Re: Axiom musings...

[Tim Daly]

A briliant essay:

In exactly the same way a small change in axioms
(of which we cannot be completely sure) is capable,
generally speaking, of leading to completely different
conclusions than those that are obtained from theorems
which have been deduced from the accepted axioms.
The longer and fancier is the chain of deductions
("proofs"), the less reliable is the final result.

https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html


On 8/8/20, Tim Daly <axiomcas@gmail.com> wrote:
> Mark,
>
> You're right, of course. The problem is too large.
> So what. is the plan to achieve a research result?
>
> There are 3 major restrictions on the effort (so far).
>
> First, the focus is on the GCD in NonNegativeInteger.
> Volume 13 is basically a collection of published thoughts
> by various authors on the GCD, a background literature
> search. Build a limited system with essentially one user
> visible function (the NNI GCD) and implement all of the
> ideas there. This demonstrates inheritance of axioms,
> specification of functions, pre- and post-conditions,
> proof integration, provisos, the new compiler, etc.
>
> Second, make the SANE GCD work in the current Axiom
> system by generating compatible code. This gives a
> stepping-stone approach where things can be grounded.
> Obviously none of the new proof ideas will be expected
> to work in the current system but it "gives a place to stand".
>
> Third, develop a lattice of functions. The idea is to attack the
> functions that  depend on almost nothing, prove them correct,
> and use them to prove functions that only depend on the
> prior layer. I did this with the category structure when I first
> got the system since it was necessary to bootstrap Axiom
> without a running system (something that was not possible
> with the IBM/NAG version). That effort took several months
> so I expect that function-lattice to take about the same time.
>
> This makes the research "incremental" so that a result can
> be achieved in one lifetime. Like a PhD thesis, it is initially
> intended as a small step forward but still be a valid instance
> of "computational mathematics", deeply combining proof and
> computer algebra.
>
> Tim
>