FOM: Foundationalism
Reuben Hersh
rhersh at math.math.unm.edu
Sat Sep 12 20:06:45 EDT 1998
Dear Steve,
I appreciate your blast at me, as giving me a welcome opportunity,
in answering it, to explain a little more about foundations and
foundationalism. May I also be permitted to acknowledge your well-known
courtesy, politesse, moderation, modesty, and strong desire to achieve
mutual understandilng with your contestant?
The issue of foundations did not arise till modern times.
The presumption that mathematics is in Heaven (Plato) or in the mind
of God (Leibnitz) answered all questions.
Kant, in his famous effort to rehabilitate Leibnitz from the
damage done by Hume, sought to justify (give a foundation to) a priori
knowledge, which meant essentially mathematics--arithmetic and geometry.
He invented universal, unavoidable "intuitions" of time and space.
Frege was a Kantian, but he disagreed with Kant about arithmetic.
Unlike geometry, he said, arithmetic is based not on intuition but on
logic. Logic was his foundation. Like Kant's foundations, Frege's
foundation was supposed to render mathematics indubitable. Later
Russell joined Frege in the logicist enterprise. He wrote, much later, "I
wanted certainty in the kind of way in which people want religious faith.
I thought that certainty is more likely to be found in mathematics than
elsewhere....As the work proceeded, I was continually reminded of the
fable about the elephant and the tortoise. Having
constructed an elephant upon which the mathematical world could rest, I
found tthe elephant tottering, and proceeded to construct a tortoise to
keep the elephant from falling. But the tortoise was no more secure--than
the elephant..."
Russell and Whitehead and Frege were good examples of people
who tried to go back and back to the indubitable. One famous difficulty
is their axiom of infinity. You can't do modern math without it. Yet
to my knowledge nobody has claimed its indubitable. Nobody knows
whether the cosmos contains finitely or infinitely many hadrons and
bosons. We could dispense with infinity, and only do the finitistic part
of mathematics. But we don't want to give up analysis, geometry, and so
on, so we accept the axiom of infinity.
There are also the axiom of replacement, which I would defy
you to claim is intuitively obvious, and the axiom of choice--need I
say more?
Hilbert, in response to the challenge of his rival Brouwer,
proposed that the foundation be, not logic, not intuition, but marks
on paper, visible,tangible pencil marks. It is widely thought that
Godel killed Hilbert's project.
Meanwhile, Brouwer found the indubitable foundation in his own head,
the languageless insight of the Creative Mathematician.
Lakatos pointed out that logicism, formalism and intuitionism
are three variants of the same philosophy--foundationalism. Namely,
that is it possible and necessary to provide mathematics with a
secure foundation, thereby rendering it indubitable.
That's what foundationalism is--the demand for a secure
foundation to make mathematics indubitable.
Being against foundationalism means denying that such a foundation
is necessary. Being prepared to accept mathematics as a particular
kind of human activity.
Now it is proposed that finding fouindations is no problem ,you
just dig back and back till you get to the indubitable place.
May I remind you of the advice Senator George Aiken of Vermont
gave to LBJ about the Vietnam entanglement? Aiken said, "Get out, and
announce victory."
If you claim that what every mathematicain actually does
is finding a foundation, you are in Russell's situation with elephants
and tortoises. You can't go down all the way, so you go down until
you decide to quit.
Was Euclid's 5th postulate indubitable? Mathematicians thought
so for 2 millennia.
Any multiplication operation must be commutative?
Likewise.
As for what every mathematician does with regard to rigorous
proof, at least 3 Fields Medal winners won fame and accaim with
incomplete proofs of important results.
Finally, politics. That of course doesn't belong on this
list, but since you felt complelled to throw in a little name calling
(after all the pleas a while back for no name calling!) I think you
can let me say how I used the scary term left-wing in my book
Left wing politics, in my meaning, is politics that increases
the influence of the bottom rung, the lower class, of any particular
society. In the U.S., for example, both the 14th and the 20th amendments
to the Constitution would be classified as left wing. Perhaps you'll
be kind enough to tell the list whether you would have suupported either
if you been around at the time. Maybe you would, if it had included
a clause outlawing taxes on upper income folks.
Reuben Hersh
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