Talk:Mathematics/Archive 13

Let's try to work on a version of the lede here instead of getting into an edit war. As I mentioned, I object to tracing the axiomatic method back to Euclid in describing modern mathematics, because this was simply not the case before Peano, Hilbert, and Co, and we don't really know what Euclid and his contemporaries did. Describing mathematics flat out as formal derivation of theorems from axioms is philosophically naive. Tkuvho (talk) 10:42, 2 May 2011 (UTC)

I agree with your statements, but I wish that you would acknowledge your previous misattribution of naiveties to me---not that my skin is thin but my respect for you is substantial.

There should be no message of Peano, Hilbert, and Co. in the lead, because they are ignored in the body, per MOS. On the other hand, we should agree that for 2000 years, there was essentially one book of deductive mathematics, The elements of Euclid (and nobody cares about the historical author, at least for this article). We should also mention Euler's Analysis of the infinite as the calculus textbook for c. 200 years.

The article's discussion of pure mathematics and putting foundations ahead of mathematics is distorted. Only in the 20th century did there appear professional mathematicians ignorant of applications and not trying to solve the urgent scientific problems of the day. It is undue weight to describe pure mathematics with so little concern with applications.  Kiefer.Wolfowitz 11:02, 2 May 2011 (UTC)

I did not say you were naive, I said that thinking of modern mathematics as formal derivation from axioms is naive. Peano and Hilbert should certainly not be ignored in the body since they are responsible for the modern axiomatic philosophy. I can't see why Benjamin Peirce should be mentioned but not Hilbert. How many mathematicians will agree with you on this choice? Certainly Euclid can be mentioned, but as I pointed out already it would be inaccurate to portray 17-19 century mathematics as dominated by anything at all resembling an axiomatic approach, except in some isolated areas. Tkuvho (talk) 11:42, 2 May 2011 (UTC)

Good: We avoid discussing axiomatics, then. (I tried to reduce the axiomatics's undue weight in the lead.)

There are several Pierces. Benjamin Peirce's definition is notable (and I provided an improved reference to the edition cited, which was edited by CSP, his son). I do not say that Renaissance & Enlightenment mathematics was axiomatic, only that Euclid was upheld as the example of axiomatic reasoning during that time (until c. 1850).

I suppose that you admit that the current definition of mathematics in this article, follows popular authors, but cannot be taken as serious. (The present article does not seem to meet the requirements of reliability, imho.) B/CS Peirce's definition can be taken seriously, on the other hand. Andrew Gleason wrote a similar definition of mathematics, if my memory is correct. Next week, I should be able to provide references to further discussion.  Kiefer.Wolfowitz 12:38, 2 May 2011 (UTC)

I agree that Einstein's definition is a bit of a joke. Hilbert's definition is more serious, and he is certainly more of a heavyweight than Benjamine Peirce. Tkuvho (talk) 16:15, 2 May 2011 (UTC)

B. Peirce's definition appears in the edition edited by his son, who is taken rather seriously by philosophers and logicians. CSP's New Elements of Mathematics has a long discussion of alternative definitions of mathematics, which is worth consideration.  Kiefer.Wolfowitz

OK, I did not know that. Thanks for pointing it out. We should make a note of that. CSP is certainly more of a heavyweight, perhaps even competing with Hilbert. At any rate, I find Hilbert's remark more informative. Tkuvho (talk) 16:43, 2 May 2011 (UTC)

I should have applauded your addition of Hilbert's quote. CS Peirce has similar remarks, claiming that every mathematician is a Platonist wanting to find the truth, and nobody can explain the advance of mathematics (and more generally science) by just pretending that it's a formal game (or establishment of consensus): This was one motivation for my changing the wording of the account of proofs---which do more than convince others.  Kiefer.Wolfowitz 17:00, 2 May 2011 (UTC)

Thanks, great. My next project is an AfD for formal science, which is a fabrication. Tkuvho (talk) 17:05, 2 May 2011 (UTC)

• Groan* I know that it is a lost cause to plead to a logician for restraint and allowances for convention and the fallible misguided masses. ;) Can you suggest another alternative phrase for mathematical logic, mathematical statistics, mathematics, and theoretical computer science, and perhaps information theory?  Kiefer.Wolfowitz 17:28, 2 May 2011 (UTC)

I have the highest respect for all these fields. How about "living sciences"? Tkuvho (talk) 17:33, 2 May 2011 (UTC)

I still have philosophical problems with this: "Since the pioneering work of Giuseppe Peano, David Hilbert, and others on axiomatic systems in the late 1800s, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. When those mathematical structures are good models of real phenomena, then mathematical reasoning often provides insight or predictions." The first sentence seems more applicable to logic than math. The second is a tautology: is the definition of "a good model" that which "provides insight and predictions?" Isn't insight subjective? — Preceding unsigned comment added by Mythirdself (talk • contribs) 18:25, 26 May 2011 (UTC)

The first sentence has undue weight on foundations, perhaps reflecting the interests of active writers. "Axiom-chasing" & foundations are a big concern for a small number of research mathematicians, roughly say the same number as pursuing applications in quantum mechanics.

A good model makes a cognitive difference, and is subjective in the sense that good mathematical models don't matter to mollusks or mud. You can find a discussion of modeling in the SIAM reprinted book on Mathematics Applied to Deterministic Problems in the Natural Sciences or von Neumann's article^[7]or C.S. Pierce's, etc.  Kiefer.Wolfowitz 19:44, 26 May 2011 (UTC)

I agree. We can work to reduce the foundational bias of the lede. If you look up the history of the page, you will see that it evolved in the right direction: the page used to start out with a categorical announcement to the effect that mathematics is a formal science, period. I assume that meant it was deriving consequences from postulates, no? Of course, most working mathematicians don't work that way. Tkuvho (talk) 19:59, 26 May 2011 (UTC)

Quine's discussion of mathematics as the most formalized part of our language (roughly) is another useful statement by an authority. I made some comments at Applied Mathematics's talk page, urging this article to discuss applications more. (Laplace and Fourier leading to Harmonic analysis, and quantum mechanics and Hilbert space, etc.)  Kiefer.Wolfowitz 20:09, 26 May 2011 (UTC)

I'm not sure quoting a famous and idiosyncratic figure like Quinne is a good idea. He's brilliant, but he has his own very Quinnian views. Something blander (hey, it's an encyclopedia not a revolution) might be more appropriate. Mythirdself (talk) 18:37, 2 June 2011 (UTC)

Hi, Perhaps the user has not understood the comments made to 'change the syntax'. The [| reference book]'s [| introduction], [| Development of Philosophy], etc. are indeed exercises in glorifying Greek Mathematics. How is this a reliable source is just one question, the point here is 'The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300 BC.' while at other places like India [| you may find the same], another source. The book therefore can only be interpreted as a reference to beginning of the systematic study of mathematics in its own right in the Ancient Greece. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 17:00, 6 June 2011 (UTC)

The edit in question involved changing "began with the Ancient Greeks" into "began in the Ancient Greeks". I don't see how that change has anything to do with reliability of whatever sources; it says the same, just with bad syntax, as far as I can tell. Marc van Leeuwen (talk) 08:59, 7 June 2011 (UTC)

Thanks for clarifying, though I later changed it to in Ancient Greece which again was removed. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 13:50, 7 June 2011 (UTC)

Hi, I would like to know why are the changes reverted after deleting 4 sources in the guise of statement that one source is unreliable without giving any proofs. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 07:10, 6 June 2011 (UTC)

Doing a spot check a second of the 4 sources says nothing about mathematics in India, it is about early dentistry. Does you have a page ref on the French work? Thenub314 (talk) 17:38, 6 June 2011 (UTC)

Your edits were reverted because most standard sources on the history of Mathematics (e.g. Boyer and Merzbach A History of Mathematics, Wiley, 1991) place Egyptian and Babylonian mathematics far earlier than Indian mathematics. The sources you are using are also unreliable/irrelevant. Subhash Kak is a well-known Hindutva nationalist whose "work" in this area has drawn widespread criticism. He is moreover not a historian of science or mathematics. Another of your sources is a paper on archeo-dentistry. What does that have to do with mathematics? As for your other two sources, would you be so kind as to quote what they actually say on the subject? Let's face it, the Indus Valley civilization didn't leave behind any mathematical or even written records of any kind, so to claim that mathematics began with it is nonsense. Athenean (talk) 17:36, 6 June 2011 (UTC)

[[:Image:Neolithic mehrgarh.jpg|thumb|right|x108px| text|Early farming village in Mehrgarh, c. 7000 BCE, with houses built with mud bricks. (Musée Guimet, Paris).]]

Again, let us see sources claiming 'Subhash Kak is a well-known Hindutva nationalist whose "work" in this area has drawn widespread criticism. He is moreover not a historian of science or mathematics', which is repeated again without proof. About the rest of the two sources, I will see where I can get details. That "Indus Valley civilization didn't leave behind any mathematical or even written records of any kind" is again another assertion without proof.

Here is the actual source [| with images!!] next to it. It does look like solid geometry, maths, simple algebra, weight and economics to me, but I may be wrong here.

Lets see if I can show a visual proof here - the images you see on the right from IVC and Mehrgarh - present structures that only a deluded would feel could be done with no sense of Geometry.

[| Early civilizations of India, an article by Bernard Sergent, CNRS researcher President of the Society of French mythology] clearly mentions "Le système de mesure indusien a été explicité. The measurement system has been explained indusien. On possède même trois règles graduées. It even has three rulers. L'une, du site de Lothal, a pour plus petite unité une longueur de 1,7 mm. One, the site of Lothal, a smaller unit for a length of 1.7 mm. Les systèmes de mesure varient extrêmement d'une civilisation à l'autre. Measurement systems vary extremely from one civilization to another. Or, le premier système de l'Inde, défini dans l' Arthasâstra de Kautilya, sans doute au IVe siècle avant notre ère, donne une unité minima de longueur de 17, 86 mm, le décuple presque exact de la mesure indusienne. However, the first system of India, as defined in the Arthasastra of Kautilya, probably in the fourth century BC, gives a unit minimum length of 17, 86 mm, almost exactly ten times the extent indusienne."

Another quote "As one scholar of Finland, Asko Parpola, comes to assume. The geometrical arrangement and oriented along the cardinal points of the cities of the Indus involves celestial observations. after the Rg-Veda, who knows at this point that myths, which he makes vague references - uses a tracking system that has absolutely heavenly no parallel elsewhere in the field Indo-European. This is the system naksatra: a set of twenty-seven points celestial defined by bright stars or constellations, which define the circuit of the moon. The only thing that looks like this is a Chinese counterpart, and also in ancient Chinese culture that are naksatra in Indian culture. Parpola, which I summarize very, reasons thus: the establishment of a system as rigorous for Astronomy in India Brahmin surprise, he cons easily explained by the Indus civilization, which had developed measurement system and was interested in heaven..." ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 19:19, 6 June 2011 (UTC)

Another source - The Shatapatha Brahmana (शतपथ ब्राह्मण śatapatha brāhmaṇa, "Brahmana of one-hundred paths", abbreviated ŚB) is one of the prose texts describing the Vedic ritual, associated with the Shukla Yajurveda. It survives in two recensions, Madhyandina (ŚBM, of the vājasaneyi madhyandina śākhā) and Kanva (ŚBK, of the kāṇva śākhā), with the former having the eponymous 100 adhyayas,7624 kandikas in 14 books, and the latter 104 adhyayas,6806 kandikas in 17 books. Linguistically, it belongs to the latest part of the Brahmana period of Vedic Sanskrit (i.e. roughly the 8th to 6th centuries BCE, Iron Age India). REF:- Keith, Aitareya Aranyaka, p. 38 (Introduction): "by common consent, the Satapatha is one of the youngest of the great Brahmanas"; footnotes: "Cf. Macdonell, Sanskrit Literature, pp. 203, 217. The Jaiminiya may be younger, cf. its use of aadi, Whitney, P.A.O.S, May 1883, p.xii."

Another source - [| engineering skill] of Harappans. A few pages about this page (backword and forward) are also interesting. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 20:14, 6 June 2011 (UTC)

Yet another source, [| grammar] and more.

Grammar? Engineering? This is insane. If you don't know that Subhash Kak is not a reliable source, you might want to start by reading the article about him. The earliest mathematical documents from India are from the first millennium BC, whereas those from Egypt and Mesopotamia are from a millennium earlier, hence much much older. Your sources prove nothing. Bricks and rulers are not "mathematics". The IVC did not leave behind a single mathematical document. Images of mud-brick huts also prove nothing. Athenean (talk) 08:00, 7 June 2011 (UTC)

Is this how Wikipedia works? Calling facts presented with secondary sources as insane and then giving random statements without and proofs? Wikipedia is becoming digital cesspool where secondary sources are not asked from supposedly high-borne who can talk anything and get away with it. Where are the secondary sources to support all the wild assertions that begin the arguments and are maintained throughout, effectively ignoring the whole discussion? ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 08:08, 7 June 2011 (UTC)

Start by reading an actual source on the history of mathematics, like Boyer & Merzbach which I have already mentioned above. And yes, showing pictures of mud-brick villages and random irrelevant sources about grammar and engineering is quite weird. What does this have to do with mathematics? Can you show me a single mathematical document from the IVC? I didn't think so. Athenean (talk) 08:14, 7 June 2011 (UTC)

Please throw some light on 'actual source' terminology. Besides it is indeed weird to know from Wikipedia that Engineering has nothing to do with Mathematics and I am alarmed by such a random statements on this page, especially after a secondary source -this- is presented exactly to clarify the same, with numbers and all. Another reference if needed [here]. About 'mathematical documents' - it would be again primary source! Please refrain to demand anything other than secondary sources on talk pages, it is against standards. By the way, where are primary sources from Greece that are understood as 'mathematical documents' and why are these considered on this pages?

About the source mentioned A History of Mathematics, By Carl B. Boyer, Uta C. Merzbach, here is a link for uninformed [| from the same source], about Early Indian Mathematics. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 11:17, 7 June 2011 (UTC)

Please read your own edit carefully. You claim that mathematics more advanced than that employed in trade and measurement appeared in 3000 BC also in India, but your source refers only to exactly this kind of less complex mathematics, trade and measurements as "weights and measures" were used fire and foremost in trade and bartering. And, besides, to support your major claim you'd need anyway a fuller discussion than this one sentence tidbit. Gun Powder Ma (talk) 12:05, 7 June 2011 (UTC)

Well, I don't consider this as 'personal shortcut to glory' when it is the people of the Indus Valley who should be credited for the whole arrangements, not I am presuming 'more advanced versus less advanced mentality' here from either side, for I am not a bigot.

A better treatment of the 'weights and measures':-

THERE IS INDEED A SCALE, symbolic(and actual usage) of calibrations I presume and 3000BC or still remote dating of IVC (another source - yes, 3000 BC, yet another source).

More about the THE SCALE- the ruler - a few more details, so also some common sense in words presented for 'better treatment'- 'indicative of numerate culture with well-established, centralized system of weights and measures' some common sense explained in words (yes, numerate culture is the word) and 'decimal system of linear measurements' suggested by THE SCALE. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 13:18, 7 June 2011 (UTC)

The claim that the "Indians" developed mathematics, alongside Egypt and Mesopotamia, as early as 3000 BC is hyperbolical for two reasons:

1. In 3000 BC, there was still no Indian civilization, but rather forerunner copper and bronze cultures. The only high culture potentially capable of producing advanced mathematics was the Indus Valley Civilization, but this had not even entered its mature period and its designation as "Indian" is problematic to say the least (and consequently avoided in the WP article, so WP consensus).
2. The only known possible script on the entire Indian subcontinent was the Indus 'script' which appeared by the mid-3rd millennium BC. This 'script', however, is still undeciphered so it is impossible to make any inferences as to its contents, including its mathematical level and use of numbers. Gun Powder Ma (talk) 09:31, 7 June 2011 (UTC)

I agree. I've just read the article by Bernard Sergent referred to above (in the French original), and it makes clear most of all that whatever written traces there are from the Indus civilization cannot be deciphered (for lack of any "Rosetta stone" for it); this would certainly exclude having any trace of a "study of mathematics in its own right". It does not exclude having any trace of mathematical activity (one could imagine finding tables of some sort that suggest such activity) but the cited article does not in fact mention any mathematics at all. It does refer to traces of Indus culture which indicate measuring and astronomical observation (and of which it is discussed in how far any heritage of it can be found in more recent cultures). This does not seem to justify in any way including "Indians" (whatever that may designate) in the sentence "More complex mathematics did not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy". With all due respect for the Indus culture, there seems to be no ground for mentioning it in this article. Marc van Leeuwen (talk) 12:43, 7 June 2011 (UTC)

The source for the above line "More complex mathematics did not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy" - just says "Kline 1990, Chapter 1."! The thinly detailed reference material stays though anything else is refuted with one reason or another.

Is this how Wikipedia works? Another reference is A History of Greek Mathematics:

From Thales to Euclid, Volume 1 - here that goes endlessly about glory of Greek Culture right from the beginning which is just fine, but that does not mean no one else thought of anything else. The nature of references here is indeed disturbing. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 13:32, 7 June 2011 (UTC)

I think you are confusing mathematics with the various rules-of-thumb used to build structures. Standard reference works agree that Greek mathematics began at an earlier date than Indian mathematics. That in no way minimizes the many important contributions of Indian mathematics. Rick Norwood (talk) 14:19, 7 June 2011 (UTC)

Where are the "Rosetta stone" proofs of these Greek Mathematics? Is there no SCALE?

So the current understanding I assume as per 'Standard reference works'(making it possible to disregard any other secondary sources), in spite of elaborate town planning (along with stone built houses, elaborate drainage system, adequate water supply etc.), great uniformity in the Harappan products throughout vast territories benefiting both producer and consumer, uniform trade mechanisms introduced across cities (including weights), and understanding on Numbers, Space and Time can hardly be passed off as rule of thumb 'just to build structures' when the line in consideration itself talks of maths 'for taxation and other financial calculations, for building and construction, and for astronomy'! ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 15:19, 7 June 2011 (UTC)

It sounds like a review of the English Wikipedia policies on reliable sources, original research and synthesis are in order. --John (User:Jwy/talk) 16:13, 7 June 2011 (UTC)

Indeed, considering the random introduction of terms like 'standard sources on the history of Mathematics', 'mathematical document', 'actual source', "'script', however, is still undeciphered so it is impossible to make any inferences as to its contents, including its mathematical level and use of numbers", 'rules-of-thumb used to build structures', 'Standard reference works' etc. are thrown around as excuses to ignore secondary sources. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 06:56, 8 June 2011 (UTC)

I thought discussions on Wikipedia would be at a higher level than the usual Internet fare. Especially for the article on mathematics. Does being educated and sharing a good cause reduce the incidence of flame-wars? — Preceding unsigned comment added by Mythirdself (talk • contribs) 01:45, 9 June 2011 (UTC)

I am sorry to inform you that education only serves to make flame wars worse. When your educated your more likely to believe your are right. Usenet was started by academics you know. Thenub314 (talk) 02:44, 9 June 2011 (UTC)

Just found a page on Ruler on Wikipedia for information. I am sure it applies to mathematics to an extent. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 19:52, 16 July 2011 (UTC)

Thisthat2011, Wikipedia talk page guidelines explicitly say, Never address another user in a headline. You seem to do this over and over again. Please desist. Please change the headlines of the section and the one below to more neutral ones. Fowler&fowler«Talk» 01:00, 19 July 2011 (UTC)

Done! ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 06:20, 19 July 2011 (UTC)

I think the Etymology from Semitic Arabic is the important thing. Because it tells the history of Mathematic just from the etymology and begin crediting some specific mathematic such as Al-Jabar, Arithmatic, Logarithma (read Loharitema), and Algorithm. It opens mind that the Greek doesn't invent this specific mathematic since The Arab invent the specific system number which lead to many specific mathematic knowledge such as Al-Jabar, Arithmatic, Logarithma, and Algorithm. See all the Al- infront of words, explain they come from Arab with Arabic names and Arabic system number. So be honest and let's tell the truth to the world this mathematic come from Arab. The Arab own the system number and put God name in Arab on mathematic knowledge. For the editor Please be honest because this is an important encyclopedy. I hope the etymolog from Semitic Arab soon move to the main page. And Please Complete the article about Al-jabar, Arithmatic, Log, Logarithma and Algorithma it self. — Preceding unsigned comment added by 202.152.202.248 (talk) 04:06, 28 July 2011 (UTC)

The New Oxford American Dictionary says that mathematics and arithmetic come from Greek. Every other source that I've ever seen agrees. I admit that this Wikipedia article is not well-referenced on the etymology. But you propose no references at all for your claims. Frankly, they sound like original research. So please back them up with reliable sources.

You also request that the Wikipedia articles Logarithm, Algorithm, etc. be improved. That is quite reasonable. However, this talk page exists to discuss the article Mathematics only, not all of the mathematics articles on Wikipedia. For that discussion, please go to Wikipedia talk:Wikiproject Mathematics. Mgnbar (talk) 19:54, 28 July 2011 (UTC)

Now that the lead sentence has (weak) citations, which is great, there is still the issue that "quantity, structure, space, change" are listed as "fields of mathematics". Mathematicians simply do not talk like this. Maybe philosophers of math talk like this? Even if so, it needs citation, and equal weight should be given to how mathematicians talk about math. Most fields of math are about specific problems (or classes of problems, or sets of axioms) that draw on more than one of these four "ingredients".

Further, if the graphics are supposed to suggest that group theory is a subdiscipline of "structure" (let's accept that it is, for now), then they also suggest that "complex numbers" is a subdiscipline of "quantity". Is "complex numbers" supposed to be complex analysis? Or is that a subdiscipline of "change"? I know that this task is difficult, but the current solution is not good. We could do better by simply citing from the American Mathematical Society's (or another comparable organization's) classifications and copying text from the appropriate Wikipedia articles. We should also describe historical views on the organization of mathematics (e.g. as indistinguishable from physics or natural philosophy?). The current solution seems neither contemporary nor historical. Mgnbar (talk) 15:59, 22 July 2011 (UTC)

This shopping-cart definition of mathematics comes from De Morgan and Hamilton in the 19th century, where it apparently comes from Kant (assuming GF from CSPeirce). It provides popularly accessible overviews of the main fields of pure mathematics, namely number theory/algebra, topology/geometry, and analysis. However, it is obviously incomplete, ignoring mathematical logic and many of the traditional applied fields.  Kiefer.Wolfowitz 17:47, 1 August 2011 (UTC)

Here is how the American Mathematical Society currently demarcates the fields of mathematics when reporting statistics on new Ph.D.s in the USA (see ). The numbers give some idea of the relative sizes of these fields. Of course, the numbers vary from year to year, the mix might be different in other countries, and one may argue whether statistics, education, etc. are fields of math proper. Mgnbar (talk) 15:55, 24 August 2011 (UTC)

More information Field ...

Field	2010	2009
Algebra, Number Theory	230	223
Real, Complex, Functional, and Harmonic Analysis	102	98
Geometry, Topology	149	139
Discrete Math, Combinatorics, Logic, Computer Science	116	141
Probability	73	82
Statistics, Biostatistics	495	483
Applied Math	229	169
Numerical Analysis, Approximations	88	93
Nonlinear Optimization, Control	27	24
Differential, Integral, and Difference Equations	102	117
Math Education	15	14
Other	6	22
Total	1632	1605

Close

Edit: Added column for 2009 results (see ). Mgnbar (talk) 18:19, 24 August 2011 (UTC)

And here is a summary of the AMS's 2010 Mathematical Subject Classification (see ). I have not changed the order of the names, but I have introduced break points, to improve readability and to try to reconcile somewhat with the coarser classification given above.

• General; History; Mathematical Logic and Foundations; Combinatorics; Order, Lattices, and Ordered Algebraic Structures
• General Algebraic Systems; Number Theory; Field Theory and Polynomials; Commutative Algebra; Algebraic Geometry; Linear and Multilinear Algebra, Matrix Theory; Associative Rings and Algebras; Nonassociative Rings and Algebras; Category Theory, Homological Algebra; K-Theory; Group Theory and Generalizations; Topological Groups, Lie Groups
• Real Functions; Measure and Integration; Functions of a Complex Variable; Potential Theory; Several Complex Variables and Analytic Spaces; Special Functions; Ordinary Differential Equations; Partial Differential Equations; Dynamical Systems and Ergodic Theory; Difference and Functional Equations; Sequences, Series, Summability; Approximations and Expansions; Harmonic Analysis on Euclidean Spaces; Abstract Harmonic Analysis; Integral Transforms, Operational Calculus; Integral Equations; Functional Analysis; Operator Theory; Calculus of Variations and Optimal Control, Optimization
• Geometry; Convex and Discrete Geometry; Differential Geometry; General Topology; Algebraic Topology; Manifolds and Cell Complexes; Global Analysis, Analysis on Manifolds
• Probability Theory and Stochastic Processes; Statistics
• Numerical Analysis; Computer Science; Mechanics of Particles and Systems; Mechanics of Deformable Solids; Fluid Mechanics; Optics, Electromagnetic Theory; Classical Thermodynamics, Heat Transfer; Quantum Theory; Statistical Mechanics, Structure of Matter; Relativity and Gravitational Theory; Astronomy and Astrophysics; Geophysics; Operations Research, Mathematical Programming; Game Theory, Economics, Social and Behavioral Sciences; Biology and Other Natural Sciences; Systems Theory, Control; Information and Communication, Circuits
• Mathematics Education

Mgnbar (talk) 16:42, 24 August 2011 (UTC)

One thing about the table which is striking is the number in probability/statistics. I've always found it odd how these are grouped under Applied mathematics in the article whereas I think they deserve a second level header. In most UK departments I've been at have been split along pure/applied/stats lines. --Salix (talk): 16:50, 24 August 2011 (UTC)

Most of these statistics degrees are trade-school theses of little theoretical interest even among statisticians. (They would be at the level a top 70 program in mathematics in the USA, say.) Few have any interest to mathematical statistics or theoretical statistics and fewer still would interest many mathematicians of either pure or applied bent, unless these mathematicians would have intellectual curiosity or scientific interests .... ;) Sensible persons generally agree that probability is a field of mathematics: Applied probability often is closer to operations research and industrial engineering than to statistics, which is the mathematical theory of scientific practice. and less of an engineering discipline.  Kiefer.Wolfowitz 12:39, 25 August 2011 (UTC)

For what it's worth, in my experience in the USA, small colleges tend to have one department (pure/applied/stats), while large universities tend to have two (pure/applied and stats) or three (pure, applied, and stats). And most of the probability theory I've seen in my life (not much, admittedly) has seemed to be quite pure rather than applied. Mgnbar (talk) 12:16, 25 August 2011 (UTC)

Kiefer Wolfowitz: In case this was not clear, I am not talking about changing the intro. I am talking about changing the section "Fields of mathematics", to use terminology that a mathematician would recognize. This can be done in a reader-friendly way. For example, the treatment of the field "Analysis" could begin "Analysis is the branch of mathematics that has emerged from the study of change." Or something about calculus. You get the idea. My point is that we can use grown-up words from the 21st Century. Mgnbar (talk) 12:16, 25 August 2011 (UTC)

Mathematicians might define analysis as the study of interesting questions, descended from infinitesimal calculus, involving topological linear spaces. But that definition would not be suitable for this article.  Kiefer.Wolfowitz 12:44, 25 August 2011 (UTC)

If that was intended as a response to what I wrote, then it misses. There is an enormous gulf between what I proposed ("change") and what you replied with ("topological linear spaces"). Mgnbar (talk) 13:27, 25 August 2011 (UTC)

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

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

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

Does it make sense?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

References:

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

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

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

Founder by Amos Elon, ISBN 0 670 86857 4

JenniferProkhorov (talk) 19:23, 19 October 2011 (UTC)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

hi.

I would replace

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

with

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

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

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

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

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

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

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

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

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

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

More formal. More painstaking.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A new suggestion:

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

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

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

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

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

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

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

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

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

.................................................................................................. — Preceding unsigned comment added by 205.125.65.84 (talk) 15:46, 26 January 2012 (UTC)

MATH is defined as the expression of logic through the use of quantity.

It is often believed to be a determining science - which means it has causal properties. This is not true. A symbol cannot have any bearing on physical or metaphysical phenomena. At its best, it expresses logic or the scale and change of entities. At its worst...well let's not talk about it...:))))

LOGIC can be expressed in different forms - Math, Words, Pictures — Preceding unsigned comment added by 117.207.152.172 (talk) 13:15, 17 March 2012 (UTC)

People have been trying to define mathematics in terms of logic, quantity, abstraction, you name it, for hundreds of years (thousands if you count Aristotle). Please see Definitions of mathematics for a sampling. —Ben Kovitz (talk) 10:07, 11 June 2012 (UTC)

The first sentence doesn't agree with the citation. The reference says that mathematics is primarily about abstraction, not simply that it is about space, quantity etc. Removing the part about abstraction significantly changes the expression and the meaning, it does not encompass the same range.Teapeat (talk) 03:47, 19 March 2012 (UTC) For example boolean logic is not really, in any normal sense, to do with space, quantity, structure or change, but is normally considered to be part of Mathematics. It is however an abstraction of logic.Teapeat (talk) 03:47, 19 March 2012 (UTC)

Given this, I am considering the verification of the reference given to have failed, and the removal of that part to be original research.Teapeat (talk) 03:47, 19 March 2012 (UTC)

This discussion went on for years. Years! Roughly from 2008 to 2010, but don't quote me on that. I was on your side, but I lost, and the current version is a compromise that took a very long time to reach. If you want to reopen this discussion, just be sure you are prepared to spend a lot of hours over the next few years trying to resolve the issue. Rick Norwood (talk) 17:40, 8 April 2012 (UTC)

"Idealization" seems better. Tkuvho (talk) 08:37, 22 May 2012 (UTC)

"Idealization" is precisely a great way of describing the creation of mathematical concepts, and not merely scientific theories. Consider, for example, ideal points in projective geometry; ideals in number theory; finally the real line itself which is indeed an idealization. This term is more descriptive and informative than "abstraction". Furthermore, abstraction currently leads to the article abstraction (mathematics) which is rather silly. Tkuvho (talk) 09:14, 22 May 2012 (UTC)

The process of generalisation and decontextualisation in mathematics is most commonly called "abstraction", not "idealization" (quick test - compare google hits for "abstraction mathematics" and "idealization mathematics" - the ratio is about 7:1). A good exposition of this process is Chapter 2 Numbers and abstraction in Mathematics: A Very Short Introduction by Timothy Gowers. But I agree that abstraction (mathematics) could do with sourcing and improving. Gandalf61 (talk) 10:36, 22 May 2012 (UTC)

The cited source is not reliable. It's a web page about a university math/comp-sci department. It's probably just copying us, and it appears to have misattributed the famous Galileo quotation to Newton. The virtues of our definition are that it accords well with the kind of definition offered in other reference works, and it reflects the body of the article that we actually have. Please see Definitions of mathematics. I would favor removing the citation and leaving our definition unsourced, but wording it in such a way that it does not come across as the final word on the subject. How about we simply mention the lack of any consensus on the definition of mathematics, and link to Definitions of mathematics? —Ben Kovitz (talk) 10:02, 11 June 2012 (UTC)

I just made some changes: a new section on definitions, and a rewording of our opening sentence so it addresses the lack of consensus on a definition of mathematics, but is still informative and reflects the body of the article. I removed the old source that was probably quoting us, and added some new sources that I think are pretty solid, with the exception of the two I found for calculus (as the study of change). It would be better to have a single good source for that. I found a pretty amusing source on the inadequacy of any definition to cover all of mathematics: a published critique of this very article in a general survey of mathematics.

There are still problems with the lead. The lead is now five paragraphs, which is too long. Much of that bulk says rather doubtful things that are not covered in the article, no doubt the residue of long-past soapboxing. I think that by merely summarizing the body of the article, the lead could easily be brought back into shape. But, I'm done for the day. —Ben Kovitz (talk) 23:22, 16 June 2012 (UTC)

MATH: Mental Abuse To Humans — Preceding unsigned comment added by Ilovenickelback17 (talk • contribs) 00:53, 29 March 2012 (UTC)

Can you provide a reliable source for this? Bzweebl (talk • contribs) 22:42, 5 July 2012 (UTC)

Please post your comments regarding recent "awkward editings" performed by me on the article page regarding:

1) Adding <br>s in order to make citations more readable;

2) Hyperlinking David Hilbert;

3) Citations made in cursive;

that have been reverted motivating that with "please to to avoid html breaks" (whatever the "double to" means).

Thanks.

Maurice Carbonaro (talk) 08:00, 20 July 2012 (UTC)

I hadn't noticed this, and it looks like you have put a similar remark on your talk page. So I will partly copy from there:

1) The breaks and indenting: I found the suddenly interrupted flow of the article with a non-bulleted indented list slightly disturbing.

2) The hyperlink from "David Hilbert": Hilbert was already wikilinked (not hyperlinked) in the article lead. This is sufficient - see wp:REPEATLINK.

3) The cursive citations: I don't really mind these. AFAIC, feel free to re-italicise them.

- DVdm (talk) 10:56, 20 July 2012 (UTC)

Fine. I have re-italicised the cursive citations. Please feel free to use talk pages more in the future. Thanks. Maurice Carbonaro (talk) 07:11, 2 August 2012 (UTC)

Math seems to be going on in multiple different directions. But if mathematicians could define a goal that all of their research amounts to (like why we're trying to solve the Riemann hypothesis and other unsolved problems, and/or what they have in common), that would be great to include here. (Hint: It's definitely not 42 or NaN.) 68.173.113.106 (talk) 21:48, 6 March 2012 (UTC)

What I mean is, some people want to investigate topology, others want to solve P versus NP, still others want to do advanced complex analysis. Personally, I'm looking into mathematical finance (even though I'm just a kid). So what do all of these approaches have in common? If at all, why is it important? (Added to 68.173.113.106 (talk)'s previous comment on 21:53, 6 March 2012 (UTC))

Read Charles Sanders Peirce's Reasoning and the Logic of Things, which has a nice forward by Hilary Putnam.  Kiefer.Wolfowitz 22:45, 6 March 2012 (UTC)

The goal is to first identify and then determine the relative size or amount of something.WFPM (talk) 22:05, 18 March 2012 (UTC) And Chemistry tells you what something is and Physics tells you what something does. And Science gives you a hierarchic list of things to think about.WFPM (talk) 22:19, 18 March 2012 (UTC)

This talk page is not a forum to discuss the ultimate goal of mathematics.

Now, it is potentially a place to discuss whether the article should talk about the ultimate goal of mathematics. My feeling, at the moment, is no, it should not. There is no agreement about whether there is an ultimate goal nor, if so, what it is. We already have enough of a nightmare regarding the "definition" of mathematics, and there we kind of have to talk about it at least a little, because pro forma we have to have something resembling a definition in the lead.

That said, if there were a recognized debate about the ultimate goal of mathematics, I suppose that debate could plausibly be covered in that article. But I really have not noticed that this topic is much discussed, even by philosophers of mathematics (and much less by mathematicians), and yes, I do know some philosophers of mathematics and have followed their discussions to some extent. --Trovatore (talk) 22:28, 18 March 2012 (UTC)

I guess I just thought that mathematics started out with counting (enumeration) and went on from there. Is that illogical?WFPM (talk) 23:09, 18 March 2012 (UTC)

See my first point: We're not here to discuss whether that's "illogical" or not. If you want to discuss the question on its merits, as opposed to what to do with the article, please look for some other forum.

Now certainly, questions on the merits could come up in the context of what to do with the article, but I do not see any serious proposal on the table, just a vague hope from the OP that something relating to the question might be included here. I have given my first impression on that question (I don't think at the moment that we should say anything, at least not in terms so grand as an "ultimate goal"); my points are certainly subject to refutation, but I don't see how your comment goes in that direction. --Trovatore (talk) 23:20, 18 March 2012 (UTC)

I don't think there is a "goal" of mathematics. It's a "tool" used in science to organize and make understandable our ideas. And the article does a good job of talking about the history of the subject matter, but not as its use as a tool in science or technology. It leaves out about the slide rule for instance. And about the evolution of mathematical ideas like the square root sample size and the unknown number approximation routine, which are all unimportant details I know, but important to the importance of an understanding of the subject matter.WFPM (talk) 00:06, 19 March 2012 (UTC) But it sure has gotten more complicated, and in some places even more complicated than the science problems we're trying to understand with it.WFPM (talk) 00:58, 19 March 2012 (UTC)

There is no goal of mathematics, neither is mathematics a "tool" for science and technology. Mathematics is what happened when people sat down and applied their intuition to abstract ideas. It just "is", and if it is anything at all it is a natural outgrowth of the human intuition. It's even surprising that it's useful for understanding the universe/real world whatsoever (and maybe it isn't). 85.229.130.18 (talk) 10:47, 4 May 2012 (UTC)

I think there has been some debate, or at least assertion of opinions by authorities, about something sort of like the goal of mathematics. Hardy and Halmos both weighed in on the value or superiority of pure mathematics over applied mathematics. I can't believe there aren't some people who pooh-pooh pure mathematics. (Well, I know there are, because they've pooh-poohed to me personally, just not in writing with recognized importance.) I think many people have pointed out that pure mathematics often "pays off" in unexpected practical ways, such as (ironically enough) Hardy's number-theory work bearing practical fruit in public-key cryptography. The article already does cover these topics to some extent, as in Mathematics#Inspiration, pure and applied mathematics, and aesthetics. It could probably be done a little more deliberately. —Ben Kovitz (talk) 10:27, 11 June 2012 (UTC)

Do we have to find an ultimate goal? Perhaps a better posed question is: Can we find traits common to all branches of mathematics? Quantity, logic, and intuition are all partially correct but not generic enough (e.g., algebra intentionally leaves quantity unspecified). How about "Mathematics seeks to construct abstract entities and discover their properties through logic based on a limited set of axioms." Of course, in trying to be generic I have introduced some terminology, and others are welcome to improve upon this. — Preceding unsigned comment added by 71.236.24.128 (talk) 22:01, 18 August 2012 (UTC)

I think math as a hobby could become a useful addition to the article if someone worked hard enough on writing it.Marvin Ray Burns (talk) 02:16, 19 May 2012 (UTC)

Can you recommend a good source or two to summarize? —Ben Kovitz (talk) 10:26, 12 June 2012 (UTC)

The Book of Numbers by Bently shows that a although a lot of early numeric discoveries were done by mathematical giants such as Pythagoras, Euclid, Newton, etc., the giants were at one time mere hobbyists.Marvin Ray Burns (talk) 01:20, 10 September 2012 (UTC)

There are good examples of math as a hobby, but none of the three names you mention seems to fit well under that heading. Rick Norwood (talk) 11:54, 10 September 2012 (UTC)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

As a graduate student in group theory, I do find it a bit alarming that this picture of a rubix cube has become the cornerstone of Wikipedia's imagery on groups. Yes, the rubix cube does form a group under composition of turns, but is it a key, or even interesting example of hows a group can function? I motion that it is changed to something more relevant to the field itself (e.g. an illustration of a dihedron group might be nice as it is the lodestone of much of most introductory texts). If I am alone on this issue, I will simply retract my argument, but I do find it a tad annoying. — Preceding unsigned comment added by 76.126.169.150 (talk) 03:46, 13 January 2013 (UTC)

Hello, i have a problem with my homework. and i was wondering if you can help me with it. — Preceding unsigned comment added by Mickenson45 (talk • contribs) 00:41, 6 March 2013 (UTC)

No, sorry. This page is for discussing proposed improvements to the mathematics article. However you can ask at the mathematics reference desk. The operative word there is "help", not "do it for you" — first, write out what you've done so far, explain where you're stuck, and ask for help getting past that point. --Trovatore (talk) 01:18, 6 March 2013 (UTC)

"Mathematics as profession" does not at all treat the right topic. "Renowned Math prizes" would be a more appropriate title. — Preceding unsigned comment added by 85.224.152.237 (talk) 20:20, 6 April 2013 (UTC)

I agree with the fact that the content of "Mathematics as profession" has nothing to do with it's title. "Math Prizes" or "Math Awards" would a more appropriate title for that content. Anyway, would be a good thing to have a section titled "Mathematics as profession" or something similar to deal with the professional jobs where mathematicians apply their knowledge. MickMurillo (talk) 21:21, 3 May 2013 (UTC)

The philosophy of mathematics, the branch of philosophy that studies mathematical assumptions, foundations, and implications, is one of the biggest branches of philosophy in the world. This branch never stops growing, from Thales, a great mathematician from Ancient Greece, to V. N. Bhat, a small mathematician from India, they have all added what little they could. Every idea and theory has helped this art grow. Every year mathematicians discover something new that helps us understand mathematics a little better. The mathematicians don't MAKE or ADD to the subject, they DISCOVER something that is already there. This is the crux of the matter, you cannot make something in math, everything is just there. There can, however, be another way to interpret the art. The Arabic numerals, which were actually made in India but were carried to England by Arabic Traders, are known as the language of math. However, if you lived in Babylon in prehistoric times, you would have a whole different way to express mathematics. The Arabic numerals aren't math, they are just a way to interpret the art. However that means that mathematics cannot be defined because there is no definite way to express it. From the plastic ladybugs used by teachers in 2nd grade to help you add and subtract, to the Arabic numerals used around the world, they are just different languages used to express math, just like all the different languages in the world that are used to express people. Math is a never ending problem, that can be used to solve problems. Just like you can never count to infinity using a language of math, you can never be finished with math. There will always be a new theory to try or a new method that explains another one of life's great questions. -- Professor Captiosus — Preceding unsigned comment added by Professor Captiosus (talk • contribs) 17:29, 20 January 2013 (UTC)

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

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of abstract objects and the logical relationships among such objects. Mathematics encompasses topics including quantity,^[14] structure,^[15] space,^[14]and change,^[16][17][18] although it has no generally accepted definition.^[19][20]

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

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

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

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

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

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

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

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

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

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

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

I agree that the laundry list of quotes should be deleted wholesale and replaced with a summary of the body of the article, per MOS:LEAD. It looks like several hours' work. Want to give it a go? I haven't felt much enthusiasm for doing this, because the body is such a poor overview of mathematics. But it would still be a fine step forward. —Ben Kovitz (talk) 13:38, 20 May 2013 (UTC)

Wrong link in lead paragraph - the space link should go to Space (mathematics) not Space. — Preceding unsigned comment added by 173.79.197.180 (talk) 02:02, 27 February 2013 (UTC)

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I'm not sure about that. The lead says that mathematics is the "study of topics encompassing quantity, structure, space, change, and other properties." So i.m.o. in this context space is meant to be taken in the more general sense. - DVdm (talk) 09:12, 27 February 2013 (UTC)

Yes. Geometry has traditionally been defined as "the science of space", not "the science of mathematical spaces" (which are not the same as space in the ordinary sense, though partly inspired by it). —Ben Kovitz (talk) 13:21, 20 May 2013 (UTC)

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

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

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

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

I'm having some difficulty understanding you. Maybe this will help: the phrase "or mathematics for its own sake" is there to clarify the meaning of "pure mathematics" for people not familiar with it. —Ben Kovitz (talk) 13:18, 21 May 2013 (UTC)