Is there a hierarchy between pure and applied mathematics?

[pivoxa15]

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.

 

[Gib Z]

The distinction between the two is becoming blurred, and not to mention many people are considering themselves to be both at the same time. I believe it is a misconception, because applied maths can be extremely difficult, on par with pure maths. Just ask the teams of physicists trying to fix up the infinities involved in string theory.

 

[morphism]

gaaaah not this nonsense again

 

[pivoxa15]

gaaaah not this nonsense again

I share your concern but we have hope of being honest and just say it.

I believe that not everything is created equal, speaking on average off course.

 

[MrJB]

gaaaah not this nonsense again

Agreed.

Questions like this makes less sense than asking: Which weighs more, a plant or an animal?
Pure and Applied Math are both rather complicated which makes them impossible to compare as a whole.

The notion that one is more difficult implies that the other is easy, and this is certainly not the case. Mathematics would be better off if the notion of superiority was dropped. Perhaps then, people would be more likely to share ideas from those different camps and more discoveries could be made.

 

[pivoxa15]

Agreed.

Questions like this makes less sense than asking: Which weighs more, a plant or an animal?
Pure and Applied Math are both rather complicated which makes them impossible to compare as a whole.

The notion that one is more difficult implies that the other is easy, and this is certainly not the case. Mathematics would be better off if the notion of superiority was dropped. Perhaps then, people would be more likely to share ideas from those different camps and more discoveries could be made.

It just means one is more difficult then the other. Easy is not implied at all. Though again it dosen't mean both words are specific but just that one is more specific then the other.

I am perhaps more concerned with undergraduate pure and applied maths. Research is always tough as there are 'impossible' problems in both fields. Just look at the mellinium problems. 4 pure, 3 applied.

As for superiority, it's a vague word. A more specific word is 'harder'.

It is definitely the case that the living standard in Africa is lower then it is in America. We shouldn't turn away from that and say it's nonsense because happiness is unmeasureable or something. We should just be honest, accept and declare it.

 

[MrJB]

How would you define a measure of the difficulty of pure or applied math?
Is there also a subjective element, as some people may be better at one than the other?

 

[Office_Shredder]

Considering more superior is meaningless, it's difficult to answer the question :p

Grammar nazi-ing aside, would you want someone to say the easier or harder subject while doing an undergraduate degree is superior?

It should be noted that research in any field is always difficult... if it was easy, someone else would have thought it up already. I think this has a kind of balancing effect, where if one subject blows past another in sophistication, people in the other start finding results easier as a result of it being a less mature subject, meaning more people start focusing on that area. I could be completely wrong, but it's a nice theory and I'm going to stick with it :)

 

[wysard]

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.

Can't speak for anyone else, but growing up in a house where Dad had switched from applied maths (Physicist for GE) to pure maths (Combnitorics Prof.) just made the math jokes more obscure. That and conversations in the car went from "Given a frictionless bearing supporting..." to "You are trapped in a dungeon by an evil magician with N choices before you.." kind of problems I think it's more about what your mind has a bent for, some of which may be more common than others, but not necessarily "harder" unless you are trying to force a square brain into a round degree.

 

[pivoxa15]

Can't speak for anyone else, but growing up in a house where Dad had switched from applied maths (Physicist for GE) to pure maths (Combnitorics Prof.) just made the math jokes more obscure. That and conversations in the car went from "Given a frictionless bearing supporting..." to "You are trapped in a dungeon by an evil magician with N choices before you.." kind of problems I think it's more about what your mind has a bent for, some of which may be more common than others, but not necessarily "harder" unless you are trying to force a square brain into a round degree.

Combinatorics is not the center stage of pure maths, it's kind of applied to me. In my department, combinatorics is definitely considered applied reserach. When I refer to pure maths, I think about the deep or abstract areas where students can even find the definitions diffcult, let alone do problems.

 

[pivoxa15]

Considering more superior is meaningless, it's difficult to answer the question :p

Grammar nazi-ing aside, would you want someone to say the easier or harder subject while doing an undergraduate degree is superior?

It should be noted that research in any field is always difficult... if it was easy, someone else would have thought it up already. I think this has a kind of balancing effect, where if one subject blows past another in sophistication, people in the other start finding results easier as a result of it being a less mature subject, meaning more people start focusing on that area. I could be completely wrong, but it's a nice theory and I'm going to stick with it :)

Fair point. But it may be the case that on average the smart grad students go into pure maths. They could have gone into applied maths with high success as well but on average the applied student may really struggle in any pure maths research.

 

[pivoxa15]

How would you define a measure of the difficulty of pure or applied math?
Is there also a subjective element, as some people may be better at one than the other?

True but I was always talking about on average.

Another genearlisation is that applied maths is about concrete examples.

Pure maths is about generalising concrete examples so more abstract. On average people find the abstract harder.

It's often the case that the pure mathematicians start with concrete examples then generalises them so they also do concrete stuff which also occurs in doing counter examples. In that way the pure mathematicians do both concrete and abstract stuff so more work for them compared to the applied mathematicians.

 

[Integral]

Fair point. But it may be the case that on average the smart grad students go into pure maths. They could have gone into applied maths with high success as well but on average the applied student may really struggle in any pure maths research.

Grad students choose their field of study out of personal preference not because it is hard or easy. I studied applied math because I wanted to pursue mathmatical modeling. My interests were differential equations and numerical methods. I have little interest in fields of pure theory, but wanted a knowledge base that would help me understand the world I live in.
Others have other goals. Intellegence at this level is common to all, and simply not a factor in these decisions.

Your generalizations and made up for argument situations are flaky at best.

 

[Gib Z]

Combinatorics is not the center stage of pure maths, it's kind of applied to me. In my department, combinatorics is definitely considered applied reserach. When I refer to pure maths, I think about the deep or abstract areas where students can even find the definitions diffcult, let alone do problems.

If you characterize all pure maths to be the deep or abstract areas when even definitions are difficult, and applied maths the rest, of course you are going to think that pure maths is harder...you sound like a pure math supremest to me.

 

[pivoxa15]

If you characterize all pure maths to be the deep or abstract areas when even definitions are difficult, and applied maths the rest, of course you are going to think that pure maths is harder...you sound like a pure math supremest to me.

My comments are general. Too general to be worthy of anything serious as most of you think but that's okay. I have my opinions because I have struggled at pure maths. In fact I am actually thinking of a more applied area to do research in because I find pure maths to be boring at times. Maybe it's because I don't get it.

 

[Gib Z]

I find pure maths to be boring at times. Maybe it's because I don't get it.

The main thing you are not getting through your head is that the two distinctions are very blurred these days. Don't think of maths as pure and applied, think of it as its different fields, which links between the fields. Some fields are purely for application, some fields have none at the current time, and some fields are a good mix. When you say "Maybe it's because I don't get it.", you are referring to 'pure' maths. Instead think you are just struggling a bit with a certain aspect of one field.

 

[JeffN]

..don't you find it a bit childish to worry about something like this? you can't possibly think that applied math is some sort of a second rate field that failed mathematicians head off into after they fail their exams..

 

[Count Iblis]

Applied math is superior, because most results were originally found using vague intuitive methods. In most cases pure maths is about proving results that are pretty much known to be true anyway.

Newton invented Calculus, pure mathematicians made it rigorous centuries later. And only in the 20th century did pure mathematicians invent infinitesimal numbers to do computations in the same way Newton did (i.e. without limits)

Dirac invented the Dirac function, pure mathematicians invented the theory of distributions to make it rigorous.

 

[pivoxa15]

Applied math is superior, because most results were originally found using vague intuitive methods. In most cases pure maths is about proving results that are pretty much known to be true anyway.

Newton invented Calculus, pure mathematicians made it rigorous centuries later. And only in the 20th century did pure mathematicians invent infinitesimal numbers to do computations in the same way Newton did (i.e. without limits)

Dirac invented the Dirac function, pure mathematicians invented the theory of distributions to make it rigorous.

It just shows that pure maths is harder as it comes after concrete examples.

 

[JasonRox]

I find Applied Mathematics way harder than Pure Mathematics!

 

[pivoxa15]

I find Applied Mathematics way harder than Pure Mathematics!

Because you spend much more time in pure maths.

Also why is it that the greatest mathematicians did pure and applied but not many only applied. Some only pure though.

 

[JasonRox]

Because you spend much more time in pure maths.

Also why is it that the greatest mathematicians did pure and applied but not many only applied. Some only pure though.

My degree is geared towards applied mathematics, so I've done enough of both.

Are you really a senior student thinking like this about mathematics?

 

[morphism]

Are you really a senior student thinking like this about mathematics?

My thoughts exactly. It's disheartening!

pivoxa15, why do you keep asking these questions and making these threads? Are you learning math only to feel superior to other people? I get the impression from most of your posts that you use math as a device to disconnect yourself from everyone and everything. In my humble opinion, this is very unhealthy.

 

[cristo]

Also why is it that the greatest mathematicians did pure and applied but not many only applied. Some only pure though.

You're talking of "great mathematicians" of many years ago, when it was a lot harder to draw a line between the two disciplines.

Some of your points are pretty bad. I agree with Integral's post (which you seem to have ignored) that grad students do not select what to study on "what is hardest," they generally pick what they enjoy the most.

Finally, I agree with the other posters above: why do you keep asking questions like this. You don't appear to be a student who is about to enter grad school, to me.

 

[pivoxa15]

My degree is geared towards applied mathematics, so I've done enough of both.

Are you really a senior student thinking like this about mathematics?

Yes. I am only trying to be honest with things.

I do realize that I should so to speak 'shut up and calculate/mathematise.'

 

[pivoxa15]

My thoughts exactly. It's disheartening!

pivoxa15, why do you keep asking these questions and making these threads? Are you learning math only to feel superior to other people? I get the impression from most of your posts that you use math as a device to disconnect yourself from everyone and everything. In my humble opinion, this is very unhealthy.

I am trying to be honest. I do feel inferior to the top pure maths students but don't generally feel superior to anyone because I know my limitations and intelligence which is average at best. I only rely on hard work to climb up.

 

[JasonRox]

Yes. I am only trying to be honest with things.

I do realize that I should so to speak 'shut up and calculate/mathematise.'

It's fine to speak up, but not about these silly things.

Like I'm not afraid to say that school is just retarded now. School is no longer about the opportunity to learn, it's now about future job security. Sure you still get the "opportunity" to learn, but that's mainly an illusion because you only have the opportunity to learn AFTER you go through all the ****ing loops they make you go through. It's pathetic.

 

[Gib Z]

The OP is ignoring everything we are saying. Cristo, I've told him the barrier between pure and applied is very vague these days at least 3 times in this thread. morphism, "very unhealthy" is a lot kinder than what I'm thinking.

 

[pivoxa15]

The OP is ignoring everything we are saying. Cristo, I've told him the barrier between pure and applied is very vague these days at least 3 times in this thread. morphism, "very unhealthy" is a lot kinder than what I'm thinking.

I am not ignoring them but that their reasons aren't good enough to convert me or is irrelevant. What is pure and applied is indeeded blurred but I have already mentioned that by pure maths I was always thinking of the pure maths that is not part of applied. By applied I am thinking of the more traditional applied subjects not for example QG.

THe thread was always about the average and in general.

 

[Gib Z]

And this whole discussion would have been settled much quicker if you looked at maths like I said in Post 16. Then you would realize there are many fields that are both pure and applied, so it makes little sense to say which in harder in those situations. And I would not have placed Quantum Gravity as a mathematics topic, unless theoretical physics and maths is all the same to you.

 

[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.