[FOM] Mathematics and precision

[Timothy Y. Chow]

Henrik Nordmark <henriknordmark at mac.com> wrote:
> However, I would not want to say that formal systems have a monopoly  
> on precision. This would put the threshold of precision way too high.  
> Most mathematics gets written in natural languages like Mathematical  
> English and NOT in some formal system.

Yes.  Let me quote from a private email that I just sent to someone who 
asked me for clarification.  By saying that mathematics is defined by 
some threshold of precision, I do not mean to specify what that threshold 
is.  Some possible thresholds might be:

1. Publishable in a reputable mathematical journal.

2. In principle formalizable in ZFC.

3. Intuitionistically acceptable.

4. Predicatively acceptable.
 
5. Finitistic and feasible enough even for V. Sazonov.
 
I'm not advocating any particular one of these thresholds.  Doing so would 
be more or less slotting myself into a familiar philosophy of mathematics.  
The argument I want to make is that all these folks are doing the same 
thing---they're defining mathematics by specifying a threshold of 
precision, whatever that threshold might be.

This is in contrast to other subjects where debates about "What is X?" are 
typically debates about what the *subject matter* is.  Physics, for 
example, is the theoretical and experimental study of the physical world.  
If you're not studying the physical world, then you're not doing physics.  
On the other hand, you can be studying the physical world and doing 
mathematics at the same time, if your study is sufficiently precise.

Tim