Re: Existence of mathematical entities (Re: Successor Axiom: on what grounds TF?)

From: Stephen Harris (cyberguard1048-usenet_at_yahoo.com)
Date: 02/13/05 

Date: Sun, 13 Feb 2005 21:55:49 GMT

"R-matrix" <random@matrix.mx> wrote in message
news:2fjt01lujb3tl513n25vo80jb92f5q5he0@4ax.com...
> On 7 Feb 2005 19:37:38 -0800, examachine@gmail.com wrote:
>
>>I really wonder the opinion of physicists, theorists and engineers who
>>might be reading this.
>>
>>Do any of the theories you use assume the "existence of mathematical
>>entities", e.g. Godel's mathematical realism?
>>
>>Are not there any worthwhile competing points of view, for instance
>>such as physicalism, computationalism, functionalism, instrumentalism,
>>etc.?
>>

Try reading Sol Feferman:

http://math.stanford.edu/~feferman/papers/ResponseToHellman.pdf

"In his provocative article for this issue, Geoffrey Hellman has
astutely attacked the philosophical grounds for predicativity
from several angles. Though I am not now nor never have been a
predicativist, I have to admit to being a sympathizer since I
am an avowed anti-platonist, at least insofar as set theory is
concerned, and I grant the natural numbers a position of primacy
in our mathematical thought."

"In The Light of Logic" by Sol Feferman page 248:
Infinity in Mathematics: Is Cantor Necessary? (conclusion)

"Independently of such detailed work which puts into question
the necessity of higher set theory for everyday mathematics*,
I am convinced that the platonism which underlies Cantorian
set theory is utterly unsatisfactory as a philosophy or our
subject, despite the apparent coherence of current set-theoretical
conceptions and methods. To echo Weyl, platonism is the
medieval metaphysics of mathematics; surely we can do better."

SH: Feferman also believes and elucidates that maybe all mathematics
for science and engineering can be done without Cantorian set theory.

http://math.stanford.edu/~feferman/book98.html

"In his concluding chapters, Feferman uses tools from the special
part of logic called proof theory to explain how the vast part, if
not all of scientifically applicable mathematics can be justified
on the basis of purely arithmetical principles. At least to that
extent, the question raised in two of the essays of the volume,
"Is Cantor Necessary?," is answered with a resounding "no."

http://math.stanford.edu/~feferman/papers/Unlimited.pdf

An early paper by Feferman on Category theory, but I don't
he believes Category theory is a better foundation than set theory.

Next in chapter 12 of "In the Light of Logic" by Sol Feferman,
"Is the Cantorian transfinite Necessary for Finitary Mathematics?"

Godel's Doctrine
"By Godel's doctrine I mean the view first enunciated in footnote 48a
of the Godel (1931) that the "true reason" for the incompleteness
phenomena is that "the formation of ever higher types can be continued
into the transfinite," both in systems explicily using types and in systems
of set theory such as ZF for which the (cumulative) type structure is
implicit in the axioms. For, as Godel says, the "undecidable propositions
constructed here become decidable whenever appropriate higher types
are added." Since the undecidable propositions are of finitary character,
Godel's doctrine says in effect that the unlimited iteration of the
power-set operation is necessary to account for finitary mathematics.

pg. 232 ... However, Godel's doctrine can be challenged when it is
read as asserting that the platonistic view of the determinateness of
the power-set operation and its iteration through all the ordinals is
_necessary_ for the derivation of _previously_ undecidable but true
II^0_1 statements. ...

According to this line of thinking, it is equally reasonable to replace
Godel's doctrine by a variant doctrine, according to which the
"true reason" for the incompleteness phenomena is that the truth
definition for the language of a formal system is not expressible in
that language (Tarski's theorem) and that by adjunction of the notion
of truth with suitable axioms, the undecidable statements produced
by Godel become undecidable." page 233 "In the Light of Logic"

Relevant Pages

  • Re: Existence of mathematical entities (Re: Successor Axiom: on what grounds TF?)
    ... at least insofar as set theory is ... Infinity in Mathematics: Is Cantor Necessary? ... the necessity of higher set theory for everyday mathematics*, ... "By Godel's doctrine I mean the view first enunciated in footnote 48a ...
    (sci.logic)
  • Re: Existence of mathematical entities (Re: Successor Axiom: on what grounds TF?)
    ... at least insofar as set theory is ... Infinity in Mathematics: Is Cantor Necessary? ... the necessity of higher set theory for everyday mathematics*, ... "By Godel's doctrine I mean the view first enunciated in footnote 48a ...
    (sci.physics)
  • Re: Existence of mathematical entities (Re: Successor Axiom: on what grounds TF?)
    ... at least insofar as set theory is ... Infinity in Mathematics: Is Cantor Necessary? ... the necessity of higher set theory for everyday mathematics*, ... "By Godel's doctrine I mean the view first enunciated in footnote 48a ...
    (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)
  • Re: Cantorian pseudomathematics
    ... What I mean is set theory was not born just by the great thought of Cantor. ... of mathematics behind, which help greatly, and which even where necessary for the theory to be imagined. ... formalization, at a level up, for meta-theory. ... scientists and engineers doesn't have to be used by engineers. ...
    (sci.math)