Logic/Syntax versus Arithmetic

[Alan Weir]

Tim Chow writes (FOM Vol. 206 Issue 29):

'So what's wrong with this?  Introduce the concept of equicardinal tokens (in effect, strings of symbols that contain the same number of symbols). 
Note how complex and resistant to definition the word "equicardinal" is but assume it is acceptable to take it as primitive.  A "natural number tipe" is then a mereological sum of all actual tokens equicardinal to a given actual token.  This gives us finite many natural number tipes.'

That's a nice point. In fact in the chapter of my book cited, I try to give a physicalistic characterisation of what it is to be an atomic constituent of a wff and of the relation  'the length of x is the same as the length of y' among wffs in terms of the relation of 'there are exactly as many Fs as Gs' which is used to characterise equicardinality in standard accounts of number. This relation is sometimes defined in terms of one:one bijections, sometimes defined in turn in terms of second-order quantification. This  is problematic for nominalists in the medieval sense, that is those who deny that there are properties. Quine was for a period a  nominalist in both modern and medieval senses, eschewing both abstract objects and properties (some philosophers assimilate the two, wrongly in my view) and indeed second-order logic whatever the range of the second-order variables.  I do not deny the existence of properties but  I think a sparse account of properties is the most plausible. Even so,  the coherence and legitimacy of asserting that there are exactly as many Fs and Gs, when there aren't many of them,  seems to me beyond question and I suggested some anti-platonist ways of doing this, including Goodman and Quine's nominalist explication of 'exactly as many as', though it is only applicable in certain cases. Hence I agree one can introduce cardinal numbers as mereological fusions of all wffs of the same length as a given one. 

I have a number(!) of problems with this. Firstly, we have strong empirical reasons to believe that on this account there are only finitely many numbers whereas we can prove that there are infinitely many. This argument will not persuade strict finitists.

Secondly, we need more than the numbers, we need to be able to characterise various relations and functions over them:- less than, addition, multiplication and so on. I do not believe a theory along Goodman and Quine's lines will be successful. Their emphasis is on having 'ontological commitment' in Quine's (in my view unclear and problematic) terminology only to concreta. They say they are not engaged in an exercise of definitional analysis of mathematical concepts. This is reasonable enough but when they say that from a purely nominalist point of view they see no problem with taking 'x has 10^1000 objects as parts' (I am using ^ to express exponentiation) as a primitive notion applicable to concreta, I demur. They themselves also say they reject any statement if it 'commits us' to abstract entities, rejecting the usual set-theoretic definition of the ancestral for this reason. So I do not accept that one could develop a form of finite number theory by taking as primitive, for example, the predicate 'x has 10^1000 objects as parts, inter-related in ways structurally isomorphic to an initial segment of omega, with some of the inter-relations playing the role of successor, addition and multiplication restricted to that segment'.

The most important problem concerns strings such as the one

10^10032+57

I introduced in a previous post to make the point that we have concrete numerals standing, platonistically speaking, for numbers far larger than the number of concrete inscriptions. This numeral token is eleven token symbols long but it stands, on the account Tim Chow suggests, for a number which does not exist, there being no concrete strings  as long as the number it stands for. 

So I see fatal problems for any attempt to develop nominalism/anti-platonism on strict finitist lines. This still leaves in play what I called 'cake and eat it nominalism'. Perhaps 'non-revisionary' nominalism or anti-platonism would be better. The non-revisionary agrees that all the (finitely many) concrete inscriptions of theorems of standard maths (in any discipline) are true. (Allowing that we make mistakes in checking proofs and classifying sentences  as theorems). The goal, which I claim can be met,  is to give a semantics in which the truth of these inscriptions can be explained without appeal to anything non-concrete.

Alan Weir
Roinn na Feallsanachd/Philosophy
Sgoil nan Daonnachdan/School of Humanities 
Oilthigh Ghlaschu/University of Glasgow 
GLASGOW G12 8QQ