[FOM] PA Incompleteness

Mon Oct 15 15:14:53 EDT 2007

Feng Ye wrote:

<<The ratio of the linear cosmological scale to the Planck scale is less 
than
10^100, and the number of fundamental particles in the universe is also
quite small (compared with numbers produced by those fast growing functions
for very small arguments). Current sciences describe *only* things within
that small range, from the Planck scale up to the cosmological scale. At
least, after excluding physics theories about things at or below the Planck
scale, all mathematical applications are applications to strictly finite
things whose physics quantities are represented by numbers that are
'ridiculously small' (in terms of absolute magnitudes or precisions).
>>

I doubt that this subject belongs in the discussion on PA incompleteness. 
But I feel compelled to react to the claim `all mathematical applications 
are applications to strictly finite things'. This seems to me to be an issue 
of physics as of yet unresolved. It seems to me you might be confusing 2 
things: the fact that we as human beings are not capable of measuring 
physical things except in finite approximations, and the possible finiteness 
of the measured objects themselves.

These are however two fundamentally different things, both 
physics/mathematically and philosophically. To me it seems that in 
constructive mathematics/intuitionism, the issue is dealt with rather 
elegantly in terms of the `potentially infinite'.

One physical phenomenon which I for instance doubt anyone has convincingly 
shown to be strictly finite is time. Brouwer used an analogy with time to 
justify our intuition of the natural numbers as a potentially infinite (but 
never finished) sequence 0, 1, 2, ... arising step-by-step in the course of 
time.

To me, it may well be that at some point in time we conclude that 
everything, even time, is strictly finite. But I find no immediate fault 
with the thought that time could be infinite, and that in the course of this 
infinite time increasingly better approximations of physical entities become 
possible, and in fact whole new families of physical entities beyond the 
current cosmological and Planck scale are seen to turn up.

The issue of elegance and understandibility of mathematics & physics also 
plays a role. In strict finitism, there is say a fixed finite set of 
permutations of finite objects. Any law in physics then is just a 
description of part of these finite states, there is really no deeper 
explanation. But infinitary mathematics gives us some understanding of why 
certain states occur, and others don't. Therefore, the strictly finitistic 
viewpoint does not look very attractive (it could still be right though).

The mathematics which already in its theory incorporates the finiteness of 
human endeavors at physics to me undoubtedly is constructive 
mathematics/intuitionism. I therefore have been feeling for a long time that 
physics should take these branches of mathematics seriously (ok now I'm also 
repeating myself, sorry).

Best regards,

Frank Waaldijk
http://home.hetnet.nl/~sufra/mathematics.html

----- Original Message ----- 
From: "Feng Ye" <yefeng at phil.pku.edu.cn>
To: "'Foundations of Mathematics'" <fom at cs.nyu.edu>
Sent: 15 October, 2007 02:03
Subject: Re: [FOM] PA Incompleteness

>I am always wondering what will be the answer if 'normal mathematics' is
> replaced by 'mathematics with potential applications in sciences', or
> 'mathematics relevant to this physical universe'. The problem is that all
> current independent propositions are related to fast growing functions, 
> but
> the scope of the physical universe that sciences are dealing with is so
> 'ridiculously small' from the mathematical point of view.
>
> The ratio of the linear cosmological scale to the Planck scale is less 
> than
> 10^100, and the number of fundamental particles in the universe is also
> quite small (compared with numbers produced by those fast growing 
> functions
> for very small arguments). Current sciences describe *only* things within
> that small range, from the Planck scale up to the cosmological scale. At
> least, after excluding physics theories about things at or below the 
> Planck
> scale, all mathematical applications are applications to strictly finite
> things whose physics quantities are represented by numbers that are
> 'ridiculously small' (in terms of absolute magnitudes or precisions).
>
> It seems unlikely that the operation patterns in Goodstein sequences, or 
> in
> Hercules and the Hydra, will be directly related to any real physical
> processes to make those independent theorems directly applicable in to 
> real
> things. This certainly does not rule out the possibility that those 
> theorems
> are applicable in some indirect ways, but I am always wondering what that
> could be.
>
> Quine and some philosophers following him called mathematics with no
> potential applications in sciences 'recreational mathematics'. Realizing
> that sciences deal with only such a small range of things in the universe,
> we may suspect that anything *essentially* beyond PA or even a weaker 
> system
> (in the sense that it strictly, logically indispensably requires the 
> logical
> power of all the axioms in the system) has to be 'recreational' in that
> Quinean sense. This is related my post last month (an initial post), that
> is, there is a chance that strict finitism is already *in principle*
> sufficient for scientific applications to things within such a small range
> of the universe dealt with by current sciences.
>
> Feng Ye
>
>> -----Original Message-----
>> From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of
> Harvey
>> Friedman
>> Sent: Sunday, October 14, 2007 10:03 PM
>> To: fom
>> Subject: [FOM] PA Incompleteness
>>
>> PA incompleteness is now a reasonably well defined and reasonably well
>> developed area within mathematical logic. After the initial results (see
>> below), the focus, for many years, has been on the integration of such
>> exotic combinatorics into normal professional mathematical activity, with
>> its normal motivation and normal culture.
>>
>
>
>
> 

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