

Some Musings on Mathematics - RiderOfGiraffes
http://www.penzba.co.uk/Writings/SomeMusingsOnMathematics.html

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RiderOfGiraffes
This was prompted by the discussion here:
<http://news.ycombinator.com/item?id=1070604>

It's not really finished, and it's certainly not complete, but I thought it
would give you something to think about. It's not necessarily mainstream, and
it may turn out that I don't believe it all myself, but I hope it's
interesting.

~~~
10ren
I like how you present the developments as filling in logical gaps - almost a
kind of deduction. It seems inevitable that some practical application using
the basic system will eventually also need one of the logical consequences. A
little bit like having different operations in a UI - eventually, each
combination will be tried by someone, and it's reassuring to find that they
are orthogonal, and do work as expected.

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jonsen
_..they have an independent right to existence._ ; mathematical structures are
not invented, they are discovered.

How far does that carry? What about algorithms? Do they also have have an
independent right to existence? Can you invent an algorithm at all or do you
in fact discover it?

~~~
lmkg
> _mathematical structures are not invented, they are discovered_

This is a doorway to a pretty abstract philosophical discussion, not a
statement of fact. I don't think that math is actually _real_ in the strictest
sense, so I'm on the fence about invent vs discover. While I think that
mathematical constructions are idealizations similar to Platonic Ideals, I
don't believe like Plato did that there's a perfect Plato-space where these
idealizations live.

When an artist arranges common-place paints (or sounds or dance moves or
whatever) into a novel combination, he is said to have created rather than
discovered a new work of art. Is there a good reason why this should not be
true when a mathematician takes common-place axioms and makes a new
arrangement of them that validates as a proof?

~~~
jseliger
Yeah -- I think this is an example of a place where the language breaks down
thanks to the different connotations that "invent" and "discover" have. It
seems like math has some properties of both, and the choice of word says more
about the person using the word than it does about the phenomena being
described.

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bitdiddle
This was a good read. I wish the author had gone a little bit further and
spoke about how many reals there are compared to the rationals, continuum
hypothesis, etc.

Although his point about riddles and puzzles is very true I think mathematics
is very much a natural science.

~~~
RiderOfGiraffes

      > I wish the author had gone a little bit further and
      > spoke about how many reals there are compared to the
      > rationals, continuum hypothesis, etc.
    

OK, later this weekend.

~~~
RiderOfGiraffes
Sorry, mis-spoke myself - I meant later in the week. I want to let this sit
and then expand it.

I intend to turn this into a growing series, related to this idea:
<http://news.ycombinator.com/item?id=672067>

There I thought I'd start with fractions, basically from the ground up, but I
think that's wrong. I think I want to engage people first, then "drill down"
on demand. The comment about wanting more about the cardinality, the C.H. and
related stuff is what I was looking for.

Perhaps I can do one a week. Or so.

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BearOfNH
_0 and the negative numbers turned up very late in our history_

I believe it's also true that the ancients distrusted polynomials (e.g.,
_x^3+7x^2+2x+9=0_ ) because who would be silly enough to add a volume to an
area to a length, etc.

If they had just done the mathematics without relating the numbers to real
world, we might have far more advanced maths today. All the more reason to
pursue topics for their own sake.

~~~
jgrahamc
Diophantus' Arithmetica contains polynomials:
<http://en.wikipedia.org/wiki/Arithmetica>

