Talk:Applied mathematics

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Why do mathematicians insist on the difference between applied mathematics and applicable mathematics?[edit]

Does anyone know why traditionally applied mathematics is divided into the three branches but not others? I think the answer is very important for mathematicians to insist on keeping the defference between applied mathematics and applicable mathematics. I just do not understand why mathematicians refuse to expand the scope of applied mathematics. The reason should be included in the article as it helps readers to understand the nature of applied mathematics.

Why do you think "mathematicians refuse to expand the scope of applied mathematics". I don't know any applied mathematicians like that. Billlion
It is because mathematicians try to keep the traditional definition of applied mathematics, and use the term "applicable mathematics" to name other branches of mathematics that have practical purposes. In other area of science, any study that has practical purposes would be named as "applied science" (e.g. applied physics). In mathematics, however, though applicable mathematics have some practical aims to achieve, unlike the common practice in science, it is named "applicable mathematics" instead of "applied mathematics". Why not name all of them "applied mathematics"? Salt 07:25, 3 November 2006 (UTC)
If we agree that mathematicians do use the term in the traditional sense, is there any need for further discussion? Wikipedia is for reporting things as they are, not trying to change them. That is to say, your point is well-taken, but I don't think this is the right audience to address it to. I'm reminded of the quote "When people in the humanities say something is classical they mean it has significant and enduring value. When physicists use the term, they mean that it's wrong." Applied science and applied math. use the term applied in slightly different ways. JJL 14:43, 3 November 2006 (UTC)
Yes, we agree that mathematicians do use the term in the traditional sense, and wikipedia is for reporting things as they are, not trying to change them. However, even if we are reporting things, we are not just reporting the superficial facts, but the reasons or origins for the facts as well. When we write about World War II in histroy, we don't just report the warfare, but also the origins of WWII; while we talk about "waves" in physics, we don't just mention they would interfere with each other, but also the reasons for this phenomenon to occur. In this article, I am NOT trying to "change" anything. I simply think that the reason for mathematicians' decision to use different terminology is as important as the simple fact inself. It is why I ask for the reasons behind in this talk page; the reasons behind should also be reported in this article. Salt 21:53, 3 November 2006 (UTC)
I don't agree that mathematicians, or people outside of mathematics, always, or even usually, use the term "Applied mathematics" in the traditional sense. It depends where you are. At many universities, you'll see the traditional use of the word. But then look at a program like Cornell University's and you'll find them using the term "applied math" in an unusually broad way--and also you'll find that different professors in the program will disagree (sometimes strongly) about the definition of applied math. This article should not try to gloss over the dissent but rather describe it. Other universities, like Duke University or the University of Maryland, College Park seem to have notions of applied math that are somewhat broader than the traditional. These schools are hardly insignificant--and it's also important that at any of these schools there is a great deal of variation as to how people define and use the term. Cazort 22:52, 1 February 2007 (UTC)
Well, I think that's why the section starts out with "There is no consensus view of what the various branches of applied mathematics are" and moves on from there! It gives the historical answer and then discusses the broader view some take. JJL 14:06, 2 February 2007 (UTC)
It's an historical division. Applied math. is applied analysis. Areas like Operations Research that have great applications came later and from different areas of math. and/or science. JJL 17:54, 27 October 2006 (UTC)
Applied math is NOT strictly applied analysis. Applied analysis is one of many branches of applied mathematics. People use the term "applied mathematics" to refer to "applied analysis" but this is an inaccurate and misleading use of terminology. While some people do use the term in this way, I think that this usage is inaccurate and should not be supported. If anything, we should say on the page that it is common for people to use the term in such a manner, but we should present a more accurate, balanced use of the term as its "true meaning". Cazort 03:11, 19 January 2007 (UTC)

Numerical Relativity[edit]

What the heck is that? I think I know what the author meant, that the computational relativistic mechanics might be applied, as oppose to relativity theory. I dont think it fits with the other headings. Can someone suggest an alternative, otherwise perhaps we should delete. Other than that the article is quite good, although I would argue that it is a stub. Billlion 13:18, 1 Sep 2004 (UTC)

Hmm. I wonder why these subdivisions of theoretical physics are even included. They really aren't directly related to the topic of applied mathematics (maybe in an applied physics article, or even engineering mathematics, but...). I don't think this article should become a list of "important subdivisions" of all the fields listed in the first paragraph, so I suggest we remove everything after "indistinguishable from theoretical physics.". - dcljr 20:28, 1 Sep 2004 (UTC)

There is a certain camp of applied mathematicians, perhaps mainly in certain universities in the UK, who think that Applied Mathematics consists only of mechanics and fluid dynamics, even some who think that asymptotic methods in fluid dynamics is the entirety of applied mathematics. Not sure how to treat that diplomatically. 21:12, 1 Sep 2004 (UTC)

Computer science[edit]

I would say computer science has more in common with pure mathematics than with applied mathematics, just going by the definitions given here. Axiom systems from which you can deduce results are the foundation of much of computer science. Often it is difficult to get a good intuition. As a result computer science has to find real world examples to map the problem to, for example the Dining Philosophers problem. This characterizes pure math more than applied math to me. 19:33, 5 October 2005 (UTC)


I would think that bioinformatics should belong into Statistics, rather than Applied Mathematics. I agree that Statistics should be considered apart from Mathematics, therefore so should be Bioinformatics. (

There is not a consensus within the mathematical community about whether statistics should be considered applied mathematics. Statistics is a branch of mathematics, and it should be included in any of the broader, more general definitions of applied mathematics. Just because it is usually represented in a separate department in Univeristies does not mean that it does not fall under the umbrella of applied mathematics. Indeed, at many colleges and universities statistics is considered part of "applied math". I think NOT including statistics under applied math is POV. In fact, I would also say that not actively discussing on the main page the way people sometimes include statistics as part of applied math and sometimes don't, would also make the article POV. Cazort 03:17, 19 January 2007 (UTC)

Creative Mathematics[edit]

Creative Mathematics

Since the traditional Mathematics focus on the algorithm like that: Problem-->Find Mathematical Model-->Find Solution, the Key Algorithm for Creative Mathematics is: Problem-->Find Mathematical Model Using AI Techniques-->Find Solution.

Dr TAM Shu Ming

New Format[edit]

The new layout, with separate sections, is a definite step forward. I think more can be done though. Right now the heart of the article is just a list of topics. Is there a better way to format it so the list isn't quite so eye-striking? Or maybe having more discussion would help. JJL 15:59, 26 May 2006 (UTC)

First Sentence[edit]

Applied math is about "the mathematical techniques typically used in the application of mathematical knowledge to other domains"?

No, it is about applying math to other domains, and is not restricted to techniques that are "typical". (Cj67 10:47, 12 July 2006 (UTC))

No, applying math. is one thing; the subset of math. known as applied math. consists of certain areas of mathematical knowledge, not of certain applications of math. JJL 15:02, 12 July 2006 (UTC)


I added the cleanup tag as the divisions of mathematics is a list where people have just added their favourite area in a nonsystematic way. This needs to be changed to a paragraph of encyclopedic prose describing the important divisions and mentioning some of the minor ones. Billlion 16:21, 24 October 2006 (UTC)

I'm thinking of a couple of ways to go about this. It seems that one useful approach would be to separate the list of topics into several groups, such as biology (mathematical biology, bioinformatics), business (mathematical economics, actuarial science, financial mathematics), computers and computing (computer science, numerical analysis, cryptography, graph theory/network analysis), and then to write one concise paragraph about each group. (Just looking at the existing list, it seems like an awful lot to fit into one paragraph.)
Does that sound like a good plan? DavidCBryant 14:27, 23 November 2006 (UTC)
Good idea. Please give it a try. Billlion 21:37, 23 November 2006 (UTC)
OK, I dropped it in there, Bill. I hope it's OK. DavidCBryant 13:25, 26 November 2006 (UTC)
Definitely an improvement! I think the Taylor series example is helpful. The Representation theory link seems to emphasize a more theoretical approach? In this context I'm more apt to think of representations by F.S., wavelets, etc. JJL 15:30, 26 November 2006 (UTC)
Thank you for the kind words, JJL. I'm new here, so I tried not to delete any of the existing links when I restructured this section -- I don't want to ruffle any feathers. Maybe representation theory doesn't belong where I put it, but I did think about it a little bit. Approximation theory deals generally with representing objects from analysis with rapidly convergent sums of, say, Chebyshev polynomials. This always seemed like applied mathematics to me, in the strict sense, because one is applying an algebraic structure (a set of orthogonal vectors in polynomial space) to simplify a problem in another area of mathematics (analysis).
Now I'm not really an applied math guy. I did take a few courses in it, but I've mostly concentrated on analysis. Anyway, the article on representation theory talks about applying the algebra of vector spaces to group theory. That also looks like applied mathematics in the strict sense, even though abstract algebra hasn't traditionally been a big topic in the applied math curriculum. DavidCBryant 16:17, 26 November 2006 (UTC)

"Segregation within Universities" vs. "Separation within Universities"[edit]

I think the word segregation should be replaced by separation. It's not a sensitivity issue; it's just that segregation is most often associated with some type of discrimination. Indeed, the first three links for the article Segregation are about racial, ethnic, and religious segregation. Separation is the better word because it does not suggest something other than separation between pure and applied mathematics. --Db099221 22:36, 14 January 2007 (UTC)

I see your point. It's not important to me to use the word 'segregation' though I think it is appropriate; however, 'separation' doesn't have the right connotation for me, preceisley because it does not suggest something other than separation between pure and applied mathematics, as you state. These depts. often (not always) choose to remain apart, in many cases because they are at odds with one another. The Dept. of Math. and Div. of Appl. Math. at Brown U. were barely civil with one another for decades. The depts. at JHU and SCU are in separate colleges and have very limited interaction (appl. math. is in the engineering school in each case). I think it's important to indicate that these depts. aren't separate for mere convenience or taxonomoy but that there is often some degree of resentment present, and that it is often the case that each prefers to be on its own. Of course, I also don't want to overstate the case in these academic turf and financing battles. 'Segregation' captures all this nicely; mere 'separation' doesn't. What's a good synonym that covers this intentional separation? JJL 16:52, 15 January 2007 (UTC)
Sequestration? Sectarianism? I'm not sure, and my dog-eared copy of Roget's Thesaurus isn't a lot of help. I'd like to drag "schism" into it somehow (as in sch[olastic]ism), but then someone would probably object on religious grounds.  ;^> Oh – I'm not a professional academician, but I have witnessed the kind of resentment to which JJL alludes. Too often the pure math guys think of themselves as intellectually superior to (sniff) mere engineers, and the applied math guys think of themselves as diligent workers (with their shirt sleeves rolled up) who are clearly more useful than the hoity-toity pure math people. DavidCBryant 20:06, 15 January 2007 (UTC)

I renamed this section "Status in Academic Departments"...maybe someone can think of a better name, but I think it's important to try to make the heading name as value-neutral as possible. Maybe something like "Categorization" or something similar? I don't know. Maybe we should also include some active discussion about hostility between departments, like the stuff you discuss--to not mention something is to make a value judgment about it! Cazort 03:36, 19 January 2007 (UTC)

scrap the section[edit]

I really don't like the "Divisions of Applied Mathematics" section. I think parts of it are POV, and I also think it is very wordy, rambling, and poorly organized! I would love to have some other people contribute to cleaning it up but in the absence of that I think I am going to butcher it somewhat. I hope I don't offend anyone too greatly, feel free to switch stuff back or engage in active debate if you don't like the results. Cazort 03:35, 19 January 2007 (UTC)

Number theory doesn't belong here, in my mind. Applied math. is, like topology, a reasonably well defined body of knowledge. It doesn't mean all math. that can potentially have applications--that would be essentially all math. It no longer distinguishes. JJL 14:27, 19 January 2007 (UTC)
I like the way it is worded now after your edits! Thanks. Cazort 18:58, 19 January 2007 (UTC)
I just went through part of the article and corrected grammatical and punctuation errors. I agree that it was a bit of a jumble before – I had tried to string a simple list of subjects that was already here into a connected narrative, and the result was not outstanding. I am curious about one thing, Cazort. What's your opinion of verbs? I think the article leans on the verb "to be" a bit too heavily, as it stands. DavidCBryant 19:53, 19 January 2007 (UTC)

Computational mathematics[edit]

I have split off all that NSF stuff that User:JJL edited out into a separate article, and added some explanations there. Computational mathematics deserves a separate article anyway, I just did not think it would be so soon. Jmath666 05:04, 3 April 2007 (UTC)

I recall that was some kind of NSF panel report in early 80s that justified the existence of computational mathematics and I think gave it the name. Much like DDDAS and I think high performance computing. If I track it down I'll add it to references. Jmath666 05:10, 3 April 2007 (UTC)

I would think that a redirect to computational science or scientific computation would suffice. It's just another example of the profusion of terms now used to describe scientific computing. What's the difference between a computational mathematician and an applied mathematician specialising in computing (or numerical analysis)? Or a computational scientist? Looking at the SIAM CS&E conference, I think computational science is becoming the accepted term. JJL 13:21, 3 April 2007 (UTC)

(indent above added) Perhaps computational mathematics should not be a simple redirect also for historical reasons. The profusion of buzzwords is unfortunate. The way it works, a group creates a new buzzword and pushes for funding for it, hence, the role of NSF and other research funding agencies in making buzzwords stick. I have taken the description of computational mathematics right out of the cited NSF website.

There is indeed no difference between someone in computational mathematics and an applied mathematician specializing in computing or numerical analysis - that's what computational mathematics means. But computational mathematics does proofs and they are essential (look at an issue of Mathematics of Computation) while computational science usually does not and just computes away (most papers at said SIAM CS&E conference), projects are often interdisciplinary and involve collaboration with an actual scientist, and correct science not the math is what's important. There is a big difference between mathematics and science. By the way, I am on different sides of that fence myself at times depending on the project. Jmath666 16:46, 3 April 2007 (UTC)

Proofs? So, is computational mathematics the same as numerical analysis? JJL 00:07, 4 April 2007 (UTC)

Numerical analysis is a subset of computational mathematics. Please look in the references in the current version of computational mathematics what the accepted descriptions of computational mathematics are. You will find references to numerical analysis in the Rheinbolt report. Jmath666 01:50, 4 April 2007 (UTC)

I am suggesting that what you and/or the NSF call computational math. isn't the final say on the matter. I think you're multiplying terms needlessly. The distinctions you're drawing are very minor. JJL 03:03, 4 April 2007 (UTC)

People keep inventing names and shifting meanings, and that's what DAB is for. What do you think about the computational mathematics page now? Jmath666 01:38, 8 April 2007 (UTC)

Pure and applied mathematics - that's just not how it happened[edit]

JJL, regarding the edit about pure mathematicians resisting applied hires, I have, sadly, seen it happen more than once. I do not want to return to it in the article, also because of WP:OR, and I do not want to cite specific examples here. But I am curious, how did it happen then, in your experience? Jmath666 01:30, 8 April 2007 (UTC)

That kind of thing happens now. But 100+ years ago, the depts. started as depts. of (pure) math. on the one hand, and depts. of appl. math./theoretical mechanics on the other. In many cases they were separate from the beginning. What you wrote indicated they split from dislike--rather, they merged from financial necessity, and occasionally now some split off again. In any event, going too deep into this isn't important here--the goal is to describe appl. math., not to compare and contrast the pure and the applied. JJL 03:01, 8 April 2007 (UTC)

It has been happening at least for several decades as far as I know. They do split and chase applied people away from dislike even if there was no merger in the past. An opposite trend exists also: theoretically inclined engineers end up in math departments these days... Thanks for the longer perspective. Perhaps a separate history article, but that would be hard to document without original research. Jmath666 03:33, 8 April 2007 (UTC)

Techno-mathematics merger[edit]

The article is currently a dictionary definition of an idea not entirely differentiated from Applied mathematics. It's a minor sub-categorisation at best, worthy of mention on this article, but with little else to form a larger separate article on. See the talk page, at Talk:Techno-mathematics for other discussion on thenotability and defintion of the term, which isn't even distinct from Applied mathematics. - Jimmi Hugh (talk) 17:52, 7 January 2009 (UTC)

Statistics versus Applied Mathematics[edit]

The autonomy of the discipline of statistics has been affirmed by the Mathematical Association of America, etc., and now appears in this article.

Teaching of Statistics by Mathematicians (without statistical expertise)[edit]

I wrote the following sentences to clarify (and document) the previous article's statements:

This unprofessional conduct violates the "Statement on Professional Ethics" of the American Association of University Professors (which has been affirmed by many colleges and universities in the USA) and the ethical codes of the International Statistical Institute and the American Statistical Association. The principle that statistics-instructors should have statistical competence has been affirmed by the guidelines of the Mathematical Association of America, which has been endorsed by the American Statistical Association.[1]

It may be too long, but it is accurate and documented. I list it here in case others want to remove it. The quoted statement also avoids euphemisms, but perhaps can be mollified further (with integrity)? Kiefer.Wolfowitz (talk) 15:42, 26 May 2009 (UTC)

Applied Mathematics and Industrial Mathematics[edit]

I think Industrial Mathematics is a name more used in Europe whiles Applied Mathematics is used by the US.Unless i'm mistaken,then we should redirect the applied mathematics to industrial mathematics.People keep asking the difference.Moorekwesi (talk) 02:50, 17 December 2009 (UTC)

Industrial mathematics is the use of mathematics in industry. In my experience (Maths at Warick in the UK, a very pure maths university), applied maths is less concerned with the application and more concerned with the sort of techniques which are often applied. The current definition at the start of this article captures it quite well - "Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains." For example, this would include solving differential equations. However, at high school I was taught differential equations as part of pure maths, so different people have different definitions. Yaris678 (talk) 11:52, 18 December 2009 (UTC)
Moorekwesi - is your question connected with your recent cut and paste to create Industrial Mathematics from Applied mathematics ? In the UK, "applied mathematics" is by far the more common term - Cambridge has a Department of Applied Mathematics and Theoretical Physics, Imperial has a Applied Mathematics and Mathematical Physics research group; there are Departments of Applied Mathematics at Leeds and Sheffield; you can get a BSc in Applied Mathematics at Edinburgh, Swansea, Birmingham etc . I can't find a UK university that has a Department of Industrial Mathematics - nearest is Oxford Centre for Industrial and Applied Mathematics. Gandalf61 (talk) 12:35, 18 December 2009 (UTC)
Staying on topic, does anyone know what happened to the Dept. of Applied Mathematics and Theoretical Physics at Queen's University Belfast? (talk) 05:13, 27 November 2012 (UTC)


A few comments

I am a bit uncomfortable with "branch" of mathematics. I think of algebra, analysis, geometry, topology, combinatorics, foundations, numerical analysis as branches (essentially major taxonomic divisions in e.g. Math Abstracts). In the mid 20th century, things were relatively simple in a typical British university -- there was a Professor of Pure Mathematics and a Professor of Applied Mathematics, and further faculty who fitted one or other category, and course in one concentration or the other. Work was unequivocally applied mathematics if it could be seen to have application to another area of endevour -- typically physics, chemistry, engineering, biology, (with hot disputes of whether statistics was separate) and beginnings of econometrics, operations research and linguistic applications. HOWEVER, the term "applicable mathematics" was being used, e.g. "Survey of Applicable Mathematics" ed. Rektorys, 1968, MIT Press, translated from Czech. It explains on page 21: "it is (on the Continent) that there flourishes the subject of 'Angewandte Mathematik' -- better described as 'useful', 'utilisable', or 'applicable' ... rather than ... literal translation of 'applied mathematics', which in Britain means something very different.

I am trying to dig back to archival material on how we separated pure and applied mathematics in university entrance (and earlier) exams, curricula and so forth. I was External Examiner in Applied Mathematics to University of Belfast for several years -- I am trying to remember commonality. Certainly the balance between pure and applied in math department was about 50/50.

When I came to the U.S. I found a totally different situation. At U. Wisconsin, I was told department was very liberal -- it had TWO applied mathematicians, who I thought rather pure by British standards. Then Brown became exceptional with a whole department of applied math.

Within a particular taxonomic category there is work that is pure, in seeming total inapplicability to anything else, and very applicable knot theory bearing on interlace patterns. Being pure or applied matter of mind set, not taxonomic category?

Now, many applied math departments changed names to applied and financial, applied math and theoretical physics, and so on

Don't know how this can be accomodated, but thought it as mentionable as some earlier discussion topics. Michael P. Barnett (talk) 15:32, 3 February 2011 (UTC)

I boldly revised the lead: The new lead recognizes that "applied mathematics" describes an activity, as well as a collection of techniques. Of course, it can be improved ....  Kiefer.Wolfowitz  (talk) 02:21, 4 February 2011 (UTC)

Suggestion for revised lede[edit]

Anyone have thoughts on replacing present lede by following lede and new section (or just keep it all as lede if not too long)

Applied mathematics has usages that include:

1. Collections of branches of mathematics, that include numerical analysis and differential equations, into groupings that are convenient in the organization of institutions (for example, in the division of effort between academic departments and sub-departments) and publications (for example, scope of monographs and journals). This usage contrasts with the branches that are grouped, typically, as pure mathematics.

2. The construction and dissemination of mathematical formulas, theorems, algorithms and methods of reasoning that have been applied, directly, to topics in the natural, engineering and social sciences, and other fields of endeavour. This usage extends to material that there is good reason to believe will have direct applications of the kind just mentioned.

3. Mathematical work conducted by research workers whose titles and / or departmental affiliations and / or media of publication categorize them as applied mathematicians, even when this appears to be in another field (for example, the mathematical chemistry of Charles Coulson, the work on relativity of Ray d'Inverno).

4. Actual applications to natural sciences and other fields of endeavour, particularly when the application serves as the name of an equation (for example: lunar motion, the brusselator, the magnetron).

Individual topics, such as vector spaces, form part of applied mathematics under usages 2, 3, and 4, even though they are part of abstract algebra that is not classed as applied mathematics under usage 1. In fact, almost every branch of mathematics that usage 1 puts into pure mathematics includes topics that are applied under usages 2, 3, and 4. Also, many topics that were considered without practical application, for decades or even centuries, now play a role in the design of computer programs.

==Terminology, continued==

The selection and sophistication of topics that are included in books that have titles containing "applied mathematics" vary considerably. This variation is accentuated between Applied mathematics for ...' books, where the ellipses stand for many different subjects. Overlap between the mathematics and the applications is increased further by the trend to university departments of "Applied mathematics and theoretical physics", "Applied and financial mathematics", "Applied and financial mathematics", and many other conjoinings of "applied mathematics".

Several other titles, some taken from popular books, overlap "applied mathematics." These include the following:

1. Mathematics of physics and chemistry: the title of a book that was a staple of mathematical methods used by research scientist from the 1960s for several decades.

2. Engineering mathematics: a popular title for books and courses for several decades.

3. Mathematical biology: the title of the major text, for several decades, on mathematics that was applied in biology,

4. Applicable mathematics has been used in English speaking countries since the publication of Survey of Applicable Mathematics ed. Rektorys, 1968, MIT Press, translated from Czech. It explains on page 21: "it is (on the Continent) that there flourishes the subject of 'Angewandte Mathematik' -- better described as 'useful', 'utilisable', or 'applicable' ... rather than ... literal translation of 'applied mathematics', which in Britain means something very different.

5. Computational mathematics comprises mathematical formulas which are encoded in computer programs, and mathods of mathematical logic used in automatic theorem proving and program verification, and mathematical analyses of complexity. Computational mathematics also looms large in computational science, that embraces the mathematics, software and scientific principles of scientific applications of computers.

6. Industrial mathematics, as used in Society of Industrial and Applied Mathematics is equivalent to applied mathematics without qualification.

7. Some branches of statistics are included by some authorities and excluded by others.

8. Likewise mathematical logic.

The choice of descriptors from amongst these, is determined by many factors. Quite often, an important concern is the need have a name that clearly differs from the name of an established activity, such as a university department or a funding program or a journal or a book, that contains applied mathematics.

There is no "one" correct usage for "applied mathematics", just are most words that are defined in a dictionary have multiple usages. And it is not the role of Wikipedia to try to legislate a "correct" usage. Michael P. Barnett (talk) 20:40, 3 February 2011 (UTC)

I don't like it. It is too complex and convoluted. Shorter sentences are easier to read and fewer sub-clauses will make your meaning clearer. Some of your claims are POV unless you can support them with sources; an example is "Quite often, an important concern is the need have a name that clearly differs from the name of an established activity ...". Gandalf61 (talk) 21:01, 3 February 2011 (UTC)

Suggestion for revised lede, draft 2[edit]

Is this better? Would examples of mathematical topics help?

Applied mathematics consists of formulas and other mathematical information that has practical application. The "other" consists of theorems, methods of reasoning and algorithms.

Mathematics faculty are often grouped in administrative units that are called "Applied" and "Pure", respectively. Most of the work that is done in the applied units has practical application. But so does some of the work in almost every branch of mathematics that is classified as "pure". Also, practical applications are found, quite often, for work that was considered pure when it was published, sometimes decades later.

The literature of applied mathematics include compendia, e.g. Pearson's<ref>''Handbook of applied mathematics: selected results and methods'', ed. Carl E. Pearson. 2nd ed. New York : Van Nostrand Reinhold Co., 1983. 1307 pages ISBN: 0442238665</ref>. Dozens of textbooks are published in the applied mathematics series of many publishers. Most of these are oriented to particular areas of application. They vary widely in sophistication and choice of material. Hundreds of journals publish applied mathematical methods and applications. Most of these focus on particular methods or particular areas of application.

The names of many university departments now join "Applied mathematics" with other topics -- for example "Applied and financial mathematics" at King's College, London, and "Applied mathematics and theoretical physics" at the University of Cambridge. Often, it would be artificial to try discussing separately an application and the mathematical methods that solved it.

Several other names connote bodies of mathematics that have practical applications. Some of these names come from books that were widely used. They include Mathematics of physics and chemistry, Engineering mathematics, and Industrial mathematics. The adjectives Theoretical and Mathematical are applied interchangeably to chemistry, physics and biology, and these terms cover work that qualifies as applied mathematics. So do Discrete mathematics and Applicable mathematics (derived from the Angewandte Matematik of continental Europe). <ref>''Survey of Applicable Mathematics'' ed. Rektorys, 1968, MIT Press, translated from Czech, page 21.</ref>. Parts of Statistics, Mathematical logic and Computational mathematics qualify as applied mathematics, too. — Preceding unsigned comment added by Michael P. Barnett (talkcontribs) 01:20, 4 February 2011 (UTC)

No, no better than before, and certainly worse than current lede. It is too long. It still contains POV assertions ("Some of these names come from books that were widely used" - how do you know that ?). And it does not summarise the body of the article, which is what a lede is supposed to do. Sorry, but Wikipedia is not the right place to write your mini essays about the structure of universities and academic courses. Gandalf61 (talk) 13:31, 4 February 2011 (UTC)

Body of article[edit]

I just wanted a lede that lets me link here from a biographical article, so I left everything else intact. It needs considerable rewriting. Just one immediate comment. The paper on "history of applied mathematics" does NOT deal with the general topic -- just a minute aspect that is not necessarily representative of topic at large. A reference to a history of mathematics (e.g. Boyer or Klein or even Cajori would be far better.) Michael P. Barnett (talk) 11:39, 4 February 2011 (UTC)

Suggestion for revised lede, draft 3[edit]

I have written the following revision, and I am posting it here, to help get a gauge (publishable in the outside world) on the future of Wikipedia. I would appreciate the further comments of the pseudonymous editor who has been so dismissive. I agree with the criticisms of the first version, but I do think the second was decidedly better, and I have put quite some time and effort into what follows. Also, my professional credentials for writing on applied mathematics, and writing for hard copy encyclopedias are matters of record. I am curious about the professional credentials of the people who have been writing this article and the other articles linked to Charles Coulson. The Discussion page of one of these contained a statement "I asked a graduate student ..." from an IP number -- not even a pseudonym. Obviously experts contributing to an encyclopedia must write in appropriate language. I know it is difficult to project to the mindset of the non-expert and how easy it is to take for granted knowledge that is consequent on expert training. I have been coping with this for many decades and several books. I SEEK critical comment on intelligibility, balance, and so forth. If this suggested revision is too long for a lede, why do so many articles have ledes that seem just as long. Where should I split it? Or reorganize it? And if it addresses the topic of the article, rather than the haphazard and extremely omissive current body, does that mean that the article is irremediable? The apparent difference between obvious interpretation of the term and the organizational use do require explanation.

Applied mathematics consists of formulas and other mathematical information that has practical application. The "other" consists of theorems, methods of reasoning and algorithms.

Mathematics faculty are often grouped in administrative units that are called "Applied" and "Pure", respectively, by reference to the branches of mathematics that they address. For example, numerical analysis is considered "applied", and topology is considered "pure". All (or most) of the work that is done in an administrative unit with the name "applied" has practical application. But so does some of the work in almost every branch of mathematics that is classified as "pure". Also, practical applications are found, quite often, for work that was considered pure when it was published, sometimes decades later. Universities that have units with the precise name "Department of applied mathematics" (in the U.S., in early 2011) include Brown[2], Colorado[3] and Washington.[4]

Applied mathematics is surveyed in compendia that include Pearson's[5] and Rektorys'.[6] The key term "applied mathematics" brings up hundreds of titles on the web sites of major publishers that include Oxford University Press,[7] Cambridge University Press[8] and Elsevier.[9] Many of these are oriented to particular areas of application. They vary widely in sophistication and choice of material. Hundreds of journals publish applied mathematical methods and applications. Most of these focus on particular methods or particular areas of application.

The names of many university departments now join "Applied mathematics" with other topics -- for example "Applied mathematics and theoretical physics"[10] at the University of Cambridge, and "Department of Applied Mathematics & Statistics"[11] at SUNY Stony Brook. Often, it would be artificial to try discussing separately an application and the mathematical methods that solved it. For example, the equation of lunar motion, the Brusselator equation and the magnetron equation are known by the applications that give rise to them.

Several names besides "applied" connote bodies of mathematics that have practical applications. Some of these come from books that were widely used. They include Mathematics of physics and chemistry,[12] Engineering mathematics, and Industrial mathematics. The adjectives Theoretical and Mathematical are applied interchangeably to chemistry, physics and biology, and these terms cover work that qualifies as applied mathematics. So do Finite mathematics[13] and Applicable mathematics (derived from the Angewandte Matematik of continental Europe -- see page 21 of the Rektorys compendium[6]) Parts of Statistics, Mathematical logic and Computational mathematics qualify as applied mathematics, too. Mathematical modeling uses methods of applied mathematics.

Topics that are covered in the compendia and other works cited above include basic arithmetic, algebra and trigonometry, plane curves, solid geometry, differential geometry, ordinary and partial differential equations (further categorized as linear and non-linear), special functions of mathematical physics, theory of the potential, vector and tensor calculus, calculus of variations, eigenvalue and eigenfunction theory, classical mechanics, linear algebra, methods of quantum mechanics and statistical mechanics, integral equations, group theory, series expansion, linear programming, combinatorics, applied probability and statistics, infinite series and products, Fourier analysis, conformal mapping, polynomial systems, differential and integral calculus of one a single variable and of several variables, orthogonal systems and vector spaces. Some of these overlap, and the list is just illustrative and by no means exhaustive.

Michael P. Barnett (talk) 15:13, 4 February 2011 (UTC)

No better; still has all the problems I have mentioned before. But that's just my opinion. I have asked for the input of other editors at Wikipedia talk:WikiProject Mathematics. Gandalf61 (talk) 10:31, 5 February 2011 (UTC)
A summary in six paragraphs is too long for a lede here. Some of the material included is clearly more suitable for treatment in more leisurely style in the body of the article. A typical lede is of at most three paragraphs and avoids detail to concentrate on major points: it's an "executive summary", doesn't require many examples at all, and should also avoid argumentative points and nuances of definition. I should say also that editing the lede is by far from the easiest route into editing typical articles—the "logical flow" of an article may become apparent through detailed editing of sections, and then what should be done about the lede ought to be read off the section structure (to some extent). Charles Matthews (talk) 11:26, 5 February 2011 (UTC)
I agree the text above would be too long for a lede. Also, at most a vanishingly small portion of mathematical logic could be called applied; it's out of place there. — Carl (CBM · talk) 13:09, 5 February 2011 (UTC)
Thanks for the two replies. I agree that the piece I wrote is too long for a lede, and that altering a lede is not the easiest way to change an article. But I thought WP does encourage the public to suggest changes that could assist readers with different frames of reference. I might be asked "what is applied mathematics": (1) by a high school student looking at college programs, (2) a mediaevalist who is reviewing a grant proposal from a topologist to apply knot theory to insular art, (3) the director of laboratory, who was trained as a biologist, needs an expert in polynomial systems to work on protein folding, and is comparing CVs.
My first concern, with the present lede, is "branches of mathematics". I think of abstract algebra and non-linear pdes as "branches of mathematics", at different levels of refinement. The boundaries of applied and pure mathematics were determined historically, and I think it essential to mention asap that categorization of a piece of work as part of pure or part of applied does not connote its applicability. Then the plethora of alternative descriptors needs mention asap.
Turning to the present lede:
1. "typically used in science, engineering, business and industry" -- what about social sciences and academe?
Brevity. However, social sciences like Statistics & Economics & Management Science & Operations Research are sometimes considered to be engineering disciplines, because of their emphases on immediate application. They are conventionally termed mathematical sciences.  Kiefer.Wolfowitz  (talk) 10:26, 7 February 2011 (UTC)
2. "a mathematical science with specialized knowledge" -- as opposed to a piece of mathematics that is not a mathematical science, or that does not require specialized knowledge?
The sociology of disciplines observes that every discipline has its core knowledge-domain, which is formalized to some extent: As your proposed (enormous) lede recognizes, "applied mathematics" has been identified as a collection of mathematical theories and techniques. However, the practice of applied mathematics like the practice of law or social work is not defined or limited to its construction of abstract theory; its contributions to solving immediate practical problems are essential, also. (Of course, the lede's wording can be improved.)  Kiefer.Wolfowitz  (talk) 10:26, 7 February 2011 (UTC)
3. "formulation and study of mathematical models" -- I think of a mathematical model as a mathematical representation of a real world situation -- does the study of methods for solving a particular pde necessarily include setting up a mathematical model for a real world system that it represents?
Please read the sentence charitably, rather than with a chip on your shoulder. Imputing "necessarily" is fallacious, and contrary to the context.  Kiefer.Wolfowitz  (talk) 10:26, 7 February 2011 (UTC)
4. "pure mathematics ... for its own sake" -- people have "sakes" -- why not "as an abstraction" or "without an immediate application in mind"?
Be bold! Copy-edit the article for clarity, etc.  Kiefer.Wolfowitz  (talk) 10:26, 7 February 2011 (UTC)
5. "applied mathematics is vitally connected with research in pure mathematics" -- I know what the writer is trying to say, but it is not very clear.
Be bold!  Kiefer.Wolfowitz  (talk) 10:26, 7 February 2011 (UTC)
6. In the "Divisions" section, the 2nd paragraph misses out quite a bit, like integration, differentiation and series expansion. What does the writer mean by "representations". This comes across as a jumble. And the qualifiers "typically", "such as" -- how are these to be generalized.
7. "As well as physics,engineering and computer science have traditionally made use of applied mathematics" -- how long has computer science been around compared with chemistry (which uses e.g. Schrodinger equation), biology (Belousov-Zhabotinsky equation)
How would this history be relevant to improving the article?  Kiefer.Wolfowitz  (talk) 10:26, 7 February 2011 (UTC)
8. Fast forward -- under Utility -- "The advent of the computer has created new problems ... numerical analysis". We seem to have moved on since Zdenek Kopal -- the opening sentence of the 1955 edition of his book "Numerical analysis" is: "Numerical analysis, in its most rudimentary forms, appears to have been one of the earliest -- if not the earliest -- intellectual manifestations of awakening human intelligence". The final sentence "Statistics is probably ..." -- to me, that reads as an unverified opinion. And "other areas of mathematics ..." is not very informative.
9. Further fast forward -- what is the parameter that defines institutions in the range between Brown University and Santa Clara University?
10. "At some universities there is tension ..." -- verifiability?
11. Scientific computing is a mathematical science -- does this sentence have information content?
12. "Computer science relies on logic, algebra and combinatorics" -- well, if computer science includes scientific computing, how about a little Fourier analysis?
13. And where, oh where, are references?
We all have limited time, and our editing occurs during triage. Please add references or ask for citations for specific claims that you honestly doubt.  Kiefer.Wolfowitz  (talk)
14. However, the comment on my User Talk page states that articles on mathematical topics are watched by excellent mathematicians who are experienced editors, with implication that their "expert opinions" do not require the point by point analysis which occurs with peer reviewed conventional publication. That creates situations which are worrisome.
It is easy to feel defensive after your first attempts at editing receive some criticism. I know that I felt defensive after my additions to Bayesian probability and Perron-Frobenius theorem were rejected as "nonsense", by good faith editors. Try to look at GA or Featured articles, or the Mathematics article, and see how the WP mathematics community operates, and see that you are not being singled out for criticism unfairly. The other editors have concerns that your lede is too long, despite its merits.  Kiefer.Wolfowitz  (talk) 10:26, 7 February 2011 (UTC)

Michael P. Barnett (talk) 00:59, 7 February 2011 (UTC)

book by Margenau and Murphy[edit]

"The first edition ... was enormously successful", from the review by Boas in the Bulletin of the American Mathematical Society, Volume 65, Number 4 (1959), page 251. I can provide quotes from reviews of the other books mentioned by name, and counts of titles over years for generic titles. But then I violate no original research. By the way, I have looked at some of your articles that have mathematical titles. Michael P. Barnett (talk) 16:22, 4 February 2011 (UTC)

Practical application of mathematical logic[edit]

"at most a vanishingly small portion of mathematical logic could be called applied". The term "vanishingly small" is close to my heart. I was advised by a pure mathematician to refer to Hardy's Orders of infinity to complete the proof that my first Royal Society paper depended on, that led to the result that supported me for the next fifteen years. According to Wikipedia articles, the lambda calculus is part of mathematical logic. It is also at the heart of functional programming. Now the number of papers that use functional programming in scientific computing annually, divided by the total number of papers on scientific computing may be less than 10^-6, which could be considered vanishingly small. But if the functional programming work is supported by millions of dollars in research grants annually, or has led to hundreds of papers in prestigious peer reviewed journals, is "vanishingly small" an appropriate criterion for exclusion from mention of applicability to real world problems? Michael P. Barnett (talk) 00:59, 7 February 2011 (UTC)

Here, I am happy to agree with Mr. Barnett. The lambda calculus and LISP are the godparents of the statistical languages (and computing environments) S (programming language), S-PLUS, and R (programming language), which join Fortran, Matlab, and SAS/IML as the primary languages for statistical computing. The theory (and practice) of computer programming languages exemplifies the applicability of pure mathematics---category theory, lattice theory, type theory, etc.  Kiefer.Wolfowitz  (talk) 09:57, 7 February 2011 (UTC)
I beg to differ. That "vanishingly small" part of mathematical logic that has relevance to computing can hardly be called "applied". There is nothing applied about the lambda calculus - it is pure mathematics. This is the dilemma of computer science - it has no applied mathematics! It tries to take pure math and directly compute with it. That is why you can't really *calculate* anything useful in computer science, despite the profusion of so-called calculi like the lambda calculus, pi calculus, mu calculus, etc. They are all just axiomatic systems of pure math. They become useful only to the extent that computer scientists can prove useful properties, but you can't calculate a program with them. Now compare this with the situation in engineering where the differential calculus comes with a body of applied math that allows it to be used to calculate useful values for circuits, controllers, flows, etc. Houseofwealth (talk) 04:31, 21 February 2013 (UTC)

Article structure[edit]

I think some basic changes are probably needed in the sections before matters can move forward. The title "Divisions" looks a bit premature, given that we are discussing "scope" issues. Searching the Web for both "modern applied mathematics" and "traditional applied mathematics" is instructive, too. There is probably not too much disagreement on the scope of traditional applied mathematics; there certainly can be some debate on what modern applied mathematics might mean, both in terms of scope and whether we are talking about a methodology or an academic discipline. Charles Matthews (talk) 11:36, 7 February 2011 (UTC)

Hi Charles, you rightly highlight the overemphasis on the knowledge of classical applied mathematics, here, and neglect of the activity of mathematics in action (when a research mathematician wants to accomplish something).
I don't understand your comment, "matters ... move forward": Would you elaborate?
On a related note: The Mathematics page suffers from an abstract discussion of MCS2010-topics, without discussion of applications. Surely, some indication that Fourier/harmonic analysis had something to do with the physics of heat, or that functional analysis and operator theory have had close connections with quantum mechanics would be in order. That page suffers not only from purism but also from undue weight put on philosophical issues, imho. There are more editors on mathematics than here, and so it seems to me that we should first reform the mathematics article, before worrying about this page.  Kiefer.Wolfowitz 22:32, 10 May 2011 (UTC)
Regarding the structure of the article, I take issue with the economics section and think it needs a revamp. The section provides little description of AM research in that area, mostly just quoting classification schemes (e.g. MSC, JEL) and textbook literature. It also uses lots of bare urls as references. Applied mathematical economics does necessarily not have more weight (I think) than other fields e.g. actuarial mathematics. If the section stays (and it should) the content should be redirected to somewhere more germane to topic (so the information isn't wasted). (talk) 03:53, 4 December 2012 (UTC)

Analysis of Mathematical Implication within psychology[edit]

I require a necessity of another individual's analysis of my theory of ,"Mathematical Implication within psychology."

 This is my analysis:

Mathematics is a language that represents quantity within change that is transcribed into written symbols so that applications in the real and virtual world can be logically observed and evaluated. In this theory a new concept is brought about in which I call Implied Mathematics. Implied Mathematics is the continuum of change that is implicated in the world around us without the use of written symbols. In lamens terms, this simply states that everything around us is Math(Math that we can't see). The question of, "How does our brains calculate our movement among the physical world", is explained by implied math. Take the process of opening a door for example. Our brain calculates how far for to reach out with an arm without missing the doorknob or without trying to go through it. Our brain tell us accurately where to go. I believe our brains are mathematical geniuses within the feild of implied math. In other words, If we knew as much Written math as we knew implied math, Math teachers would be unnecessary. Implied math is developed by practicing motorskills. Sports and Martial arts especially can contribute to improvements in our brains ,Implied Math Skills

  All I ask is for someone's thesis.    

-M.H.Winter -- (talk) 21:48, 14 May 2011 (UTC)

  1. ^ The unprofessional teaching of statistics by mathematicians (without qualifications in statistics) has been addressed in many articles:
    • Should Mathematicians Teach Statistics?
    David S. Moore The College Mathematics Journal, Vol. 19, No. 1 (Jan., 1988), pp. 3-7 Published by: Mathematical Association of America Stable URL:
    • Mathematics, Statistics, and Teaching
    George W. Cobb, David S. Moore The American Mathematical Monthly, Vol. 104, No. 9 (Nov., 1997), pp. 801-823 Published by: Mathematical Association of America Stable URL:
    • The State of Undergraduate Education in Statistics
    Richard L Scheaffer, Elizabeth A Stasny. The American Statistician. November 1, 2004, 58(4): 265-271. doi:10.1198/000313004X5770.
  2. ^ Department of Applied Mathematics, [1]
  3. ^ Department of Applied Mathematics, [2]
  4. ^ Department of Applied Mathematics, [3]
  5. ^ Handbook of applied mathematics: selected results and methods, ed. Carl E. Pearson. 2nd ed. New York : Van Nostrand Reinhold Co., 1983. 1307 pages ISBN: 0442238665
  6. ^ a b Survey of Applicable Mathematics ed. Rektorys, 1968, MIT Press, translated from Czech.
  7. ^ Oxford University Press, search results for applied mathematics,[4]
  8. ^ Cambridge University Press, search results, criteria: applied mathematics[5]
  9. ^ Elsevier: Search Results for "applied mathematics"[6].
  10. ^ University of Cambridge Department of Applied Mathematics and Theoretical Physics (DAMPT)[7]
  11. ^ Department of Applied Mathematics & Statistics[8]
  12. ^ H. Margenau and G.M. Murphy, The mathematics of physics and chemistry, van Nostrand, New York, 1943.
  13. ^ J.G. Kemeny, J.L. Snell, and G.L. Thompson, Introduction to Finite Mathematics, [9]