FOM: naive or brainwashed?

Martin Davis by way of martind@cs.berkeley.edu Martin Davis martind at cs.berkeley.edu
Thu Mar 12 15:35:50 EST 1998

Dear Reuben,

Thanks for your civilized and friendly message (no sarcasm). Some comments.

>Thanks for including me in your interesting posting.  I'm surprised
>that anti-foundationalism is now an industry, but if so, all to the
>good.  And I cherish your image of me "leading the charge," sword
>in hand?  or just a pen?  word-processor in hand seems visually wrong.

>From Tymoczko's anthology:

"Hersh's essay begins the challenge to foundationalism."

>May I apologize to Frege, Hilbert, and Brouwer, if you think I
>trivialize them?  I am glad to acknowledge all three as great
>thinkers.  Of course it wasn't my purpose to expound at length
>their contributions to mathematics and logic, although if you
>take a look at my book I think you will have to admit that I
>presented more than mere slogans.  I contended that all three
>programs to restore indubitable foundations to mathematics had
>not attained their goals, and are no longer being pursued for
>that goal.  I don't think that is a controversial statement.

I've read your book - cover-to-cover, and have learned some things from it.
(Thanks again for sending me a copy.) In my posting I agreed that you were
right about the above. What I believe was irresponsible on your part IN A
POPULAR BOOK was to give no hint that the foundational programs to which you
refer do live on and have produced much of value. And yes, I think you did
trivialize them.

>You also list Godel as a great man whom I dissed.  Well, I
>included Boolos proof of Godel's incompleteness theorem.  But
>I did quote Godel mainly as the outstanding Platonist.  If
>Volume 3 of his collected works had been available, I would
>have quoted him in more detail.  I might not have tried to
>refute him in detail, since a critique of Platonism in general
>is one of the main threads of my book.  I don't see that I
>trivialized Godel.

By not even mentioning the issues I listed in my posting under the head
"G\"odel's Legacy" you do trivialize his foundational ideas (of which
there's plenty in volume 2).

>"Reuben Hersh makes the remarkable discovery that mathematical activity
>is a social process."
>
>May I be forgiven for guessing that the "remarkable" is sarcasm?

Right!

>To me it's interesting that there are two popular responses to my
>unremarkable observation (not discovery.)  A fair number of fom'ers
>respond with indignation, fury, mockery, etc.  And then a few
>others respond, "So what?  Of course!  What else is new?"
>
>Can you account for this interesting duality?  Everybody
>agrees I'm all wet, but from two opposite reasons.

There are two points not at all opposite.

1. The simple fact that mathematical activity is social is obvious.
2. Noting that fact doesn't help in the least in resolving deep foundational
issues.

True in the past as well as the present. The Cauchy-Weierstrass program
giving us our epsilon-delta foundations for analysis. If only they had
understood the social context, how much better (quicker?) this would have
been accoumplished! (Yes, sarcasm.) The general notion of function beyond
what was analytically representable: Dirichlet, Riemann, Borel. Ditto. The
difficult debate over AC?
And finally, G\"odel's legacy?

>It's not a remarkable discovery.  It is an important
>fact, which philosophers of math (with a few exceptions whom
>I credit) has consistently ignored.  Not only ignored hitherto,
>but insists on continuing to ignore, even after I suggest
>some attention be paid.  As you know, since (I think) you have
>read my book, I attempt to show how this commonplace fact
>can be useful in unraveling some philosophical puzzles.

Not a successful attempt!

>I hope I won't be overstepping the bounds of friendship if
>I remind you of our conversation last summer in your house in Berkeley.
>You told me that you are a Platonist about the natural numbers,
>an agnostic about the rest.

This is the second time you mention on fom the half-facetious comment I made
when we talked. I don't have a problem with your bringing it up, but I would
have preferred not going into this murky and complex area. But OK: In fact I
regard the real objective character of the cumulative hierarchy up to \omega
+ \omega as so clear that it is difficult for me to understand those
(including distinguished professionals like Sol Feferman) who evidently
don't share this belief. (That's why I asked: am I naive or brainwashed.)
But (G\"odel's Legacy again) the hierarchy doesn't stop there - and once one
goes beyond that place there are only way-stations no place to stop. If we
think we have it all, the paradoxes (or, if you prefer, diagnolization)
intervenes and laughs in our face.

Of course, this is a thoroughly unsatisfactory state of affairs begging for
serious foundational research. If your approach is helpful with this, please
explain how. (By the way, I've brought these matters to your attention, not
only when we talked in Berkeley, but also much earlier after your talk I
attended at CUNY.)

> As you mentioned in your posting
>about Lagrange's theorem, it's clear to you that such a theorem about the
>natural numbers was true before there were people,
>and I suppose before there was an Earth or a Sun.  Anyway,
>I said that to me, as a materialist, such a view was hard to
>connect or relate to my overall materialist philosophy.  You
>said that you also didn't know how to make such a connection.
>To me, that's a reason why Platonism is untenable.  To
>you, evidently, this gap does not prevent you from continuing
>as a Platonist.

One consequence of Lagrange's theorem is that any bunch of pebbles can be
rearranged into four piles each of which is arranged as a square array. As a
materialist do you really believe that it is conceivable that bunches of
pebbles on a jurasic beach were immune to this law just because there were
no mathematicians around to formulate it?

>Understand, I am not dreaming of convincing you of anything or
>changing your mind about anything.  Just trying to get you
>to see why someone in his right mind might find Platonism indigestible.

Oh, I do understand that very well.

>In my view, locating math in social interactions is a materialist
>explanation of the nature of math.  I give long lists of other
>social realities just to drive home the obvious but often
>ignored fact that reality does include social reality--naturally,
>based on a physical basis.  Just as the mind is based on the
>brain, but is not the same thing as the brain.  (Surely that familiar fact
>doesn't require justification?)

As Marvin Minsky says: "The mind is what the brain does." As to math, see above.

>It's not a matter of finding a physical location for 7.  It's
>a matter of explaining the sense in which we can say it
>exists.  Its objectivity is not an explanation of that.

I believe that for mathematics the fact that an entity's property's are
objective (there for us to find out about, not for us to fix) is what we
need to know to say that that entity EXISTS. No more and no less. I'm happy
to leave to philosophers whatever deeper ontological issues there may be.
But I dispute that "locating math in social interactions" sheds the least
light on WHY there is that objectivity.

>Good luck with your talk.  I hope you send me a copy.

Thank you. I'm not sure that I'll write anything.

Martin



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