Re: Cantorian pseudomathematics

david petry said:

Tony Orlow wrote:
david petry said:

I believe I have stated very precisely what the criteria are. A
statement is meaningful iff it makes predictions about the results of
computational experiments. A computational experiment can be put in the
form "Turing machine T, with input M, will halt within N steps".

I am also claiming that we can build a new formalism for mathematics
such that the axioms themselves are computationally meaningful, and
such that the laws of logic we use preserve computational meaning. Then
every grammatically valid sentence in our formalism will be meaningful.

David, it sounds like you want a statement to be a symbolic string that is
manipulable through formal rules so that it produces a quantity.

I don't actually understand that statement of yours.


I am trying to get a handle on exactly what you mean by computationally
meaningful, and the best guess that comes to mind, with your mention of Turing
machines, is that you want to see a construction in the form of symbolic
manipulation of strings which constitute statements, to generate the elements
of a set. Maybe that's still not clear.


So, what primitive notions
do you need to add to this grammar for that purpose? Have you identified
specific problematic constructions that have no computational meaning?

The use of the unrestricted existential quantifier quickly leads to
statements which have no computational meaning. In other words, if you
tell me that something exists, I want you to tell me precisely how I
can find it. If you can't do that, I will claim that you don't really
know what you are talking about.


You mean having axioms that say, "there exists such-and-such a set", or "there
exists some smallest infinity"? How would you restrict the use of the
existential quantifier to avoid this, grammatically?


Are there any systems of symbolic manipulation which you don't consider
computational, or any computations which don't involve symbolic tranformations?

I have trouble grasping what you are really asking here.


Again, I am trying to grasp exactly what you mean by computation, and it sounds
like logical symbol manipulation, for which you would like to suggest some
additional rules. So, I am saking what you think the relationship is between
manipulation of strings of symbols, and "computation". What does a Venn diagram
of the two look like to you?


What I am advocating is that we
would have to derive from the proof an upper bound on how much
computation we need to do in order to compute the integer, in order to
say that we have a way to compute it.

By "how much
computation", what do you mean, exactly. How do you measure computation in your
scheme.

A Turing machine takes one step at a time, and we can count those
steps.


So, one symbol processed, and one state transition, per unit of computation.


As I already pointed out, using lists would be preferable to using sets
here.

A list being just a set with order, right?

No. A list can have elements repeated.


Okay, a string is a list of symbols, then? But then, there is always the
alphabet, the set of symbols from which to construct a string. Similarly, I
would think that your lists would have elemenets drawn from some set of
possibile values. Perhaps each successive element of a list could come froma
successive unique set of values. A sort of dimensional thing, no?


Here's what I am claiming: we can build up constructive mathematics
independently of classical mathematics. Then, we can notice that
classical mathematics implies the existence of a universe much "bigger"
than the universe of constructive mathematics, and hence there is
necessarily something about classical mathematics that is fantasy,
where "fantasy" is defined to be anything beyond the world we can
observe (i.e. the constructive universe).

Hmmmm..... How do you know that what you view as non-constructive has just not
had its construction discovered/defined yet?

Well, constructively speaking, the burden is on someone who claims
something exists to prove that it exists, and not on me to prove it
doesn't.



Okay, but you have to define what you mean by "prove", I suppose, by stating
axioms which are considered foundations of a system able to prove such things,
right? I am trying to address the same area, myself. I am leaning towards a
dimensional/spatial notion of sets and set membership. So, I am curious as to
what notions you think are a good starting place for your axioms.
:)

--
Smiles,

Tony
.

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