
Some Musings on Mathematics - ColinWright
http://www.solipsys.co.uk/new/PureMaths.html?HN_20150815
======
jveld
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.

~~~
tikhonj
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.

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

------
chetanahuja
> 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.

~~~
rivalis
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.

------
machinelearning
"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.

------
globuous
> 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...](http://www.amazon.com/The-Universal-Book-Mathematics-
Abracadabra/dp/0471270474) pp. 151 (according to wikipedia, I haven't read the
book)

------
twelfthnight
> 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).

~~~
MrManatee
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).

~~~
JadeNB
> 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'?"?

~~~
conceit
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.

~~~
JadeNB
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.)

~~~
conceit
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.

------
j2kun
> 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.

------
heimatau
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.

~~~
johncolanduoni
> 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.

------
Wiskerz
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.

~~~
Retra
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.

~~~
Wiskerz
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!

------
conceit
> 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?

~~~
ColinWright
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.

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

~~~
srtjstjsj
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.

------
dmvaldman
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.

~~~
jeffreyrogers
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.

------
makeset
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.

------
michaelZejoop
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.

~~~
jeffsco
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.

~~~
spacehome
> There's no one right view

How can you possibly know that?

------
Mz
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.

~~~
ColinWright

      > 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."

~~~
Mz
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.

~~~
ColinWright

      > 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.

~~~
Mz
Thank you.

