Conference rusure::math

    Anyone who believes s/he knows enough to make general statements about
    "mathematics" and "science" is grossly ignorant.
    
    Prof. Marcus sounds like a get-rick-quick-artist.
    
    Dick

    >What do you make of this? Is some or all modern mathematics divorced from
    >reality? Is the science of mathematics "disintegrating" (no puns, please)?
    >Do you know of any other discussions of this type?
     
    I agree, *some* math work have no useful direct impact on life, that 
    i think every one can agree on. but i think you find that in other
    areas too, not just math. 

    some people *must* publish papers to earn a living, or to stay employed
    or to earn a tenure etc.., so you are bound to see stuff that is just
    written for the sake of writing a paper..
    
    i read somewhere, that only 10% of science writings is actually new and
    useful, the rest is repetitions, or rehashing of ideas, or just too 
    abstract and out of this world to be of any use.

    but again, back to math, most pepole do math because it is just 
    fun ... they leave the rest to figure a good place to apply to..

    /nasser

    and yes, i agree with .1 too :-)

.1> Anyone who believes s/he knows enough to make general statements about
.1> "mathematics" and "science" is grossly ignorant.
    
    That's interesting, Dick. Are you a skeptic? Is there anything at all
    that may be said with certainty? 
    
    I'm truly interested in your (everyone's) opinions.
    
    Thanks,
    
    Dwayne
    

Hardy's book A Mathematician's Apology dwells on the assertion that math is
useless (but nonetheless lovable).
Tom

For an excellent book which deals with these and many other epistemological and 
philosophical problems within mathematics see:

	Kline, Morris. Mathematics, The Loss of Certainty. NY, Oxford 
	University Press, 1980.

Kline cogently addresses the classic problems, as well as explores new ground
with plenty of historical perspective. Topics covered range from the genesis of 
mathematical truth, the "mathematization" of science, Logicism versus 
Intuitionism, Formalist and Set-Theoretic foundations, the "illogical" 
development of mathematics, plus many more!

If Russell & Whitehead's Principia Mathematica has ever kept you awake at 
night, this book is for you!

James. (An old logicist who wishes G�del's margin had been too small..) :-)

    re .4
    
    My statement explains why I wouldn't cross the street to hear the
    professor's lecture or watch his tape.  It explains why I believe he is
    one more in a collection of popularizers trying to make a quick buck
    off the general population's ignorance of mathematics.  But, in the
    spirit of providing some understanding of my (admittedly arrogant)
    statement, I submit the following:
    
    There are so many specialties in both math and science that any
    allegations about relationships between the two domains can be tested
    empirically.  If the author just makes generalizations like "math today
    is irrelevant to science", what is he really saying?
    
    Here are two possiblities, together with some of my thinking about why
    the implication I attribute to the professor is false.
    
    a) The professor could mean 
    
    	"There is no scientific application of the work in fractals or
    catstrophe theory".  
    
    Both are less than thirty years old and so should qualify for "recent"
    mathematics.  A number of workers in scientific fields which deal with
    quasi-organized phenomena (meteorology, embryology, etc) have
    introduced the ideas of fractals and catastrophes in their attacks on
    these difficult problems.
    
    b) Another possible meaning might be
    
    	"The fields which use recent mathematics are not scientific"
    
    For example, the technology for graphic compression was strongly
    influenced by fractals.  You could define technological development as
    outside of science, if you wanted, but that wouldn't change the fact
    that many of more recent developments in math do have applications.
    
    I'm not going to try to refute the statement attributed to the good
    doctor (whose name escapes me) any further.  I hope I have clarified
    why I made the dogmatic statement.
    
    Dick

"I have never done anything useful. No discovery of mine has made, or is likely
to make, directly or indirectly, for good or ill, the least difference to the
amenity of the world. I have helped to train other mathematicians, but
mathematicians of the same kind as myself, and their work has been, so far at
any rate as I have helped them to it, useless as my own. Judged by all
practical standards the value of my mathematical life is nil, and outside
mathematics it is trivial, anyhow."

    "One thing I have learned in a long life: that all our science,
    measured against reality, is primitive and childlike-and yet it
    is the most precious thing we have."
    
    by....
    
    
    Albert Einstein.

    re .9
    
    As has been pointed out by Eric Temple Bell, within fifty years of
    Hardy's death, his "unapplicable pure mathematics" was being used in
    several ways.  I'll look up the specifics tonite and post them here.
    
    Hardy had no practical intentions, that much is clear.  But every
    generation makes whatever use it can of the knowledge it inherits,
    regardless of the originator's ideas.
    
    Dick

re -1

Hear, hear! As far as I can remember my history of Mathematics, when matrix
algebra, quaternions, or Boolean algebra were invented, nobody could see any
use for them at the time. Did that render them useless?

Peter

There's no question that some mathematics have been invented with no apparent
use, which indeed eventually became useful.

Think about the *GREAT* mathematicians, though. In this list, I personally
would include Newton, Gauss and Euler. They invented their mathematics for the
purpose of describing, explaining and predicting reality. They were scientists.

Newton invented his to describe planetary motion.

Euler invented his to describe planetary motion (lunar), for hydrodynamics, and
for analytical mechanics.

Gauss invented his for astronomics, and for the physics of magnetism and
electricity.

They were reality-focused.

Oh yeah, and what about Fourier's study of heat diffusion? Mathematics invented
to describe and predict reality.

In contrast to reality-based mathematics, consider the modern versions which
start with arbitrary axioms. Head games. SOMETIMES a use is found for them. But
I claim that that's irrelevant. The purpose was fantasy - like writing fiction.
That the fiction became true in a few cases is merely serendipitous and the
mathematicians often couldn't care less. (In fact, some have claimed that
applying their mathematics defiled it.)

I would love to see another great mathematician in my lifetime. I predict he
will NOT come from the roles of academic mathematicians. If he comes, he will
be a scientist who had to create his mathematics, just like Newton, Euler and
Gauss.

Dwayne

> Think about the *GREAT* mathematicians, though. In this list, I personally
> would include Newton, Gauss and Euler. They invented their mathematics for the
> purpose of describing, explaining and predicting reality. They were scientists.
    
    I believe this to be more a function of the times in which these people
    lived, rather than any specific desire on their part.
    
    For example, Gauss invented non-Euclidean geometry but said nothing of it.
    One can only speculate why, but I think his silence was due to anticipated
    ridicule on the part of his contemporaries.
    
    Euler did a lot of work with primes and factoring, something considered
    useless until quite recently, and which is REALLY important to certain
    government departments.  No "predicting reality" back then, but he did
    it nevertheless.
    
    I think the evidence quoted in .13 is too selective.
    
      John

    Riemman geometry was invented in 1800's and used by Einstein in 1916
    to formulate the differential equations over the space-time continumum(sp?)
    for his general theory of gravitation.

    i dont know if at the time Riemman came up with his geometry, that
    there was a use for it?

    I agree though, many math today has no apparent purpose, but who knows
    some of that might have a use for years from now?

    /nasser

I interpreted Hardy to mean, that in symbols it's math, and if you use 
numbers it's arithmetic.  Math can never be applied.
It's tenuous, but that's my take.
yrs
Tom

    You don't have to "plug in numbers" to do applied mathematics.  If you
    can prove that a certain differential equation cannot be solved in
    closed form, you have made a very practical discovery, which if heeded,
    will save hours of futile searching.
    
    Dick

RE: .14 by John

>    I believe this to be more a function of the times in which these people
>    lived, rather than any specific desire on their part.

Interesting point of view, John. I *think* you're saying that the science of
their day required them to invent their mathematics, and that the science of 
today does not require that kind of invention. You may be right. How sad if
true.
    
>    For example, Gauss invented non-Euclidean geometry but said nothing of it.
>    One can only speculate why, but I think his silence was due to anticipated
>    ridicule on the part of his contemporaries.

If your speculation is true, it supports my argument that the mathematics
of that time was in general reality-oriented (yes, I know that non-Euclidean 
geometry has real-world applications; but that wasn't the motivation for 
its invention).
    
    Euler did a lot of work with primes and factoring, something considered
    useless until quite recently, and which is REALLY important to certain
    government departments.  No "predicting reality" back then, but he did
    it nevertheless.

Did he consider work with primes and factoring useless? I'm not sure. I think
I remember reading something about his motivation in his work on the theory
of numbers. I'll have to see if I can find it. I'd be very surprised if he
didn't see practical applications to combinatorial mathematics, though.

There's no dispute that Euler worked in both applied and pure mathematics.
    
>    I think the evidence quoted in .13 is too selective.

Perhaps. I'm not claiming that Newton, Gauss, Euler et al were exclusively
applied mathematicians (although that could easily have been inferred by my
previous notes). Let me see if I can express myself better.

Mathematics used to be defined as the science of measurement. As such, it 
was a valuable tool for man. It allowed him to describe the world and predict
the future. As "the queen of the sciences," it made possible all the 
creations of engineering and technology.

Mathematics has recently become "the study of numbers, their form, 
arrangement, and associated relationships, using rigorously defined
literal, numerical, and operational symbols" [American Heritage Dictionary,
2nd College Edition]. Note the total absence in this definition of any
reference to reality. It has become the equivalent of poor science fiction or
fantasy: entertaining, but not limited by reality.

Pure mathematics is not the queen of the sciences, but the joker. It's a deck
of wild cards. No one knows whether any piece of it will become worth 
anything or not. Rather than being objective, increasingly mathematics is 
becoming subjective.

It's this trend that I see foretelling the dead-end of mathematics as 
anything valuable. Science will be thwarted by the limits of existing 
practical mathematics (as wonderful and as powerful as it is). The Newtons 
of the future will not invent The Calculus, but will be involved in a 
self-gratifying, self-righteous mental masturbation of deductive symbol
manipulations.

I hope I'm wrong. I hope scientists will be able to drag mathematicians back
down to earth. But I'm doubtful. That's why I believe that future practical
mathematics will increasingly be invented out of necessity by scientists 
while the "mathematicians" are busy diddling themselves.

Dwayne

    re .18


    Your dictionary definition is so far off the track that it is misleading. 
    The mathematics of the twentieth century cannot be limited to numbers 
    and quantitative topics.  Indeed, neither could that of previous
    centuries.  Geometry only addressed numeric relations after Descartes.
    
    Mathematics is an aesthetic pursuit, so its subjectivity is essential,
    just as you and I might not agree on the beauty of a Picasso, for
    example. Its objectivity is attributable to its logical processes, not
    to its subject matter.

    The "value" of mathematics is essentially subjective.  Your definition
    and mine my differ, so I am not convinced by your "foretelling the
    dead-end of mathematics as anything valuable."
    
    Science will NOT "be thwarted by the limits of existing practical
    mathematics."  Mathematics has never limited science, but only those
    scientists who knew too little mathematics.
    
    Your diatribe sounds very much like what one of my sophomore
    engineering students who was having trouble with symbolic logic might
    generate.
    
    >I hope scientists will be able to drag mathematicians back down to
    >earth. 
    
    That is unlikely.  When scientists frame questions that mathematicians
    find intriguing, the latter pursue them.  When scientists just continue
    to collect the pedestrian data that don't test their hypotheses,
    mathematicians will look elsewhere for inspiration.
    
    fwiw,
    
    Dick

	It is agreed here that mathematics has aesthetic value.
	My version of Hardy is not "mathematics should be stopped, it is
	stupid and useless", it's "mathematics has it's own value, and
	aesthetic value, but pure mathematics is never applied.  Applied
	mathematics is arithmetic."
	Tom

    re .19 by Dick


>    Your dictionary definition is so far off the track that it is misleading. 

Agreed. The AHD2nd is a poor dictionary. This fact does not alter my argument
that mathematics is progressively becoming subjective.
  
>    Mathematics is an aesthetic pursuit, so its subjectivity is essential,
>    just as you and I might not agree on the beauty of a Picasso, for
>    example.

I absolutely disagree. Mathematics is an offshoot of epistemology, not 
aesthetics. It's a study of what's true, not what's beautiful. Mathematics
is no more a matter of aesthetics than is physics, chemistry or biology.

I also disagree that subjectivity is essential to the pursuit of aesthetics.
But since it's not germane to the topic, I'll not pursue that rathole any
further.

>    Its objectivity is attributable to its logical processes, not
>    to its subject matter.

No. Objectivity pertains to the relationship of consciousness to existence. 
It depends upon the view that reality exists independent of the mind (and 
therefore, of logic). The objectivity of mathematics is measured by how well 
it may be used to describe reality and predict future events or states.

>    The "value" of mathematics is essentially subjective.

The value of mathematics depends on the situation to which it's applied and
for whom it's applied. In other words, it depends on the objective 
circumstances. If it were subjective, it wouldn't matter to me what axioms
you chose to use to build the bridge I'm about to cross. I assure you, it 
would.

Is it a subjective matter whether to use Euclidean or non-Euclidean geometry
to describe the curvature of space? No. Mathematics must be objective to
be of value.
    
>    Mathematics has never limited science, but only those
>    scientists who knew too little mathematics.

In other words, mathematics has always outpaced science. Why, then, did Newton
need to invent The Calculus? Wouldn't you say that science would be limited
today had he (or someone else) not invented it?
    
>    Your diatribe sounds very much like what one of my sophomore
>    engineering students who was having trouble with symbolic logic might
>    generate.

I'm not sure what your point is. Is the similarity between your student's 
objections and my diatribe germane to this discussion?
    
>    When scientists frame questions that mathematicians
>    find intriguing, the latter pursue them.
    
Are today's "pure" mathematicians interested in soiling their hands with 
real-life problems? What kind of scientific question would intrigue a modern
mathematician? (Not rhetorical questions!)

Dwayne

re .18 ("Pure Mathematics is subjective")

Reading your note I get the impression that you seem to think it is enough for
a mathematician to define an arbitrary set of axioms and proceed from there.
This is what you lament as the lack of connection to reality.

Perhaps the little I learned in Metamathematics will help you; the part usually
referred to as Model Theory which deals with systems of proofs. When confronted
with an arbitrary set of axioms - and here you and I agree: they are arbitrary
until proven otherwise - as a Metamathematician I need to ask two questions:

1. Is this set of axioms consistent?
2. Is this set of axioms complete?

The way to answer these questions and give proof of the answers is neither simple
nor easy, so what follows is a gross oversimplification, nevertheless indicative
of what happens.

Basically, in order to establish the consistency and completeness of a set of
axioms, I need to another system, which has already been proven to be complete
and consistent, and with which I can establish a one-to-one correspondence.
This other system is usually another mathematical system, but not always,
otherwise we would suffer infinite regress. When it is not an mathematical
system, it is a set of observable phenomena in the real world. Here is your
anchor to reality: there is nothing more consistent and complete!

Let me give you an example. If I can establish a one-to-one correspondence 
between the concepts used in my axioms and the workings of a left-luggage
office on a railway station, I can examine:

1. the truth of my axioms when mapped to this office, and
2. the coverage of my axioms when mapped to this office

If the axioms are true for the left-luggage office, then they are consistent.

If there is nothing that I can say about left-luggages, using the concepts of
the axioms, that are not implied by my axioms, then the axioms are complete.

--------------

Having said that, let me apologise to those in the know for such an inadequate
and oversimplified description of model theory.

But also let me ask those that would dismiss Mathematics as intellectual jacking
off to inquire more assiduously into its disciplines before castigating it.

Peter

.21> I absolutely disagree. Mathematics is an offshoot of epistemology, not 
.21> aesthetics. It's a study of what's true, not what's beautiful. Mathematics
.21> is no more a matter of aesthetics than is physics, chemistry or biology.
    
    "It's a study of what's true, not what's beautiful."  Now there's a
    quote to haggle over coffee on a long winter's evening.  Surely the
    study of Fermat's Last Theorem, not known to be true in general, is
    more a study of beauty than of truth.  Mathematicians often decide to
    study problems that are aesthetically beautiful, unlike physicists who
    are presented with hard, cold realities that they must develop theories
    to explain.  I claim the more energy is spent investigating elegant 
    unsolved Mathematics problems (such as FLT, the Goldbach conjecture, etc.) 
    than in all of physics.
    
    So if we define Math as "what Mathemeticians do" I would say it is
    primarily an aesthetic pursuit.  (Is Computer Science "what hackers do"?)
    
.21> The value of mathematics depends on the situation to which it's applied and
.21> for whom it's applied. In other words, it depends on the objective 
    
    Stipulating that to be true, don't we have a "time value of money"
    problem here?  Today's Knot Theory may be totally inapplicable for the
    problems we perceive today, but who knows -- tomorrow's astrophysics
    may depend on knowing how to transform one Knot into another.  Is it
    your contention that we NOT study Knot Theory today just because nobody
    needs it today?  Do you wish to make the contention that whenever we
    decide we need advances in Knot Thoery, then we somehow materialize an
    army of experts to work on the problem at hand?  Isn't that sort of the
    bottom line conclusion from all this discussion?  It's the conclusion
    I draw, anyhow.
    
> Are today's "pure" mathematicians interested in soiling their hands with 
> real-life problems? What kind of scientific question would intrigue a modern
> mathematician? (Not rhetorical questions!)
    
    Some of today's problems, such as algebraic coding theory, combine
    beauty and practicality.  Trouble is, sometimes it takes just a touch
    of math and you've got enough infrastructure to live with.  A code that
    only works with (say) 51200-bit strings can often be "made do with", so
    no further math is required.  Or, there may not be a known 15x15 Latin
    Square, so the experimenter make do with 14x14 or 16x16 instead.
    
      John

	Oh I think physics is aesthetic as well.
Tom

RE: .23 by John

>    Surely the
>    study of Fermat's Last Theorem, not known to be true in general, is
>    more a study of beauty than of truth.

If a monster computer, having cranked away for years, finds a falsifying case 
for FLT, how much more study would mathematicians put into it for the sake
of aesthetics?

I put to you that the study of FLT is a search for truth and knowledge, not
for beauty and aesthetics.

Dwayne

...and I put it ot you that if FLT were 

	x ^ 0.78345562 + y ^ 3477762 = z ^ 13.5678

or something equally ugly, nobody would have spent time to study it.


Peter

    re .25 and previous,
    
    Dwayne,
    
    We will just have to disagree.  There is too little common ground for
    the discussion.  Obviously, we don't associate the same concepts with
    the words "mathematics", "science", "true", "beautiful".  It would take
    longer than I have to reach understanding between us.
    
    fwiw,
    
    Dick

.25> If a monster computer, having cranked away for years, finds a falsifying 
> case for FLT, how much more study would mathematicians put into it for the 
> sake of aesthetics?

    Besides the excellent riposte in .26, let us stipulate that a computer
    search reveals a falsifying case, say for N = 9091827216977716341729.
    Does everybody pack up and move on?  No, the search continues for other
    Ns as well as analysis of the number 9091827216977716341729 and the
    bases raised to N.  Plenty of work for generations.
    
    I point out that another Fermat-related problem regarding primality of
    Fn :== 2^(2^n+1) did not die suddenly when it was discovered (in 1732,
    by Euler(!)) that F5 was composite.  F6 was factored in 1880, F12 in 1877,
    etc., etc.  The problem lives on, not because it is useful but because
    it is beautiful.  The same will apply to a falsified demo of FLT.
    
      John

    OK. I can see no one's getting anywhere with this. And, I'm spending
    way too much time on it. So, I'm going to have to leave it as it
    stands.
    
    Thanks for the stimulating discussion!
    
    Dwayne
    

    .18
    
    The greatest mathematician of this century was John (Janos) Von
    Neumann. Von Neumann made contributions to Logic and Set Theory, which
    were not very applied, yet the results are very germain to the modern
    theory of computability and complexity of computation.
    
    Von Neuman also made contributions to theory of self replicating
    automata at a time when genetic chemistry was not as advanced as it is
    to day. These very results are now being used to understand the role of
    enyzymes in genetic replication.
    
    Von Neuman also worked in very applied areas such as the development of
    computers (electronic variety) and in the area of quantum mechanics. 
    
    His approach was always that of a mathematician, rather than a
    physicist. Non the less his contributions are among the formost
    mathematical creations of all time.
    
    

    Re .30
    
    While agreeing that von Neumann is an excellent example of a 20th
    century mathematician in the mould of Gauss, if we're voting for
    greatest 20th century mathematician then I'd like to put in a vote for
    Kurt G�del on the grounds that his main contribution to maths was to
    break the mould of the deterministic belief that any problem can be
    solved if only enough time and resources are spent on it.
    
    Brian

    Re .31:
    
    And I would contend that G�del's contributions while being the most
    important single mathematical revelation of this century, were just
    that: "single".  G�del never really did much beyond this narrow field.
    
    My own nomination for greatest mathematician of this century, and by
    many accounts the last of the "Truly Great" mathematicians (in
    traditional terms) was David Hilbert.  Of course to be fair, Hilbert
    was only half in this century and half in the last.
    
    --  Barry

    talking about "greatness" seems very subjective, there ought to be
    criteria for these things.

    There are many "great" mathematicians , I dont think it is fair to
    pick one or two just because they are more known than others.

    I think it is the human nature that asks to have just one, as the
    greatest, because it is not as dramatic to say these 23 people are
    all together as equally great, as when one says this one is the
    greatest.

    so, who is the greatest mathematicians this century, my answer there
    is no one, there are many, each in their own way and field in the
    "greatest" .

    I can come up with 20 names of mathematicians this century, you'll
    be hard pressed not to say that any of them is not the the "greatest".
    
    /Nasser

    

    "..even the most remarkable relationship between mathematics and
    physics. Mathematics is not a science from our point of view,
    in the sense that it is not a natural science the test of its
    validity is not experiment. we must incidentally, make
    it clear from the beginning that if a thing is not a science, it is not
    necessarily bad. for example , love is not a science. so, if something
    is said not to be a science, it does not mean that there is something
    wrong with it; it just means that it is not a science."

    from The Feynman lectures on physics, volume 1.