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.