On Tuesday, January 29, 2019 at 5:23:36 AM UTC-6, Bruno Marchal wrote:
>
>
> On 28 Jan 2019, at 14:55, Philip Thrift <cloud...@gmail.com <javascript:>> 
> wrote:
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> 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> wrote:
>>
>>
>>
>> *Varieties of finitism*
>> http://www.mbph.de/Logic/Finitism.pdf
>> Manuel Bremer
>> http://www.mbph.de
>>
>> *Annotated bibliography of strict finitism*
>> http://jeanpaulvanbendegem.be/home/papers/strict-finitism/
>> Jean Paul Van Bendegem
>> 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.
>
>
>
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>
>
> 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
>
>



I am familiar with the theory of potential infinity (currently being 
pursued by Hamkins).


I am skeptical that it is coherent. *You can't be a little bit pregnant.*




[image: Joel David Hamkins]
<https://twitter.com/JDHamkins>
Joel David Hamkins
@JDHamkins
<https://twitter.com/JDHamkins>
·
Jan 26
<https://twitter.com/JDHamkins/status/1089170702119436290>
I'll be giving the Jowett Society lecture here in Oxford on 8 February: 
"Potentialism and implicit actualism in the foundations of mathematics".

*Abstract.* Potentialism is the view, originating in the classical dispute 
between actual and potential infinity, that one’s mathematical universe is 
never fully completed, but rather unfolds gradually as new parts of it 
increasingly come into existence or become accessible or known to us. 
Recent work emphasizes the modal aspect of potentialism, while decoupling 
it from arithmetic and from infinity: the essence of potentialism is about 
approximating a larger universe by means of universe fragments, an idea 
that applies to set-theoretic as well as arithmetic foundations. The modal 
language and perspective allows one precisely to distinguish various 
natural potentialist conceptions in the foundations of mathematics, whose 
exact modal validities are now known. Ultimately, this analysis suggests a 
refocusing of potentialism on the issue of convergent inevitability in 
comparison with radical branching. I shall defend the theses, first, that 
convergent potentialism is implicitly actualist, and second, that we should 
understand ultrafinitism in modal terms as a form of potentialism, one with 
surprising parallels to the case of arithmetic potentialism.

Jowett Society talk entry 
<https://jowettsociety.wordpress.com/abstract-for-joel-david-hamkins/> | my 
posts on potentialism <http://jdh.hamkins.org/tag/potentialism/>

- pt

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