the RM and Godel

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mountain man - 25 Jun 2005 07:01 GMT

I seem to recall there is no "solid ground" in mathematics
in that the formalisms of mathematics cannot lead to
anything resembling "absolute truth".

Here is an interesting article on the history of mathematics
http://www.cs.auckland.ac.nz/CDMTCS/chaitin/cmu.html

How do RM theorists view the work of Godel, Turing and
Chaitin? What are the implications of Godels theorem of
incompleteness, or Chaitin's random truth, to the RM?

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Pete Brown
IT Managers & Engineers
Falls Creek
Australia
www.mountainman.com.au/software

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Jan Hidders - 25 Jun 2005 10:34 GMT

> I seem to recall there is no "solid ground" in mathematics
> in that the formalisms of mathematics cannot lead to
> anything resembling "absolute truth".

That's a grave oversimplification and a very misleading statement. A
slightly less oversimplified version would be that if you formalize the
theory of natural numbers (or sets, for that matter) you cannot have
"the truth, the whole truth, and nothing but the truth" because you have
to choose between either "the whole truth" or "nothing but the truth".
Most tend to choose the "nothing but the truth".

> How do RM theorists view the work of Godel, Turing and
> Chaitin? What are the implications of Godels theorem of
> incompleteness, or Chaitin's random truth, to the RM?

Practically zero. Note that there is Goedels *completeness* result for
first-order logic (i.e., the flat relational model) that tells us that
for uninterpreted predicates we in fact can and do have a complete
axiomatization. So whether there is going to be a problem in this
respect depends upon what you take as your domains, and that decision's
not really part of the RM anyway. But even if the problem would occur,
that would be practically meaningless in practice. Would it stop us from
proving things? No. Would it stop us from being able to ask certain
queries or reason correctly about them? No.

-- Jan Hidders

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mountain man - 26 Jun 2005 23:53 GMT

>> I seem to recall there is no "solid ground" in mathematics
>> in that the formalisms of mathematics cannot lead to
[quoted text clipped - 6 lines]
> choose between either "the whole truth" or "nothing but the truth". Most
> tend to choose the "nothing but the truth".

The article which supported the statement concerning lack
of absolute truth in mathematics was provided as:
http://www.cs.auckland.ac.nz/CDMTCS/chaitin/cmu.html

>> How do RM theorists view the work of Godel, Turing and
>> Chaitin? What are the implications of Godels theorem of
[quoted text clipped - 9 lines]
> No. Would it stop us from being able to ask certain queries or reason
> correctly about them? No.

Thanks for the direct responses to these questions.

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Pete Brown
IT Managers & Engineers
Falls Creek
Australia
www.mountainman.com.au

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Dan - 27 Jun 2005 15:42 GMT

> I seem to recall there is no "solid ground" in mathematics
> in that the formalisms of mathematics cannot lead to
[quoted text clipped - 6 lines]
> Chaitin? What are the implications of Godels theorem of
> incompleteness, or Chaitin's random truth, to the RM?

Mathematics has never claimed "Absolute Truth". It is a system of
logical deductions and in some small cases inductions based on
"reasonable" axioms. But what is surprising is that many "pure" maths
have eventual applicability to the natural world. For example, number
theory now forms the basis for most advanced cryptography.

Human constructed models, mathematical or not, are only an approximation
of nature and hence open to constant improvement.

Pure mathematics on the other hand is a playground for the mind. No
need for applicability.

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