[FOM] Logic/syntax versus arithmetic
Timothy Y. Chow
tchow at math.princeton.edu
Tue Feb 18 21:38:31 EST 2020
Alan Weir wrote:
> Alternatively, idealisation can be done metamathematically. For example,
> there are, or at any rate 'could be' in a fairly ordinary sense,
> concrete metamathematical proofs that every sentence of delta_zero
> arithmetic (a system we have concrete proofs exists, in the same sense
> as we have concrete proofs of the infinity of the primes) is either
> (formally) provable or else is refutable. That is enough, I claim, for
> the nominalist to hold each such wff is either true or false.
Since you seem sympathetic to my objections to Leng, but also see a way
for a nominalist to "hold that a wff is either true or false," maybe you
can explain to me what "concrete" and "abstract" are supposed to mean.
Nominalists complain that numbers are abstract and not concrete. I don't
fully understand what they're saying but I sort of do, so let me carry on.
If a nominalist is going to be saying that a wff is true, then presumably
the nominalist affirms that wffs exist? That wffs are sufficiently
"concrete" to satisfy a nominalist? But in what sense is a wff, or even
more fundamentally the symbol "0", concrete? Suppose that I provisionally
grant that ink, being physical, is "concrete". I can write down "0" on a
sheet of paper using a pen with ink. Naively, this might seem to be a
demonstration that the symbol "0" is concrete. But I don't think it is.
The ink on the page is not the symbol "0". The symbol "0" is an
abstraction. If I write "0" on a blackboard with chalk, or display a "0"
using pixels on a computer screen, then these are distinct concrete
physical entities. If I allow myself the ability to *abstract*, then I
can regard these physical entities as various concrete representations of
the abstract symbol "0", but abstraction is precisely the activity that
the nominalist is questioning. If there is no such thing as the abstract
object `the symbol "0"' then what is it that links the ink on the page
with the chalk on the blackboard with the pixels on the screen?
In fact my objections can be taken further, because quantum mechanics
raises questions about whether physical objects are any more "concrete"
than mathematical ones. Given what we know about physics, is an electron
"concrete"? But without even going that far, I remain puzzled by why
nominalists regard symbols as more "concrete" and less "abstract" than
numbers. I think I've mentioned before on FOM that the first edition of
Kunen's textbook on set theory asks the question, "What is a symbol?" and
answers it by defining a symbol in terms of numbers. So I'm not the only
one who fails to find it obvious that symbols are more concrete than
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