[FOM] Certainity vs Boundedness

Timothy Y. Chow tchow at alum.mit.edu
Tue May 13 19:32:31 EDT 2008

A. Mani <a_mani_sc_gs at yahoo.co.in> wrote:
> It is possible to write nice bounded proofs for bodies of vague and 
> imprecise knowledge and those become part of the branch of science in 
> question.

Depending on what examples you have in mind, I might argue that these are 
not counterexamples to my idea that bounded proofs are what distinguish 
mathematics, but examples of mathematical knowledge becoming a part of 
scientific knowledge.  After all, it is commonly acknowledged that (for 
example) "physics is very mathematical," which I take to mean that a 
significant portion of physics knowledge is really mathematical knowledge.

> Apparently you mean "potentially bounded", rather than "bounded"... then 
> also "so many Mathematicians have been wasting their time in doing 
> non-Mathematics like trying to solve apparently unsolvable problems".

I don't mean to *limit* mathematics to bounded proofs.  Investigating 
conjectures in an open-ended fashion is clearly part of mathematics, and 
is also unbounded in any reasonable sense.  I mean just that mathematics 
is unique in that its primary standard of justification is a proof whose 
correctness requires only a bounded amount of verification.

> Is the intended concept of proof a 'formal one within a language'?

I don't believe that it matters.  Whether or not you are a formalist in 
this sense, you can still (potentially) agree that proofs are bounded.

> Is your Mathematics, the Mathematics of which Platonic universe ?

I don't believe that this matters, either.  As far as proof checking goes, 
it doesn't matter whether or not you believe that the theorems are talking 
about platonic realities.

> For which isms in Mathematics is your proposal intended for?

All of them.  Intuitionists, for example, have a slightly different notion 
of proof, but they still have a clear notion of proof.  It plays the same 
central role and the verification of the correctness of a proof is still 
bounded, regardless of whether the proof is intuitionistic or classical.

> Is your proposal totally against 'applied mathematics' being part of 
> Mathematics ?

As I said above, the intent is not to limit mathematics to be "nothing but 
proofs," but to argue that what makes mathematics distinctive is the 
bounded nature of its proofs.  Applied mathematics spans more than one 
field and so it will have characteristics of more than one field.  The 
mathematical part of applied mathematics has proofs just like any other 
branch of mathematics.  This does not stop it from having other 
characteristics that distinguish it from "pure mathematics."

> Computer scientists write bounded proofs? 

To the extent that they do, their work is mathematical.


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