Re: Did Natural Selection extend to the laws of logic?

From: Marchal <marchal.domain.name.hidden>
Date: Thu Sep 21 02:32:16 2000

John Bailey reminds us davidsmyth's question:

>Could there be a universe where the rules concerning number theory
>would be different than in this universe?

He gives us a nice Reference (John Foy quant- ph/0001074
An ``Epistemological Derivation of Quantum Logic)" + the well
known ``Machines, Logic and Quantum Physics"
David Deutsch and Artur Ekert and (don't forget!) Rossella Lupacchini.

I was about sending a reply, but then Brent Meeker send an answer
which is better than mine:

>Logic and number theory and in fact any mathematical theory are
>independent of
>physical reality since they are simply sets of axioms and their
>consequences according to some rule of inference.
>
>Euclidean geometry didn't change in 1823; only the idea that it necessarily
>applied to the physical world changed. There are many physical entities that
>number theory does not apply to. It's just that the the non-applicability is
>so obvious that one doesn't usually think of number theory applying. For an
>example that sometimes trips up 4th graders; consider that there are two
>school
>clubs, one with 15 members and one with 20 members. They have picnic
>together.
>How many are at the picnic?
>
>Similarly the application of logic is empirical. Ordinary predicate calculus
>has a limited range of applicability - you get into trouble if you quantify
>over predicates. Logic ordinarily deals with propositions which have
>true-false values and connectives (and, or,...). But it could be extended to
>include things like 'red implies not-green'. All of this is independent
>of the
>physical world. It's just that we like to adapt our language so as to
>make it
>easy to talk accurately about the world.

I want just add some comments.

1) It is well known (by logician) than
any axiomatisation of number theory admit so called non-standard model.
In these model addition and multiplication are not computable. This show
that we can consider curious structure in which number obeys different
rules. But this has nothing to do with the fact that in all universe
the standard rule apply. In fact without the standard logic you cannot
even give meaning to the non standard structures. Algebra has been
invented to care about non isomorphic structures.

2) Those like Putnam who have try to use quantum logic to give a realist
account of quantum mechanics have failed. Quantum logic just highlights
the many worlds feature of quantum mechanics (and/or the non realism
of the isolated system).

3) Deutsch, Ekert and Lupacchini wrote in the mentionned paper:

<<Though the truths of logic and pure mathematics are objective and
independent of any contingent facts or laws of nature, our knowledge of
these truths depends entirely on our knowledge of the laws of physics.>>

I could understand that some physicist believe (unlike me) that
our knowledge of these mathematical truth depends on the laws of physics.
But that such knowledge depends on our *knowledge* of the laws of
physics: well I cannot make sense of that at all.
It looks like young children should apply Schroedinger equation on their
own brain to understand that 1+1 gives 2.

Bruno
Received on Thu Sep 21 2000 - 02:32:16 PDT