> On 12 Mar 2021, at 14:42, Tomas Pales <litewav...@gmail.com> wrote:
>
>
>
> On Friday, March 12, 2021 at 1:30:55 PM UTC+1 Bruno Marchal wrote:
>
> I translate this by “an object is an element of a set together with some
> structure or laws. OK? So vectors, numbers, maps, can all be seen as
> (mathematical) object.
>
> Yes.
>
> (And with mechanism, we can then deduce that there is no physical object,
> although the mind can easily approximate them by some “object” (build by the
> mind).
>
> Not sure what you mean by "physical”.
The observable. It concerns measurable numbers, in the usual repetitive sense
of physics, like temperature, momentum, position, clock, etc.
> I regard as physical those mathematical objects that are in spacetime (and
> spacetime itself is a mathematical object too, a 4-dimensional space with one
> dimension somewhat different that the other three).
This looks like making the physical into a mathematic structure. With
Mechanism, the physical universe is not a mathematical structure among others,
but an invariant in the mind of all Turing machine. It is the arithmetical (or
Turing equivalent) seen from inside. Physics becomes (again) a branch of
Theology, albeit here Digital Mechanism makes the theology into a branch of
computer science/mathematical logic, and even into a branch of arithmetic. It
is testable by comparing the observation with the physics “in the head of the
universal Turing number/machine.
The ontology is simple (just the natural numbers together with the two laws of
addition and multiplication).
>
> OK. In math we use often set theory, intuitively (or formally) to define, or
> better to represent, the different object we want to talk about.
>
> It is known that arithmetic (the natural numbers) can be used too, for most
> of the usual mathematics (including a lot of constructive real objects, and
> more, but not all real numbers)
>
> Reality may be bigger than arithmetic
The internal phenomenology of arithmetic is indeed bigger than the arithmetical
truth. But the ontology is not, and eventually we have to limit the
arithmetical truth (something infinitely complex) to its partial computable
part. “God” is the sigma_1 (partial computable) part of the arithmetical
reality (which is the union of the sigma_1, and all sigma_i (which are less and
less computable, necessitating stronger and stronger oracles (in Turing sense).
> and then we need set theory to capture it, no?
Only in the phenomenology. It is a theorem of arithmetic (+ mechanism) that
most universal number believe in set and infinity axioms, due to the
unbound-able complexity of the arithmetical reality when "seen from inside” (a
notion made precise using the mathematic of self-reference (Gödel, Löb,
Solovay).
> Well, we may never know if reality is bigger than arithmetic because it's
> impossible to prove that even arithmetic is consistent, let alone something
> bigger.
We cannot prove anything about Reality, not even that there is one, beyond our
personal consciousness. But we can try theories, and we learn something when
and if they are refuted.
Now, Arithmetic, with a big A, that is, the standard model of arithmetic is
consistent per definition. Also, we can prove the consistency of arithmetic
with no more axioms that we use in Analysis, and a theory like PA (Peano
arithmetic) is believed to be consistent by all mathematician (except Nelson).
But with Mechanism, even PA is too much for the ontology, and we can use only
RA (Robinson Arithmetic). This is PA minus the induction axioms. The usual
induction axioms have to be added only in the machine’s phenomenology. RA (aka
Q) is believed to be consistent by all mathematicians, including Nelson and
even the ultrafinitist.
>
> “Concrete” is a tricky term which does not survive Mechanism, which reverse
> not just physics and psychology-theology, but also abstract and concrete.
> Just 0, s0, … are concrete, but a physical object like a table becomes
> abstract. It looks concrete phenomenologically, but that is because we have
> millions of neurons making us feel that way.
>
> By "concrete" object I mean an object that is not a property.
But what is an object?
> For example, the general triangle (an abstract object) is a property of all
> concrete triangles such as ones I can draw on a piece of paper.
(Hmm… I cannot use at the ontological level notion like piece of paper. I can
explain (or see my papers) that a piece of paper is something quite abstract.
That it is looks concrete is basically an illusion, requiring long
computational histories, and many neurons...
> But a concrete triangle is not a property of anything. Same with tables; the
> concrete table in your room is not a property of anything but the abstract
> table ("table in general") is a property exemplified in all concrete tables.
I would need to know what you assume to exist, and what you derive from that
assumption. I do not assume an ontological physical universe. I do assume a
physical universe, but the goal will be to explain it without that assumption,
and I show why we have to do that when we assume that a brain or a body is
Turing emulable at a relevant level of description.
>
> We cannot really invoke “reality” as its very nature is part of the inquiry.
>
> I regard as reality all objects (that are identical to themselves, of course).
I take x = x as a logical truth about identity. So every thing is equal to
itself, and so, self-identity cannot be a criteria of (fundamental) existence.
But the collection of all sets equal to themselves, {x I x = x} is typically
not a set, despite that collection is equal to itself.
You seem to assume everything at the start, but without defining things, that
will lead easily to inconsistencies. A square circle is equal to itself,
arguably.
>
>> A red car that is blue is a red car that is not red. Violation of law of
>> identity, therefore nothing.
>
> Fair enough, at least with a content relative to the metaphysics, or basic
> ontology we assume at the start.
>
> Without respecting law of identity, logical explosion will erase all
> differences between object and non-object, existence and non-existence,
> turning everything into nonsense. Paraconsistent logics arbitrarily deny law
> of identity in some circumstances and arbitrarily block explosion in some
> circumstances. They are meaningful and corresponding to reality only to the
> extent they affirm the law of identity.
I can agree. Paraconistent logic is useful to make theories (and semantics) for
natural languages, but is useless for fundamental science, where we should
better start from simple things on which everyone agrees (except Sunday-type of
philosophers).
Bruno
>
>
>
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