[FOM] Absolute truth vs. relative meaning and formal nature of mathematics
V.Sazonov at csc.liv.ac.uk
Mon Feb 6 14:46:28 EST 2006
Quoting aa at post.tau.ac.il (Arnon Avron):
> If there is no certain truth
> even in mathematics then certainly there is no certain truth anywhere else.
Yes, we are living in not so solid world, and mathematics is not
the exception - just from a general point of view, and I strongly
believe that any philosophy of mathematics that ignores this
fact is non-adequate, if not wrong.
> So for me the most crucial problem of FOM is: is there absolute
> truth in mathematics, and if there is - what theorems of mathematics
> can truthfully and safely be taken as meaningful and *certainly true*.
> Predicativism (at least for me) is all about this question.
I do not understand what is such a thing as absolute truth
neither in the real world, not in mathematics. I think,
the main usage of the word absolute is in a negative context
that such and such thing is NOT absolute. Wherever anything
is called absolute, it is actually some relative assertion.
For example, I could definitely assert myself that mathematics
is absolutely free in its constructions, thereby completely
realising that at least one (but the most crucial! and even
defining) restriction of this freedom exists - mathematical
Nevertheless, there is (seemingly) nothing else so certain as
mathematics. But what is the source and the real meaning of
this apparent certainty? To my understanding it is its well
known (but, by some highly unclear to me reason, often
ignored) rigorous character which can be characterised in
more contemporary terms as (potential) formalisability.
In no other human activity its formal (in a rather wide but
sufficiently definite sense of this word) character does not
play so high role (except in computer programming, with a
related, but somewhat different, engineering accent). I should
add that formalisms on which mathematics is based not only do
not exclude, but are strongly based on intuition, imagination,
creating a thought picture, analogies, etc., etc. Correctness
of mathematical formal or formalisable proofs and constructions
can be checked in (potentially) mechanical way or even by
computer. When we repeatedly run a complicated computer program
(say, transforming latex to postscript) we always get, by some
miracle(!?), exactly the same result. This is not because it is
some a priory fact - we check this by experiment. Essentially
in the same way mathematical proofs, even not completely
formalised, are either correct or not.
Is not this the real source for our feeling that mathematics
is so certain? Mathematics is just deliberate relying our thought
on formal mechanisms.
And where in the above picture the "absolute truth" of mathematical
theorems appears, and where at all do we need to objectivise the
appearing thereby illusion of "absolute truth"? And what does this
truth ever mean if it is not just asserting derivability of a theorem?
Our intuition? But this is a vague, unreliable and quite fluid,
amoeba-like thing? Only mathematical proofs are (highly) certain
and make our intuition sufficiently solid by putting it in a this
As to another related discussion on the uniqueness, or certainty
of the concept of natural numbers initiated recently by Arnon Avron,
on what means `and so on'. Yes, even children know this idea
`and so on'. But do they and we all, mathematicians ("damaged"
by the education), understand this in exactly the same way?
Do or should all of us have the same intuition on this subject?
For example, I ASSERT, not as a mere speculation, that there is a
simple, but rather unusual FORMAL system of axioms and proof
rules in which a (semi)set of natural numbers 0,1,2,... < 1000
is FORMALLY definable which is PROVABLY closed under successor
and is also upper bounded by the number 1000. Quite intuitive
informal examples of such semisets from our real world are well
known (e.g. presented by P. Vopenka).
Where is the absolute truth here?
Quoting Joseph Vidal-Rosset <joseph.vidal-rosset at univ-nancy2.fr>:
> A Tolerance Principle à la Carnap could maybe solve the problem from a
> strict scientific point of view: "truth" in mathematic or mathematical
> logic depends on the axiomatic system (there is no Moral in Logic).
Can not this be interpreted as a formalist point of view on
mathematics (choose and develop any formal axioms and proof
rules which formalise most adequately any kind of intuition)?
I also should note that I disagree with the traditional
identifying formalist view with a meaningless play with symbols.
This caricature version of formalist view is of course highly
non-coherent with mathematical practice. On the other hand,
an INTERPLAY between formal game with symbols and our intuition
is exactly what is happening in mathematics, and I believe that
what the philosophy of mathematics should do is explaining
how and why this interplay works, instead of never ending
speculations on "mathematical truth". Just looking at what
mathematicians do (introducing axioms, definitions, formulating
theorems and presenting their proofs) shows that they are actually
interested not in truth, even if they sometimes mention this word
(as a figure of speech only). They are interested in techniques,
they create tools of thought. They are more like to engineers.
It seems to me wrong idea to restrict mathematics to dealing
with objects of such and such kind (numbers, sets, etc.) and even
to the first order logical language formally appealing to the idea
of truth. Much more general idea of *any intuition deliberately
regulated by formal rules* seems to me philosophically more
appealing in describing the nature of mathematics, especially
taking into account that we cannot predict the future of
mathematics and with which kind of objects will it deal and by
using which kind of logic, etc. It appears that the most general
and distinctive feature of mathematics is its formal nature.
All other details should follow from (or should be based on,
or should not ignore) this evident fact.
In particular, it follows that mathematics is (almost) absolutely
free. The only "restriction" on its considerations is:
to be meaningful and rigorous (if it is mathematics at all).
Again, where is the "truth" here? For example, the formal theorem
Yf=f(Yf) in the type-free lambda calculus (ANY function f must have
a fixed point) - is it true or meaningful? Recall also that Dana
Scott suggested so called domain theory where the isomorphism (or
homeomorphism) D = [D -> D] is possible and where the above fixed
point theorem can be interpreted. Is this about absolute truth or
about an ADDED MEANING to the lambda calculus (highly interesting
and important meaning with applications to denotational semantics
of programming languages)? The proper question about D is even not
whether it exists, but what is the idea of its construction, what
is the meaning of doamain theory. Not truth, but meaning. Imagine
also that we know by some miracle (from the God?) that, for example,
P=NP "holds". Is it interesting? What will we do with this "knowledge"?
Will this mere "fact" without any additional idea essentially change
the direction of activity of mathematicians related with this problem? These
"truths" are simply irrelevant to mathematics. We know that
Fermat theorem was proved, and so what? I mean - for those (many
of us?) who did not try to undrestand the idea of proof?
Finally, I completely agree with Harvey Friedman (his last posting)
that mathematics, in general, cannot be reduced to only one
formal system, be that ZFC or something stronger. Analogously I
cannot imagine that there could be a unique most general kind of
intuition on which mathematics could be based (a replacement of
the idea of absolute truth). Mathematics is rather a lot of various
formal systems and related intuitions some of which can have relatively
universal character, but never absolutely universal.
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