Re: Deep Thoughts # 7: A New Kind of Mathematics

From: Vlad (vtt0000_at_mail.utexas.edu)
Date: 06/24/04 

Date: Thu, 24 Jun 2004 15:10:54 -0500

Chairman of the Ozzy Osbourne Appreciation Society wrote:

> Charlie-Boo wrote:
>
>> David C. Ullrich <ullrich@math.okstate.edu> wrote
>>
>>
>>> This makes at least three time that you've simply ignored
>>> a specific objection to your grand scheme (the whole approach
>>> makes sense at best only for categorical theories, which I
>>> stated using smaller words), showing that whatever it does
>>> it can't possibly have the sort of universal applicatbility
>>> that you claim.
>>
>>
>>
>> Where did I say it was "universally applicable"?
>>
>> If anything, I said the opposite. When asked, "Does the programming
>> language that you have in mind program the behavior of a mathematical
>> "machine"? or a physical machine, like a PC?", I responded, "I have in
>> mind (initially) a physical machine (Turing Machine), i.e., the
>> universal set is recursively enumerable." and referred only to number
>> theoretic functions and relations.
>
>
>
> That response must have been made in reference to my question,
> which may or may not have been what David was after. I suspect
> it isn't, but anyway...
>
> First, I'm skeptical that Turing machines are considered to
> be a physical machines. The reason why I asked is because
> a Turing machine is a mathematical construction which
> presupposes the existence of the naturals; since there
> are enumerations of states.
>
> So, I was wondering: why? if we've already presupposed the
> existence of a Turing machine (and consequently the naturals)
> would we then want to derive a set of primitive axioms
> in order to prove the existence of the naturals?
>
> It doesn't seem elegant at all to me, but confusing and
> backwards.
> <snip>

This is exactly my feeling about this. I do not understand why
is deriving properties of a program more elegant, than formalizing
directly an untiuitve notion which we can describe informaly
in some natural language (or only have as an idea in our mind).
Turing Machines were proposed by Turing as a formalization of
what we (humans) do when we have to perform some algorithmic
task, complete with an infinite amount of paper and abiltiy to scribble on
it. So to construct a Turing Machine we already have to know what
we want to formalize. For example the proposed program that generates
the natural numbers was written with the intention of doing this, so
it is not surprising that we can extract properties of the natural numbers
from it. What I do not understand is how is having the program helping?
We could have just formalized straight away the notion we want to
reason about (for example the natural numbers with Peano's axioms).

Furthermore I do not believe that the following is true. This is a quote
from Charlie-Boo's message which started this thread.

>> writing the program for any particular intuitive phenomenon is trivial
>> and derivation of the axioms from that program is automatic.

First, writing a program for a "particular intuitive phenomenon" is
probably just as hard as formalizing the phenomenon in the usual
language of mathematics and second, derivation of properties of
a program is by far not automatic.

Maybe the author of the above ideas should formalize them as a Turing
Machine (this should be trivial), extract the properties of
this Turing Machine (automatically) and then surprise the world with
A New Kind of Mathematics together with a proof of this. As a guide for
his undeavor he can use Principia Mathematica (I do not think it
can be found in the popular science section of a bookstore) -- the
"hackings" of a guy called Bertrand Russel, who was also proposing
foundational principles of how to do mathematics, but never called them
"A New Kind of Mathematics".

Vladimir Trifonov

Relevant Pages

  • Re: Deep Thoughts # 7: A New Kind of Mathematics
    ... > existence of a Turing machine (and consequently the naturals) ... is deriving properties of a program more elegant, than formalizing ... A New Kind of Mathematics together with a proof of this. ...
    (sci.logic)
  • Re: Skolems Paradox
    ... Whatever philosophy of mathematics you espouse, it remains a rather trivial mathematical fact that and are not equivalent. ... In this case it's a theorem of ZFC that implies, but only for the rather uninteresting reason that both of these are provable in ZFC and all provable statements imply each other. ... One could take the view that there is one and only cumulative hierarchy, the class of all sets, and hence CH is either true or false in the ordinary mathematical sense, but hold that our ideas about the cumulative hierarchy are not detailed enough to determine whether CH or ~CH holds. ... If someone were to ask me whether there exists a set, I would give the natural answer: "sure, for example, there's the empty set, the set of naturals, the set of countable ordinals and so forth". ...
    (sci.logic)
  • Re: Orlow cardinality question
    ... > 1's between naturals violates by being finite in size. ... elements in an infinite set. ... > source of much dispute up to this day. ... mathematics as applied to the real world. ...
    (sci.math)
  • Re: Zenkins paper on Cantor (reply of Dr. Zenkin)
    ... > construction when things talked of are supposedly infinite objects? ... we define, for all naturals i, f= i; and, for all naturals j, g ... > think this was one of the steps of reasoning which Dr. Zenkin ... anyone else here seems to be talking about - mathematics. ...
    (comp.theory)
  • Re: Zenkins paper on Cantor (reply of Dr. Zenkin)
    ... > construction when things talked of are supposedly infinite objects? ... we define, for all naturals i, f= i; and, for all naturals j, g ... > think this was one of the steps of reasoning which Dr. Zenkin ... anyone else here seems to be talking about - mathematics. ...
    (sci.math)