[FOM] Numbers vs writhmetic. was: n-th order ZFC
Staffan Angere
Staffan.Angere at fil.lu.se
Thu Jul 14 05:11:59 EDT 2011
One way to hold arithmetic true, but to deny the existence of numbers, is to use a non-Tarskian semantics. For example, one can make use of a truthmaker semantics, in which the truth of statements depend on something else than the existence of the things mentioned in them. Arithmetical statements can then be taken to be true in virtue of, say, certain conventions adopted by most mathematicians, or the possibility of producing certain proofs, rather than in virtue of the properties of some abstract objects (the numbers). We can get objectivity this way, since mathematics is interpreted as being about certain objective features of reality (i.e. conventions, proofs, etc.), but the objects in question do not have to be mathematical.
There is a well-known problem with accepting the truth of arithmetic without accepting the existence of numbers, of course: it seems that "there are prime numbers over 2" implies "there are numbers". So internally to mathematics, it is quite trivial that there are numbers. This is basically what Carnap says in "Empiricism, Semantics, and Ontology". Mathematics, however, is not as such a metaphysical theory, so even "there are numbers" can be given truth-makers that do not involve any objects of the kind we usually think of when we discuss numbers.
Some variant of this approach is, in my opinion, a quite reasonable standpoint to take. Tarskian semantics is certainly not the only type of semantics imaginable, even if it is one that seems highly intuitive to most people.
Staffan Angere
University of Lund
________________________________________
Från: fom-bounces at cs.nyu.edu [fom-bounces at cs.nyu.edu] för W.Taylor at math.canterbury.ac.nz [W.Taylor at math.canterbury.ac.nz]
Skickat: den 13 juli 2011 08:26
Till: Roger Bishop Jones
Kopia: fom at cs.nyu.edu
Ämne: [FOM] Numbers vs writhmetic. was: n-th order ZFC
Quoting Roger Bishop Jones <rbj at rbjones.com>:
> ... one can believe in the objective truth of arithmetic
> without also believing in the existence of numbers.
I would like to hear more about this, if possible, as it touches on
a dichotomy that I have been hearing a lot about in the last few years.
If one (a) DOES accept the objective truth of arithmetic,
but (b) does NOT accept the existence of numbers,
then wherein resides the objectivity of the arithmetic?
If one has no semantics for arithmetic (which seems to be what (b) says),
then on what grounds is truth to be defined for arithmetic?
(I do not intend to be combatively rhetorical, I would just like
to understand this combination position better.)
-- Wondering William
fom at cs.nyu.edu
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