Harvey Friedman wrote:
> I just read Holmes 2:18PM 3/25/99: properties, in which he responds to Sazonov.
>
> Their interchange, which has been going on for some time, centers around
> issues concerning objectivity and meaning, often going under the names of
> realism, platonism, formalism, finitism, etcetera.
>
> I have always found such discussions unsatisfying, where people rarely seem
> to change their mind. I, personally, have always felt more comfortable
> trying to produce results that clearly bear on these issues. I try to sense
> what is relevant and doable - and try to do it.
The issues you mentioned (realism, platonism, formalism,
finitism, etcetera) play too crucial role in f.o.m., however,
by my opinion, there is usually not so much of clarity in their
description and discussion here in our forum and elsewhere. I
completely agree that more rational manner of these discussions
should be found.
By the way, is the term realism (= platonism?) the standard one?
This term seems to me rather unsatisfactory and ambiguous
because corresponding position has nothing to do with reality.
It is rather quasi-realism.
> At a later stage in my career, I may well move towards dealing more
> directly with such matters. But I am still inclined to think that,
> personally, I have a better chance of doing something of permanent
> intellectual value under my present mode of operation. Nevertheless, I like
> to read what people have to say, as it frequently suggests new kinds of
> results.
>
> With that preamble, let me suggest a somewhat different thread for the FOM.
> WHAT KIND OF MATHEMATICAL DEVELOPMENTS WOULD BEAR ON THE ISSUES THAT ARE
> BEING DISCUSSED?
>
> Just to get the ball rolling, let me throw out an obvious one. THAT NEW
> KINDS OF INCONSISTENCIES ARE DISCOVERED.
As I know, nothing analogous to Russel's paradox was found in
contemporary mathematics. However, there is one kind of concrete
and related with our discusion inconsistency which I consider
interesting and even crucial. I wrote to FOM on an attempt of
formalizing the intuitive concept of feasible natural numbers
(cf. my paper "On feasible numbers" via my homepage). In a
reasonable and quite precise sense it is consistent
formalization, but for some concrete arithmetical sentence A
(involving unbounded quantification) it is provable, against
expectation, both A and its negation ~A.
This is rather a semantical than formal contradiction because we
could get the formal contradiction (just `false' which would
imply _any_ legal statement) from A and ~A (with ~A considered
formally as A -> false) only by using modus ponens or cut rule.
But the latter is restricted in this formalization (to avoid
formal contradiction). The first reaction may be that
formalization having these features is non-satisfactory. But I
am afraid that such unusual features are unavoidable for any
reasonable formalization of the feasibility concept.
Thus, for formalizing such new concepts we probably should
reconsider our ordinary views on mathematical formalization of
anything, on truth and on contradiction and, in general, on
f.o.m., platonism, formalism, etc.
Also this is an illustration to my discussion with Holmes.
I wrote:
> What does it mean "well-defined"? Say, is any property given by
> an arithmetic formula involving unbounded quantifiers (if you do
> not like my first example) well-defined? Are you sure that such
> a property is either true or false for each natural number?
Indeed, the above mentioned arithmetical sentence A in the context
of a theory of fasible numbers has a problematic or at least very
unusual truth value (A is _both_ true and false). How can
Platonism with his dogmatic inclinations on truth help in
understanding this situation? It will probably reject the very
idea of feasibility as non-coherent, non-mathematical or the
like instead of making reasonable attempts to formalize it.
Randall Holmes wrote:
I think that it is very telling that
> a mathematician when he is actually working will "put on his Platonist
> hat"
I agree only with "local" and with the weakest form of
Platonism, relativised to each concrete formal system and
subject matter. The traditional Platonism intends to be
global (and therefore meaningless): some unique
mathematical universe for all possible mathematical
considerations and absolute truth in this universe.
-- this suggests that other views are hard to reconcile with the
> actual nature of the subject (which is not a knockdown argument that
> a realist view is correct!).
Local Platonism needs no global one. For each concrete
mathematical theory (whatever it be) we have corresponding
intuition ("local" truth concept, "a small Platonist hat",
if you want). What is really hard and actually impossible
to reconcile, it is achieving some unique, global, limit,
absolut Platonistic truth. We may have only _illusion_ of
it, no more.
John Mayberry wrote:
> The enemies of truth are to be found in unexpected places. For
> example, in the introduction to his volume on set theory Bourbaki says
> "Mathematicians have always been convinced that what they prove is
> "true". It is clear that such a conviction can only be of a sentimental
> or metaphysical order, and cannot be justified, or even ascribed a
> meaning which is not tautological, within the domain of mathematics."
> This is not just nonsense, it is pernicious nonsense. It pollutes the
> stream of intellectual life. What on earth does Boubaki think the point
> of presenting a proof is, if it does not represent at least an
> *attempt* to establish the truth of its conclusion? Actually Bourbaki
> tells us what the point is: "The *axiomatic method* is nothing but the
> art of drawing up texts [a post-modernist buzz word] whose
> formalisation is straight-forward in principle". Of course he leaves us
> completely in the dark as to what the value of having such a formalised
> text might be (or of just knowing "in principle" that we could produce
> one). This is formalism in its most egregious manifestation. It is an
> anti-intellectual, indeed, an irrational, doctrine.
I think that mathematics deals with arbitrary formalisms having
any meaning (formal or informal, truth like interpretation,
any kind of semantics, etc.). I am not an advocate of Bourbaki,
but I cannot imagine that anybody in his right mind will
seriously assert that mathematical formalisms are absolutely
meaningless. That is why I do not see anything especially
dangerous in the above, essentially anti-platonistic,
anti-globalistic citations. I cannot imagine that, say, Choice
Axiom (and what is proved with its help) is TRUE or FALSE in
some absolute sense of these words. I like this axiom and rather
consider that it is REASONABLE or USEFUL. The same for
Induction Axiom, etc. What we also could say about axioms and
theories considered in mathematics it is that they are
APPLICABLE, what is somewhat closer to the vague idea of truth
and is much better. In general, any (reasonable) mathematical
theory has _its own_ meaning relativised to the corresponding
subject matter. Of course we cannot describe in advance any
possible meaning of any possible mathematical theory, as
Platonism seems intends to do (of course without any success).
It is the opposite "absolutistic" opinion that I consider
"anti-intellectual" and "irrational".
> It is with this kind of thing in mind that I insist on making
> the distinction between set theory itself and its 1st order
> formalisation in ZF.
Can you make this distinction clearly? Do you believe that CH
has a definite truth value (not depending on your own or on
some group of peoples subjective opinion)? I hope you will not
tell me that all "true believer" mathematicians will at last
_agree_ on some specific truth value of CH. (I hope heretics
will not be subject to inquisition in this case.) I believe
that science should be, first of all, OBJECTIVE and APPLICABLE
(in the Nature or may be in the same or other science). Here
is TRUTH, not in our religious or quasi-religious BELIEFS.
This position, as I understand it (I am not expert on Bourbaki
and say mainly on the position as I describe it), does not
reject truth, it rejects absolutism and subjectivism in f.o.m.
Stephen G Simpson wrote:
Bourbaki is a good example of the pernicous consequences of
> polylogism in mathematics. According to Bourbaki, the logic used in
> mathematics can never be anything except a meaningless formal game.
> For Bourbaki, this logic happens to be a variant of ZFC, but it could
> be anything.
Do you assert that mathematics should be always based on the
unique (first-order classical) logic? What if for formalizing
some new concept you will need to use a different logic (with
some specific axioms and proof rules) which is not reducible in
any direct or evident way to FOL (unlike e.g. Intuitionistic
Logic which is reducible to FOL via Kripke models)?
Vladimir Sazonov
--
Computer Logic Lab., | Tel. +7-08535-98945 (Inst.),
Program Systems Institute, | Tel. +7-08535-98953 (Inst.),
Russian Acad. of Sci. | Tel. +7-08535-98365 (home),
Pereslavl-Zalessky, | e-mail: sazonov at logic.botik.ru
152140, RUSSIA | http://www.botik.ru/~logic/SAZONOV/