FOM: certainty; discussion with Reuben Hersh

[Vladimir Sazonov]

Reuben Hersh wrote:
>
>     On Sun, 3 Jan 1999, Vladimir Sazonov wrote:
>
> > You wrote:
> >
> > >         It seems to me, so far as I know, that mathematics is the
> > > only non-physical  activity that maintains unanimity.  Some of
> > > the physical sciences  maintain virtual unanimity, that's
> > > why the definition of mathematics requires both the non-physical
> > > and the unanimous features.
> >
> > First, what is the reason or machinery by which that unanimity
> > in mathematics is achieved?
>         That's a big question.  Part of the answer is the determination
>         of mathematicians to maintain unanimity.  Almost always, a
>         persistent disagreement is resolved by introducing a distinction.  For
>         example, the intuitionist "revolution" has become a particular specialty
>         in "foundations."

I think that these are *formalisms* which maintain unanimity in 
mathematics. What can be said against a formal proof? On the other 
hand, there may be discussions whether this or other formalism is 
more appropriate for some applications. 

> > Second, we know very well that water-nymph has a fish tail
> > instead of legs.
>
>         minor linguistic point:  "mermaid" would be more colloquial
>         English than "water-nymph".

Thanks! I realize that my English is rather primitive.

> Is this non-physical and unanimously accepted
> > fact a mathematical truth?
>
>         No, I guess not.  My "non-physical, unanimity" was meant
>         only to distinguish math from other scientific and academic
>         disciplines.  There is an academic discipline (mythology)
>         that works on mermaids.  Not much unanimity there.

Unfortunately, however I suggested this example, I am not a 
specialist on mermaids and on mythology.  Anyway, unanimity 
seems to me something too amorphous to characterize and explain 
the nature of mathematics. We need to find such a "root" of 
mathematics which would, in particular, lead to this unanimity 
with a guarantee. 

> >
> > What is the difference between water-nymph and, say, the bottle
> > of Klein (both being imaginary objects)?
> >
>         I guess Klein's bottle being defined with precision, permitting
>         more definite reasoning about it.

"Precision", "definite" means formalizability, using definite 
(well recognized by mathematicians) rules of reasoning. I think 
there are no such rules for thinking on mermaids.

> > >From another your reply to me:
> > >
> > > I take it you declare disagreement with me on two counts:  one, my
> > > statement that mathematics is not mental; two, my underestimating
> > > in your opinion the role of deductive proof.
> > >
> > > You cite Lobatchewsky as proof that mathematics is located in
> > > the individual mind of individual mathematicians.  For no one
> > > he knew understood his new geometry, he worked it out all alone.
> >
> > Let me be a bit more precise: *New* math. is usually *created* by
> > the individual mind of individual mathematicians.
>                 True.>
> > >
>   The social world of mathematics includes, not only presently
> > > active mathematicians, but also the accumulated archive of books and
> > > articles, and even the word of mouth tradition inherited from
> > > mathematicians of the past.  Did Lobatchewsky, or could anybody,
> > > invent non-Euclidean from scratch, all out of his head?  Certainly
> > > not.  The very name non-Euclidean indicates that it is a continuation,
> > > an outgrowth, of Euclidean geometry.  Not, of course, in the sense of
> > > continuing with Euclid's postulates and axioms.  But rather, by
> > > changing axiom #5, it then pursues the same questions and analogous
> > > theorems as in Euclid.  Lobatchevsky had a university math educuation,
> > > that is to say, he was socialized into the mathematical culture of his
> > > time.  His work on geometry was a natural continuation of the geometry
> > > he had been taught.  If you doubt this, remember that Gauss and Bolyai
> > > independently arrived at te same thing.
> >
> > I absolutely agree and do not remember that I said something
> > contrary. Lobatchewsky, Gauss and Bolyai started to work with
> > the same formal system of Euclid investigating the role of its
> > V-th postulate.
> >
> > >         The mathematical culture throws up questions and problems, and
> > > often several mathematicians
> > > seize on the same problem, either knowing or not knowing of each
> > > other.  When I say math is not mental. I mean it is not part of
> > > the private, un-transmittable inner world of individuals.
> >
> > I understand that mathematics deals with and creates formalisms
> > having any meaning and intuition.
>
>         Could you clarify this sentence?

* Any useful algorithms like that of multiplication of decimal 
numbers.

* Axioms of Euclidean geometry even with naively and unconsciously 
understood criteria of a rigorous proof. (I do not think that 
missing Pash's axiom was so tragic.) 

* Formal rules how to differentiate and integrate.

* Kantorian (even contradictory) set theory.

* ZFC, PA, PRA, FOL, Intuitionistic FOL, etc.

Mathematicians create these clever "games" as instruments to 
make their intuition (on physical reality or on their own 
imaginary worlds) more organized, rational and stronger. Say, we 
can feel something on the mechanical motion. But without special 
"levers" (in the form of appropriate formalisms submitted by 
mathematics) our intuition and thought is too weak! All the 
history of science demonstrates how surprisingly powerful becomes 
our thought with using these instruments. Mathematicians are 
like engineers making useful devices for peoples. The 
instruments cannot be true or false.  They may be *applicable*. 
Mathematicians investigate possibilities of these instruments 
(by deducing some thoughtful theorems), how these instruments 
may be related one with another (interpretation of one formal 
theory in another.) It unifies these instruments of thought, 
trying to find some unique foundation (instruments which are as 
universal as possible). 

Sometimes mathematicians just play with these instruments, say, 
to find a proof of FLT. Such a training may have in principle 
unexpected useful side effects. (However, with FLT it seems the 
situation was somewhat contrary: its proof was a side effect of 
other more thoughtful mathematical activity.)

> It is due to formalisms why we
> > can transmit socially our mathematical knowledge in so reliable,
> > stable way. That is why we can have so strong social consensus
> > on mathematical notions and theorems. The intuition is more
> > private and ephemeral, but it is also (partially!) transmittable
> > *together* with and *due to* formalisms. Mathematical intuition
> > cannot even exist in pure form completely separate from a
> > formalism.
>
>         I think this is a strong overstatement.  Isn't "a straight
>         line is the shortest distance betwen two points" mathematical?  Yet
>         it seems that a cat chasing a rabbit and a rabbit escaping from
>         a cat have some kind of knowledge of this mathematical fact, without
>         benefit of formalism.  "2 is more than 1" is a mathematical fact
>         understood by an infant offered candy, but this seems separate
>         from formalism.

What can we do with this phrase "a straight line is the shortest 
distance betwen two points" in isolation from our possibility to 
make logical conclusions? Almost nothing! 

> > > It is
> > > social, in that it is part of a shared, transmitted body of
> > > knowledge and skills.  Any piece of nonmaterial culture--law,
> > > religion, political ideology, folklore, literature, music, etc.
> > > is realized by individual consciousness of those participating in it.
> > > But its existence and features are not dependent on any individual's
> > > consciousness.  If you and I die tomorrow, War and Peace and Crime and
> > > Punishment will survive.  And so will Euclidean and non-Euclidean
> > > geometry.  They survive because they are not limited to any individual
> > > consciousness, they are art of a social consciousness.
> >
> > And these geometries are
>
> just formal systems (rules of a game)
>
>         yes they are formal systems, among other things, but no, they are
>         not *just* formal systems.  The Beltrami and Poincare and Klein models
>         are visual, they permit us to see facts of hyperbolic geometry directly
>         without formalisms.

Note that you broken my phrase. They are indeed not *just* formal 
systems. Also I have nothing against visuality. It helps very much. 
But mathematics assumes something more because we cannot "travel 
a long way" only on the base of visuality and "ungoverned" intuition. 

>  with some explanations of their meaning. Of course *now*, after
> > their creation, they are not limited to any individual
> > consciousness and they will (can) survive.
> >
> > > What about the role of deductive proof, which you think I
> > > underestimate?
> > >
> > > I can explain this by rejecting your stataement that until
> > > a conjecture has a deductive proof, it isn't mathematics.
> >
> > I did not say this about a conjecture! Any *statement* is a part
> > of math.  if it is formalizable. A *theorem* is a part of math.
> > if, additionally, its proof is formalizable in an appropriate
> > formal system.  A *theory* is mathematical if it is presentable
> > in the form of a formal system (with any, possibly very informal
> > and even very non-traditional meaning).
>
>         I think this is the point where we really disagree.
>         Isn't it true that any text, mathematical or not (e.g., Alice in
>         Wonderland) can be
>         considered formally--i.e., a sequence of uninterpreted symbols?

First, mathematics consists not of texts, but of formal systems 
of *rules*. Second, these rules should have interpretation, 
meaning, etc.

>         On the other hand, the notion of infinitesimal was not formalized
>         until the 1950's, yet it was a part of the mathematical repertory
>         of Archimedes and Cavalieri.

Any kind of intuition (which is to be subject to formalization) 
is a part of the mathematical repertory. What Archimedes and 
Cavalieri did was by this or other way formalized 
(formalizable). Any good mathematician works in such a way that 
his ideas are eventually formalizable. This is the difference 
with non-mathematical way of thinking. 

>         I think your position would be rejected by very many
>         analysts, algebraists, geometers, number theorists...
>         Pure mathematicians certainly stress the importance of
>         rigorous proof, but the rigorous proof they write is not formalized,
>         as you yourself admit below in your role of referee.

Not necessarily formalized, but formalizABLE! 
What does it mean "rigorous" otherwise? 

>         It is not true, with perhaps rare exceptions, that formalizing
>         a mathematical text decides whether it is correct.  On the contrary,
>         if a mathematical text is ever formalized, it would usually be
>         after one
>         is convinced it is correct; for instance, to implement it in a
>         computer program.

Good point! Computers also strengthen our thinking and 
intellectual abilities. They are also some specific physical 
"active" formal systems.

>         It's a pity that logicians sometimes have treated mathematics
>         as just applied logic.  That claim cannot stand up against
>         even a brief acquaintance with mathematics in real life.

It seems that Euclid did not know explicitly FOL. Nevertheless, 
I consider his geometry (even having some gaps) as a formal 
system. Also formalisms need not be necessarily based on 
traditional formal logics like FOL. Who knows what we will see 
in mathematics in the next century? 

>         Please understand,  I am a long-time sincere admirer
>         of the astonishing achievements of modern logic.  Logic,
>         like analysis, algebra, geometry, or number theory,
>         is an admired, important mathematical specialty.  It need not
>         claim to be the generator or chaperone of all mathematics.
>
> > > I would agree if you said, it  isn't part of proved mathematics.
> > >
> > > Is the Riemann conjecture part of mathematics?  It hasn't been
> > > proved, so you evidently claim it isn't. What is it part of, then?
> > > Music?  Nuclear physics?  Bourgeois ideology?  Don't be silly.
> >
> > Why do you not assume that I have some minimal sufficient level
> > of intellect? Why do you interpret my views so primitively?
>
>         I apologize.  No insult is intended.  I am sorry to say, sometimes
>         I get carried away with my rhetoric.

OK!

> > > I suppose you believe that Fermat's last theorem has been proved.
> > > Before Wiles cleaned it up, it wasn't part of mathematics.  Surely
> > > all the partial results that had accumulated over the centuries, in
> > > unsuccessful attacks on FLT, were mathematics, for they did prove
> > > something or other.  But the conjecture or problem that they were
> > > struggling to prove, that wasn't mathematics?????
> > >
> > > To me, mathematics includes all the mathematical work that
> > > mathematicians do.  Making conjectures, gathering evidence
> > > for or against conjectures, coming up with defective proofs,
> > > finally making a complete proof.
> >
> > Completely agree!
> >
> > > You say none of that was
> > > mathematics, until the final stage is reached.
> >
> > Let me be again a bit more precise. (I hoped that what I mean is
> > easily recoverable.) Mathematical activity has various stages,
> > (some of which are so intimate that may be even not
> > transmittable to other mathematicians; the same mathematician
> > may even forget how things have been worked out by himself).
> > Anyway, if it is not assumed that what is done by a
> > mathematician will be culminated *at last* (probably with the
> > help of future generations of mathematicians) by *a* formal
> > statement or *a* formal proof in *a* reasonable formal system
> > (which is probably even unknown in advance) then this hardly
> > could be called a mathematical activity. This may be a
> > philosophy, theology or anything else, but not a mathematics.
> > Or, it could be called a mathematics only to that degree to
> > which it is or may be formalizable.
> >
>
> > > You say all mathematics is striving for deductive proof.  May
> > > I point out that this notion is only 100 years old, or so.
> >
> > What is a *formal* proof? It is a proof where only (or mainly) the
> > form of reasoning determines its correctness. (Of course, we are
> > usually working in mathematics only with such formal systems in
> > which formal deducibility guarantees *some* agreement with the
> > intended meaning of the system.)
> >
> > Since some historical period of time what is called mathematics
> > was based on formal proofs (formal systems) in this
> > (sufficiently broad) sense. Of course, there were periods of a
> > crisis in achieving mathematical rigour when *new* mathematics
> > arose. There were "gaps" in otherwise formal proofs. These gaps
> > corresponded to some intuitively plausible (but too numerous to
> > be explicitly fixed) axioms or hypotheses.  But mathematicians
> > always had a feeling of discomfort with non-rigorous proofs or
> > with proofs in poorly fixed formal systems and made attempts to
> > overcome such a crisis.  This was actually a process of
> > construction of some new formal system(s) with a small number
> > of axioms having some reasonable intuitive meaning (ZFC or the
> > like). At present mathematicians (especially mathematical
> > logicians) have somewhat better understanding of what is "formal"
> > than Euclid had, but the difference, however important, is not so
> > crucial for this discussion (for a while).
> >
> > It may be objected that there were intuitive concepts of
> > mathematical truth and mathematical (platonistic or social)
> > world which "finally" were axiomatized by ZFC (let incompletely).
> > I would say differently.  There were numerous interrelated
> > "small" formal systems of rules and axioms (such as the rule for
> > derivative of composition of functions, etc. which may be used
> > independently on our "final" understanding, if any, of what is
> > derivative) which had application/interpretation in practice
> > (say, in engineering or astronomy).  It was very important to
> > find (if possible) a unique reliable formalism with a plausible
> > unique sufficiently simple intuition behind it which could unify
> > these numerous fragments.  And mathematicians succeeded with
> > building set theory as foundation of (contemporary) mathematics!
> > But in principle it is possible that it would be found only a
> > (small?) number of some incomparable formal systems with badly
> > unifiable intuitions. Who knows, may be this is the future of
> > mathematics?
> >
> > > Newton didn't do mathematics.  Fourier didn't do mathematics.
> > > What about Oliver Heaviside?  When criticized for not having
> > > rigorously justified his heuristic method of solving differential
> > > equations, he answered, something like, "Does my ignorance of
> > > the details of the process of digestion prevent me from enjoying my
> > > dinner?"
> >
> > Very nice! I like it! But can anybody assert that eventual
> > rigorous justification (= formalization) is unnecessary in
> > mathematics?
>
>         THe great majority of mathematicians would say yes to
>         "eventual rigorous justification," no to formalization.
>         Your = sign begs the question.

First, nobody is forced to write completely formal proofs. 
Second, what does it mean "eventual rigorous justification" if 
it is not formalization (let in principle)? Also we should not 
ignore that for all contemporary mathematics it is essentially 
known that it is formalizable in principle (if to forget the 
issue of feasible formalization) in ZFC on the base on FOL. 
Thus, I think that careful checking correctness, say, of the 
proof of FLT by specialists, even if they will ignore FOL, will 
give us a guarantee that there (potentially) exists a completely 
formal proof of FLT in ZFC. 

> Say, in physics we have a reality and experiment
> > as criteria.  It seems that instead of standards of mathematical
> > rigour based on clearly presented formal systems and corresponding
> > objective criteria you suggest social consensus. (How can we know
> > that a consensus in a mathematical question have been achieved?)
> > I would feel myself extremely uncomfortable if having no criteria
> > independent of social mathematical groups when working, say as a
> > referee of a mathematical paper.
>         You are chosen as referee because you are recognized
>         as an accepted part of a relevant mathematical specialty.
>         Your own judgment is therefore automatically a
>         judgment within the criteria or tastes of that specialty.
>         Which specialty happens incidentally to be a group of people.
>         I.e., a social group.)
>
> We all know that mathematical
> > statements and proofs are usually presented not very formally.
>
>         Usually?  Almost always!
>
> > Therefore additionally we need very much solid ground of a
> > formalism with respect to which we will check the intuition
> > and therefore achieve the desirable consensus.
>         Check with non-logician referees for journals in
>         analysis, algebra, etc, etc.  How many use formalization
>         in making their decisions?  I guess less than 1%.>

Again, not formalization, but formalizability. Also let me recall 
that what is formal may be understood with different degrees of 
detalization. It should be only sufficient to the subject we are 
considering. (However, there may be subjects or situations very 
sensitive in this respect.) Finally we have sufficient training 
and experience to recognize formalizability even by feeling. This 
is quite enough for "working mathematicians".

> > > Applied mathematics today overwhelmingly depends on computer
> > > solutions of differential equations.  Solutions that are
> > > rigorously justified only in simple special cases,
> > > which are not at all the cases that the applied mathematician is
> > > interested in.
> > >
> > > So you say applied mathematics isn't mathematics?
> >
> > As I wrote, mathematics deals with formalisms having a meaning,
> > i.e. with applicable formalisms. Thus mathematics is always
> > applied science (somewhere, say, in mathematics itself). This
> > does not prevent any related experiments with computers, etc.
> >
> > > >From the applied viewpoint, a rigorous justification of
> > > an algorithm is a plus, it adds some confidence to the
> > > calculations with that algorithm.  If the algorithm is
> > > one that has been used dozens of times or hundreds of
> > > times, all over the world, the extra confidence from the
> > > rigorous proof is not decisive.
> >
> > This is engineering rather than a proper mathematics. However
> > I see no contradictions between these two kind of activities.
> > Mathematical (eventually formal) activity should be in good
> > agreement with corresponding (say, computational) practice.
> >
> > > You could say, when you say mathematics, you mean pure
> > > mathematics.  I revert to the earlier discussion of
> > > Fermat's last therem.
> > >
> > > After all, this disagreement is largely one of emphasis.
> >
> > I hope! But the emphasis may be sometimes essential. I believe
> > that the formal nature of mathematics is its main feature and
> > it explains very well corresponding effects of social consensus.
> > (Who can disprove the rules of chess game?)
> >
> > > I'm in favor of rigorous proof, in fact I never published
> > > a research paper that wasn't entirely rigorous.
> >
> > I had no doubts on this. One thing is what we are doing everyday
> > and another thing is our reflection on this.
> >
> > > But in judging the role of rigorous proofs, I take into account the
> > > overall picture of what mathematicians do.
> >
> > What essential of this overall picture did I lose (except quite
> > unnecessary illusions of Platinistic world and absolute
> > mathematical truth in which you seems also do not believe)?


Vladimir Sazonov