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

[Lucas Kruijswijk]

Dear Timothy,

In answer of your question. I am a computer scientist and have
logic as hobby. So, I am the perfect person here to give my
thoughts.

I do not want to make a specific choice between "integer" or
"symbol". However, I do think that the construction of one object
from other objects is quite cumbersome and unnatural in the logic
I have learned. The most simple form is of course the 'pair' operator.
With the pair operator, of course, you can do the other constructs.

If you take FOL + PA (containing addition and multiplication) it is
a tricky construction, to make a pair. I am also a little bit frustrated
here, because in the books I had to learn on the university, there
were theories about Godel incompleteness etc. but none of the book
did a proper job to explain how to do things like a pair. I only learned
that from an other book in some side note.

Furthermore, I do not think you can take "integer" or "symbol" out
of its context where it is used. In Object Oriented programming
you learn that an object consists of data and its methods.

One can also take the metaphore with fundamental physics. In classical
Newton physics the different entities like time and distance are entities
at their own. However, if you relativistic physics, then you can not
take time separate from distance (in classical physics, light has a speed,
in fundamental physics light defines distance). Similar, I think you
can't take "integer" or "symbol" as an notion at its own.

For that reason, I consider the Turing Halting problem as the most
"clearest mathematical concept". It contains data and operations
together. It appears on many places in different shapes, but its meaning
is not disputed and clear.

You might argue that a Turing machine is more complex concept than
an integer. However, as said, I think that the integer must have a context.
If you put an integer in the context of FOL + PA, than such system is more
complex than the Turing machine.

Based on above argumentation, I believe that the Turing machine and the
Halting problem will gain more status within the fundaments of mathematics
the next 50 years.

Regards,

Lucas Kruijswijk


----- Original Message ----- 
From: "Timothy Y. Chow" <tchow at alum.mit.edu>
To: <fom at cs.nyu.edu>
Sent: Sunday, January 02, 2011 7:32 PM
Subject: [FOM] Which is clearer, "integer" or "symbol"?


> 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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