Logic in physics and mathematics

[Pentcho Valev]

r.mccall at staff.mcgill.ca wrote:

> I wonder if this posting to philphys I sent a week ago has been received.
>
> Storrs McCall
>
> ----- Forwarded message from r.mccall at staff.mcgill.ca -----
>     Date: Sat,  5 Feb 2005 19:30:45 -0500
>     From: r.mccall at staff.mcgill.ca
> Reply-To: r.mccall at staff.mcgill.ca
>       To: philphys at philosophy.elte.hu
>
> Dear Dr Valev,
>
> The problem you seem to be calling attention to is not unknown, but well-known
> in other domains, notably mathematics.  Consider whether there are, or are not,
> irrational numbers a and b such that a^b is rational.  Let rt2 be the square
> root of 2.  Using classical logic, the number rt2^rt2 is either rational or
> irrational.  If it is rational, the problem about a^b is solved affirmatively.
> If it is irrational, consider (rt2^rt2)^rt2.  This is rt2^2 which is 2, and
> again the answer is affirmative.  Needless to say, this argument would not be
> accepted by mathematical constructivists.  Are you suggesting that physics may
> have to divide over what is logically acceptable in the way mathematics has?
>
> Storrs McCall

Dear Dr. McCall,

Your messages don't reach the list but I don't know the reason. Perhaps you should
change the way you address them.

I don't think physics should divide over what is logically acceptable - rather, an
agreement can be reached in the end. Logic in physics differs essentially from
logic in mathematics in that, in an argument, apart from explicitly employed
premises (axioms), there is an indefinite number of implicit ones coming from
common sense, scientific practice etc. Due to their implicitness they form a
"twilight zone" where almost all logical abuses occur. Two examples:

1. A true (implicit) premise is "overlooked" and replaced with its false negation.
For instance, reversible machines cannot perform a cycle in the absence of an
operator - someone or something belonging to the surroundings and undergoing
INDEFINITE changes in the process. If this true premise had been made explicit, the
second law of thermodynamics would have never been established.

2. A false explicit premise naturally contradicts true premises from the twilight
zone but since it is too dark there a harmony rather than contradiction is
officially announced. For instance, the postulate of constancy of speed of light
plays this role in relativity.

Pentcho Valev

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