FOM: Absolute truth

[John Mayberry]

Vladimir Sazonov has raised some very important issues in his posting 
of 30 March. There's a lot of what he says that I don't really disagree 
with. In particular, I think that his attempt to clarify the notion of 
"feasibility" and its bearing on the foundations of arithmetic is 
useful and interesting. But it seems to me that he has got himself into 
an awful muddle about truth, and the muddle is not unique to him, but 
is quite widespread.
 	A mathematician is a kind of professional sceptic: he demands 
that propositions be *proved* and terms be *defined*, and his standards 
of proof and definition are much much stronger than those of any other 
science. Benjamin Peirce put it rather well: "Mathematics is the 
science which draws necessary conclusions." It is their attempt to draw 
*necessary* conclusions that distinguishes mathematicians from 
physicists, for example.
	 Mathematicians have *ideal* standards of proof and definition, 
although in particular cases they may, indeed, do fall short of the 
ideal. But, of course, you can't prove every proposition and you can't 
define every concept. The mathematical propositions that are accepted 
as true without proof, and the mathematical concepts that are accepted 
as meaningful without definition constitute the foundations upon which 
mathematics rests.
 	There is nothing irrational or anti-intellectual, *per se*, in 
doubting whether a given proposition is true or a given concept is 
clearly determined. (There is nothing irrational in Sazonov's doubting 
that the notion of natural number is clearly determined in this sense.) 
What is irrational and anti-intellectual is *declaring*, as Bourbaki 
did that there is no such thing as a true proposition or a determinate 
concept. If you really believed that, why would you take up mathematics 
as a profession?
 	Sazonov says "I cannot imagine that, say, the Choice Axiom (and 
what is proved with its help) is TRUE or FALSE in some absolute sense 
of those words." But there is no "absolute" sense of those words, when 
they are used in their primary meanings, which are the only ones 
relevant here. Truth and falsehood do not admit of degrees. *Certainty* 
admits of degrees, and *absolute* certainty may indeed be unattainable, 
or even undesirable. But what we are certain (or uncertain) *of* is the 
truth or falsity of propositions. We can't even make sense of the 
notion of certainty unless there is something for us to be certain or 
uncertain about. "Certainty" is a psychological term, and therefore 
does not belong to logic. And allowing psychological notions to leach 
out into our logical ones is productive of the most profound confusion. 
	We can see this confusion at work in Sazanov's own remarks: "I 
like this axiom [i.e. the Axiom of Choice] and rather consider it as 
REASONABLE or USEFUL." Now "reasonable" is ok here - you might think it 
is *reasonable* to suppose the axiom to be true - you might even think 
that if you thought it to be false, or unsuitable in the role of an 
*axiom*. But whatever could it mean to think it *useful*? Useful for 
what? We are talking about *proof* here. Any proposition occurring as 
an ultimate presupposition in a proof is utterly useless unless it is 
true, and even being true is not enough for a proproposition to be 
taken as an *axiom*. We may think Goldbach's conjecture is true; it may 
*be* true. But we cannot just take it as an axiom.
 	Of course there is nothing prima facie irrational in supposing 
that the Axiom of Choice is neither true nor false: it might not be a 
determinate proposition at all, for the concepts in terms of which it 
is couched may be ill defined in some sense. (I don't think this is the 
case here, by the way.) But to suppose that there is no such thing as 
being true or being false, to suppose that truth claims are 
"sentimental" or "metaphysical" as Bourbaki claims to do 
("metaphysical" was a term of abuse in Bourbaki's youth), is to give up 
rationality altogether. In mathematics, which is essentially about 
*proof*, you clearly cannot adopt *axioms* on the sole ground that they 
are useful. If you do, who is going to accept the conclusions of your 
proofs?

John Mayberry
School of Mathematics
University of Bristol

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John Mayberry
J.P.Mayberry at bristol.ac.uk
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