such as quantity, structure, space, and change. It developed, through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the study of the shapes and motions of physical objects. It involves choosing appropriate axioms and definitions, and using deductive reasoning to prove theorems from these. In this way, a large constantly expanding body of knowledge has developed, covering a broad range of topics.

I think that this version is better than the existing first paragraph in the following ways. It provides a better summary of the existing body of the article. It gives a description of mathematics, while avoiding the trap of trying to define mathematics, and is consistent (I think) with both the formalist and and realist viewpoints, thus obviating the need to mention this distinction in the intro. It places less of an emphasis on logical rigor, and thereby more on intuition and creativity (although somewhat obliquely). It indicates that mathematics is not static but an ever growing body of knowledge. Does anyone prefer the present version over this one? If not I suggest that we replace it with the above and go from there. Paul August ☎ 17:10, 31 May 2006 (UTC)

I prefer the new version for the following reasons:

• It doesn't give two potentially opposing views in the first paragraph
• I'm not convinced that a lot of mathematics is or has been logically rigorous in practice
  • (look at eg renormalisation in Quantum Theory)
• I think intuition comes first and axioms come later

However, it still doesn't cover uncertainty very well - which of the four areas would you put this in? Stephen B Streater 18:07, 31 May 2006 (UTC)

I agree with all of your points above. But can we please save how best to incorporate "uncertainty" for later? Paul August ☎ 18:41, 31 May 2006 (UTC)

Yes - I'll think about it too. It's a relatively new area, I know, but one which has become very important for Mathematics in every day life - particularly information theory (eg mobile phones) and Financial Mathematics. Perhaps it needs a separate type of mention from the others. It would be unfortunate if the article became so rigid there was no natural place to include it. Stephen B Streater 19:30, 31 May 2006 (UTC)

This change is OK by me. It's a definite improvement over what's currently there. JJL 19:07, 31 May 2006 (UTC)

How about the following for the first paragraph:

Mathematics is a discipline that studies topics such as quantity, structure, space, change, and chance. It developed, through the use of abstraction and logical reasoning, from counting, calculation, measurement, the shapes and motions of physical objects, and outcomes of games. The method of mathematics is deductive reasoning from carefully stated axioms and definitions.

The variety and usefulness of mathematics is covered in the second paragraph.

Rick Norwood 21:09, 31 May 2006 (UTC)

Rick do you have objections to my proposed version? If so can you say what they are exactly? And do you prefer the version currently in the article over my proposal? Paul August ☎ 21:36, 31 May 2006 (UTC)

The "outcomes of games" phrase is a nice way to work in uncertainty/chance in a historically accurate manner (e.g., Galileo and the question of rolling a 3 vs. a 4 with three dice, (Hacking, The Emerg. of Prob.)). Adding it to the current suggestion is fine by me. The "method of math." sentence still seems to slight modeling/intuition/computation/etc., though. Between the two I would take the first version, not the second one. JJL 21:38, 31 May 2006 (UTC)

So perhaps I will restate my reason for including chance early on. Namely that a lot of people don't appreciate that Mathematics is not all black and white. A mention of chance in the first sentence is likely to draw people into the rest of the article. Stephen B Streater 22:31, 31 May 2006 (UTC)

I agree with the paragraph suggested by Paul. The second paragraph (which I think could use a little work after this too) will follow easily from this and uncertainty, when the article discusses it, shouldn't be too difficult to be incorporated. Rick, your version doesn't mention that mathematics is still being developed and I don't like the last sentence, to the layman I think it might suggest that mathematics is just the application of a set of formulas. --darkliight 23:44, 31 May 2006 (UTC)

I like Paul's version best so far, although I think some mention of chance or uncertainty should be added. Referring to games seems like a nice way to do this. One minor quibble: "constantly expanding" sounds a bit like marketing speak - how about "a large and growing body of knowledge" instead? -- Avenue 00:55, 1 June 2006 (UTC)

I like Paul's version too. I would disagree with adding anything about uncertainty, that would not be consistent with the article as it goes on below. Oleg Alexandrov (talk) 01:37, 1 June 2006 (UTC)

Yes, on second thought I agree that uncertainty should be left out of the preamble until such time as it appears in the main article. -- Avenue 09:37, 1 June 2006 (UTC)

It's pretty good. I still think it may slight non-axiomatic ways of thinking a bit, though. For example not everyone who tries to figure out whether the continuum hypothesis is true, thinks of this process as looking for appropriate axioms; some try to resolve it by direct argument, though obviously of a sort not formalizable in ZFC. --Trovatore 01:47, 1 June 2006 (UTC)

They are not mathematicians anyway, at least not according to the definitions in Misplaced Pages ;) --Lambiam 03:06, 1 June 2006 (UTC)

Then the definitions in Misplaced Pages are wrong and need to be changed. Don't have time to check that at the moment. But I insist on this point: This sort of argumentation is mathematics, and any definition that excludes it is too restrictive. --Trovatore 06:08, 1 June 2006 (UTC)

Yes, it is quite restrictive: knowledge = theorems. Note (when you have time) that the def. also applies only to research mathematicians; most industrial mathematicians as well as mathematics teachers are excluded.

Was Newton not a research mathematician? I say he was, and he worked non-axiomatically. So (perhaps more to the point) did Cantor. It's not just "quite restrictive"; it's "too restrictive", meaning an incorrect definition. --Trovatore 05:19, 2 June 2006 (UTC)

The article Mathematician has no mention of axioms or axiomatic approaches. When I wrote "quite restrictive" I was referring to the second sentence: "In other words, a mathematician is a person who contributes new knowledge to the field of mathematics, i.e. new theorems." --Lambiam 08:14, 2 June 2006 (UTC)

Oh, I see what you're saying. That strikes me as a bit too aggressive an exegesis of that passage. Mathematicians do other things than prove theorems, and those other things are still mathematics (and still knowledge), but you'd hardly call someone a mathematician if he didn't prove theorems, and most mathematical knowledge is usually conveyed in the form of theorems. So I don't find that particular passage objectionable and don't think it contradicts what I'm saying, on any common-sense reading. --Trovatore 16:16, 2 June 2006 (UTC)

Isn't Erdős' definition better: A mathematician is a device for turning coffee into theorems? I wonder actually if whoever contributed the sentence to the article meant "e.g." rather than "i.e.". --Lambiam 16:54, 2 June 2006 (UTC)

At least one person has claimed at some point that A mathematician is a device for turning coffee into theorems. We'd need a cite that Erdős actually drank coffee to go further, of course. Stephen B Streater 18:42, 2 June 2006 (UTC)

The article on Alfréd Rényi states that he was "probably" the source of the quote generally ascribed to Erdős. --Lambiam 23:21, 2 June 2006 (UTC)

I would say that somone who applies theorems, but doesn't prove them, is still a mathematician. Stephen B Streater 18:39, 2 June 2006 (UTC)

I think a problem with the proposed versions so far is something that has been suggested by a few people so far and that is that an impression is given that the method of mathematics is logical deduction. It slights actual practice in creating new mathematics where rigor is left until the very last stages. Some great mathematicians even eschew this last step (everyone knows about Euler, but lesser known is that Fields Medalists have been prone to do this also) Since it seems an important part of this article is to disspell common misconceptions about mathematics and mathematics research, I don't think it's a good move to create such a misconception from the get-go. --Chan-Ho (Talk) 03:12, 1 June 2006 (UTC)

Chan, do you think the version in the article is better than my proposed one? Paul August ☎ 03:28, 1 June 2006 (UTC)

I don't think that's a very fruitful question, as both are problematic. But if I was forced to pick one or the other, I'd rather have the current version. See my response to Lambian, below. --Chan-Ho (Talk) 05:49, 1 June 2006 (UTC)

The present version in the article requires logical rigor, or else recognizes only results justified by deductive reasoning. So it is clearly worse in the aspect deemed problematic than Paul's version. Someone, a long time ago, in a galaxy far far away, suggested a version that said: "mathematical results can be justified by deductive reasoning", not implying that they actually are. But that is now history. --Lambiam 04:29, 1 June 2006 (UTC)

Paul's version is actually worse with respect to the issues I raised than the current introduction. I wasn't saying that logical rigor is not needed in mathematics; I would say it's an essential part of it. (Of course, standards of rigor change over time but that's a tangent). According to the modern day view of mathematics, in fact, all theorems are supposed to be ultimately justified by deductive reasoning (although perhaps the level of details or foundational details are not universally agreed to). The current intro does, as you pointed out, de-emphasize the role of conjectural and speculative aspects, but then again, so does Paul's version. In addition, Paul's version is seemingly describing the process of mathematics by the language he chose. It reinforces the misconception that theorems come after axioms and definitions in the discovery process. I don't believe that was intentional on his part, but I believe that the false impression is conveyed. --Chan-Ho (Talk) 05:49, 1 June 2006 (UTC)

Are we talking about the same versions? --Lambiam 05:54, 1 June 2006 (UTC)

Paul's version is a big improvement on the current version. McKay 04:32, 1 June 2006 (UTC)

Perhaps we can think of a version which is better than both. I don't buy the argument that the current article doesn't reflect chance, so nor should the heading. The current article is going to be rewritten to meet Featured Status, and by that time it will include Chance. We are starting with the first paragraph, so let's get this right now, rather than having to come back and fix it up all over again. And I agree with all the people who think that rigor is often only the final step, and it can be largely absent from the creative stage, and that this should also be reflected in the introduction. Stephen B Streater 08:36, 1 June 2006 (UTC)

I do see your point, but after thinking about it, I don't think it's such a good idea to fragment the article with hints of information that is coming soon. The article as it stands has very few references, let alone discussion, of uncertainty/chance. This information should be presented, then the appropriate changes made to the introduction -- not the other way around, imho. Yes I do agree that it should be mentioned in the introduction, just not until it discussed within the rest of the article. Feel free to get the section started here, and myself and no doubt everyone else here will be happy to contribute, or at least comment. I'd start it myself, but this topic really isn't my speciality. --darkliight 09:01, 1 June 2006 (UTC)

Well, let me also give it a try. Starting with the "Paul August ☎ 21:32, 28 May 2006 (UTC)" version, which I like better than the subsequent attempts, I arrived at:

Mathematics is a discipline that studies topics including quantity, structure, space, and change. It evolved, through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the study of the shapes and motions of physical objects. Mathematicians aim to justify their results by rigorous deduction from axioms and definitions.

I changed "developed" in "evolved" in the second sentence, to emphasize that maths is not finished. I tried to address Chan-Ho's point that axioms and definitions do not come first. I agree that some mathematicians do not strive for rigour, but that's only few and I think that mathematicians as a whole, do try to find a proof. To me, "rigorous deduction" is a pleonasm because deductive reasoning is per definition rigorous, but it seems that others disagree. I don't like the "large body of knowledge", because it does not give much information: every discipline has collected a large body of knowledge. -- Jitse Niesen (talk) 09:26, 1 June 2006 (UTC)

I think you addressed my concerns with your version. --Chan-Ho (Talk) 10:48, 1 June 2006 (UTC)

I like Jitse's version. JPD (talk) 11:02, 1 June 2006 (UTC)

I agree the "large body of knowledge" comment was a bit vacuous, but I think Jitse's version is not clear enough about mathematical knowledge continuing to grow. How about expanding the last sentence to:

Mathematicians extend their knowledge by making conjectures about these topics and proving them by rigorous deduction from axioms and definitions.

Avenue 11:36, 1 June 2006 (UTC)

Jitse's version, like most of the earlier attempts, is an improvement over what is now in the article. Some remarks.

• For purely stylistic reasons I'd replace "including" by "such as".
• The style guide for the lead section states we can use up to four paragraphs for our elevator pitch. At present there are three, of which one hardly counts, and the others are not fat. Something that is nice to have in the lead section but too much for the first paragraph might go in one of the next paragraphs.
• I miss the aspect that mathematics is not dead (but in fact blooming as never before). Jitse's version leaves room for the possible misconception that the evolution of mathematics was completed somewhere between Pythagoras and Hilbert.
• Some other ingredients that I liked in earlier attempts are not represented. They might be made to make a reappearance perhaps in the next paragraphs. One is the aspect that mathematics is also a body of knowledge, with ingredients such as definitions, theorems, and methods. Then there is the depth aspect of that body of knowledge (results building upon results), and its breadth (which is more than sheer size). Although related, the breadth is not quite the same as the wide scope of applicability.
• And if we still have space in the lead, there might be a hint of what mathematics is as an activity. I think everyone will agree that it would be good if we manage, somehow, to convey an impression of the creative aspects involved.
• Altogether, here is my shopping list for aspects/ingredients to go in the lead section, not necessarily in the best order: Maths by topics studied (QSSC); emergence from counting and such by abstraction; maths as a body of knowledge (definitions, theorems, proofs; depth, breadth; interrelatedness); application, also giving rise to new fields in maths (science, engineering; ...; optimization, scheduling, planning; decision making); maths for the sake of maths; continuing bloom; maths as an activity (creativity; language; proving, conjecturing, experimentation; having fun).

Although this looks like a lot, I think it may all fit without making the section feel crammed. Most of this is there already. --Lambiam 12:44, 1 June 2006 (UTC)

This all looks good to me. I think it is important to illustrate not just the range of subjects, but the range of types of things encompassed by Mathematics. As "chance" doesn't seem to have consensus for the first sentence yet, perhaps we could include it later in the first section instead - particularly as we have multiple paragraphs available. I think it is qualitatively different from all the current "pillars", and covers many topics, of which I've already listed about a dozen. I don't see other similar sized holes in the article at the moment. Stephen B Streater 13:02, 1 June 2006 (UTC)

I like this version better--especially the "Mathematicians aim to justify..." language. I do disagree with the comment following the paragraph that "I agree that some mathematicians do not strive for rigour, but that's only few"; it seems to me that this reflects a bias towards academic mathematicians. There are a great many bachelor's and master's level mathematicians out there computing, modeling, approximating, optimizing, programming, etc., who will never prove anything in their professional (post-college) lives. After all, the students taught by the Ph.D.'s go somewhere, no? The lead paragraph still leaves out that mathematics involves creativity, and that one can "do math." without proving things. JJL 13:47, 1 June 2006 (UTC)

To answer Paul's question, my suggestion was just an attempt to get prob & stat into the picture. The reason I left off your last sentence, Paul, is because the second paragraph says this, and there is no need to say it twice.

If we're voting, I prefer Jitse's version.

The discussion of whether mathematics must be proved, or can be arrived at by other methods (pattern recognition?, intuition?, messages from God as Ramanujin claimed) belongs in philosophy of mathematics. Rick Norwood 14:43, 1 June 2006 (UTC)

The discussion belongs here, if we aim to define or even describe mathematics. There needs to be some agreement about what math. is. When I see someone with a B.S. and M.S. in math. developing and coding an algorithm to implement a simulation for an astronomer, I think that person is doing math.--namely, doing the astronomer's math.--even though he or she isn't proving anything, and may even be applying algorithms that aren't proven to work (at least, under the circumstances/assumptions of the problem at hand). If we don't agree on issues like this, then how can we have a paragraph starting "Mathematics is..."? JJL 14:57, 1 June 2006 (UTC)

My point is that we are not going to come to an agreement. This whole subject was debated here at length about a year ago, and the current first paragraph is the compromise we reached. I don't have any objection to either Paul's or Jitse's first paragraphs, but to think we are going to resolve a question that has been hotly debated by mathematicians for at least a century is unrealistic.

As for your example, I would say that the person with a degree in math is doing computer science, and the earlier example of renormalization seems to me to be physics, not mathematics. Come the revolution, comrade, mathematics will consist of theorems, rigorously proved. But in this imperfect world, we have to allow for the usage that mathematics is pretty much anything with numbers in it. Rick Norwood 16:31, 1 June 2006 (UTC)

The question of rigor could be addressed by acknowledging that rigor is variable. Nobody doing mathematics uses the style of Principia Mathematica; on the other hand, I think even Ramanujan proved some things. If we say this, it becomes immediate that some degree of rigor is part of mathematics. Septentrionalis 17:14, 1 June 2006 (UTC)

Close to a consensus?

The version with the most support at the moment seems to be Jitse's version:

Mathematics is a discipline that studies topics such as quantity, structure, space, and change. It evolved, through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the study of the shapes and motions of physical objects. Mathematicians aim to justify their results by rigorous deduction from axioms and definitions.

(That includes Lambiam's trivial suggestion of changing including to such as).

Two more comments from me..

• Should the word result be used to describe something that hasn't been justified yet? Maybe a word like ideas should be used in place of results, which might even help get the idea accross that mathematics isn't a process of following rules, it has a creative aspect to it too.
• I think the growth and size of mathematics today, should be left to another paragraph, but still included in the introduction.

--darkliight 05:26, 4 June 2006 (UTC)

I'd be reasonably happy with this, and suggest a couple of amendments as in my mind, mathematics includes using the existing body of knowledge:

• "Studies" sounds a bit odd here. I suggest the alternatives "includes" or "encompasses".
• I'm not sure about "justify". Perhaps "prove"? Also they may prove other people's results, so leave out the "their", possibly replacing it with "new" or "mathematical". Similarly "aim" could be replaced with "prefer". Think of Ramanujan.

Mathematics is a discipline that encompasses topics such as quantity, structure, space, and change. It evolved, through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the study of the shapes and motions of physical objects. Mathematicians prefer to prove results by rigorous deduction from axioms and definitions.

Stephen B Streater 08:43, 4 June 2006 (UTC)

I'm sure the word prefer is inappropriate in the last sentence -- surely aim is better here? Also, since the sentence begins with mathematicians, not A mathematician .., then doesn't that imply they work on each others work? So I think aim to prove their results/ideas/work by.. fits better here. --darkliight 09:20, 4 June 2006 (UTC)

I think "aim" works here, and agree that "their" is not the same as "their own" so your wording for the last sentence works fine for me (with results or ideas rather than work - they might do it for play). Stephen B Streater 11:07, 4 June 2006 (UTC)

I'm happy with the darkliight version and actually prefer "studies" over "encompasses", or, following the example of Physics, Chemistry, and Biology, we could say "deals with", a bit flatter but stll active. We could continue flogging this horse to death, but I hope instead someone will be "bold" and actually update the first paragraph. I don't think we'll get more consensual than we are now. Then we can spend the next week on the second paragraph. --Lambiam 15:15, 4 June 2006 (UTC)

Yeah, I like Jitse's version too. Anybody willing to add it to the article? Oleg Alexandrov (talk) 15:51, 4 June 2006 (UTC)

I think the above is a huge improvement, too. I'm particularly glad to see the claim about many mathematicians defining math as the body of knowledge justified by deductive reasoning removed. "Mathematicians aim to justify..." is just right. Here's another tweak:

Mathematics is the study of topics such as quantity, structure, space, and change. It evolved, through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the study of the shapes and motions of physical objects. Mathematicians aim to justify their results by rigorous deduction from axioms and definitions.

Regarding the problems that a "result" that isn't justified isn't yet a result, by one definition of result, and the fact that not all mathematicians are so concerned with rigor, please keep in mind that this is only an overview. Its job is only to summarize what is already written later in the article, without going into so much detail and precision.

--Ben Kovitz 16:51, 4 June 2006 (UTC)

OK -- I've tried to add something meeting the above (I removed the Oxford comma and made the suggested adjustment from studies to encompassing). Feel free to change back if you wish. --Richard Clegg 16:55, 4 June 2006 (UTC)

Megalithic mathematics

Neolithic monuments on the British Isles are constructed using Pythagorean triples.

This, at a minimum, requires a source. There is a long-standing conjecture that Neolithic peoples could have used Pythagorean triples for a right angle; it was stated about Egypt, and I believe Morris Klein is responsible for it. I know of no evidence for it anywhere, and the British Isles is a most unlikely place to find some: ropes will not survive, and few of the British megaliths have right angles. Septentrionalis 17:14, 1 June 2006 (UTC)

Applied mathematics

The list under this heading seems a bit biased toward areas of application (e.g., Mathematical physics, Mathematical economics) rather than methods of applied math. (e.g., Representation theory, differential equations, approximation theory). To an extent this is also true of things listed at applied mathematics. Some fields, like fluid dynamics, are in both categories. But I think it fits in more with the other headings to list subfields of math. like ODEs, PDEs, etc. preferentially over Mathematical biology and such, or to list more of the subfields at least. JJL 21:22, 1 June 2006 (UTC)

Maybe part of the reason for this is that it is slightly ridiculous to refer to representation theory as applied maths, even though it is used in mathematical physics. JPD (talk) 09:55, 2 June 2006 (UTC)

Could it be that JJL was talking about some Branches of representation theory ? Brian W 00:19, 3 June 2006 (UTC)

Well, I don't see JLL referring to representation theory as applied mathematics, just as a method of it. That's certainly a reasonable statement. Even if one excludes what you might consider rather "esoteric" applications in mathematical physics, there are still plenty of applications to very concrete situations involving cryptography, crystallography, error-correcting codes, etc. --Chan-Ho (Talk) 10:54, 3 June 2006 (UTC)

I see putting representation theory under the heading Applied mathematics as referring to it as applied mathematics. My point was that while something like mathematical physics may be appropriate under such a heading, it wouldn't be right to put representation theory as a whole there. (Representation theory as a whole is not represented by that link at the moment, either!) Anyway, I think the reason the list is "biased" to areas of application, is because the methods are also methods/areas of pure maths. I did not mean to imply that mathematical physics is the only application of representation theory, and the question of whether cryptography, error-correcting codes or even mathematical physics are considered "applied maths" is another issue altogether. It would probably be a good idea to put differential equations in there, however. JPD (talk) 15:16, 3 June 2006 (UTC)

I don't care about rep. theory (which I assumed, without checking, would go to a subset of approx. theory--I was thinking of Fourier series representations, not group theory) in particular. Clearly, as it stands it should not be in applied math. I meant more about methods/areas like approx. theory vice areas of applicability like math. bio.--applied math. over applicable math. JJL 05:01, 5 June 2006 (UTC)

Sounds like a good idea to me. JPD (talk) 10:23, 5 June 2006 (UTC)

Jargon

The section Notation, language, and rigor talks about mathematical jargon, but as has been mentioned, this refers to words. Perhaps code would be more appropriate to refer to the notation itself, or perhaps language. I feel this sentence stops a bit abruptly with music, and would like to bring it back to finish with something about the mathematical notation. Stephen B Streater 11:43, 4 June 2006 (UTC)

New intro

Looks good to me. Rick Norwood 16:59, 4 June 2006 (UTC)

Yes I like the new intro as well. I've been away since Thursday in New York, glad to see we've made some progress here ;-) Paul August ☎ 20:38, 4 June 2006 (UTC)

I too have been away this weekend in NY, and I too like the new intro. As to discussions further down the page such as concepts vs. topics, I don't much care, but "discipline encompassing the study of concepts" does seem a bit wordy to me. As to the fact that it doesn't clearly state that the results of such studies may be put to practical use, that doesn't bother even an applied mathematician like me! JJL 14:49, 6 June 2006 (UTC)

Congrats, this time the intro looks very good (oh yes, I am watching you all since a long time). Now, it's time to do some work over the logic related articles. Brian W 20:53, 4 June 2006 (UTC)

Wow, the new intro is worlds better. —Mets501 20:59, 4 June 2006 (UTC)

Well done everyone! --darkliight 01:15, 5 June 2006 (UTC)

RE: Opening paragraph. Good grammar can be as precise and pleasing as good notation. We should try to be consistent in use of either nouns, or present participles in series. Being good in both enhances clarity. Bcameron54 01:40, 5 June 2006 (UTC)

Hi there. I like the opening paragraph as well, with one reservation. Could we change the word "topics" to "concepts"? I generally think of a "topic" as something which might be covered by a textbook, chapter, or a class (such as geometry or group theory). Quantity, structure, space and change are general concepts which are used in a variety of topics. I didn't want to make the change without getting some feedback first though, so if I hear nothing back for a day or so I'll go ahead and do it. capitalist 02:24, 5 June 2006 (UTC)

I agree, "concepts" is better. -- Jitse Niesen (talk) 04:02, 5 June 2006 (UTC)

Comrade Jitse, the comissar is watching! No, we don't agree with those capitalists/monoplists/bourjois! Not at all. Oleg Alexandrov (talk) 04:14, 5 June 2006 (UTC)

Bah! Then I will now launch a hostile takeover! I will buy Misplaced Pages! Oh, wait. My board of directors has just voted me out. Never mind. :0) capitalist 04:11, 6 June 2006 (UTC)

Agreed here too. I made the change, but needed to add the words the study of to make the sentence flow. If anyone disagrees with that feel free to remove them. --darkliight 04:51, 5 June 2006 (UTC)

I think restricting it to study is a bit odd because it excludes use. A musician plays music as well as writing it. But no one else seems to be concerned, so I'll leave it until I have a better argument. Stephen B Streater 06:52, 6 June 2006 (UTC)

Music is the study of notes. But what about actually playing the music? Mathematics is the study of quantity. See my point? Stephen B Streater 08:30, 6 June 2006 (UTC)

Mmm, maybe I'm not seeing your point, but to be fair, we're not describing a mathematician or the use of mathematics, we're attempting to define the discipline of mathematics. Our opening sentence is, to the best of our collective ability, a definition of mathematics. We've said what we think it is. Now, a mathematician certainly uses mathematics, and this should be mentioned throughout the article, but the use of mathematics in the definition of mathematics is not required, surely? --darkliight 10:49, 6 June 2006 (UTC)

The word "study" has connotations of learning. To use your point, mathematicians study topics just as much as they use them, but you don't seen to mind the mention of studying. I would prefer a less restrictive form in the definition, such as "analysis of". This includes learning and use. Stephen B Streater

For me the connotation is not that strong. To solve a problem, unless it is routine, I study it. I study the issues and I study the approaches, for doing which I may repair to my study room. (Maybe I'm taking "mathema" too literally. :) ) --Lambiam 12:37, 6 June 2006 (UTC)

Suppose you are solving the equation 2x=4. Would you say this involved study? Study to me would be learning or working out how to solve the equation ax+bx+cxdx+e=0. But applying the formula would not be study, but would still be mathematics. Stephen B Streater 13:57, 6 June 2006 (UTC)

What's the difference between the two solutions, other than length? -lethe 14:52, 6 June 2006 (UTC)

I don't see applying a formula as study, but I do see it as mathematics. A random dictionary here (v1.0.1 of Dictionary on my Mac) says: study is "the devotion of time and attention to acquiring knowledge on an academic subject". On this definition, study excludes using the knowledge. I think mathematics encompasses the use of mathematical knowledge, not just learning it. So applying a formula is Mathematics as much as studying it is. Stephen B Streater 15:25, 6 June 2006 (UTC)

The above was definition 1. Definition 2 gives: "a detailed investigation and analysis of a subject or situation" - again not actually applying knowledge. Stephen B Streater 15:31, 6 June 2006 (UTC)

While I'm here, it gives mathematics as: "the abstract science of number, quantity, and space. Mathematics may be studied in its own right ( pure mathematics), or as it is applied to other disciplines such as physics and engineering ( applied mathematics). • the mathematical aspects of something : the mathematics of general relativity." - again not restricted to the study of things. Stephen B Streater 15:33, 6 June 2006 (UTC)

I'll just mention that I'm prepared to change my view if that's not how everyone else sees it. Stephen B Streater 15:36, 6 June 2006 (UTC)

The word "mathematics", especially in the abbreviated form "math" or "maths", is both the study of mathematics and also a subject taught in school. The same is true of "history", but we expect an encyclopedia article on history to focus on the professional meaning, not the schoolboy meaning. Solving 2x=4 is something a high school student would call "math" but an encyclopedia article will naturally focus on the more professional meaning. Rick Norwood 15:07, 6 June 2006 (UTC)

I'm weighing in on the side of removing the phrase "the study of". Does a discipline encompass concepts or does it encompass the study of those concepts (or both)? I think a discipline encompasses concepts. Mathematics is about quantity; it is not about the study of quantity. A discipline about the study of quantity would involve concepts such as how people learn, how they form abstractions or whatever. So a discipline like Cognitive Theory might encompass the study of quantity or the study of cells. But Mathematics encompasses the concept of quantity and Biology encompasses the concept of cells. Then again, I could be just babbling again... capitalist 02:37, 7 June 2006 (UTC)

We could say : "is a discipline that deals with concepts such as ...". The articles Physics, Chemistry and Biology all have "deals with". --Lambiam 04:15, 7 June 2006 (UTC)

This meets my constraints. I would also be happy with removing "the study of" completely, as this is more concise and is more accurate to me, in that mathematics doesn't "do" anything, it's more just sits there encompassing various areas. It's mathematicians who deal with things. I think it was changed from this, though I didn't catch the reason. "Mathematics is a discipline encompassing concepts such as ..." Stephen B Streater 06:10, 7 June 2006 (UTC)

Matching the sciences articles sounds good to me. JJL 12:50, 7 June 2006 (UTC)

It looks good, everyone looks happy with it and so I was bold and changed it ... again :) --darkliight 16:48, 9 June 2006 (UTC)

Thanks. Stephen B Streater 17:21, 9 June 2006 (UTC)

More intro work.

Since we're on a bit of a roll with the intro, here are a few more things I think are worth discussing:

• Should the word results be replaced with ideas or something similar in the last line of the first paragraph?
• Since the last line of the first paragraph is alluding to proof, I thought this sentence could be expanded a bit suggesting that ideas can sometimes take hundreds of years to prove, with many old ideas still yet to be proven and many new ideas still coming in. I'm not sure how to word it exactly, but I think it would help people realise that mathematics is a still growing body with many unsolved problems remaining.
• Ideas for the second paragraph? I was thinking this paragraph could be used to introduce the history of mathematics and lead into the third paragraph.
• Third paragraph? I was thinking this could be used to describe the current state and current uses of mathematics, similar to what the second paragraph does now.
• Any ideas on what to do with the line about the abbreviation of mathematics? I don't think it warrants its own paragraph, but I don't think it really belongs in the etymology section either .. or does it?

Anyway, I'm just trying to keep the ball rolling. Cheers --darkliight 08:29, 17 June 2006 (UTC)

A quick reaction to the very last bit: both Math and Maths redirect to Mathematics, and it is customary to mention commonly used alternative designations in bold in the intro (usually even in the very first sentence, but in this case that is not a good idea). --Lambiam 11:10, 17 June 2006 (UTC)

I agree with Lambiam, we need to keep the brief mention of "math" and "maths" where it is. Darklight's other ideas sound good, but of course the devil is in the details. Why not try making the changes one at a time and see what happens? Rick Norwood 13:02, 17 June 2006 (UTC)

I'm not too keen on ideas. You cannot prove an idea, you have to formulate it first. However, I agree that results is not good either when talking about statements that are not yet proven.

I am curious how the history can be summarized in one paragraph, and I'm not sure that would be sufficiently important to put in at the top. But it may well turn out to be better than I imagine. Perhaps you can write a rough start to give us an idea?

I'd love to see the "math" / "maths" paragraph go. It is indeed customary to mention other terms. However, that is so that readers will not arrive at the page in bewilderment. For instance, if you type in Burma, you end up at Myanmar, and if you don't know that these names refer to the same country and it isn't mentioned in the article, you'll be very confused. I don't think the possibility for confusion is great in the math(s) / mathematics case. -- Jitse Niesen (talk) 13:55, 17 June 2006 (UTC)

A friend and I had an attempt an the last sentence:

Mathematics is the discipline that deals with notions such as quantity, structure, space and change. It evolved, through the use of abstraction and logical reasoning, from counting, calculation, measurement and the study of the shapes and motions of physical objects. Mathematicians aim to justify/verify their concepts by rigorous deduction from axioms and definitions, which in some cases can take hundreds of years to accomplish.

• Instead of ideas, we used the word concept in place of the word results and ideas. It seemed to fit well, but the word was already used in the first paragraph. Another version I thought of was, reading almost straight of Jitse's comment: Mathematicians aim to formulate and justify/verify their ideas by rigorous deduction from axioms and definitions, which in some cases can take hundreds of years to accomplish.
• To save confusion we changed concepts in the first paragraph to notions.
• What do people think about the word verify in place of justify? While they both work, I think justify is little neater ... it has a more of a creative aspect to it and doesn't have the connotations of plugging numbers into a formula to get your answer.
• To convey that mathematics is still a progressing discipline, we've simply added which in some cases can take hundreds of years to accomplish. to the end of the last sentence. We could also add with many ideas still unproven or unjustified or unverified, but it starts to drag along a bit.

I tried keeping changes to a minimum, my main goal here removing the word results and hinting that mathematics is not complete. Ofcourse these are just some ideas for now, and I'll write something up for the second and third paragraphs next week. Cheers --darkliight 10:28, 19 June 2006 (UTC)

I suggested verify instead of justify because verify is a more precise, objective, specialised and idealised form of justify, and connotes truth (being derived from the Latin word for true). While the ideal may not be realised as frequently as we would like, an introductory paragraph should be as idealised as possible.

I also proposed somehow introducing the word explore, to suggest both the creativity and ongoing evolution of mathematics. My poor sleep-deprived brain came up with a somewhat clunky Mathematicians use rigorous deduction from axioms and definitions to explore and verify these concepts, a process which in some cases can take hundreds of years. Perhaps someone can work this into a prettier sentence.

-- Cuadan 13:04, 19 June 2006 (UTC)

I agree with Darkliight's dislike of verify as suggesting some trivial activity. Replacing idea with concept does not help with the problem I stated above: what does it mean to verify/justify concepts? However, there are some good ideas. How about:

… Mathematicians explore these and related concepts, aiming to formulate statements and establish their truth by rigorous deduction from axioms and definitions. This process may take hundreds of years.

To be honest, I'm not so happy with the last sentence. There may also be a better word for "statements". -- Jitse Niesen (talk) 13:35, 19 June 2006 (UTC)

I like Jitse's sentence — how about "conjectures" instead of "statements"? — Paul August ☎ 14:56, 19 June 2006 (UTC)

This sounds good, but the This process may take hundreds of years. part is just hanging there like a quick afterthought. Would it be too much to include it in the previous sentence? With Paul's suggestion we'd have ..

… Mathematicians explore these and related concepts, aiming to formulate conjectures and establish their truth by rigorous deduction from axioms and definitions, a process that may take hundreds of years.

My main concern now is that it may give the impression that a new conjecture might take hundreds on years to formulate, again implying that maths is mostly complete. Any idea's how to reword it to avoid that confusion? Cheers --darkliight 06:30, 20 June 2006 (UTC)

For the reasons Darklight gives and others I don't think tacking that phrase on the end is a good idea. So to be clear I would support the sentence:

… Mathematicians explore these and related concepts, aiming to formulate conjectures and establish their truth by rigorous deduction from axioms and definitions.

Like Jitse, I'm not particularly happy with the sentence: "This process may take hundreds of years." The idea that some conjectures take a long time to be proven true or false, is not important enough, in my view, to warrant being in the lead. And while it might be a way to help convey that mathematics is continuing to grow, I don't think it is the best way. Better might be to deal with this more directly with something like:

… Mathematicians continually explore these and related concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.

I've also added here "appropriately chosen" to convey the idea that the axioms and definitions are subject to change as well.

Paul August ☎ 15:53, 20 June 2006 (UTC)

Agreed about the hundreds of years sentence. I like your version, it gets the point accross and removes the need to for the word results nicely. Good job. Cheers --darkliight 03:45, 21 June 2006 (UTC)

I haven't been around much for a few days, but was thinking of "confirm" instead of "justify/verify". I think the new formulation has removed the need for this solution, but I'll leave it here in case it comes in handy later. Stephen B Streater 20:43, 20 June 2006 (UTC)

Paul's latest suggestion looks excellent to me, so I implemented it. -- Jitse Niesen (talk) 04:16, 21 June 2006 (UTC)

What point is being made with the word "continually" in "continually explore"? I presume it is not supposed to contrast with "occasionally explore". Is anything wrong with just "explore"? McKay 08:01, 21 June 2006 (UTC)

The word "continually" really doesn't work, here. Just "explore" is better.

Unless you want to make it, "to boldly explore where no mathematician has gone before". ; )

Rick Norwood 12:36, 21 June 2006 (UTC)

I take Rick and McKay's points. I've removed the word "continually". The point was to emphasize that the development of mathematics is ongoing, to help dispel the common misunderstanding that mathematics is a dead discipline. Perhaps "continue to" would have been better. However I think the lead probably deals with this adequately as it is. Paul August ☎ 13:48, 21 June 2006 (UTC)

Popper again

Still unsure about this falsafiability. I found the following quote from the paperPopper as a philosopher of mathematics

Popper is not usually regarded as a philosopher of mathematics. As mathematical propositions fail to forbid any observable state of affairs, his demarcation criterion clearly divides mathematics from empirical science, and Popper was primarily concerned with empirical science.

which could the the require citation. --Salix alba (talk) 11:54, 28 June 2006 (UTC)

This definitely doesn't back up the claim about what Popper himself believed. It may be relevant enough to include in the article, but previous discussions show that some people don't agree with the assertion, so it is only one POV. JPD (talk) 12:03, 28 June 2006 (UTC)

To establish falsifiability all that is required is a few clear examples that illustrate why certain mathematical expressions cannot be implicitly trusted as being representative of something real. I submitted such an example this morning but was rv'ed almost immediately by JPD. Archived discussions about this seem to circle around the semantics of Popper's position on Math, rather than tackling the validity of the idea that science (our knowledge of the real world) is a very restricted subset of the realm of mathematics. My rather trivial example, for what it's worth follows: ${\displaystyle ...{n^{2} \over m^{2}}+{o^{2} \over l^{2}}+{p^{2} \over k^{2}}+{q^{2} \over j^{2}}+{r^{2} \over i^{2}}+{s^{2} \over h^{2}}+{t^{2} \over g^{2}}+{u^{2} \over f^{2}}+{v^{2} \over e^{2}}+{w^{2} \over d^{2}}+{x^{2} \over c^{2}}+{y^{2} \over b^{2}}+{z^{2} \over a^{2}}=1}$.

Although this formula might satisfy the requirements of mathematical logic and could be differentiated and integated, such a multidimensional construct would be quite impossible in nature. Imre Lakatos, has used similar reasoning in his review of mathematical logic.Geologician 15:51, 28 June 2006 (UTC)

I don't think this quite addresses the point. Previous discussions have also featured disagreements concerning whether mathematics satisfies Popper's criteria or not. The question is not whether mathematical expressions can be trusted as being representative of something "real" (by this I assume you mean physical), but what it means to be falsifiable. In this example, it is not clear what it means to say whether the "multidimensional construct" is possible or impossible in nature. JPD (talk) 16:03, 28 June 2006 (UTC)

The expression implies the existence of at least 13 dimensions, more than enough for most versions of string theory. Even if there are 26 dimensions the mathematical realm is not limited by that constraint.Geologician 16:41, 28 June 2006 (UTC)

If you interpret dimension as degree of freedom, then the physical world has quite a few. Stephen B Streater 18:46, 28 June 2006 (UTC)

Constructive idea Stephen. Say there are 12 degrees of freedom :

sticking with euclidean geometry, these apparently could occupy the same dimension space described in string theory. So math expressions that extrude beyond that realm of reality remain 'Beyond the Pale'. Geologician 09:37, 29 June 2006 (UTC)

That diagram is meant to be 12 degrees of freedom of a hydrogen molecule. There are a lot more complicated systems out there than a hydrogen molecule. Your argument seems to rely on a particularly restrictive idea of what can represent something physically real and doesn't at all address other viewpoints. JPD (talk) 11:15, 29 June 2006 (UTC)

I am purposely avoiding semantics and retricting my argument to reality. Reality is about expressions like 'If you knock your head aginst a brick wall you will get hurt" "Water is wet" Simple things that human beings have known since the dawn of time, but more recently may have eluded the attention of some academics. Geologician 14:02, 29 June 2006 (UTC)

I fail to see how the question of whether maths is a science can be dealt with without semantics. The notion of science is not something I can knock my head on, so we can't restrict ourselves to that sort of "reality". JPD (talk) 16:18, 29 June 2006 (UTC)

And you don't need to go to string theory for physically meaningful examples of spaces with 26 or more dimensions. A classical model of the solar system will live in a phase space with at least 9x6=54 dimensions (and more if you want to model satellites, minor planets, comets etc.). Gandalf61 12:25, 29 June 2006 (UTC)

Introducing phase space is a red herring. Phase space is merely an aid to _visualization_ of (states of) matter that itself is possibly confined to the restricted dimensions allowed by string theory. Just because something can be visualized doesn't mean its relevant to reality. For example, I can quite easily visualize a moon made of green cheese. Pressure, temperature and composition are not dimensions, simply mathematical constructs that describe the state of matter in space and time. Let's concentrate on attempting to define limits beyond which mathematical expressions cease to be relevant to the real world. Geologician 14:02, 29 June 2006 (UTC)

With all due respect, I think it is insisting that expressions are only relevant to the "real world" in terms of your "dimensions", and even discussing relevance to the physical world are the red herrings here. See Salix alba's comment below. JPD (talk) 14:31, 29 June 2006 (UTC)

Geologician - if you read beyond the introductory paragraph in the phase space article you will see that phase space is not the same as a phase diagram in physical chemistry. The concept of phase space has nothing to do with "visualization"; it is not concerned with states of matter; it has not connection at all with string theory, which it pre-dates by over a century; and the co-ordinates in phase space are positions and momenta, not pressure, temperature and composition. And it is intensely relevant to answering questions such as "will Halley's comet impact the Earth in 2061 ?", which seems like a "real world" problem to me. Gandalf61 15:50, 29 June 2006 (UTC)

Math is all about visualization. Position and momentum are simple functions of mass, time and distance so they fit neatly within the constraints of string theory dimensions. (Direction is not a dimension) Geologician 23:27, 29 June 2006 (UTC)

Hmmm. Well, either you really have a really flawed understanding of the basic terms and principles of mathematics and physics, or you are pretending so in order to provoke a response. In either case, I see no point in continuing this discussion. Gandalf61 07:51, 30 June 2006 (UTC)

Care is needed to distinguish mathematics from mathematical physics. The latter attempts to address science (our knowledge of the real world)(Geologician above) whereas Pure mathematics does not - Hardy's apology being a prime example of the lack of real worldness.

For lack of real-worldness we don't need to consider anything as exotic as higher dimensional spaces, some of the most basic abstractions of mathematics: say the concept of a surface do not correspond to anything physical, as no physical entity will have zero thickness. There is a good section in one of Feynman's books on distinction between abstract mathematical entities and real world entities. Whether that most basic concept of the real numbers is more than a useful abstraction is a good question: is space infinitely divisible? Is this statement falsifiable?

In any case we still have

Karl Popper believed that mathematics was not experimentally falsifiable and thus not a science.

without any direct evidence that he was so crude in his thinking. A better statement might be

Some authors have used Poppers's concept of experimental falsifiability to distinguish mathematics from the empirical sciences. Ref Glas

--Salix alba (talk) 13:33, 29 June 2006 (UTC)

That sounds much better to me, although it probably would be less "weaselly" to mention some of these authors in the text, as well as giving a reference. JPD (talk) 13:52, 29 June 2006 (UTC)

In response to JPD's 14:51 suggestion re. Salix alba's comment above. I am well aware of the distinction between Pure and Applied Math. However surely Popper's concept of experimental falsifiability is relevant only to the latter. Applied math (or mathematical physics) seeks to use the most appropriate tools available in the mathematical grab-bag to describe physical phenomena. These phenomena are prepared for mathematical analysis by making measurements. Measurements convert the observation into real numbers curtailed to a discrete number of significant digits. It is at that point that Salix alba's concept of infinitely divisible space breaks down. Numbers only become relevant to the real world when something can be measured. And no result of a calculation can attain more precision than the component measurement made with the least precision. Regrettably no scientific calculator software facilitates the tracking of the least significant digit. My point is that mathematics tend to promote the idea that it is "the Queen of the Sciences" but fails to deliver when called to account. Geologician 15:12, 29 June 2006 (UTC)

The article makes it quite clear that the use of the word "Sciences" in that description is not the meaning most often used today. We are talking about mathematics, not mathematics as used in physics. Your claim that Popper's concept of falsifiability is not relevant to mathematics in general is exactly what is disputed. We can't put examples into the article in a way that assumes your POV. I probably agree with you more than it seems, but I think it is important for the article to be in NPOV terms, using wording such as that which Salix alba suggests. JPD (talk) 16:18, 29 June 2006 (UTC)

Popper used the word in the modern sense so we shouldn't be distracted when we consider Salix alba's proposition that: Karl Popper believed that mathematics was not experimentally falsifiable and thus not a science. Apparently no-one has yet come up with a citation to support this, however 'falsifiability' derives from statements like "no empirical hypothesis, proposition, or theory can be considered scientific if it does not admit the possibility of a contrary case." As mathematics is a collection of these components, one cannot test the entire collection by Popper's criterion, one has to consider each component individually. Otherwise the whole of mathematics stands or falls by the reliability of one of its components. Therefore it is perfectly legitimate to point out that statements like "the area of a circle of radius 1.7 cms is 9.079 sq.cms" is in significant error and the true area is actually 9.1 sq cms. Consequently the key proposition that the area of a circle is pi times the radius squared is also wrong, if it lacks the important caveat that the result cannot have more significant figures than the radius. And so we can proceed with a hatchet though the entire corpus of mathematical formulae insofaras they relate to the real world. QED Geologician 20:53, 29 June 2006 (UTC)

In mainstream mathematics the hypotheses, propositions and theories are not empirical. While it is possible to do "experimental mathematics", in which conjectures may well be falsified, the claim that in Euclidean geometry the area of a circle with radius r equals πr has an entirely different status. Should the measurements of a physical embodiment of a circle reveal something different, three possible explanations are: (1) our measuring apparatus is defective or insufficiently precise; (2) we are measuring an imperfect physical model of the ideal circle; (3) the local geometry is not Euclidean. The experiment will not put the claim in doubt. It is increasingly unclear what you are trying to say. --Lambiam 21:53, 29 June 2006 (UTC)

P.S. A quote from Einstein (according to Wikiquote): As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. (I'd like to add that as far as anything refers to reality it is not certain.) --Lambiam 21:58, 29 June 2006 (UTC)

You'd like to, but can't be entirely certain ;-) Stephen B Streater 08:24, 30 June 2006 (UTC)

Custodian of the realm of numbers

It is proposed that mathematics re-establishes itself as custodian of the realm of numbers. There is universal evidence that numbers are being misused in every scientific discipline, usually though ignorance of the limitations of real numbers when used in calculations founded on measurements and estimates. Mathematicians should insist that scientists remember that significant digits inevitably curtail the significance of their results. For example, if this surveillance were in place we would not see absurditiies such as those in Terry Quinn's 2003 paper in Metrologia, (p105) giving the wavelength of various radiations to a precision of 14 or 15 significant figures. In 1.1 for example lamda is given as 236 540 853. 54975 femtometres. The trailing digits after the decimal point imply that this measurement was accurate to one ten-thousanth part of the diameter of an electron. The standard metre bar was only ever measured to seven signifcant figures; later the metre was established by convention with nine significant figures. Where did the additional five significant figures in Quinn's results come from? Geologician 10:28, 30 June 2006 (UTC)

What has this got to do with the article??? JPD (talk) 10:33, 30 June 2006 (UTC)

If mathematics' responsibilities as Custodian of the realm of numbers are spelled out in the article it will be less likely to shirk them, as is apparently the case. Geologician 12:08, 30 June 2006 (UTC)

Your question is irrelevant to this article, but here is the answer. The metre was not "established by convention with nine significant figures". It was defined exactly in terms of the time standard, which in turn is defined exactly in terms of a particular physical process. The accuracy of the wavelength measurements reflects the accuracy of measuring radiation frequency in terms of the time standard. McKay 12:56, 30 June 2006 (UTC)

Mckay's comment returns us rather neatly to exactly the point I have been making in the previous section. Karl Popper said "no empirical hypothesis, proposition, or theory can be considered scientific if it does not admit the possibility of a contrary case." As McKay so helpfully points out, here at the very heart of physics is clear evidence that metrology cannot be regarded as a science. This is because it is impossible to calibrate any future measurement against a standard that has merely been defined. When a universally respected yardstick such as the standard meter rod existed, an investigator could check his laser measuring device against a physical length. As this has apparently been abandoned the wavelength of light has itself become the standard and does not admit the possibility of a contrary case. Wavelength is itself dependent upon relativistic effects. Therefore there is no future possibility of determining whether there is any actual drift in the velocity of light with time. In short no electromagnetic method of distance measurement is subject to calibration against a physical distance that exists on Earth. Can nobody else smell a rat? Geologician 22:36, 30 June 2006 (UTC)

I am not sure exactly what you think "mathematics" is, if you say it "promotes itself as the Queen of the Sciences" and "shirks its responsibilities", and I am even less able to see how changing this article is likely to have any significant effect on how it acts. Please stick to issues relevant to the article. JPD (talk) 13:07, 30 June 2006 (UTC)

Perhaps a perusal of WP:SOAP#Misplaced Pages is not a soapbox would be in order. Stephen B Streater 15:29, 30 June 2006 (UTC)

Okay. Misplaced Pages was not made for opinion, it was made for fact. Fine. I have been pointing to a series of facts that appear to support the need for a paragraph in the Mathematics Article that outlines the realm of responsibility of mathematics in the real world. Of course there will always be mathematicians who prefer to conjure with abstractions in an airy-fairy world, and good luck to them. Perhaps they might avail of an equal opportunity to explain in Wiki why the rest of us owe them a living. Geologician 23:02, 30 June 2006 (UTC)

Mathematics is a subject, not a person or organisation. A subject cannot have responsibility any more than a word can. Stephen B Streater 23:09, 30 June 2006 (UTC)

I much prefer to avoid semantic discussions, but a discipline's name usually provides clues to its intended realm of responsibility. For example, the Wiki on Bioinformatics begins with a nice clear description of its relevance to and place in the real world. The start of the Mathematics article talks about concepts and abstractions without ever mentioning numbers, which are its vital link to the real world. Geologician 09:42, 1 July 2006 (UTC)

Mathematics is an area of knowledge which gives power without responsibility. There is no "intended realm of responsibility" intrinsic in Mathematics. Numbers are only one link between Mathematics and the real world. On the other hand, many parts of Mathematics have no link to the real world. Stephen B Streater 18:03, 1 July 2006 (UTC)

Of course Math has a realm of responsibility. Check out The Mathematical Atlas If you are a dyed-in-the-wool semanticist you can call it the mathematical landscape. You will find 'Significant Figures' tucked away obscurely in Area 62 (Statistics) rather than upfront where it belongs. Geologician 11:23, 2 July 2006 (UTC)

I might have misunderstood what you mean by responsibility. What do you mean by responsibility? Stephen B Streater 15:37, 2 July 2006 (UTC)

Exactly the same as Concise Oxford Dictionary (var2). Geologician 17:42, 2 July 2006 (UTC)

Are you saying that Mathematics has some sort of duty? Stephen B Streater 18:22, 2 July 2006 (UTC)

I refuse to be drawn further into semantics. Read with more care what I have already written. Geologician 20:47, 2 July 2006 (UTC)

Maybe the article should also state that mathematicians must be guardians against the abuse of very large numbers, like disgustingly rich people can only be that rich because their bank accounts are specified with very large numbers. If we spell this out in the article, we will witness a new dawn of humanity. --Lambiam 22:38, 30 June 2006 (UTC)

We should be taking out cut somewhere. If only we were lawyers ;-) Stephen B Streater 22:43, 30 June 2006 (UTC)

I think there are some mathematicians who do have this sense of duty. Its maybe more common in statistics where missuse of numbers can have a large effect on policy. I've seen talks by statisticians devoted to how statistics are missused and ways to combat this problem. See for example A Mathematician Reads the Newspaper. --Salix alba (talk) 21:04, 2 July 2006 (UTC)

Yes, I agree, many mathematicians have private senses of duty. I have of sense of duty concerning every good act I do. However I don't really see what this discussion has to do with this article. Paul August ☎ 21:25, 2 July 2006 (UTC)

The introduction needs to emphasise the importance of number as the foundation in reality of all mathematics. Even crows can count. Thus when other disciplines err when they intrude into the realm of mathematics then mathematicians should not stand idly by. For example when folks whose background was in the discipline of religious studies began revising the tenets of evolution, others whose background was in the realm of paleontology were quick to point out the relevant fallacies. Likewise when physicists make serious errors in their assumptions about what thay can do with numbers, mathematicians are bound to defend their territory with a rejoinder. In the case of metrology (mentioned above), this apparently hasn't happenend. Geologician 08:59, 3 July 2006 (UTC)

I agree with this too, but I will look back at what Geologician has written in more detail as he suggests. Stephen B Streater 21:46, 2 July 2006 (UTC)

You and Geologician seem to be using the term "responsibility" in two different senses, and thus have been talking past each other. Geologician is using the term to mean the set of concepts covered by the subject of mathematics. This is the same sense in which we use the word when we say, "The division's area of responsibility stretched from the edge of the river to the town of Metz." It is a way of determining what the boundaries of some subject are. The Mathematical Subject Classification System (MSC) is a good example of this. You have taken it in the sense of a "duty" as in, "We all have a responsibility to feed our pet aardvarks." capitalist 02:36, 3 July 2006 (UTC)

Perhaps we could use the word scope to avoid ambiguity. Stephen B Streater 06:34, 3 July 2006 (UTC)

Geologician is using "responsibility" in all sorts of ways and using a dislike of "semantics" as an excuse for not being clear about what he is saying. Of course the intro should emphasise the scope of mathematics in relation to the "real world", which it already does. Quantity is actually a better description of what we are talking about than number, and space, change and structure are just as important. The possible responsibility of mathematicians to correct people's misuse of numbers is another matter, and is completely irrelevant to the article. JPD (talk) 10:47, 3 July 2006 (UTC)

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