[FOM] FOM currents [A.Robinson, Platonism and Formalism]
Vladimir Sazonov
V.Sazonov at csc.liv.ac.uk
Mon Oct 13 17:18:30 EDT 2003
Harvey Friedman wrote:
>
"Dean Buckner" <Dean.Buckner at btopenworld.com> wrote
> >> [A] completely standard piece of mathematics that you forced
> >> ordinary language considerations on, is that "in any sequence of sets of
> >> integers, some set of integers is missing."
> >
> > This is a standard and elementary theorem of ordinary language, too!
My Reply to Dean Buckner:
There could be NO theorems "of ordinary language". Theorems may by
only in mathematics and in a suitable mathematical language
(formalism). If somebody (does not matter who) proves a theorem
(if it is really a theorem), he/she is at this moment a mathematician.
Why
> > should I make complaints about it? My complaint is about the idea that any
> > set of integers constitutes ALL integers.
As I do not know the context, I cannot comment on this.
Friedman:
>
> The set of all integers has every integer in it (as an element). Yes, there
> are some people who are uncomfortable with this idea.
that the set of natural numbers is existing (actually or potentially)
in an absolute way?
Friedman:
> Not Cantor, and not
> the overwhelming number of mathematicians. But certainly there are some.
E.g. Abraham Robinson. Recently his formalist position was mentioned
by Haim Gaifman with reference to his 1964 paper``Formalism 1964''
in {\it Logic Methodology and Philosophy of Science}, Y. Bar-Hillel
ed., North-Holland 1965, pp.228-246.
>From that context it was unclear to me what exactly was the position
of Abraham Robinson. Now I have got this paper and can cite it
with some my own comments in-between (and with no pretension
on a most adequate presentation of the ideas of Robinson;
at least not everything in this paper is sufficiently clear
for me):
Abraham Robinson:
"My position concerning the foundations of Mathematics is based on
the following two main points or principles.
(i) Infinite totalities do not exist in any sense of the world (i.e.,
either really or ideally). More precisely, any mention, or purported
mention, of infinite totalities is, literally, {\em meaningless}.
(ii) Nevertheless, we should act {\em as if} infinite totalities
really existed. [VS - we can imagine them without any beliefs
in their absolute existence.]
..........
K.G\"odel, who may be regarded as the outstanding platonist of our
time, has emphasized the similarity between the investigation of
physical objects on one hand and of mathematical objects on the
other. ... I am in sympathy with this point of view to a very
limited extent.
[VS - I recently argued in FOM what is the difference between physical
objects, even sush abstract as electrons, and mathematical ones.]
......
By contrast, I feel quite unable to grasp the idea of an actual
infinite totality.
.....
It follows that I must regard a theory which refers to an
infinite totality as {\em meaningless} ...
.....
An opponent to my position might put....because my brain suffers
from a particular limitation.
.....
As far as I know, ***only a small minority of mathematicians,
even those with platonist views*** [VS - my emphasis], accept
the idea that there may be mathematical facts which are {em true}
but unknowable.
[VS - does not it mean that the overwhelming majority of
mathematicians are just formalists in their "working days"?]
.....
At any rate, it seems to me that the present situation in Set Theory
favors the Formalist.
.......
...the formalist holds that direct interpretability is not a
necessary condition for the acceptability of a mathematical theory.
.....
... it is a fact that the organized world of Pure mathematics
is regulated to a very large extent by our ***vague intuitive ideas***
[VS - my emphasis] on mathematical beauty and pure mathematical
importance.
.....
To {\em understand} a theory means to be able to follow its logical
development and not, necessarily, to interpret, or give a denotation
for its individual terms. [VS - I understand - any precise or even
only "unique" platonistic denotation is unnecessary; only a vague
idea suffices.]
In Hilbert's view the formal or uninterpreted part of a theory
belonged entirely to Mathematics.
........
Since the formalist cannot rely on semantic interpretation of
his theories [VS - I think, because it is based only on our vague
intuition], their complete formalization is essential to him.
This is part of the original formalist approach due to Hilbert."
etc.
Vladimir Sazonov:
Abraham Robinson is very polite. Nevertheless, he used the quite
strong word "meaningless". Also, I do not think that his "brain
suffers from a particular limitation".
By the way, was not this doubt of Robinson in meaningfulness
of the unique N his starting point to invention of Nonstandard
Analysis? Does anybody know how have he came to this idea?
It does not matter that he used quite standard mathematical
tools (ultraproducts or whatever else).
Friedman [VS - again]:
> Not Cantor, and not
> the overwhelming number of mathematicians. But certainly there are some.
>
> E.g., Sazonov.
I ("together" with A.Robinson) already commented above on
"overwhelming".
I feel myself very comfortable in this company of David Hilbert
and Abraham Robinson. What is very strange to me that the greatest
Kurt Godel was a Platonist. When first I studied his mathematical
results I had no idea about his general views and was highly
surprised later. It seems to me that his (working days) results
contradict to his (weekend) philosophy.
> But note that Sazonov's postings have some connection with
> the foundations of mathematics,
Thanks for not considering my efforts as irrelevant.
> even though - in my opinion - he has not
> convinced anyone to be any more skeptical about the set of all integers than
> they were before they read his postings.
Who knows? There are some people who explicitly in FOM or
privately wrote or told me on analogous views. Not so many
people are writing to FOM. Those who have some superbeliefs
cannot be convinced. What I consider important, that MEANINGLESS
or MYSTICAL views in science or in foundation of science should
meet some counteraction. They are not so harmless as it could seem.
Even if to put aside the question of the vagueness of N
(whether it is understood potentially or actually), WHAT DOES
IT MEAN that it exists UNIQUELY.
"What does it mean" is the most important scientific question
as we know this from Albert Einstein and other greatest scientists.
We are, indeed, using this question everyday, although in some
specific cases, related with our beliefs, forget.
What is still unclear to me, is your own, Professor Friedman,
opinion. You seems to deviate from some of the above questions.
The reason looks good - let people discuss, the concrete
mathematical activity is more important. But what is YOUR
personal opinion on mysticism in the form of Platonism?
By the way, I recall some your quite concrete arithmetical
sentences in terms of sin and exponential which you predicted
will be never proved or disproved in any natural extension
of PA (that is, even by using some strong extensions of ZFC).
[I do not remember precisely, but this seems was your point.]
This looks rather plausible. Large cardinals can hardly have
any relation to resolving such kind of statements.
Also, some people believe that P=?NP will be eventually resolved
in PA. But let us assume temporary that it is not the case.
Will then large cardinals settle this problem? The character of this
problem is such that this problem is hardly related in any way
to stronger versions of ZFC. On the other hand, this (or a very close)
problem can be formulated in a rather weak fragments of PA, or
just as a finite (non-asymptotic) form, like the problem "who wins
in 8x8 Chess game?".
We could take, for example, a finite, may be second-order,
(recursive) arithmetic of a segment {0,1,...,1000}
or something like this and ask whether second order quantifiers
can be eliminated. Who knows, may be this is a direction where
this problem will be properly understood and settled down?
(Here we may need a formal theory feasible numbers because
we should consider only feasible proofs to demonstrate the
possibility or inpossibility of feasible second order
quantifier elimination or some independence or non-constructibility
result. Say, to demonstrate that whites in Chess have no
constructive winning strategy we cannot avoid some formalized
feasibility concept.)
P=?NP might be quite independent of PA (like CH on ZFC), but
resolvable (in some way) in quite different approaches assuming
some new views on natural numbers. This might be considered as
a fantasy. But even your own, somewhat artificial but convincing,
candidates (with sin, etc.) on undecidable arithmetical statements
show that it is quite imaginable that such effects like independence
of CH are possible in PA without any hope to be resolved forever
(except as trivially taking them as axioms).
Should we still believe in the uniqueness of N?
Or, is it worth to continue to believe? Why to believe at all
in anything in mathematics or in science? This seems to me
a very inappropriate term and way of behavior in science.
Likewise the term "realism" (equal? to Platonism) in the philosophy
of mathematics is highly misleading. I consider myself a realist
in the normal meaning of this word, and my realism has nothing
general with "realism" = "Platonism" how philosophers are using it.
The very occupation of this quite respectable term in quite
opposite way demonstrates that this philosophy is wrong.
Terminology should help to understanding, not to disappointment.
Does not it mean that philosophers just have no clear idea
on the proper meaning of this word, or rather try to use it
to make it easier to convince us in their wrong ideas.
I refer to your examples (arithmetical candidates on a strongest
independence). What is YOUR opinion? Why have you presented them
at all?
> Harvey Friedman
Best regards,
Vladimir Sazonov
More information about the FOM
mailing list