J.P.Mayberry at bristol.ac.uk
Fri Mar 26 09:42:31 EST 1999
On the question whether the second order formulation of the
proposition "Every binary relation contains the graph of a function
defined on its domain" is valid in standard second order semantics,
Steve Simpson asks whether I would be irritated by the answer "it
depends on the real world". I'm not in the least irritated by that,
though I would have put it rather differently: "It depends on whether
the Axiom of Choice is true." Most mathematicians do believe it is
true, but that doesn't *make* it true. Most mathematicians also believe
that the Axioms of Infinity and Power Set are true, but that doesn't
make them true either. If either of the latter is false then that has
very serious consequences indeed for mathematics. But the question
whether they are true is a serious question and you shouldn't allow
yourself to be "irritated" by it.
Surely it's obvious that the questions about Infinity and Power
Set have a profound effect on how we look at second order semantics,
and first order semantics too, come to that. Indeed, unless we accept
Power Set there is no such thing as second order semantics.
But the fact that Choice does or does not hold in this or that
model of 1st order ZF or is or is not provable in this or that
consistent extension of 1st order ZF is, strictly speaking, quite
beside the point. To be sure, such facts may be relevant to what we
*believe* about the Axiom of Choice. But we know that the *truth* of
the Axiom doesn't depend on these things once we know that it is
formally independent of 1st order ZF. Of course there is a certain kind
of formalist who would say that once you know that AC or CH is formally
independent of 1st order ZF then that's an end of the matter. You now
have two different versions of set theory: Cantorian set theory, in
which CH is true, and Non-Cantorian set theory in which it is false.
But only one of these theories can have a consistent second order
extension, and *that* is the correct one.
I must confess, however, that I *am* irritated by people who
talk about "worlds", real or otherwise, if they do so in order to
avoid talking about truth. Of course a lot of such talk is harmless,
because it isn't an attempt to be mealy mouthed about truth - I'm sure
that is the case with Steve Simpson: I can't imagine him being mealy
mouthed about anything. It's precisely the attack on truth that I find
profoundly irritating in post-modernism and relativism in their various
guises ("Irritating" isn't really strong enough here: "irrational"
would be closer to the mark.)
The enemies of truth are to be found in unexpected places. For
example, in the introduction to his volume on set theory Bourbaki says
"Mathematicians have always been convinced that what they prove is
"true". It is clear that such a conviction can only be of a sentimental
or metaphysical order, and cannot be justified, or even ascribed a
meaning which is not tautological, within the domain of mathematics."
This is not just nonsense, it is pernicious nonsense. It pollutes the
stream of intellectual life. What on earth does Boubaki think the point
of presenting a proof is, if it does not represent at least an
*attempt* to establish the truth of its conclusion? Actually Bourbaki
tells us what the point is: "The *axiomatic method* is nothing but the
art of drawing up texts [a post-modernist buzz word] whose
formalisation is straight-forward in principle". Of course he leaves us
completely in the dark as to what the value of having such a formalised
text might be (or of just knowing "in principle" that we could produce
one). This is formalism in its most egregious manifestation. It is an
anti-intellectual, indeed, an irrational, doctrine.
It is with this kind of thing in mind that I insist on making
the distinction between set theory itself and its 1st order
formalisation in ZF. Formalisability in ZF is not the *primary
criterion* for validity of mathematical proofs, although any ordinary
mathematical proof that is valid will be so formalisable. If an
insurance company offered a special deal on life insurance to American
citizens over the age of 30, then "being qualified to serve in the U.S
Senate" would be a necessary and sufficient condition to qualify for
the policy. But it would not be the *primary criterion* so to qualify.
The company would be most unlikely, and ill advised, to market it as a
policy for potential U.S. senators.
School of Mathematics
University of Bristol
J.P.Mayberry at bristol.ac.uk
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