On 20 Sep 2016, at 02:56, Russell Standish wrote:

On Mon, Sep 19, 2016 at 04:18:21PM +0200, Bruno Marchal wrote:

On 18 Sep 2016, at 23:54, smitra wrote:

https://www.youtube.com/watch?v=nK6XawDE8_U


I was hoping seeing Nelson's more or less recent proof of the
inconsistency of Peano Arithmetic, which was enough convincing to
require some good logician to find out the precise error, that
Nelson immediately acknowledges.

Now, this was different and not entirely uninteresting but it is
just obvious that by assuming there is a physical universe, and that
it is finite, and by defining some notion of physical numbers, you
can show that the axioms of Peano will not work for those physical
numbers.

It is just that what PA, and Euclid, Pythagorus, Gauss, Fermat,
Dedekind, Peano, etc.  talk about is something which has nothing to
do a priori with the sort of numbers this physicalist believer
(apparently) is talking about. Of course, with computationalism they
become related: the "physical numbers" would belong to partially
sharable 3-p-computations/1-p dreams of the natural numbers.

But I guess this was a bit of a joke, lol.

Bruno


Norm is one of the preeminent latter-day ultrafinitists. He is quite
persuasive when he talks.

It is physicalist ultrafinitisme. It is the move out of the step seven's consequence, but although step 8 does not make it impossible, it makes it into a quite magical move equivalent to the usual creationist trick. It is at the diametral opposition of the "everything" is simpler than this highly particular thing.



And he has done a lot of work in presenting
"trigonometry" using rational functions that avoid the need for
infinite quantities.

That is finitism. Ultrafinitism avoid the existence of big numbers.

The baby arithmetic (RA without its third axiom) is still enough Turing universal to provide a universal dovetailing, yet, is consistent with the axiom that there is only one number.

Now, that model with only one number is hardly interesting, and the universal dovetailing has better not to use (too much often) that axiom, if some made God would have added.



I gather that ultrafinitism still lacks a
rigorous foundation, however...

God created the numbers from 0 to 24.
24 did all the rest.

I am not sure that ultrafinitism in mathematics (or in theology and probably in physics) makes much interesting sense. Computationalism can be considered as a form of finitism, but ultrafinitism would make it meaningless.

Bruno




Cheers

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Dr Russell Standish                    Phone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Senior Research Fellow        hpco...@hpcoders.com.au
Economics, Kingston University         http://www.hpcoders.com.au
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