FOM: Irritation

John Mayberry J.P.Mayberry at
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.

John Mayberry
School of Mathematics
University of Bristol

John Mayberry
J.P.Mayberry at

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