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During the recent arguments over what pure mathematical logic and
formalisms can or can't achieve, I found the following passage of my father's to
be useful. I might as well post it.
From page 5 of Life, Itself, Robert Rosen
wrote:
"Mathematics over the past century has given little
evidence that it is concerned with eternal, timeless, and hence unarguable
truth. On the contrary, contemporary mathematics is filled with (no pun
intended) chaos and turbulence, bespeaking a profound internal instability.
Historically, we are still witnessing what (it is hoped) are the transients
arising from two profound shocks: the overthrow of Euclid and the discovery of
inconsistencies (paradoxes) in Set Theory.
To be sure, most practicing mathematicians, like most
practical (empirical) scientists, go on about their business, indifferent to
such matters, convinced to the depths of their soul about the reliability of
what they do; like corks, they believe they will float on calm and troubled
waters alike. But I am speaking of foundations, and it is here we shall
look.
The two great shocks of which I spoke above have
coalesced, beginning in the early years of [the 20th century], into a frantic
concern with consistency, with a demand that a system of inferential
entailments (e.g., set of axioms or production rules, operating on a set of
given propositions or postulates) be free of internal or logical contradictions.
How can we be sure that a system of entailment, e.g., a mathematical system in
the broadest sense, is consistent? I shall be concerned with one particular kind
of answer given to this question, an answer championed by David Hilbert, which
can be summed up in a single word: formalization.
Hilbert and others thought they had traced down the
ultimate source of all the difficulties in mathematics. They pointed out that
propositions in mathematics are nominally about something; i.e., they
have meanings that involve referents outside themselves. Thus, for instance, in
Euclid, the word "triangle" is not just an array of letters to be
manipulated in a certain way; it refers to a rich and vivid kind of
geometric object. And even beyond that, it even refers to things in the
phenomenal world. In that sense, any Euclidean proposition containing the word
"triangle" can be thought of as describing a percept or quality
manifested by this external referent.
Thus, according to this analysis, mathematical truth had
come to involve two distinct aspects, one pertaining to how we are allowed to
manipulate the word "triangle" from one proposition to another, and another
pertaining to the actual referents of that word. I will call the former
syntactic truth, and the latter, semantic truth. Hilbert and his
colleagues argued that it was precisely by allowing semantic truth into
mathematics at all (i.e., in the admissibility of regarding a mathematical
proposition as the description of a percept or quality of something, allowing a
mathematical proposition to refer to something) 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 syntactic 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; (3) a finite set of production
rules for turning given formulas into new ones. In such a purely syntactic
system, consistency is guaranteed.
[ Statement by Kleene on formal axiomatics, which I will not
bother to type in]
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 formal analog of "hardness" and quantitation in science 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. 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 of
discourse needs to consist of nothing more than meaningless symbols pushed
around by definite 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.
Gödel's Theorem
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
inmost 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 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
(i.e., any attempt to capture every percept through a formalization of
any finite set of percepts) 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 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."
End of excerpt.
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