The Logic Off Topic list

Thoughts from the mind of Kool Musick, 10-11-2001:

>Hi Hendrik Jan,
>
>>Yes, he's right (hey GA: see!  again I agree!  That's already 3 times
>>this week!)
>Sigh. Not entirely.

Also: sigh...  Why do you always have to write complete lectures in 
response to some posting?  True art is the art of "leaving out". 
Less words would have equally sufficed -- or would maybe even have 
been better, as such long postings tend to lose the readers 
attention, and hence don't help in getting your point across.

>Both pure and applied mathematicians, of course, have to develop the 
>important mathematical skills required that enable them to solve 
>problems, but they tend to adopt different strategies. Since you 
>teach mathematics yourself, you will be intuitively aware, at the 
>very least, of this fact, although you may not have focused on those 
>differences because they aren't that important in that a wrong 
>answer is just a wrong answer, period.

I'm very well (and more than just intuitively) aware of the 
differences between pure and applied mathematics, both in education 
and in formal research.

The subject however was not the difference between pure and applied 
maths, and I don't see how this has any bearing on the topic. 
GAmoore simply said that most mathematicians believe that mathematics 
is beautiful in and of itself, without the need to refer to 
applications of mathematics.
That, again, doesn't mean mathematicians can't appreciate a beautiful 
application.  They just are able to appreciate the beauty of pure 
abstraction _as well_, contrary to most non-beta's who see no beauty 
in pure abstraction.  They just see "square" stuff about formulas, 
theorems, proofs and numbers.

A testimony to this is the fact that _any_ half-decent mathematician 
knows what's meant when talking about a "beautiful proof".  The proof 
of the 4-colour-map theorem (or whatever it's called in English) is 
generally regarded to be an ugly proof, and people actually spend 
time to come up with a more esthetically satisfying one -- even 
though the plain _fact_ of the theorem has already been proven.  To a 
"layman" this most often seems absurd.  A proof is a proof, and it's 
all just boring stuff anyway, so why bother?

>These days, mathematicians can try to prove things that would have 
>been impossible for them even to try to think about proving in 
>earlier epochs simply because the nature of the language -- and 
>therefore the nature of proof -- has changed so much.

No, this is not true.  What constituted a proof 500 years ago still 
is a proof today.  What has changed is the scope of the field, and 
hence the amount of "tools" you can use to construct a proof.  The 
concept of what a valid proof is though has remained unchanged. 
Otherwise: why would we still study Euclid's "Elements" today (and 
not just for historical reasons, but to learn math from it)?

>After all, there's a very good argument for saying that the whole 
>idea of 'proof' transcends mathematics.

Yes.

>Philosophers also try to use logical argument. So do lawyers.

Yes, "try to use" is the correct expression :-).

>As you well know, many great scientists and mathematicians have been 
>deeply religious, and have built impeccable proof structures built 
>upon their religious assumptions whether they be Islamic, Jewish or 
>whatever.

Yes, I know.  And what is the point of that observation?

>Kool Musick quoted:
>>  >"One would have to have completely forgotten the history of mathematics so
>>  >as to not remember that the desire to know nature has had the most constant
>>  >and also the happiest influence on its development".
>>  >Henri Poincare
>
>HJ said:
>
>>Not to the point.

Note; I didn't say it isn't true what Poincare says here.  I just 
said it's not to the point.  I.e. it doesn't refute GA's claim that 
most mathematicians are able to appreciate "pure abstract beauty".

>Well ... IMO it is exactly to the point, and I will try to explain why.

OK, go ahead.

[...]
>What Poincare had done was change the game yet again by changing the nature
>of proof, and by changing what was an acceptable solution. Pretty much --
>creating a new branch of mathematics with new ways to tackle the solutions
>there was regarded as a 'correct' solution. I think this a pretty sensible
>and felicitous approach, don't you? But ... Poincare took the approach he
>did because he troubled to acquaint himself with the history of the subject
>... which was also actually the first thing that Andrew Wiley did when he
>decided that he was really rather interested in solving the Fermat problem.
>Surely a sensible thing to do.

Yes, of course.  However, I still fail to see the point you're trying 
to make.  Maybe you can write it down in one or two sentences.  I 
feel that this stream of words only confuses matters instead of 
clearing things up.  It might just be my non-native-English status 
that's playing a role here, but I fail to get the real sense of what 
you're trying to say.

>  >   GA says, rightfully, that most math-heads believe
>  >in, or are able to appreciate, the beauty of mathematics "in itself"
>  >-- contrary to most mathematical "laymen" who think mathematics is
>  >just a boring bunch of formulas and incomprehensible stuff, without
>  >any inherent "quality" or esthetics.
>
>And ... my sole point is that this is a very modern approach. Earlier
>epochs did not really stress its importance in this kind of way, although
>they surely at least implicitly recognized it.

Ah, so this is your point!  OK: so what?  Nobody said anything about 
the past, nobody claimed that mathematicians all over the centuries 
have always been in this or that mindset or whatever, and nobody 
argued that the way mathematics is viewed or appreciated has always 
been the same.

What or who exactly are you arguing with?  Again: this only seems to 
confuse matters instead of clearing anything up (as there initially 
was nothing unclear, as far as I can see).

And, as a sidenote to your comment above: the word "importance" was 
never used.  It's about appreciation.  Something quite different.

>  Those were smart people and
>I'm quite sure that if we went back and pointed these things out to them
>they'd get the message within milliseconds. A separate issue would be
>whether or not, having recognized it, they'd look on it as particularly
>important.

"Important": see above.

>However, since it does appear that, statistically, most of the 
>mathematicians who have ever lived are probably alive today (I once 
>read that, but cannot remember where so cannot validate the 
>assertion with a reference),

:-)  Funny fact.  I don't need any validation.  Some fact should just 
be taken at face value -- they're too good to allow them to be 
refuted by boring facts and validations :-).

>I am more than happy to concede this point to the both of you and 
>accept that 'most mathematicians who have lived' should take 
>precedence over 'most of mathematical history'.

I don't see why one should take precedence over the other, but then: 
I don't see the relevance of either option.

Again I fail to see what this has to do with the observation GA made. 
Every mathematician of whom we know anything at all (i.e. excluding 
Euclid and the likes) has been able to recognize the beauty of pure 
abstraction.  Otherwise they simple wouldn't have been able to become 
great mathematicians -- the whole stuff would just be too boring. 
Even when doing so-called applied maths, a large piece of the work 
(more than 50%) is still "just boring fiddling with numbers and 
formulas".  No-one in his right mind would do that sort of stuff if 
(s)he wasn't able to see the beauty of it.


>  > >"The scientist does not study nature because it is useful; he studies it
>>  >because he delights in it, and he delights in it because it is beautiful.
>>
>>Which is not at all opposed to what GA says.  "Delight in nature"
>>doesn't exclude the possibility to delight in pure abstract form as
>>well.  Lots of people delight in nature, but very few delight in pure
>>abstractions.
>
>And ... this is not at all what I was trying to say.

No, but it is what GA was trying to say.  And _that's_ what we're 
talking about.

GA, if I misinterpreted your words, feel free to step in and say so.

>With you personally, however, I am perfectly willing to try to 
>present the reasons as to why I sent in those quotes for I know that 
>if you come back to me again it will be with  something that is not 
>only intelligent, but that is also relevant to the points that I am 
>trying to make; and that it will be in a genuine spirit of 
>give-and-take.

Well, thanks for your trust, for starters.  Honestly.  But... if your 
point was _only_ to say something like "ok, mathematicians nowadays 
might be delighted by pure abstraction, but that hasn't always been 
so -- here are a few quotes to suggest otherwise", then 1) you should 
have said so from the onset, and 2) your first sentence in your reply 
to GA shouldn't have been "???? !!!! ****".  The latter suggested (to 
me at least) that you didn't agree with GA's remark.  And that, I 
still belief, is in error.

>  > >"The profound study of nature is the most fertile of all sources of
>>  >mathematical discoveries".
>>  >Jean Baptiste Fourier
>Well ... I hope you will understand that in my view that quote was actually
>spot on,

No-one denied its truth, as far as I can see.

>Laplace, Lagrange, Fourier, Euler and the like would not have 
>recognized, for example, that there was a SEPARATE field of study 
>called 'mathematical logic' although all of them would probably have 
>recognized instantly what that particular subject was up to.

A non-issue, since mathematical logic is just a bit over 100 years old...

>as to the issue of what numbers actually are, not even Gauss really achieved a
>viable definition. To do that, you have to go to other arenas of 
>mathematics, and mathematical logicians are even now trying to find 
>and plug holes in the definition of number.

And so... ?

>What else can I say but that pure mathematicians do not, and cannot have,
>defined objects that they can study 'in the abstract' -- nor methods that
>they can use to treat those objects -- until pure mathematics in itself
>exists as a properly defined subject area.

This is rethoric.  Of course, if you think that pure mathematics is 
not a properly defined subject area, then there cannot even be such a 
thing as a "pure mathematician", and thus, obviously, these 
non-existent subjects can't have defined _anything_ -- simply because 
they (the subjects) don't exist.

In a more practical sense, this still doesn't hold.  There's no real 
reason a pure mathematician cannot define abstract concepts and 
figure out ways how to deal with them.  Set theory, especially the 
study of infinite sets, is a purely abstract thing.  Cantor figured 
out a way how to deal with them (and met great resistance initially). 
No problem there...

>  > >"Mathematical Analysis is as extensive as nature herself".
>>  >Jean Baptiste Fourier
>>
>>Which, again, is not an argument.
>Fourier did not offer it as an argument, to be honest.

No, but you seemed to be doing so.  For the record: I happen to agree 
with the above quote.  I just fail to see its relevance.

>I think when he said it he was trying to
>convince the wife of one of his friends that what he spent the whole of his
>days doing was really worth while and really a beautiful thing to do.

"A beautiful thing to do" -- exactly what GA said...

>The above is not a LOGICAL argument ... but it is certainly to me a convincing
>argument that mathematics is a beautiful subject that is well worth
>pursuing.

Again: exactly the point GA tried to make.

>  That was all Fourier meant by it. If it does not strike you in
>that way or seems ridiculous, then that's quite fine by me, to be frank.

No, not at all.  I agree wholeheartedly with it in fact.

>As far as I could see, the major point at issue was what has been the primary
>source of mathematical investigation, of mathematical proof, mathematical
>discovery, and of mathematical appreciation throughout its history.

No, not at all.  Either _you_ missed the point completely, or I did...

>  All I
>tried to say is that, contrary to the assertions made in GA Moore's
>original statement, the primary and principal object of study throughout
>mathematical history has been nature itself -- albeit when seen in a very
>particular kind of way.

Let me quote the first 3 lines of your reply to GA:
>  >Actually most mathematicians believe the beauty of mathematics in and of
>  >itself without any regard to real world applications.
>???? !!!! ****

Is there anything about "the primary and principal object of study throughout
mathematical history" in there?  I don't see it...

>But ... it has been nature.

Yes, of course it has been.  No-one in his right mind would deny that.

>It could not have been something abstract like number in and of 
>itself, because the definition of number that we have today is very 
>current, and it is only very recently, within the last 150 to 200 
>years, that it has been possible for mathematicians to think in 
>those kinds of terms.

It might have been some other abstract subject.  The history of one 
particular concept (i.e. the number-concept) is of no relevance. 
However, it (indeed) was _not_ some abstract subject.  Whether it 
could have been this or that is, again, of no relevance.

>If that's the way the two of you want to play it then I bow my head humbly

Stop pretending a kind of humility that's refuted by the length of 
your letters, the effort you take to be exact & precise, and the 
lengths to which you go to ensure you're ahead of any kind of 
"attack" that could follow.  This is something I don't like at all. 
Sorry.  And as for bowing one's head: bollocks.  No-one has to bow 
for anyone.  Ever.  If you fuck up, you just say "sorry" and that's 
it.  However, this is not about fucking up -- it's about different 
interpretations of what the subject at hand is.  Stooping low and 
bowing heads is a non-issue in such matters.
And "the two of us" don't want to play anything.  I don't know what 
you're suggesting here, but it doesn't give me a warm & cozy feeling.

>and respectfully concede the point, but this qualification simply 
>was not made clear in the original assertion. I took it to mean that 
>it was a statement that was being made as one to be accepted as true 
>for all mathematicians through all historical epochs. That's all I 
>can say, and I apologise in that case for having opened my big mouth.

1) I think you read way more things into GA's msg than were there. 
He simply said "most mathematicians appreciate pure abstraction". 
That's all.  No reference to present, future or history, no 
broad-sweeping claims, nothing of the kind.  Much like "most butchers 
don't trust beef from the UK" (mad-cows disease, in case you didn't 
know).  Of course this doesn't mean all butchers from all times all 
over the world have always distrusted English beef...

2) Even if GA meant it in such a general way, then you still haven't 
provided any convincing argument to refute such a statement.  The 
fact that mathematics has been driven forward by the need to solve 
practical problems, and the fact that many, if not all, 
mathematicians were interested in all kinds of applications is still 
_no_ argument against the simple statement that they're able to 
appreciate the abstract beauty of mathematics.  One doesn't exclude 
the other.
For all we know, Euclid or Phythagoras may well have been smitten 
with the beauty of abstractions.  Well, in fact Pythagoras most 
likely _was_.  His number based cosmology is, in a sense, highly 
abstract.

But let's not go there, as it's not the issue...

>"The whole problem with the world is that fools and fanatics are always so
>certain of themselves, but wiser people so full of doubts".
>Bertrand Russell.

Very wise...

>I hope it's OK if I contact you privately should I want further
>clarification on anything.

Of course.  Although I don't see what there would be to clarify.  As 
far as I can see this is a very simple matter which got complicated 
by the fact that you read things in GA's message that just weren't 
there.

>I thank you most sincerely for your kindness.

Oh, come on, cut the crap...

>I also thank you for the complete lack of personal attacks in your email.

Did I do better this time?  LOL!  :-)))

Seriously, I know I probably sound a bit "irked" in this letter. 
That's not meant that personally -- I hope you know me well enough by 
now.  I just tend to get that way when people make things 
unnecessarily complicated, as you seem to be doing in this case.  I 
know, a flaw in my character probably...  Trust me, I'm working on it.


keep cool ;),
HJ

-- 
     Hendrik Jan Veenstra
     email: mailto:h@...
     www:   http://www.ision.nl/users/h/index.html