Is there a hierarchy between pure and applied mathematics?

[yasiru89]

Superiority need not be too abstracted a concept in itself and comfortably distances itself from falling into measure with difficulty, perhaps leading to the reluctances on this thread. I've said this many times before but find reason here to trouble you with it again; the course set for mathematics is set by itself, along a line of greatest progress, it does trouble people- this classification of pure and applied mathematics, they choose to sulk and brag about the 'good old days' when mathematical effort wasn't specialised and the problems addressed were those that were fancied by the influential mathematicians of the day. They always dodge the simple fact that mathematics has become far too vast even without its trivialries to succumb to individual conquest so easily. For us it is these trivialries that need garner concern, for applied mathematics is most definitively concerned with applicability; whereas in earlier times a problem brought about theories we have come to a point where the abstraction and theory work their way into problems. As such problems are commonplace and their solution and all the rest attached with the matter- which is irrefutably the highlight of applied studies becomes a triviality.
Pure mathematics on the other hand seeks boundless expansion, from time to time it lends application, as the vice versa of past generations(eg- Fourier from heat conduction to analytical Fourier series), but for the most part it considers abstraction and hence presents the greatest of human intellectual achievement.

 

[J77]

Only students or disgruntled mathematicians who need an excuse why they're not getting any grants would form a difference between the two

 

[mathwonk]

well, archimedes was an applied mathematician of some success, having staved off almost single handedly the attack by marcellus on syracuse, by his machines of defense, for over a year.

and his discovery of the volume of a sphere proceeded by first understanding how it balanced with an appropriately placed cone and cylinder. So he used principles of applied mathematics to accomplish goals in pure mathematics. so it seems he derived his primary inspiration from applied mathematics, or applied physics.

He is said to have valued those "pure" results which he distilled from applied principles, more highly however according to his system of values, in which things of practical use were considered mundane, less exalted. maybe this classical greek attitude is the one you are perceiving persisting today, against the roman goal of applying practical engineering to conquer the real world.

Perhaps it is the violence which so often accompanies our use of applied mathematics, e.g. machines of war, which by association lowers that science in our estimation. But if you want to estimate the intelligence possessed by, or needed for, applied mathematicians, archimedes alone completely settles that question, having anticipated by 2,000 years the greatest mathematical ideas in existence, limits and infinitesimal calculus.

 

[Coin]

The question is just plain silly, because it's impossible to tell the difference between pure maths and applied maths except in retrospect. Subjects in pure maths have a sneaky way of turning out to unexpectedly have some kind of incredibly useful application. Meanwhile subjects in applied maths can occasionally instantly transform into uselessness as new methods are developed which supersede them.

The archetypal example here as for why the argument for or against pure vs applied math isn't even worth fighting would be G H Hardy, who famously in his book "A Mathematician's Apology" gloated at the superiority of pure math-for-maths-sake over applied math, in particular repeatedly stressing that his preferred fields of pure math had a moral superiority because they could not be used to wage war. And what were Hardy's preferred fields? Well, group theory and the number theory of prime numbers-- the former of which is today critically important as an applied field for everything from quantum physics on up, and the latter later turning out, in part I'm told building on Hardy's work, to be a fundamental building block in the military's very favorite field of math, cryptography. Meanwhile Hardy himself is in many circles best remembered for the Hardy-Weinberg principle, an important and fundamental result in population genetics which Hardy discovered more or less by accident-- his version of the principle was formulated in a page-and-a-half letter to the editor of Science, responding to an algebra problem which Hardy had assumed obvious and which had been forwarded to him by a guy he played cricket with.

Studying the life of Hardy, it starts to seem like you can't avoid contributing to applied math even if you try with all your might.

 

[yasiru89]

The question is just plain silly, because it's impossible to tell the difference between pure maths and applied maths except in retrospect. Subjects in pure maths have a sneaky way of turning out to unexpectedly have some kind of incredibly useful application. Meanwhile subjects in applied maths can occasionally instantly transform into uselessness as new methods are developed which supersede them.

The archetypal example here as for why the argument for or against pure vs applied math isn't even worth fighting would be G H Hardy, who famously in his book "A Mathematician's Apology" gloated at the superiority of pure math-for-maths-sake over applied math, in particular repeatedly stressing that his preferred fields of pure math had a moral superiority because they could not be used to wage war. And what were Hardy's preferred fields? Well, group theory and the number theory of prime numbers-- the former of which is today critically important as an applied field for everything from quantum physics on up, and the latter later turning out, in part I'm told building on Hardy's work, to be a fundamental building block in the military's very favorite field of math, cryptography. Meanwhile Hardy himself is in many circles best remembered for the Hardy-Weinberg principle, an important and fundamental result in population genetics which Hardy discovered more or less by accident-- his version of the principle was formulated in a page-and-a-half letter to the editor of Science, responding to an algebra problem which Hardy had assumed obvious and which had been forwarded to him by a guy he played cricket with.

Studying the life of Hardy, it starts to seem like you can't avoid contributing to applied math even if you try with all your might.

Oh please, not the half witted extrapolations again! If you had read A Mathematician's Apology properly you'd realize that it was meant to shut dimwitted arguments from the likes of you up for good! "Could not be used to wage war", what a pathetic, naive interpretation of what was really being said. It deserves no defence on my part or any other's and yet I shall present what shall amount to a justification to so poor a statement.

Kicking off, well of course most pure disciplines are bound to find application sometime or other; I mean really, if this whole discussion can't get past that there IS no point going at it. What matters though is the attitude which is brought to this expansion of the subject's horizons. An applied mathematician would, quite usefully I might add find numerous utilities and develop these there-on. Straightforward yes, but I shall not undermine difficulties as has been the case for most arguments for and against here.
A pure mathematician on the other hand approaches with a mindset of conquest- another won but many more rising above the horizon. The purpose is intellectual achievement, nothing more, nothing less- perhaps a bit too abstracted but irrelavantly that is a vice of habits. For him the fight is fought for the sake of his mathematics, he cares not the uses it may however distantly be put to; and if it were he may well lose interest. He evolves as pure mathematics does, towards the edge and beyond as the front runner. This was never about contribution and to speak of such is grave folly. Hardy for one was no fool to think of an untainted to forever be his association, his was merely the apology, again written at the risk of misinterpretation simply for the sake of it. It shall no doubt be comforting to him that his was motivation for progress and whatever use his results may be put to and the consequences thereof rest firmly upon the shoulders of those who, in full circle once more - applied them.

 

[morphism]

This thread should be locked.

 

[J77]

This thread should be locked.

Yeah -- these pure vs applied discussions can get messy...

 

[mathwonk]

to paraphrase a teacher friend of mine at the start of school year, the problem is letting all these people post their opinions on here.

 

[animalcroc]

People should not be bashed for asking certain things even if one finds them silly. The last thing we need is enforced uniformity and if that was the case there would have never been progress in world. There are ways to disagree without unprovoked insults or rudeness.

 

[pivoxa15]

One clear reason for me why I think pure maths is superior (to put it succinently and bluntely) to applied maths is the concentration one must exert when doing pure maths sometimes to the extent of headache. However with applied maths, it's sometimes just a 'trick' one needs to use or just trying different things. So there is much more an element of deep thinking in pure maths. A lot of it is obviously to do with the fact that pure maths is much more abstract.

Thinking things from the barest of fundamentals is also attractive and gives the feeling of superiority to me.

However that said, there are plenty of long standing unsolved applied problems. What does it take to solve them?

 

[Gib Z]

animalcroc, pivoxa15 asked us if we think "pure" is superior to "applied". If we voted no/equal, he asked why. We told him why, then he ignores that and continues as if we didnt make a decent point. look through the entire thread and just see how he does this. His last post just gave us an example of how he still thinks pure vs applied. We've told him its hard to differentiate between them anymore, a lot of pure mathematics can be applied in some way. Both complex analysists and computer scientists would like to see the Riemann hypothesis dis/proven, is the Riemann Hypothesis now in "applied maths"? To date no 'trick' has solved it.

pivoxa15 is a mathematical supremist, who a) thinks mathematicians are superior to any other scientists and b) "pure" mathematicians are better than the rest.

Considering this forum has many people who are primarily interested in biology, chemistry, etc etc, I recommend he finds a "pure" mathematics only forum, I'm sure he would like that.

 

[yasiru89]

Can't say I'm entirely free of sin either then Gib Z, I for one am positive on both counts, though with regard to other scientists I shall exclude psychologists and, if so categorised, philosophers;
just as well though, since the distinction I've forwarded has been largely ignored as well. Perhaps I need to apologise for not being politically correct but honesty has had its moments!

 

[ice109]

animalcroc, pivoxa15 asked us if we think "pure" is superior to "applied". If we voted no/equal, he asked why. We told him why, then he ignores that and continues as if we didnt make a decent point. look through the entire thread and just see how he does this. His last post just gave us an example of how he still thinks pure vs applied. We've told him its hard to differentiate between them anymore, a lot of pure mathematics can be applied in some way. Both complex analysists and computer scientists would like to see the Riemann hypothesis dis/proven, is the Riemann Hypothesis now in "applied maths"? To date no 'trick' has solved it.

pivoxa15 is a mathematical supremist, who a) thinks mathematicians are superior to any other scientists and b) "pure" mathematicians are better than the rest.

Considering this forum has many people who are primarily interested in biology, chemistry, etc etc, I recommend he finds a "pure" mathematics only forum, I'm sure he would like that.

pivoxa is a ______ . many things go there. i choose dummy.

 

[Holocene]

Okay, I am a total newbie here, just finishing up a study of algebra 1.

I have a question:

What exactly is the primary difference between pure math and applied math?

I take it that "applied" math means using mathematics as a means to define or understand aspects of the physical world, while pure math is strictly a study of mathematics in and of itself?

...if that makes any sense?

 

[HallsofIvy]

In recent years, it has become fashionable to divide mathematics into three general areas- "pure" mathematics, "applied" mathematics, and "applicable" mathematics.
"Pure" mathematics is mathematics that is done for the sake of the mathematics itself- it does not depend upon whether or not that mathematics can, at some later time, be applied to a non-mathematical problem. "Applied" mathematics refers to applying mathematics to some non-mathematical problem. "Applicable mathematics" refers to mathematics that does involve non-mathematical applications but is being done specifically to give techniques that can, immediately, be applied to non-mathematcal applications.

Notice that I am saying that whether or not a mathematical theory can, at some future day, can be applied to some non-mathematical problem does not affect it being "pure" mathematics. Also, notice my reference to "non-mathematical" applications. In Norbert Wiener's classic "The Fourier Transform and Certain of its Applications" the "applications" are only to mathematics.

Finally, I must say that pivoxa15's statement,

One clear reason for me why I think pure maths is superior (to put it succinently and bluntely) to applied maths is the concentration one must exert when doing pure maths sometimes to the extent of headache. However with applied maths, it's sometimes just a 'trick' one needs to use or just trying different things. So there is much more an element of deep thinking in pure maths.

is just silly. That is not a "clear reason", it is a meaningless reason. It says that "pure" mathematics is superior to "applied" mathematics because some problems in pure mathematics are hard and some problems in applied mathematics are easy! Some problems in pure mathematics are very easy and some problems in applied mathematics are very difficult. I rather suspect that the "multi-body" problem requires as deep thinking as any "pure" mathematics problem.

 

[lfallo1]

LOL @ saying it doesn't matter... I wasn't ALLOWED to take "Applied Math" with a COSC minor at Towson. So i ended up taking a tougher workload, and graduating with a Pure Mathematics degree + the COSC minor. The Pure Math major was identical (aside from 3 classes... my 3 extra classes being monsters). Either way, it really isn't paying off after a few months of applying for jobs b/c most places have a fetish for the word "applied."

Alot of places also seem to think the "pure" guys are too abstract.

It sucks for me, but IMHO i'd say applied is the better way to go, unless you plan on getting your MS or PHD.

 

[lfallo1]

I find Applied Mathematics way harder than Pure Mathematics!

Hilarious. At most colleges, the difference is but 2 or 3 classes at most... And the Pure classe at THE VERY LEAST cancel themselves out.

In my experience, the classes at my college (that Applied didn't have take) were Real Analysis, Algebreic Structures, and Applied Combinatorics... If someone finds proving the backbone of math easier than taking Stat II & Math Models, then more power to 'em. But FWIW, NO ONE wanted to take Pure Math over Applied (frankly, neither did i but i got screwed due to Towson).

But like i said, companies love the word "Applied." So take the easy route, and go "Applied."

 

[Diffy]

I always get the impression that pure maths is more superior meaning harder, grander than applied maths and that the smart people on average go into pure maths. Is this a misconception?

Also rarely is it that applied mathematicians switch into pure maths but the vice versa is plentiful.

I do find pure maths harder then applied maths.

Maybe this is just me, but, your post almost has an elitist tone to it.