[FOM] Which is clearer, "integer" or "symbol"?

[Oran Magal]

I have no idea as to present-day practice, but I would offer a
historical remark which might be useful. If you think of 'symbol' as
used in such contexts as 'a predicate is an incomplete symbol', then I
would say 'integer' is probably clearer. Or, if you think, like
Kronecker, that "Die ganzen Zahlen hat der liebe Gott gemacht, alles
andere ist Menschenwerk", then it would indeed make sense to assume
the integers as a primitive concept and one's starting point for
explanation of other concepts.

The alternative, that 'symbol' is in some sense simpler, whereas
'integer'  a complex concept that can be explained in terms of simpler
ones, is based -- I think -- on thinking of symbols in a
quasi-concrete manner, as marks or inscriptions. A relatively familiar
locus for such usage is Hilbert's Programme, in "Uber das Unendliche"
(1926) and elsewhere. There, it is argued that a basic cognitive
capacity for identifying, manipulating and performing certain
operations on symbols (e.g., concatenation) is presupposed by language
and congnition quite generally, and  antecedently to logical inference
and any mathematical reasoning. If one buys into such a line of
thinking, one can see why the natural number sequence is an
explanandum, while symbols, thought of as the objects of this most
basic cognitive apparatus, are explanatorily primitive.

My impression, which is not sufficiently well-informed, is that the
first view (starting with the integers), has a longer history than the
latter. A full historical account would not start with Kronecker or
Hilbert, though, but with predecessors of both. I hope this sheds a
bit of light on the question.

Oran Magal
McGIll University

On Sun, Jan 2, 2011 at 1:32 PM, Timothy Y. Chow <tchow at alum.mit.edu> wrote:
>
> It is my impression that, at least among people without formal training in
> logic and foundations, there has been a gradual shift over time from the
> point of view that an "integer" is the clearest mathematical concept, to
> the point of view that a "symbol" is the clearest mathematical concept.
> I am wondering if other FOM readers have a similar impression, and if so,
> whether any solid historical evidence can be accumulated in support of
> my claim.
>
> To give you an idea of what I'm talking about, in Kunen's book on set
> theory and independence proofs, he asks the rhetorical question, "But what
> is a symbol?"  The implication is that the concept of a symbol might not
> be totally clear.  Kunen then addresses the issue by *defining* a symbol
> to be an integer.  Again, the implication is that the concept of an
> integer is clearer than that of a symbol.
>
> On the other hand, it has been my experience that nowadays many
> mathematicians, computer scientists, physicists, etc., who have some
> casual interest in logic and foundations have the opposite point of view.
> "Symbols," "strings," and rules for manipulating them are considered
> unproblematic.  Integers, on the other hand, are mysterious, and suspect.
> The suspicion may be fueled by a naive form of anti-Platonism, or by
> confusion about the status of statements such as "PA is consistent."  What
> prevents such people from seeing that analogous skepticism can be directed
> towards symbols and strings I'm not sure, but perhaps familiarity with
> computers has something to do with it.
>
> So to recap my question...is it just me, or has there really been a
> sociological shift over the past several decades?
>
> Tim
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