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

From: Charlie-Boo (chvol_at_aol.com)
Date: 06/25/04 

Date: 25 Jun 2004 08:53:21 -0700

Vlad <vtt0000@mail.utexas.edu> wrote
> Chairman of the Ozzy Osbourne Appreciation Society wrote:

> > 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.

I understood his question to be referring to the distinction between
PCs/Turing Machines (that operate on recursively enumerable sets e.g.
integers and character strings) vs. mathematics, which deals with
aleph-1 sets such as the real numbers, which cannot in general be
represented in computers (because any such representation would form
only an aleph-0 set.)

> > 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 is really not a matter of presupposing anything. It is a matter of
generating by computer what has been done by hand for 2,000 years
(axioms of various branches of mathematics), with all of the
advantages
inherent in the use of a computers: ease, precision, optimization,
reduction in the number of primitives (Occam's Razor), the unification
of Computer Science and Mathematics, occasional machine crashes.

> > It doesn't seem elegant at all to me, but confusing and
> > backwards.

Using computers is backwards? Yikes! Do you value theoretical
Computer Science? The unification of two areas of study into one
(discovering and removing overlaps)? A reduction in the number of
primitives in a formal system?

> 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).
> We could have just formalized straight away the notion we want to
> reason about (for example the natural numbers with Peano's axioms).

What are the axioms of addition that uniquely define that function?
Of less than? It is not a trivial task. But it is fairly trivial to
write programs to calculate these functions. And if a system can
extract properties of the function that a program is calculating, we
can automate the discovery of the properties of these and other
primitives of Mathematics.

> 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.

See above. And the derivation of properties of an arbitrary program
is the new task. It replaces the individual tasks of trying to
imagine adequate axioms for each function. It is the "big picture"
approach. In effect, you are working at a higher level of
abstraction. Each inference that applies to programs applies to the
many primitives of Mathematics that can be easily programmed.

There is similarity and overlap between the axioms of the individual
primitives of Mathematics and their derivation. But Computer Science
provides the opportunity to unite all of these specific instances of a
recursive function into a single general formalism that shows us what
the general rules are that explain all of them.

> 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".

No, he decided to use Latin because it sounds more impressive!

Charlie Volkstorf
Cambridge, MA

PS Computers have been used to find many mistakes in PM and can also
be used to create smaller (less redundant) sets of axioms and rules of
inference. But to me, programming recursive functions and Turing's
proof of the unsolvability of the Halting Problem shows me that
Russell's idea to prohibit self-reference to fix the Russell Paradox
is a very bad idea! After all, even the definition of the natural
numbers requires self-reference.

> Vladimir Trifonov

Relevant Pages

  • Re: Deep Thoughts # 7: A New Kind of Mathematics
    ... represented in computers (because any such representation would form ... (axioms of various branches of mathematics), ... reduction in the number of primitives, ...
    (sci.math)
  • Re: Mathematician Using Nil With Students
    ... however I haven't seen anyone demonstrating that his axioms are ... It might be a useless system, but mathematics doesn't care about ... you can make it defined by doing "tricks" like extending the complex plane ... analysis without "fudging" is a well known shortcoming of computers. ...
    (borland.public.delphi.non-technical)
  • Re: Cantorian pseudomathematics
    ... I don't know of theorems of set theory that have ... According to my axioms of the ... > is the set of "computational axioms of the rest of mathematics"? ... > To have an axiomatized theory, you need a logistic system, primitives, ...
    (sci.math)
  • Re: Cantorian pseudomathematics
    ... I don't know of theorems of set theory that have ... According to my axioms of the ... is the set of "computational axioms of the rest of mathematics"? ... To have an axiomatized theory, you need a logistic system, primitives, ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... most of the standard axioms would get scrapped ... you claim that set theory is ... theory in which to express virtually all of mathematics. ... S (call this function 'omega pre S'). ...
    (sci.math)