[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]

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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