On Sunday, September 1, 2019 at 4:34:19 AM UTC-5, Bruno Marchal wrote:
>
>
> On 30 Aug 2019, at 20:53, Philip Thrift <[email protected] <javascript:>>
> wrote:
>
>
>
> On Friday, August 30, 2019 at 10:10:22 AM UTC-5, Bruno Marchal wrote:
>>
>>
>> On 30 Aug 2019, at 04:33, Alan Grayson <[email protected]> wrote:
>>
>> If there are infinities in mathematics, but not in physics or in nature,
>> is that a problem? AG
>>
>>
>> Is that an interesting problem? I guess so.
>>
>> Some theories in mathematics assume an axiom of infinity, like in set
>> theory, analysis, etc.
>>
>> That has often led to paradoxes, but they have been “solved” by diverse
>> means. So most such theories are considered not being problematic. We can
>> also show that, even restricted on the arithmetical truth (which has no
>> axiom of infinite, as all natural numbers are conceived to be finite),
>> adding an axiom of infinity lead to stronger provability abilities. The set
>> theory ZF proves much more than the arithmetic theories PA, even on just
>> the numbers relations. Yet ZF, and actually all effective theories are
>> limited on a small spectrum of the arithmetical reality. The omega-initial
>> segment of ZF mirrors PA faithfully.
>>
>> In physics, the universe itself could be infinite, without having any
>> infinite things in it, like the model of Arithmetic (all numbers are
>> finite, and the set of all numbers is just a meta-concept, not
>> representable directly in the theory, but still manageable (you can prove
>> in PA that there is an infinite of prime numbers, by proving
>>
>> For x (prime(x) -> It exist y (y bigger than x) and prime(y)).
>>
>> Are there actual infinite object in the universe?
>>
>> I can prove that if mechanism is false, then there as such object. With
>> Mechanism, the mind is infinite, and physics is somehow the Mind seen from
>> itself internally. That might favours an infinite physical universe. Does
>> our substitution level depend on Planck Constant? Open problem.
>>
>> With mechanism, the axiom of the infinite is inconsistent on the
>> ontological level, but is a theorem on the phenomenological level. It
>> shortens the proofs, and provide many tools to to handle mathematically the
>> semantic, the notion of limit, many form of approximation, even to learn
>> just about the natural numbers or the combinators.
>>
>> Mechanism provides a testable account of the mind-body relation, an
>> account which does not assume more than elementary arithmetic, and which
>> doesn’t involve any other ontological commitment. So let us see. The
>> quantum structure, and time, admits a “simple" arithmetical interpretation,
>> but space, dimension, energy remains in the shadow.
>>
>> Science has not yet decided between Plato's and Aristotle’s conception of
>> reality, and not all people are aware of the hypothetical nature of those
>> options, nor that Digital Mechanism, or even just the Church-Turing thesis,
>> leads us back to Pythagorus and Plato.
>>
>> The natural numbers realised the infinities without making them into
>> existing things, or beings.
>>
>> Bruno
>>
>>
>>
> Also, some may think that because theoretical physics (field theories) is
> expressed in a language that includes a mathematically continuous (real
> number) background R^4 of spacetime and the methods of multidimensional
> calculus (tensor calculus, etc.), that because *mathematically
> infinite-divisibility* is present and *infinitary definitions* (like
> "limit") are present, that these "infinities" of the mathematics are real
> in the actual world.
>
>
> I agree. Infinities are useful does not entails that infinities exist in
> “Reality”.
>
> That is why I can be more open to your fictionalism for second order
> arithmetic, analysis, set theory, more than for arithmetic (that would make
> Mechanism and even just computer science into fiction)..
>
> Bruno
>
cf.
Joel David Hamkins @JDHamkins https://twitter.com/JDHamkins
http://jdh.hamkins.org/
(I think he has to be a fictionalist.)
@philipthrift
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