[FOM] The empirical foundations of deductive logic and the axiomatic method (& applications to, e.g., ODEs & stochastic processes) [repost, with revision and expansion, due to technical problems with original posting]
Vladimir Sazonov
V.Sazonov at csc.liv.ac.uk
Wed Sep 14 13:11:00 EDT 2005
Quoting Richard Haney <rfhaney at yahoo.com>:
> I am generally rather amazed and somewhat puzzled by the degree to
> which mathematicians typically have confidence in (deductive) logic to
> obtain new mathematical "knowledge". This perplexing state of affairs
> seems especially poignant, for example, in the area of the mathematical
> theory of stochastic processes.
..........
> The validity of deductive logic (as a methodology for deriving
> empirically testable facts) seems to be an empirically-based general
> fact similar to the laws of conservation of mass-energy and of linear
> and of angular momentum. We seem not to know of any exceptions to
> these "laws"; however, because they are empirically-based, there
> remains the possibility -- true to scientific experience generally
> regarding supposed "laws" of nature -- that these laws could be invalid
> or false in some extreme (or perhaps not so extreme) circumstances not
> yet specifically explicated and tested. Newton's law of gravity is
> such an example in the history of science. As with Newton's law of
> gravity, this "empirical validity" of deduction might be expected to
> fail as to accuracy and/or precision under certain extreme (or perhaps
> not so extreme) conditions.
Even not so extreme - a play with the ordinary physical stones;
see below.
> Does anyone know of any specific scientific study, especially in modern
> times, of such empirical questions concerning deductive logic and the
> axiomatic method? Any insightful comments on this issue would be
> especially of interest.
I will present a negative answer below showing that deductive
logic in its traditional form can fail in some physically
meaningful context.
Let me recall Hilbert's comparison of Geometry as Mathematics
vs. Geometry as Physics. It looks like you want another comparison:
Logic as Mathematics vs. Logic as Physics. Yes, we can consider
Logic as Physics in the case of Logic applied to physically
presented finite models where we can experimentally confirm
physical truth of the deduced tautologies. But in general (say
for the case of logical lows applied to infinite sets) we can
only rely on applicability of currently existing mathematics
(which is based on formal logics) in various application domains.
No other empirical confirmation is possible in principle. Logic
assumes dealing with abstract, in general not empirical entities
whatever they are. But empirical refutation of some logical
principles in a *specific* application domain is possible.
I already wrote on a very simple version of formal arithmetical
theory (a kind of Arithmetic considered as Physics).
It consists of several universal axioms like forall x,y (x+y=y+x),
forall x,y (x+1=y+1 => x=y), etc., all being physically true
(in terms of experiments with stones like you described in
your posting) one of which is not so traditional, but also
universally quantified and physically true. It states that
forall n log_2 log_2 n < 10. (Read it as "for all physical
sets (of n) stones...". (Astro)physics also confirms the truth
of this axiom.) This theory should inevitably be based on
a restricted version of classical first order logic. The point
is that without this restriction a quite short contradiction
would be *physically* deducible from these (physically true!)
axioms. See the details of deriving such a contradiction in
http://www.cs.nyu.edu/pipermail/fom/2005-May/008964.html
This shows that in some application domain of arithmetical
nature (counting manipulations with physical stones) the
ordinary non-restricted first order logic is not applicable:
it leads to a contradiction from physically meaningful and
true assumptions. In particular, unrestricted application
of modus ponens leads to a contradiction. (Intuitively, a
"small error" is accumulating with multiple application
of modus ponens and giving rise to a contradiction.
But formally, there is no concept of error or measurement
here, or any technique of fuzzy logic - only a first order
theory in a language of arithmetic.)
The restriction is actually well known: only cut-free proofs
should be allowed. This restriction is only a technical tool
to achieve a kind of formal consistency. There may be other
appropriate and/or more intuitively appealing restrictions
(say, to allow cuts for formulas with bounded quantifiers
only, or to restrict using some kind of abbreviations like
2x for x+x, etc.).
>
..........
> And so it seems that mathematics, at least mathematics in a formalistic
> sense,
Is there any other? (Whichever philosophy we rely on, mathematical
statements, proofs and definitions should be rigorous and formalisable.
There is no choice.)
might be defined as the science of deterministically repeatable
> processes.
I prefer: M = science on formal systems describing (or having
some interpretation in) any kind of reality. Its goal is not
this reality, but rather formalisms themselves as tools for
approaching to the reality.
Vladimir Sazonov
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