The Logic Off Topic list

Hi Hendrik Jan,

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

Being a person greatly interested in the HISTORY of the subject, one has to 
bear in mind the appreciation of differences in the ways of thinking that 
seem 'normal' to us today; but that were not really accepted and would be 
hard to recognize and/or would seem strange to thinkers who existed prior 
to the time those differences became accepted. The 'real' differences 
between pure and applied mathematics, and between these and 'mathematical 
logic', and in fact between this and the 'history of mathematics' are in 
fact quite hard to state even today. It is also only recently that they 
have become properly recognized as differences important enough to be given 
names within the discipline.

Nowadays, we see the main differences between the pure and the applied 
variety, which is mainly the topic under debate here, as resting not only 
in the topics studied, but also in the approach to be taken to how given 
problems are to be solved. 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. Pretty much, though, when you emphasize for example the 
specialized mathematical language of algebra as being the preferred method 
for learning and understanding mathematical concepts and as the preferred 
way to solve problems, then you are working more within the field of pure 
mathematics. In order to learn and progress in this field, it is necessary 
to present, more directly, the mathematical theories underlying those 
concepts (and also in a systematic and orderly way), to learn how to find 
exact value solutions to given equations, and to use formal mathematical 
reasoning and given models in the solving of problems. To return to applied 
mathematics, one teaches there more by the setting of challenging and 
interesting problems and activities that tend to be taken from real-life, 
or else that are devoted to the completion of a specific project set by the 
teacher. One encourages students to develop their mathematical skills in 
the mathematical operations under review more by getting them to do it 
often enough so that they understand the concepts. One also teachers more 
using graphs, constructing scales, diagrams and tables -- and these days 
using computers and spreadsheets. These are usually necessary when long and 
complex mathematical calculations are required. To the applied mathematics 
teacher the graphing and the spreadsheet operations are opportunities for 
the students to develop a very real competency in algebra, for example. 
This does not mean that pure mathematicians never use graphs! That is an 
absurdity, but I hope I have said enough for you to see that there is a 
difference between these approaches, even though the fundamental aim is the 
same -- to do mathematics well.

I have focused more broadly on the differences, though and in the 
applications, between pure and applied mathematics, but of course each 
specified division within mathematics does have its unique form of 
reasoning. Statistical theory has its particular form of reasoning and 
standard of proof; probability theory has its particular form of reasoning 
and standard of proof and so on and so forth. Many problems can in fact be 
solved through a variety of methods. I used to have a truly wonderful book 
that had over 200 methods for proving the Pythagoras Theorem which nicely 
demonstrated the very different kinds of methods of reasoning and proofs 
contained in mathematics.

But ... no matter that there may be different methods, what they all have 
in common, of course, is proof. This is easy to say, but the word 'proof' 
has had different meanings in different contexts, and has also had 
different meanings in historical epochs. What does seem to be broadly true, 
though is that proof is what mathematicians use to communicate a kind of 
quality of essential validity -- truth -- concerning a given piece of 
knowledge.

However, as with any language, the specific forms that 'proof' has taken 
has changed throughout the history of the subject -- as also, inevitably, 
the topics which that unique language can discuss. Mathematics has changed 
and the things mathematicians can prove has changed, as also the things 
that mathematicians try to prove has changed. 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. 
And essentially, in this context, what has changed is the nature and the 
perception of what it is to communicate, and what it is that can be 
communicated. But what does seem to have distinguished mathematics over the 
ages is the attempt to communicate a result or a discovery -- the 
excitement felt by the scribes in the Rhind Papyrus, for example, in 
presenting their information -- by using an argument or sequence of 
statements the aim and the end of which is to utterly convince the reader 
that the result present is truly valid. Of course, even coming by THAT 
understanding, by that kind of attempt to distill the essence of 
mathematics, requires a very modern perception of what it is that 
mathematics try to do within all the various branches, as opposed to what 
other disciplines try to do. It's hard to be sure, in fact, that even in 
the preceding statements, the real 'thing' or 'essence' that distinguishes 
mathematics has in fact been successfully captured. After all, there's a 
very good argument for saying that the whole idea of 'proof' transcends 
mathematics. Philosophers also try to use logical argument. So do lawyers. 
It's there in rhetoric both within and outside 'religious' arguments. Where 
many religious arguments fall down is in the original assumptions, and not 
in the argument that it built upon those assumptions. 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. If in the opinion 
of the reader the argument presented by that kind of a thinker (and Newton 
for example was very religious) falls down then the inconsistency can often 
be traced back to the speciousness (in the critic's eyes) of the initial 
assumptions.

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.
Well ... IMO it is exactly to the point, and I will try to explain why. 
Poincare was very well versed in the history of mathematics. It was what 
led him to his first great discoveries in maths, such as when he tackled 
the three body problem. He also clearly understood the nature of proof and 
the differences between pure and applied mathematics. When he presented his 
paper to the King of Oscar II of Sweden to claim his reward, his paper 
contained a section on why there was no 'exact' solution available. In this 
regard, though, he was a vast improvement on Euler in that Poincare 
outlined the reasons why, and pretty much put the issues concerned with 
treating dynamical systems on the map. All that was then left for the King 
to decide was whether or not Poincare had given 'the best solution 
possible', and whether or not a 'best solution' was good enough given that 
not even Poincare had given an exact solution, and had only really claimed 
that an exact solution was impossible. Luckily for Poincare, one of the 
judges was Weierstrass who wrote to the King's ambassador (I think it was) 
and said: "You may tell your Sovereign that this work cannot indeed be 
considered as furnishing the complete solution of the question proposed, 
but that it is nevertheless of such importance that its publication will 
inaugurate a new era in the history of Celestial Mechanics. The end which 
His Majesty had in view in opening the competition may therefore be 
considered as having been attained".

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.

>   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. 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. In that kind of way ... earlier mathematicians did see the 
difference between solving for the physical motions of the planets and 
solving for the motions of the planets by interpreting their behaviour with 
respect to the lives of mortal humans ... but by and large they just didn't 
see those differences as being particularly important. That's the 
historical record. Things in this regard have changed so much that nowadays 
most (but not all) respectable mathematicians distance themselves as far as 
possible from astrology. In fact many mathematicians take great exception 
to the very idea that their tools (i.e. physically determining the 
positions of the planets) are used in that kind of way at all and wish that 
those who do it would stop it. Yet ... do I really need to construct a list 
of all the mathematicians in history who made astrological predictions 
seeing it as not only their duty, but a very real part of their job 
description?

Forgive me if I erred, but all I was trying to say was that this was a very 
modern attitude and not really representative of the attitude to the 
subject that has generally prevailed. 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), 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'. Since my stock 
in trade is the history of mathematics, I hope you will accept that this is 
a pretty big concession for me to make and that you will accept it in the 
spirit in which it is offered.

> >"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. Please excuse me for 
the fact that I had deemed it wisest to say nothing at all further about 
the underlying point that I was trying to make, but my experience of the 
L-OT list is that many people upon it are not really interested in the 
sharing and back and forth of information, but are rather more interested 
in always being right themselves. I tried to hint elliptically at what I 
was trying to say hoping that anyone who was really interested would work 
it out for themselves from the scant information I gave in those few 
quotes. I did not see the point in trying to elaborate any further on the 
matter because my experience is that it would have led nowhere but rather 
to another rather pointless exchange of emails. 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. I just didn't want to write any words at all of my 
own and find myself getting absolutely nowhere at the end of it by which I 
mean having had my store of knowledge increased. I do not enjoy it when I 
can palpably feel the little store of knowledge that I have being sucked 
out of me by things that I am reading. I am a bit weird that way.

> >"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, because Fourier lived before the times when the kinds of 
distinctions that you and I make were not really being made. (I had 
actually felt a bit like including a quotation from Vetruvius in my earlier 
email, actually, but refrained because what would have been the point, I 
had said to myself?!) I hate to belabour the point, but Fourier was a 
working mathematician in his day, and the attitude he evidences in the 
above statement was the generally accepted one of his day. Mathematicians 
simply did not make, in his day, the distinctions that we now make so 
easily. (I am willing to amend this too -- did make them but regarded them 
as 100% trivial if that would be better!). 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.


>"Phenomena" could as well refer to abstract phenomena -- i.e. "pure"
>mathematics.
Completely accepted -- because that's pretty much what all phenomena become 
once a mathematician starts looking at them from the point of view of the 
language of proof, which is the mathematicians' stock in trade. However, a 
directly related issue is the SOURCE of the phenomena that arouse the 
interest of a mathematician to study them so that they can prove something. 
And ... even within number theory as it was long practised, that source was 
the world of nature. Even Peano (1858-1932) was forced to keep the counting 
of things in mind when he tried to define that slippery concept we know as 
'number'. I gave his dates simply to try to indicate the recency of that 
phenomenon.

What was so outstanding in its day about Gauss' Disquisitiones Arithmeticae 
was the very way in which it helped to emplace number theory at the very 
heart of mathematics, and pure mathematics at that, as perhaps the prime 
exemplars of what mathematicians could and should reason about in and of 
themselves, and without regard for externals. It was a powerful revelation, 
and one that mathematicians quite rightly seized on. The methods and topics 
that Gauss directed himself towards in that kind of way helped to clarify 
what pure mathematics was ... as also did the methods and modes of study he 
directed at the physical problems he equally well delighted in. But ... 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.

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. By common consent, that process 
is generally regarded as starting properly with Gauss. But even there, you 
are dealing with somebody in whom mathematics of both the pure and the 
applied varieties took such an easy residence that it would be very foolish 
indeed to try to separate them out in that particular case. It is also 
regarded as not that constructive to try to separate them out too closely 
when dealing with the subject in earlier epochs ... even though it is often 
of great value to US to do so. Nowadays, it is easy to see their 
differences, and one can observe even at grade school level in which 
direction a given mathematician is the more likely to go simply by 
observing the topics they are interested in and the methods they prefer to 
use to solve the kinds of problems they seem to have a natural affinity for.

> >"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. He offered it as a 
way of describing a good picture. 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. If 
you really want me to I can try and check out the exact context, but I'd 
rather you didn't ask, to be honest, because I have other things to do. 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. 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.

>Sorry Kool, but I think you missed the point this time...
And ... I have done my very best to indicate why I don't think I have. 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. 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. But ... it has been nature. 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. As far as I can see the 
only way this is not so is if one adopts the 'brute number' approach -- 
i.e. to insist that since most of the mathematicians who have ever lived 
are either alive today or else have lived within the last hundred years 
then that should be the deciding factor in this matter. If that's the way 
the two of you want to play it then I bow my head humbly 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.

Come to think of it, it's probably best if I keep my big mouth shut from 
here on in if this is the kind of palaver it's going to cause every time I 
open it.

If I have failed to present my case properly then it is simply because I 
have not learned as much from doing mathematics as you have ... but please 
do accept it from me that my love for the subject is absolutely no less 
than yours. I just became interested in a different side of it, that's all.

I do accept your contention that I might somehow have missed the point ... 
and I accept that contention enough to revisit the issue because I am 
genuinely not trying 'always to be right' or to such a degree that I ever 
refuse to change my mind and therefore stop learning anything. I would 
really much rather be wrong and then grow from it than to insist that I am 
always right and thus ossify. And that really is the truth. What's on 
earth's the point in anything else?

"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.

I am not particularly wise, but I do have enough doubts about things, I 
hope, to be open to more things.

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

I thank you most sincerely for your kindness.

I also thank you for the complete lack of personal attacks in your email. 
It has certainly been a refreshing change from some of the stuff I have had 
to wade through recently. I look forward, therefore, to reading anything 
else you might have to say be it on this matter or on anything else.


Kool Musick
Keep Musick Kool


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