[FOM] First-order arithmetical truth

Timothy Y. Chow tchow at alum.mit.edu
Wed Oct 25 10:25:48 EDT 2006

Vladimir Sazonov wrote:

>you seemingly do not notice my points.
>But why do you ignore the words "sheet of paper"? They are crucial here!
>but I do not see that you can follow the lines of thought either by me 
>or Francis Davey or the other person you cite above.

Apparently you also choose to ignore my explanation of why I chose to 
ignore certain parts of your message.

However, I believe that I've said all I can say on the original topic 
(namely, the point that Arnon Avron was trying to make to Francis Davey).  
You and I also appear to be in agreement on that point.  Therefore I'll 
say a couple of things about the new topic that you repeatedly insist on 
bringing up.

>Naive formal systems are something from our real life and (for example, 
>computer) practice. They are quite rigid and reliable and even can be 
>represented physically by computer systems. The main point is that they 
>are REAL unlike ABSTRACT mathematical numbers and abstract 
>(meta)mathematical formal systems (quite similar by the nature to 
>mathematical numbers).

I don't see how (naive) formal systems are more real than (naive) numbers.  
Say you take a pencil and move it in a certain fashion relative to a piece 
of paper.  Is the resulting physical object a formal system?  If so, then 
it would appear that this particular formal system

- weighs about 4.5 grams
- is solid at room temperature and pressure
- will ignite if heated to several hundred degrees

Or perhaps you will point to a particular spot on your hard drive and say 
that that is a formal system?  Then that formal system has very low 
thickness, and an area of a few square millimeters.

If you happen to claim that the two formal systems are *the same*---by 
which I take it to mean that they are the same physical object, since you 
are emphasizing the physicality of the formal systems---then I will be 
puzzled.  I would already be slightly puzzled if you were to show me two 
separate pieces of paper and say that they were the same physical object; 
there are many contexts in ordinary life---say, in a court of law, where 
physical evidence can be very important---in which one piece of paper is 
decidedly *not* the same as another piece of paper that is photocopied 
from the first.  But I am baffled when you say that the section of your 
disk drive is the same physical object as the paper.  What is the relevant 
*physical* property that they have in common?

I cannot see that the process of abstraction that allows you to equate the 
two (naive) formal systems is any more "concrete" or "real" than the 
process of abstraction that allows most people to work with (naive) 


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