[FOM] Platonism and Formalism

[Karlis Podnieks]

----- Original Message ----- 
From: "Torkel Franzen" <torkel at sm.luth.se>
To: "FOM" <fom at cs.nyu.edu>
Cc: <torkel at sm.luth.se>; "Vladimir Sazonov" <V.Sazonov at csc.liv.ac.uk>
Sent: Sunday, September 28, 2003 4:14 AM
Subject: Re: [FOM] Platonism and Formalism

...

> In the present case, a metaphysically sensitive
> person will reject the idea that "the number"
>
>       25195908475657893494027183240048398571429282126204
>       03202777713783604366202070759555626401852588078440
>       69182906412495150821892985591491761845028084891200
>       72844992687392807287776735971418347270261896375014
>       97182469116507761337985909570009733045974880842840
>       17974291006424586918171951187461215151726546322822
>       16869987549182422433637259085141865462043576798423
>       38718477444792073993423658482382428119816381501067
>       48104516603773060562016196762561338441436038339044
>       14952634432190114657544454178424020924616515723350
>       77870774981712577246796292638635637328991215483143
>       81678998850404453640235273819513786365643912120103
>       97122822120720357
>
> (which is the RSA $200,000 challenge number) has a determinate
> factorization which we don't as yet know, and perhaps will never
> know.

...

> Since you mention absolute time, let us note
> that Einstein spent no time or effort arguing about the mystical
> character of the idea of absolute time.
>

This may be true for Einstein, but not for Ernst Mach - one of Einstein's
favorite readings before 1905. In my terms, Mach proposed (or, at least,
used in his arguments) the following methodological principle (Mach's
principle?), which proved to be very useful in modern physics:

If some theoretical construct cannot be interpreted empirically even in
principle, then this construct should be moved from the theory as far as
possible (sometimes, we are not able to move it far enough - we do not wish
to leave our theory empty).

Following this way, Einstein arrived to the relativity, and a group of other
distinguished people - to the quantum theory. (However, following this way
himself, Mach rejected "atomystics".)

Couldn't we try applying this principle to arithmetic? (This is not a new
idea.)

We never will be able to generate the above decimal number by iterating the
successor function. Computers manipulate natural numbers in the binary
notation, never converting numbers into the unary notation. It seems,
the range of natural numbers used in cryptography could be regarded
as a more complicated mathematical structure than "initial segments of N"?

In formal theories, we cannot even formulate directly the "fact" that each
natural number can be generated by iterating the successor function. The
induction principle and even the "standard interpretation of PA" are only
indirect formulations. Perhaps, this is a good feature, because there is no
such "fact"?

Again, this is not new - for an account, see a FOM posting
by Robert Tragesser:
http://www.cs.nyu.edu/pipermail/fom/1998-April/001874.html

But why nothing generally significant has come out of this?

Best wishes,
Karlis.Podnieks at mii.lu.lv
www.ltn.lv/~podnieks
Institute of Mathematics and Computer Science
University of Latvia