Mathematical Logic

D OC donaloc at hotmail.com
Tue Jul 8 07:35:07 MDT 2003


It's a long time since I did a course on Advanced Logic at University. I
remember just bits and general outlines. As I've stated before, my interest
at the time was in QM - specifically its mathematical representation. The
discussion on logic and 'fuzzy logic' is of some interest.

The first thing is that the whole area of mathematical logic becomes akin to
a series of parallel worlds in which one axiom applies or doesn't apply. It
becomes like a meaningless game, very difficult, but one which is engaged in
for completeness sake alone - it would appear. As for QM and its
mathematical explanation - I'm afraid it didn't make its way onto our Logic
syllabus. The closest I ever got to that was its inclusion in a pure
mathematical course dealing with Operator theory and Representations. There
was quite some distance between that and the stuff which we dealt with in
Formal Logic. In the latter, we were told to assume the general Goedel
Theorem, so I never got to do anything but a proof in a simple situation
(about 4 pages long).

As I stated before a few times, there was much in differential geometry
which might parallel a dialectical appreciation of reality. Maths is well
capable of modelling conflicting (and contradictory) forces - even basic
electromagnetic theory has a capacity to explain conflicting tendencies. It
might be the most practical manner of finding a mathematical equivalent for
dialectical modelling. Having said that, within themselves these particular
forces are relatively 'pure' and non-contradictory, well aside from the fact
that they are time-bound and are influenced by qualitative changes occuring
from without. (Although this would be different to Mao's comprehension of
dialectics as per his essay on the subject).

What might be termed 'classical logic' is therefore not really where
mathematics is really 'at' in the higher levels. The notion of implication
or exclusion are merely assumptions for particular languages of logic -
there is no single (natural) logic which is assumed to reflect our reality.
Once you realise that there are many different types of logic, its not
really on to talk about logic as if its such a coherent whole but a set of
studies, most of which have no real relation to reality.

As for QM, well, to try and get a 'logical' basis for the theory (as the
term is used by most pure mathematicians i.e. derivation through a big
'proof') is just a joke. In (applied) mathematical terms, the theory starts
out by plucking an equation (the Schrodinger Equation) from thin air. The
unknown constants are found through experimentation. The whole model is
chosen simply because it provides the best fit with experimental data.
Attempts by Pure Mathematicians to deal with QM, start with consideration of
different types of operator fields, then we get to deal with those with the
properties of the quantum operators. In effect, when mathematicians are
taught QM, they learn it backwards in comparison to how it was discovered by
the physicists (who were much more concerned with modelling behaviour than
with systematic beginnings). As for a logical proof for QM, there is none as
yet. Goedel kind of scuppered hope of that or similar anyhow which his
theorem disproving the existence of global, closed mathematical languages.

Finally, someone mentioned getting into Wittgenstein at an earlier stage. It
must be almost 6 years since I read most of his books and I concluded at the
time that the worth of his early work is extremely limited, except in an
illustratory manner - something he recognised himself subsequently. The
conclusion he reached about the inability of mathematics to define the
limits of language is clear. The conclusion had a deep impact on me, as I
had just experienced a considerable let-down in terms of my natural
positivist reductionism through my experience of University Mathematics and
Physics. As a result of this journey of learning, I lost faith in
mathematics as anything but a human-made approximation to reality or else an
intellectual game which occasionally finds unexpected (practical) uses. To
engage in it at this period of time, is a waste, when there is so much else
which needs done. All the same, it has been interesting to read the
discussions on the apparent necessity of a dialectical explanation for
quantal behaviour under the 'standard interpretation'.

Is mise
Domhnall.

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