Some Musings on Mathematics | Hacker News Reader

Some Musings on Mathematics

solipsys.co.uk

48 pointsColinWright45 comments

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 richer, but closely related structures available by adding additional axioms or constraints. For example, add commutativity to group theory and you get abelian groups. Add metrics to vector spaces and you get topology (sorta).

This is useful because the world is full of complex systems that emerge from simple rules. Therefore, when we observe that some system in the real world displays the characteristics of a known mathematical structure, we inherit a bunch of free knowledge about that system.

In practice, most math falls on some spectrum between the above definition and "applied math". Historically speaking, it's a pretty modern idea (although its pedigree begins with Euclid). Before the mid-nineteenth century, mathematicians had indeed been chasing puzzles like "how to find the roots of polynomials" and "can you square a circle using constructions?" for several centuries. And puzzles are certainly not dead, as the millenium prize clearly shows. Number theory also doesn't play nice with this definition. I intentionally ignored that - if I start thinking about the ontology of numbers I risk losing quite a bit of sleep :p.

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.

> 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.

The real question the "outsiders" are asking are not because they deeply care why you (the "pure" mathematicians) spend your time any particular way. The real question behind the question is why should the larger society support this endeavor (by way of research grants, universities etc.. things which are mostly funded out of public funds at large). And it's not unique for mathematicians either.... all academics have to face that question at some point.

Not always. Sometimes "outsiders" just want some kind of insight as to why you do what you do all day; they're curious. A mathematician interested in engaging with the public has to try and identify which question is being asked. It's actually slightly dangerous to assume that your interlocutor is only interested in utility, because this type of answer can come across as disingenuous or dismissive.

I don't think that whenever someone asks an artist or writer what the "point" of their work is that they're looking for a justification for allocating public funds to it. They might want that, but they also might want some kind of insight into or identification with the intrinsic motivation for the work.

"The interesting thing is that stuff from pure maths ends up being useful anyway, even when they were studied just because they were interesting at the time."

I agree with this statement. In fact, I am inclined to believe that it is not by chance that the results in pure mathematics become useful in the real world.

Much of math is dealing with hypothetical situations before they arise. These hypothetical situations may have real world analogs, or they may just be solutions to other abstract problems. While its real world utility is often obscured due to the lack of an immediate direct application, the truth is that by definition, the results derived will be useful if and when an apt situation arises. Granted, many results may never see the light of day, but are still potentially useful in inspiring solutions that require a novel way of thinking.

To judge mathematics based on its current real-world utility would be extremely short sighted. We probably would not have much of the technology we have today if mathematicians in the bygone era decided that they would stop developing it due to the lack of a real world application.

Now there are definitely certain things which can in no conceivable way be applied to the real world, such as the concept of infinity, but those concepts seem to complete a comprehensive framework for us to think about problems.

> Pure Number Theory is motivated by applications in cryptography,

> Pure Calculus is motivated by applications in ballistics and weather forecasting,

> Pure Combinatorics is motivated by analysis of computer networks and data processing,

> Pure Statistics is motivated by life assurance, insurance and gambling,

> Pure Linear Algebra is motivated by optimization problems and Google's Page Rank algorithm.

Math is really sweet, math allows you to make so much, just look around you. I see it as the most powerful, low level, oldest (probably) API in the world. And it's mostly free and open source ! Math gives you tools and tells you in what context they work and don't work. Then it's up to you using it to make or understand something cool. Actually, math purposely tries to abstract itself as much as possible from reality in order to give you a robust framework to work with.

I think this quote of David Hilbert's response upon hearing that one of his students had dropped out to study poetry made me understand why pure math and applied math were two distinct fields: "Good, he did not have enough imagination to become a mathematician" [1]

Like honestly, who cares about whether or not all simply connected closed 3-manifold are homeomorphic to a 3-sphere. But the understanding it brings us about the behavior of manifolds in particular contexts is very real. Whether it's useful or not isn't a pure mathematician's problem though :D (but the truth is that it probably is, just that somebody else will make use of that)

[1] http://www.amazon.com/The-Universal-Book-Mathematics-Abracad... pp. 151 (according to wikipedia, I haven't read the book)

> 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). Perhaps this should be rephrased "because there might be an answer and it would be unbearable not to know it if it did exist."

3) The argument is circular (calling not knowing bad doesn't answer why knowing is good).

1) Since he seems to be talking about mathematicians, that's actually likely - but not necessarily relevant.

2) Mountaineers climb mountains. Musicians make music. Artists make art. Developers make code. To varying degrees, everyone suffers from the same problem. And people who are exceptional in their fields are far more likely to find not pursuing their passion intolerable.

3) Considering how useful pure maths turn out to be - often in completely unexpected ways - I'm quite happy to leave pure maths types to do what they do.

Even if the hit rate were one gamechanger a century, that's still an exceptionally good return. (The actual return seems to be much higher than that.)

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).

> If a single yes/no question is clearly stated, then it does have an answer.

Without having to reach for fancy counterexamples, how about the Empedoclean "Is the answer to this question 'no'?"?

Simple answer on the metalevel: It's not clearly stated, because it's self referential. It's not a question by the GP's standards.

Well, to be sure, if the definition of 'clearly stated' is made to exclude counterexamples to the assertion, then there are no counterexamples. :-) I don't think many people would include 'not self referential' in the definition of 'clearly stated'; for example, "is this question more than eight words long?" is pretty clear.

To put my words in context, I was thinking about mathematical questions, such as "does there exist a curve that goes through every point in a square?". Everyone has some kind of intuitive understanding of what's a "curve", but mathematicians need a precise definition in order to make this question clear enough for mathematical study. (There is such a curve: the Hilbert curve.)

It's not at all easy to recognize when a question is stated "clearly enough", and your questions provide an excellent example of this. The question "is this question more than eight words long?" refers to question itself (as a string), and in general, this kind of self-reference is perfectly harmless. The question "Is the answer to this question 'no'?" refers to its own answer, and this kind of self-reference is potentially problematic.

Also, it occurs to me that the self reference can be avoided by an epistemologically insignificant dodge: "Are you about to say 'no'?" (This dodge is, I admit, a little dodgy; it admits things like "I wasn't planning to" as a perfectly sound answer. However, I'm taking it as read that we understand all questions as yes / no questions.)

Implicitly, the "no" references the answer to the question itself, hence it's still selfreferential.

In other words, the fallacy is to assume the following response was fixed to be an answer to the question. If I answered "no" it would be obvious that i wasn't planning to say "no" prior ("about") to the question. There is just no context in which the question would make sense the way you suggest. It's non-sense.

> It can be of no practical use to know that Pi is irrational, but if we can know, it surely would be intolerable not to know.

The need is to know why it is irrational.

Two quotes that sums up this article for me:

"The whole point of pure maths is that there are problems to solve, and you're working to solve them."

"So why don't we do what we did before? When we couldn't solve equations like x+8=5 we invented the negative numbers. When we couldn't solve equations like 3x=5 we invented fractions. When we couldn't solve equations like x2-x-1=0 we invented the algebraic numbers."

To me, I look at math as a collaborative language of precision in dealing with units. Math isn't static. It's collaborative, it's inventive. It's desire is to accurately communicate measurements. I'm still in my undergrad for Applied Mathematics but that's my two cents.

Math look static until you reach college and leave the mainstream number + operations fed in previous school stages. Most people will never encounter the need for higher abstractions.

> To me, I look at math as a collaborative language of precision in dealing with units.

Although this might apply for early number theory, how does it fit with more abstract mathematics that doesn't deal with numbers, let alone units, at all?

However, I completely agree math (as in the human practice) is a collaborative and inventive process. The directions in which it progresses is often determined by what mathematicians consider "interesting", and not some purely analytical and sterile reasoning.

Meaning is not something that is particularly an end goal. That is why there is specific differenciation between the syntax (structure) and its semantics. Once you isolate them, one could think of the structure itself and manipulate its related symbols, without worrying about their meanings. Sometimes surprisingly, some of these results map back to the meaning of the structure, but that is purely a byproduct.

The manipulation of symbols it itself meaningful, because the symbols are real things that exist in nature. You may be working at a different layer of indirection, but it makes no difference to the usefulness of your work, because nature itself does no indirection (as far as we can tell); your results are always applicable to something real.

Even if many mathematicians are so poor at explaining it that they'll just pretend it isn't true; that mathematics doesn't need a purpose. It does -- it just always has one.

As far as my limited knowledge permits, it may be that meaning does motivate the creation of the said symbols. A problem therefore can motivate the creation of a full set of symbols: Seven Bridges of Königsberg and Graph Theory; Calculus finds its root meaning in the necessity to formalize change.

I also agree that the manipulation operates at an indirection but has meaning, I think the greatest example to that is problem reduction, that operates at a higher level of indirection but eventually maps back to a completely different meaning.

But it is often times that the manipulation of the symbols is simply inspired by meaning, in fact, you can define abitrary operations on any system (by doing S -> S in a relation) without any useful semantic relevence to anything. Maybe we are cherrypicking the meaningful ones? But maybe the fact that you can do so implies a different meaning. Well I guess we will never know. If we define symbols to be part of nature then surely "The opposite of nature is impossible."

Mathematics is effectively the study of things, therefore it has purpose, it can even study itself!

> Meaning is not something that is particularly an end goal.

No, of course it isn't. The goal is where meaning gets you to.

Unless you are studying the meaning of meaning, in which case the goal is the (second) meaning. (i.e https://en.wikipedia.org/wiki/Proof-theoretic_semantics ) I may be wrong though.

> 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?

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.

1. Because at least one claimed that there wasn't any such trail back.

2. The problem of not everyone having noticed yet how elated one is. \s

I think there is no positive solution to this geometric puzzle.

Can you find a dissection where some pieces don't touch the center point?

Can you prove that it's impossible to have the center in the interior of a piece?

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.

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.

Pure mathematics is like teenage sperm donation. Even if it does end up being useful to someone eventually, that's really not why they did it.

To me, personally and practically, mathematics became impactful, and even beautiful when I came to the realization that it was a universal modeling notation for comprehending our world. Math did not come easy to me, but as I persevered through my aerospace engineering degree I came to appreciate its importance to, and role in modeling. There is so much churn in systems modeling approaches such as SysML and UML et al, that I appreciate math as a modeling notation all the more. It remains pretty stable and elegant as it is. Thanks for the links; I'll check them out.

The thing you realized is also what the guy is arguing is not the case. There's no one right view, but that's the point of his argument.

> There's no one right view

How can you possibly know that?

I had more math from 8th-11th grades than most people with Bachelor's degrees. I had no math my senior year because there were only three of us qualified for the next level math course, so they made it a zero hour class. I have a serious medical condition that was not yet diagnosed. There was no way I was going to go to school an hour earlier and take an extra class when I already had more math than I needed by far to graduate high school. The result was that when I started college, I was given a placement test and they said I could repeat trig or I could take calculus. Having had no math the previous year, I ended up dropping out of calculus. Years later, I tested out of college algebra and ended up getting the highest grade in intro to stats, something above 100 after they curved the grade, while explaining it to all my classmates because I seem to have long had a knack for explaining math.

Which is a long way of saying that although I am nothing impressive for the HN crowd, I am no math dummy either. And I don't feel I understand the point being made. I say that to say this: If the desire is to explain something to people who aren't very serious math people to begin with, then it probably isn't succeeding.

It sounds to me like the desire is to explain something to "outsiders," which is the only reason I am commenting. Because I sure as hell have no fantasy that anyone cares whether or not I get it.

  > 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 succeeding.

Here's the bit that contains the main point:

  The whole point of pure maths is that
  there are problems to solve, and you're
  working to solve them.  On the way you
  might generate all sorts of stuff that
  has no real relevance to the real world
  at all.

  The interesting thing is that stuff from
  pure maths ends up being useful anyway,
  even when they were studied just because
  they were interesting at the time.

The rest is examples, structure, and motivations to try to help support that point.

Does that help? If not, then ask something more detailed.

  > Because I sure as hell have no
  > fantasy that anyone cares whether
  > or not I get it.

Given that I wrote it and posted it here, why should I not care whether you get it or not. The whole reason for writing it is to help people "get it."

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.

  > Let me restate my point: If you are trying
  > to help "laymen" get some point, it isn't
  > very clearly written.

That wasn't obvious to me from your first comment - thank you for the clarification. I will read it again with your comment in mind, and see what I can do to improve it.

  > Singling me out ...

I didn't single you out - I have responded to every comment that seems to need a reply. There is one other that I replied to first, the others don't seem to need replies from me. Yours did.

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

If you think I've talked down to you then I apologise. I don't believe I have, but clearly our ideas about what constitutes the nuances behind the words are different. I tried to take you exactly at your word and reply in a factual manner. I regret that you see it otherwise.

  > Note to self: Stop trying to help
  > other people.  It's a bad habit
  > that only comes back to bite me.

I have your email from earlier in my queue to be answered, and I will get back to you.

Thank you.

> 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.

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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