On Friday, January 18, 2019 at 7:36:34 AM UTC-6, Bruno Marchal wrote:
>
>
> On 17 Jan 2019, at 21:02, Philip Thrift <cloud...@gmail.com <javascript:>> 
> wrote:
>
>
>
> On Thursday, January 17, 2019 at 12:45:31 PM UTC-6, Brent wrote:
>>
>>
>>
>> On 1/17/2019 12:22 AM, agrays...@gmail.com wrote:
>>
>>
>> *Later I'll post some questions I have about your derivation of the 
>> Planck length, but for now here's a philosophical question; Is there any 
>> difference between the claim that space is discrete, from the claim or 
>> conjecture that we cannot in principle measure a length shorter than the 
>> Planck length? *
>> *TIA, AG *
>>
>>
>> The theory that predicts there is a shortest measured interval assumes a 
>> continuum.  There's no logical contradiction is this. But physicists tend 
>> to have a positivist attitude and think that a theory that assumes things, 
>> like arbitrarily short intervals, might be better expressed and simpler in 
>> some way that avoids those assumptions.  This attitude does not assume the 
>> mathematics itself is the reality, but only a description of reality; so 
>> there can be different descriptions of the same reality.
>>
>> Brent
>>
>
>
>
> *A* theory that does this assumes a continuous mathematics.
> But that doesn't mean *every* theory has to.
>
> As Max Tegmark's little lecture to physicists says:
>
>     Our challenge as physicists is to discover ... infinity-free equations.
>
>
> http://blogs.discovermagazine.com/crux/2015/02/20/infinity-ruining-physics/#.XEDdLs9KiCQ
>
> Unless he is wrong in his premise, of course!
>
>
>
> That assumes non-mechanism, and thus bigger infinities. Tegmark is right: 
> we cannot assume infinity at the ontological level (just the finite numbers 
> 0, s(0), s(s(0)), …). But the physical reality is phenomenological, and 
> requires infinite domain of indetermination, making some “observable” 
> having an infinite range. The best candidate could be graham-Preskill 
> frequency operator (that they use more or less rigorously to derive the 
> Born rule from some “many-worlds” interpretation of QM.
>
> Bruno
>
>
>


I think it is possible some of this can be approached with what is referred 
to as *higher-type computing*, where 

higher-type computing is about

-  *the characterization of the sets that can be exhaustively searched [1] 
by an algorithm, in the sense of Turing, in finite time, as those that are 
topologically compact*

- *infinite sets that can be completely inspected in finite time in an 
algorithmic way, which perhaps defies intuition*

[1] Exhaustible sets in higher-type computation
     https://arxiv.org/abs/0808.0441
[2] A Haskell monad for infinite search in finite time
    
 
http://math.andrej.com/2008/11/21/a-haskell-monad-for-infinite-search-in-finite-time/

from Martin Escardo's page
     http://www.cs.bham.ac.uk/~mhe/

 - pt

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