On Thursday, March 19, 2020 at 7:46:10 AM UTC-5, Lawrence Crowell wrote:
>
> On Wednesday, March 18, 2020 at 12:47:56 PM UTC-5, Philip Thrift wrote:
>>
>>
>>
>> On Wednesday, March 18, 2020 at 8:42:19 AM UTC-5, Lawrence Crowell wrote:
>>>
>>> On Wednesday, March 18, 2020 at 4:13:36 AM UTC-5, Bruno Marchal wrote:
>>>>
>>>>
>>>> On 17 Mar 2020, at 16:14, Lawrence Crowell <goldenfield...@gmail.com>
>>>> wrote:
>>>>
>>>> I pretty seriously doubt these things will enter into physics.
>>>> Computations with Cantor's aleph hierarchy of transfinite numbers seems
>>>> pretty far removed from anything really physical.
>>>>
>>>>
>>>>
>>>> OK. It is just abstract recursion theory, has been with Turing, and it
>>>> concerns divine creatures, and the goal is to show that even those
>>>> “divine”
>>>> (infinite being) cannot effectively recover the arithmetical
>>>> truth/reality.
>>>> I doubt too that such machine can be brought to Earth, and they do play
>>>> some role in the internal phenomenology of the “terrestrial machine”, a
>>>> bit
>>>> like infinite sums can play a role in physics, like the zeta
>>>> renormalisation in superstring theory.
>>>> Note that most of those divine (infinite) machine are still obeying to
>>>> the same self-reference logics (G and G*), and would not change much in
>>>> the
>>>> mathematical derivation of physics from arithmetic. Note also that the
>>>> first person indeterminacy is connected to the machine + random oracle, so
>>>> that the “sigma_1” predicate is really sigma_1 in all oracles, and this
>>>> makes it possible to use some strong set axiom (like “projective
>>>> determinacy”) to assure the existence of the measure, and probably of the
>>>> needed generalisation of Feynman integral in arithmetic. (This needs
>>>> indeed
>>>> a generalisation of the Church’s thesis, called the hyperarihmetical
>>>> church’s thesis in the classical textbook by Rogers).
>>>>
>>>> Bruno
>>>>
>>>>
>>>>
>>> When we get into subtleties of ZF set theory we get into the application
>>> of axioms, eg the axiom of replacement, axiom of infinity, axiom of choice
>>> etc, that have a limited bearing on most standard mathematics. This means
>>> physics is even further removed. Transfinite numbers and the question of
>>> א_0 ≤ c ≤ 2^א_0 or the continuum hypothesis has a role with Robinson’s
>>> numbers. These are in a sense the reciprocals of transfinite numbers or the
>>> continuum. This has some bearing in the foundations of calculus, though I
>>> am far from familiar with this. I am not sure what possible role this could
>>> have for physics.
>>>
>>> Infinite mathematical quantities are common in physics. The 2-dim CFT is
>>> Virasoro which is a Witt algebra extended to infinite dimensions with a
>>> kernel. This does not though mean physics predicts the measurement of
>>> infinite quantities A Hilbert space may have an infinite number of states,
>>> but that does not mean we expect to measure a quantum state with N → ∞.
>>>
>>> LC
>>>
>>>
>>>
>>
>> Permitting entities in its language is a )perhaps the major) problem with
>> the current approach to theoretical physics and which most physicists seem
>> oblivious to. It is a form of double-speak to say there are no infinities
>> but then make theories based on them ... unless they are introduced in a
>> theoretically appropriate manner, which has yet been done.
>>
>> The continuum threatens us with infinities:
>>
>> But even if we take a hard-headed practical attitude and leave logic to
>> the
>> logicians, our struggles with the continuum are not over. In fact, the
>> infinitely
>> divisible nature of the real line—the existence of arbitrarily small real
>> numbers—is a serious challenge to physics.
>>
>> We have seen that in every major theory of physics, challenging
>> mathematical
>> questions arise from the assumption that spacetime is a continuum. The
>> continuum threatens us with infinities. Do these infinities threaten our
>> ability to
>> extract predictions from these theories—or even our ability to formulate
>> these
>> theories in a precise way? We can answer these questions, but only with
>> hard
>> work. Is this a sign that we are somehow on the wrong track? Is the
>> continuum as we understand it only an approximation to some deeper model of
>> spacetime?
>>
>> Only time will tell. Nature is providing us with plenty of clues, but it
>> will take
>> patience to read them correctly.
>>
>> *Struggles with the Continuum*
>> John C. Baez
>> https://arxiv.org/abs/1609.01421v4
>> https://arxiv.org/pdf/1609.01421v4.pdf
>>
>>
>> @philipthrift
>>
>
> In a funny way quantum mechanics may rescue this situation. The Fermi and
> Integral spacecraft measurements of EM radiation from distant burstars
> indicates spacetime is smooth down to 50 times smaller than the Planck
> length. This suggests space or spacetime is smooth "all the way down" in a
> calculus ε - δ sense. This would appear to lead directly into questions on
> how it is that an uncountable infinite number of entities might enter into
> physics. Of course in these measurements there was no Heisenberg microscope
> looking in on space. Further, space and spacetime are likely an
> epiphenomenology of large N-entanglements of quantum states. In this sense
> there is no physical meaning to space or spacetime in this sense.
>
> LC
>
Following John Baez (Struggling with the continuum: "Is the continuum as we
understand it only an approximation [sic] to some deeper model of
spacetime?") and Max Tegmark ("use only finite computer resources by
treating everything as finite") leads one to get away from the 1800s ε - δ
model to modern programmatic alternatives:
Within https://en.wikipedia.org/wiki/Automatic_differentiation
*Operational calculus on programming spaces*
https://en.wikipedia.org/wiki/Automatic_differentiation#Operational_calculus_on_programming_spaces
"Operational calculus on programming spaces provides differentiable
programming with formal semantics through an algebra of higher-order
constructs. It can thus be used to express the concepts underlying
automatic differentiation."
ref: https://arxiv.org/abs/1610.07690
In this work we present a theoretical model for differentiable programming.
We construct an algebraic language that encapsulates formal semantics of
differentiable programs by way of Operational Calculus. The algebraic
nature of Operational Calculus can alter the properties of the programs
that are expressed within the language and transform them into their
solutions.
In our model programs are elements of programming spaces and viewed as maps
from the virtual memory space to itself. Virtual memory space is an algebra
of programs, an algebraic data structure one can calculate with. We define
the operator of differentiation (∂) on programming spaces and, using its
powers, implement the general shift operator and the operator of program
composition. We provide the formula for the expansion of a differentiable
program into an infinite tensor series in terms of the powers of ∂. We
express the operator of program composition in terms of the generalized
shift operator and ∂, which implements a differentiable composition in the
language. Such operators serve as abstractions over the tensor series
algebra, as main actors in our language.
@philipthrift
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