From the Praeludium of "Life, Itself":
Robert Rosen wrote:"The two great shocks of which I spoke
[the overthrow of Euclid and the discovery of inconsistencies (paradoxes) in
Set Theory] have coalesced... into a frantic concern with consistency, with
a demand that a system of inferential entailments (e.g., a set of axioms or
production rules, operating on a set of given propositions or postulates) be
free of internal or logical contradictions. Hilbert and others thought they
had traced down the ultimate source of all the difficulties in
mathematics... Hilbert and his formalistic school argued that it was by
allowing semantic truth into mathematics at all... that all the difficulty
arises.
Hilbert and his formalistic school actually asserted much
more than this. They argued that what we have called "semantic truth" could
always be effectively replaced by more syntactic rules. In other words, any
external referent, and any quality thereof, could be pulled inside a purely
syntactic system. By a purely sysntactic system, they understood: 1. A
finite set of meaningless symbols, an alphabet; 2. A finite set of rules for
combining these symbols into strings or formulas; and 3. A finite set of
production rules for turning given formulas into new ones. In such a purely
syntactic system, consistency is guaranteed...
The idea that all truth can be expressed as pure syntactic
truth, which is the essence of the formalist position in mathematics, I
claim to be the analog of Rutherford's position in science, the formal
analog of "hardness" and quantitation.
The formalist position is, first of all, an _expression_ of a
belief that all mathematical truth can be reduced to, or expressed in terms
of, word processing or symbol manipulation [computable] .
Hence the close association of formalization with the idea of
"machines" (Turing machines) and with the idea of algorithms. These embody
purely automatic procedures, which require no thought, no perception,
indeed, no external agency at all.
Second, the formalist position, that the universe needs to
consist of nothing more than meaningless rules of manipulation, is exactly
parallel to the mechanical picture of the phenomenal world as consisting of
nothing more than configurations of structureless particles, pushed around
by impressed forces.
The formalist position seems, on the face of it, very
attractive. For, by asserting that all truth is syntactic truth, it tells us
that 1. We lost no shred of mathematical truth in the process of
formalization, and 2. We are automatically guaranteed that mathematics is
consistent. We pay for these benefits by giving up the idea that mathematics
is "about" anything, i.e., that its propositions express percepts or
qualities, but on the other hand we are "informally" free to interpret these
propositions in any way we want. These are, of course exactly the same
attractions that the "hard" or quantitative sciences offer in the phenomenal
world.
The celebrated Incompleteness Theorem of Gödel effectively
demolished the formalist program. Basically, he showed that, no matter how
one tries to formalize a particular part of mathematics (Number Theory,
perhaps the innermost heart of mathematics itself), syntactic truth in the
formalization does not coincide with (is narrower than) the set of truths
about numbers.
There are many ways to look at Gödel's theorem. Indeed, the
Theorem itself has provoked an enormous literature, as might be expected.
For our purposes, we may regard it as follows: one cannot forget that
Number Theory is about numbers. The fact that Number Theory is about
numbers is essential, because there are percepts or qualities (theorems)
pertaining to numbers that cannot be expressed in terms of a given,
preassigned set of purely syntactic entailments. Stated contrapositively; no
finite set of numerical qualities, taken as syntactical basis for Number
Theory, exhausts the set of all numerical qualities. There is always a
purely semantic residue, that cannot be accommodated by that syntactical
scheme.
Gödel's Theorem thus shows that formalizations are part of
mathematics but not ALL of mathematics. Mathematics, like language itself,
cannot be freed of all referents and remain mathematics. Any attempt to do
this... must already fail in the Theory of Numbers.
On the other hand, Number Theory is still mathematics,
still a system of inferential entailment in itself. It is only that it is
not a purely syntactic system, not entirely a matter of word processing or
symbol manipulation, independent of any external referent. In other words,
Number Theory is not a closable, finite system of inferential entailment.
These facts, as embodied in Gödel's Theorem, do not make us give up Number
Theory as a part of mathematics nor even give up formalization as a strategy
for studying certain kinds of mathematical systems. They express, rather,
the limitations of formalization; it is not, as Hilbert thought, a
"universal" strategy.
My father quoted something by S.C. Kleene, from the book
"Meta-mathematics", which I didn't include here but which very succinctly
illustrates the ultimate purpose this kind of thinking was trying to
achieve. What they were trying to achieve was to make mathematics
computable, and there was a similar drive in science to make all phenomena
similarly computable. Science became equated with that notion (i.e., "it's
not science if you don't formalize it into something computable"), which is
what my father took issue with. As he said in the final paragraph, above,
you can do a lot with that limited world, but it is not all there is. And,
if you wanted to do something in mathematics that is not available from
within that limited world, you can go outside that axiomatically "perfect"
world and still be using mathematics. He said the same is utterly
true in science.
Judith
PS: It was brought to my attention that I forgot, on a recent post, to
delete an automatic signature that my email program puts on all email I
create. It is a signature that reflects something about the times, this
being an election year (and about me, I imagine)... But it is not
appropriate on posts to the list and that's why I delete it before I send. I
missed that one! My apologies to the
list.