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Some Musings on Mathematics

solipsys.co.uk

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Re: Some Musings on Mathematics

#31

Post by jeffreyrogers »

dmvaldman wrote: post #30

People without at least a masters in math are in no place to make commentary on math. 98% of the time it makes me cringe. That being said, pure math is when you invest in the tool, applied math is when you invest in the problem. There is a very similar relationship in programming.

The author has a PhD in math. And I thought the article was very good, actually. Many pure mathematicians don't care at all about applications, but the problems are just interesting in and of themselves. Take the twin primes conjecture for example, which recently got a lot of coverage as weaker forms of the conjecture were proved. Why does something like that matter at all? It doesn't. Maybe it will eventually have some sort of cryptographic application years from now, but probably not, and the people who are working on it certainly aren't motivated by that.

So pure mathematician's aren't making tools purposefully. They're playing around with ideas just for the fun of it.

Re: Some Musings on Mathematics

#32

Post by rivalis »

dmvaldman wrote: post #30

People without at least a masters in math are in no place to make commentary on math. 98% of the time it makes me cringe. That being said, pure math is when you invest in the tool, applied math is when you invest in the problem. There is a very similar relationship in programming.

If you're talking about the linked post, I think the person who wrote it is Colin Wright, and he has a PhD from Cambridge in Combinatorics and Graph Theory.

And anyway, at least they're talking about it... Is being ignored because math is "boring" better? Maybe yes, maybe no. I suppose it depends on how "evangelical" you want to be with respect to mathematics.

Re: Some Musings on Mathematics

#33

Post by DougMerritt »

Mz wrote: post #17
ColinWright wrote: post #16

Earlier quoted context omitted.

> Which is a long way of saying > that ... I am no math dummy ... Actually, what it makes clear is that you never encountered any real pure math. You only ever encountered what gets done in school under the heading math. Not the same thing. In fact, very far from the same thing. > I don't feel I understand the > point being made. ... If the > desire is to explain something > to people ... then it probably > isn't suc…

Let me restate my point: If you are trying to help "laymen" get some point, it isn't very clearly written. Singling me out and talking down to me in public in no way improves your actual article. Note to self: Stop trying to help other people. It's a bad habit that only comes back to bite me.

> Singling me out and talking down to me in public in no way improves your actual article.

I think I see why you took it that way, but I am quite sure that that is not how Colin meant it, because I have seen him bend over backwards to educate, over the last decade or two.

He said:

> what it makes clear is that you never encountered any real pure math.

That sounds bad to you at first blush, ok...but the thing is that it is literally true, not an insult, and not just of you, but of almost everyone who is not an outright mathematician, because "math" is not what people think it is.

As a very general truth, it includes you, without at all picking you out from the general population.

To a mathematician, the college 101 courses like calculus and calculus-based statistics (and their prerequisites like arithmetic, algebra, geometry, trig) are not "math", they are more like the alphabet is to an English Literature major -- absolute necessary, but too low level to be directly about the subject at hand.

"Math" is about reasoning, not about e.g. knowing formulae to use on a pocket calculator.

Colin's blog post implies that people with a mathematical turn of mind will itch to know answers.

And I take it to be aimed at people who can sympathize with that itch: to know/understand answers to abstract problems.

I don't speak for Colin, but I myself welcome you or anyone who itches to know the answers, and I am unapologetic about defining outsiders as those who don't care.

It's not about your coursework scores, it's about what one's interests include, really.

Disclaimer: Colin is a real honest-to-god mathematician, I myself am merely a philomath.

Re: Some Musings on Mathematics

#34

Post by gech »

ColinWright wrote: post #9
conceit wrote: post #7

> The truth is far simpler. Mathematicians are solving puzzles, and some of those puzzles don't come from the real world at all, and can't be motivated in that way. Why is it that mathematicians are unable to see the recursion: Pure maths is maths applied to maths, so it is applied maths, nevermind how often this operation had to be repeated until the result would be substantiv?

There are two approaches to answering your question. Firstly, what makes you think that mathematicians are unaware of the trail back to "the real world"? Secondly, I'd be interested in knowing what you think this real world issue this problem might be descended from: Dissect a circle into congruent pieces, such that the centre point is contained in the interior of one of the pieces.

Cutting up a pineapple evenly without including the core because no one wants to eat that part?

Re: Some Musings on Mathematics

#35

Post by tikhonj »

jveld wrote: post #29

Here's how I explain "pure math" to people, based on my experience taking a highly theory-oriented linear algebra course and thinking that "this stuff couldn't possibly be useful." Boy, was I wrong... I think of (pure) mathematics as exploring the structures generated by simple rules . You start with some system of axioms, maybe those of group theory or linear algebra, and you see where it takes you. Often, there are…

I think number theory fits perfectly: the definition of the natural numbers is dead simple and, well, eminently natural but results in an intricate structure we're nowhere near understanding.

Re: Some Musings on Mathematics

#36

Post by srtjstjsj »

gech wrote: post #34
ColinWright wrote: post #9

Earlier quoted context omitted.

There are two approaches to answering your question. Firstly, what makes you think that mathematicians are unaware of the trail back to "the real world"? Secondly, I'd be interested in knowing what you think this real world issue this problem might be descended from: Dissect a circle into congruent pieces, such that the centre point is contained in the interior of one of the pieces.

Cutting up a pineapple evenly without including the core because no one wants to eat that part?

That is not an application of the problem posed above -- the core would still be in one of the pieces, and an easier solution would be to core the fruit first and then divvy up the annulus that remains.

Re: Some Musings on Mathematics

#37

Post by jveld »

tikhonj wrote: post #35
jveld wrote: post #29

Here's how I explain "pure math" to people, based on my experience taking a highly theory-oriented linear algebra course and thinking that "this stuff couldn't possibly be useful." Boy, was I wrong... I think of (pure) mathematics as exploring the structures generated by simple rules . You start with some system of axioms, maybe those of group theory or linear algebra, and you see where it takes you. Often, there are…

I think number theory fits perfectly: the definition of the natural numbers is dead simple and, well, eminently natural but results in an intricate structure we're nowhere near understanding.

good point - for some reason it hadn't clicked that number theory only involves the integers and not, ya know, complex numbers.

Re: Some Musings on Mathematics

#38

Post by MrManatee »

twelfthnight wrote: post #27

> Why do we care that there are only five Platonic Solids? The true answer is because there is an answer, and it would be intolerable not to know it I see a few holes with this argument: 1) Who is this "we"? I'm sure not all non-"non-mathematicians" agree with this sentiment. 2) If a mathematician is still looking for an answer to a question, they don't yet know if an answer exists or not (see: Godel, halting problem…

With regards to 2), there's a difference between trying to find an answer to a single yes/no question, and trying to find a general method for solving an infinite class of such questions.

If a single yes/no question is clearly stated, then it does have an answer. This just a tautology - that's what it means to be clearly stated. For example, does there exist integers x, y, and z such that x^3 + y^3 + z^3 = 33? No one knows, but at least the question is clearly stated. The answer is definitely yes or no.

More generally, x^3 + y^3 + z^3 = 33 is an example of a Diophantine equation (an equation between multivariate polynomials with integer coefficients and integer variables). Suppose we are not so interested in this particular equation, but instead want to find an algorithm that can always tell if a Diophantine equation has a solution. This kind of question might not have a solution, since such an algorithm might not exists. And in fact, it doesn't (Hilbert's tenth problem).

Re: Some Musings on Mathematics

#40

Post by Mz »

DougMerritt wrote: post #33
Mz wrote: post #17

Earlier quoted context omitted.

Let me restate my point: If you are trying to help "laymen" get some point, it isn't very clearly written. Singling me out and talking down to me in public in no way improves your actual article. Note to self: Stop trying to help other people. It's a bad habit that only comes back to bite me.

> Singling me out and talking down to me in public in no way improves your actual article. I think I see why you took it that way, but I am quite sure that that is not how Colin meant it, because I have seen him bend over backwards to educate, over the last decade or two. He said: > what it makes clear is that you never encountered any real pure math. That sounds bad to you at first blush, ok...but the thing is that…

I am sure there was no malice aforethought in his words. I also am well aware he is a real mathematician and serious educator. He spoke in his own defense very aptly.

I am aware most people here will understand his side far better than mine and will generally be more sympathetic to him than to me. Let me suggest he understood my side far better than you did (and another who came to his defense, then deleted the remark) and his reply was far more respectful, diplomatic and appreciated.

I'm sure you meant well and I appreciate the good intentions behind your words.

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