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Some Musings on Mathematics

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Re: Some Musings on Mathematics

#11

Post by Wiskerz »

Retra wrote: post #8
Wiskerz wrote: post #5

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 p…

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!

Re: Some Musings on Mathematics

#12

Post by 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.

Re: Some Musings on Mathematics

#13

Post by Wiskerz »

conceit wrote: post #10
Wiskerz wrote: post #5

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.

> 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.

Re: Some Musings on Mathematics

#14

Post by agumonkey »

heimatau wrote: post #4

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…

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.

Re: Some Musings on Mathematics

#15

Post by 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.

Re: Some Musings on Mathematics

#16

Post by ColinWright »

Mz wrote: post #12

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 t…

  > 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."

Re: Some Musings on Mathematics

#17

Post by Mz »

ColinWright wrote: post #16
Mz wrote: post #12

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 t…

> 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.

Re: Some Musings on Mathematics

#18

Post by 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.

Re: Some Musings on Mathematics

#19

Post by ColinWright »

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.

  > 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.

Re: Some Musings on Mathematics

#20

Post by Mz »

ColinWright wrote: post #19
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.

> 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 r…

Thank you.
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