> On 28 Jan 2019, at 14:55, Philip Thrift <cloudver...@gmail.com> wrote:
> 
> 
> 
> On Monday, January 28, 2019 at 5:46:12 AM UTC-6, Bruno Marchal wrote:
> 
>> On 26 Jan 2019, at 08:36, Philip Thrift <cloud...@gmail.com <javascript:>> 
>> wrote:
>> 
>> 
>> 
>> Varieties of finitism
>> http://www.mbph.de/Logic/Finitism.pdf <http://www.mbph.de/Logic/Finitism.pdf>
>> Manuel Bremer
>> http://www.mbph.de <http://www.mbph.de/>
>> 
>> Annotated bibliography of strict finitism
>> http://jeanpaulvanbendegem.be/home/papers/strict-finitism/ 
>> <http://jeanpaulvanbendegem.be/home/papers/strict-finitism/>
>> Jean Paul Van Bendegem
>> http://jeanpaulvanbendegem.be <http://jeanpaulvanbendegem.be/>
>> 
>> Apparently, strict finitism requires a (likely) paraconsistent modality 
>> (implying there may be inconsistencies). If one takes strict finitism 
>> seriously (Tegmark would have to be a strict finitist, if he isn't kidding 
>> people about what he said) then of course physics (and mathematics, of 
>> course) would be radically different.
> 
> 
> Computationalism (digital mechanism) is consistent with strict finitisme, but 
> rather unsound.
> 
> Mechanism is a finitisme, but it keeps the potentially infinite of the 
> classical and intuitionist thinkers. 
> 
> But with mechanism, we cannot put the induction axioms in the ontology, so we 
> cannot prove that there is no biggest natural numbers in the ontology. From 
> outside, we know that this is consistent only because we believe in some 
> infinite objects, making strict finitisme consistent, but rather 
> arithmetically unsound.
> 
> Nothing in Tegmark suggests that he would espouse anything like “strict” 
> finitisme, but when he moved to computationalism, he might become a finitist.
> 
> The best book (beside my work :) ) on the subject of mechanism and finitism 
> is the book by Judson Webb, 1980. 
> 
> WEBB J. C., 1980, Mechanism, Mentalism and Metamathematics : An essay on 
> Finitism, D. Reidel Pub. Company, Dordrecht, Holland.
> 
> Bruno
> 
> 
> 
> 
> Max Tegmark writes that the mathematics of physics in the future needs to be 
> "infinity-free".
> 
>       
> http://blogs.discovermagazine.com/crux/2015/02/20/infinity-ruining-physics/
> 
> (Almost like it's a sin to have infinities around.) That sounds like strict 
> finitism to me.

Not it is not. It is finitism, not strict finitism. Actually you need sting 
actual infinities to make sense of strict finitism. Finitism need only 
potential infinite to be define, but strict finitism needs actual infinities at 
the metalevel. To define what is a machine, or what is “finite", you need 
potential infinite. Strict finitisme makes sense … thanks to big infinities at 
the meta-level. Absolute strict finitism cannot be Turing universal.







> 
> As Manuel Bremer's paper above shows, a strictly-finite arithmetic is 
> (likely) inconsistent,

It is not valid. RA can be strictly finitist, but again, at the semantical 
level this is made possible by the existence of a non standard natural number 
which are bigger than all standard numbers.




> which is OK, since "inconsistent mathematics can have a branch which is 
> applied mathematics”.

Inconsistency treatment I useful for natural language, and human psychology, 
but in most applied math, we need consistent theories. In metaphysics, 
paraconsitency is a red herring. It hides the problems instead of solving them. 
But now, self-reference makes inconsistency consistent, and G/G* has a small 
quasi-para-consistent part, useful indeed for the embedded machine (in sheaves 
of arithmetical computations).

Bruno




> 
>     https://plato.stanford.edu/entries/mathematics-inconsistent/
> 
> - pt
> 
> 
> 
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