FOM: General intellectual interest

Harvey Friedman friedman at
Thu Dec 18 14:11:06 EST 1997

Reply to Thayer 2:54PM 12/18/97.

Correct me if I am wrong: you are a computer consultant who treats
mathematics and philosophy as a part time hobby. Is that right? It shows.

Professional F.O.M. is a subtle and highly developed subject, about which
you should be trying to learn - if you can - and not be offering up the
extreme opinions of an amateur. Phrasing your biases in terms of questions
is much much more suitable - and welcome.

I have to be harsh like this since the over 245 people on this list might
mistake you for something beyond an opinionated hobbyist. The fom of course
very much welcomes amateurs; we are very anxious to have them on the list.
But usually it is difficult to get them to say anything.

But not in your case! My advice to you is to refrain from taking up space
in the fom with your "opinions" just because you want to vent some
misguided emotions. Maybe you view the fom as a badly needed release.
Obviously there are no risks to be taken by exposing yourself in this way
in an e-mail list, since you make your living in a totally unrelated way.
Ask some polite questions. But do not masquerade as a professional of any
kind. In fact, on each and every subsequent e-mail, remind us of your lack
of qualificatons. That's the way most students would work with the fom. If
it's good enough for students, it's good enough for you.

HAVING SAID THIS, I will dissect your nonsense just to demonstrate how it
can be done. But I am unlikely to waste my time any further with you -
unless you clean up your act and behave more as a student of F.O.M.

Thayer quotes from Lou:

>>Otherwise I am not conceding anything. Sure, "suspect" is vague, so is
>>"general intellectual interest", and perhaps even more, the significance
>>of "general intellectual interest".

>I will go farther: it is blatant nonsense as used by Harvey and Steve.

Your failure to grasp these concepts - which are used by everyone implicity
in their choice of professional research - is quite harmless. This is
because you are not a professional intellectual. Should you become one of
significance, you will undoubtedly have to use these concepts at some level
- within your abilities. Lou uses these concepts even though he may deny it
- but at an undeveloped level.

>"general intellectual interest" do most of Harvey's "interesting technical
>results"  (to quote Anand) actually have.

I'll try to get someone else to answer this for me. I work almost
exclusively for "general intellectual interest," so you can be assured that
I have a lot to answer with. But I will remain at least slightly polite and
refrain from doing so directly.

Anand, you are mentioned here, but I'm not quite sure you want your name
being used in such a negative context. Any comments, Anand?

>How many people even understand
>large cardinals, much less care about them?  Far fewer than are interested
>in number theory, be it abc or FLT.

Everybody who has gone through high school geometry is familiar with
theorem/proof at some level, far more than number theory, abc, or FLT. The
realization that mathematics needs more axioms after a long period of
complacency is quite substantive, gripping, interesting, intriguing, to a
very large group of people. The interest in number theory, abc, or FLT is
very recreational, in constrast. And it is not even what mathematicians
really value. They value instead things much more abstract and difficult to
communicate - where such things as FLT are consequences. The "realization
that mathematics needs more axioms .." and the fact that "we have the extra
appropriate axioms in hand..." is not only eminently communicable, it is
really a central matter in contemporary F.O.M. that is being emphasized.

The formal deatils of the large cardinal axioms are besides the main point.
Just as for the solution to Hilbert's 10th problem, the formal details of
computability are besides the main point.

>I suspect that the folks who partake of the botanicals that grow in Doctor
>Cantor's garden have OD on them: we know since Goedel that number theory is
>incomplete, indeed incompleteable.

Of course, this is a repulsive venting of emotions against - I suppose -
set theorists. Did you have trouble learning set theory? Most people do.

Any competent professional is aware that the incompleteness of number
theory raises more vital questions than it answers; especially, "what kind
of number theory cannot be answered with the usual axioms for mathematics?"
I know of absolutely no competent professional mathematician who is not, in
some way, moved by at least the statement of such questions. The answers
are not in.

>So details as to what large cardinal
>hypotheses are required to prove the graph minor theorem, or
>Paris-Harrington are about as exciting as improving the exponent on
>logloglogloglog(n) in the error term for some number theoretic function.

Congratulations! You have uttered a declaration that is both extremely
flattering to me, personally, and shows your gross amateurish bigotry far
better than anything I could say. Firstly, the Paris-Harrington theorem was
discussed - at least - in Science Magazine and Scientific American. That
should make you pause - if anything can. I never saw anything there about
improving the exponent on logloglogloglog(n) in the error term for some
number theoretic function. I really am coming to believe that you can't
tell the difference between the PH theorem and this kind of number
theoretic result! So why would I continue to talk to you? An exercise in
how to handle intellectual nonsense. After all, nonsense collection is
important in programming - this you may know(?) - and also in cleaning up
the neighborhood.

What is so flattering about this statement is that you are ignorant and
silly enough to believe that I proved that large cardinal hypotheses are
required to prove the graph minor theorem. Given the history of the graph
minor theorem, if I showed that it is necessary and sufficient to use large
cardinals, then, in my humble opinion (taking the context into account in
which this was hypothetically done) this would be arguably the greatest
single mathematical acheivment by a single individual of all time. It would
completely change our whole view of the nature of mathematics, and capture
the imagination of much of the academic community in a way that transcends
even Godel.

Well, this is what I like about your ignorance - it has amusement value.
Back to reality.

>The real "general interest" stuff has been mostly discussed by Tennant and
>Pratt in their discussion about mathematics on Quasar 1036, and Shipman et
>al in the recent discussion of "convincing proof".

Want to see what Pratt, Shipman, and Tennant think of being used in this
negative context? A deep aspect, by the way, of convincing proof, is
already illustrated by the necessary use of nonconstructive - or even
impredicative methods. This is a different aspect, one which you are
probably unaware of. And, yes, it has general intellectual interest.

>The two biggest foundational questions in mathematics are:

Now, Mike. I am repulsed by the sight of seeing something like this from a
hobbyist. You know next to nothing compared to the professionals on this
list, and have the nerve to write this! There is absolutely nobody -
including me - who would start a paragraph like this. How about starting
with "Among?" You should be ashamed of yourself.

You then go on and give a couple of reasonable questions with entirely
uninformative formulations. You know nothing about what it is like to
actually accomplish something professionally in this context, and so you
don't have a clue that the real problem is to break such questions down
into parts for which something dramatic can be really done. That is the job
of a professional - not a hobbyist.

>Next to this sort of discussion (which may be too vague for Harvey) details
>of which uninteresting set theory statement follows from the existence of
>which implausible infinite entity seems remarkably like medieval theology -
>which is probably not of much intellectual interest to most people.

I don't even know what set theory you are referring to - you certainly don't.

I close with some relevant postive content. Apparently we now have a very
clean simple finitary theorem, which has been "strongly certified" by a
very well known person in Ramsey theory, which can only be proved by going
well beyond the usual axioms of mathematics. And its connection with
standard topics in computer science are quite clear, and expected to grow.
I expect to be lecturing on it early next calendar year.

Mike - maybe you can chew on that - in your hobby time!

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