FOM: What is mathematics?

[Vladimir Sazonov]

Gordon Fisher wrote:
> 
> Vladimir Sazonov wrote:
> 
> > Insall wrote:
> > >
> > > Miguel Lerma wrote:  ``Proofs may be an important part of mathematical
> > > activity but not essential in defining mathematics.''
> >
> > What else, if not proofs, rigor and formal reasoning, distinguishes
> > mathematics from other sciences? Intuition, abstractions? They are
> > widely used in any science, e.g. in philosophy, (and often in so
> > non-mathematical manner!). WHAT ELSE???
> 
> Something here hinges on what counts as formal reasoning, or
> as rigor.  

Recently I described school (=Euclidean?) geometry as 
sufficiently rigorous. However, everyone here knows that 
Hilbert presented much more rigorous version in his well known 
book. I am sufficiently flexible in understanding rigor and 
formal reasoning and seems have demonstrated this in my postings. 
Moreover, it is actually not so clear what is a formal proof, 
even now, having formal predicate calculus on which normal 
mathematical proofs are assumed to be based. We write real 
(usually very short) formal proofs very seldom. We rather 
say about potential formalizability. But what does this precisely 
mean? In what exactly sense potentially? I have something to say 
(and already wrote to FOM some time ago), but let me stop now. 


Example:  Is Newton's _Principia_ a work of
> formal reasoning?  

I did not read and strictly speaking cannot judge. But I 
believe that he had something sufficiently rigorous in what 
was related with mathematics. 

To be sure, there are mistakes here and
> there in that work, but then any kind of formal reasoning is
> susceptible to that, humans being what they are.  


I see here no problem to discuss. 


Have any
> of the scarce mistakes been found by recasting contents of
> the _Principia_ into axiomatic form, and then into some
> symbolic form in the manner of symbolic logic?  

This is absolutely unnecessary. We have a very good standard 
of mathematical rigor without complete formalization. Only 
potential formalizability (whatever it means) plays the role 
in contemporary mathematics. Also partial formalization usually 
suffices. 

How many
> physicists or engineers have made egregious errors in
> applying Newtonian mechanics and dynamics, where it is
> applicable, because (and _only because_) the reasoning
> involved was not _sufficiently_ formal in a good textbook
> on Newtonian dynamics? 

Mistakes were, I am sure, recognized, and the whole theory 
(its mathematical part) formalizable in a reasonable sense. 
In that sense which was enough for real, reliable applications. 
I do not understand what is the reason for paying so much 
attention to mistakes. Do you want to allow them as something 
which should not be eliminated? [Just rhetoric question.]


> On a different tack, is there such a thing as a formal system
> presented, say, in notations of symbolic logic, or so-called
> mathematical logic, which doesn't have a close relationship
> with ordinary languages?

As our intuitions are closely related with the ordinary language 
and mathematical formal systems should be based on (or should 
formalize) some intuition, there is nothing strange that our 
formalism are sufficiently close (but not identical) to the 
natural language. I do not see what bothers you. 


> > Formalisms are used also in other sciences, but in that case
> > it is usually interpreted as application of mathematics.
> > Say, rigor is not so important in physics, unlike facts,
> > experiments and general understanding the nature.
> >
> 
> Here, do you suggest that the intuitions on which mathematicians
> base their formal systems (formal in some strict sense) 


Not necessary in the strictest sense. Just formal enough. 
Say, like in school geometry where a lot of things is 
done on the base of a visual feeling, but some essential things 
are proved with sufficient rigor to recognize it as such and to 
be able to repeat it in new and new situations. 


are different
> in nature from those on which physicists base their expositions? 

They may be sufficiently close, when physicists (very fruitfully!) 
apply mathematics. 

> So that they don't need rigor?  Or perhaps is it the case that
> physicists can't be sure of anything until their mathematical
> apparatus is imbedded in formal systems which justify their
> use, and provide a kind of certainty physicists can't obtain
> without such formal systems having been constructed?

As I already wrote, the main criteria for physics is correspondence 
with the experiments and with the whole picture of the real world 
they have and try to develop. Thus, they may have some 
great results without so strong relying on mathematics 
and on its rigor. Probably we could say that they have their 
own understanding of rigor (as in any other science). 
But this is something non mathematical, of a different character. 
However, mathematics with its (formal) rigor is very often used 
in physics. This is because mathematics strengthen our thought 
and therefore can be applied, in principle, everywhere. 


> 
> >
> > By the way, let me give a small addition to my favorite definition
> > of mathematics as science on formal systems, IN DEFENSE OF INTUITION
> > and the like:
> >
> > Only those formal systems are considered mathematical which are
> > based on and formalize some intuition or abstraction, etc. And only
> > those intuitions, abstractions, informal considerations are considered
> > mathematical which are formalizable. We have two inseparable sides
> > of the same "medal" - Mathematics. And this is a proper
> > FORMALIST VIEW on mathematics as I suggest to understand it
> > (unlike it is interpreted by many others).
> >
> 
> Agreed.
> 
> >
> > Any separation of a pure intuition or a pure formalism is fruitless
> > when discussing the general nature of mathematics.
> > Moreover, pure mathematical intuition is just impossible.
> 
> I wonder.  Remember Poincare's story about how he came
> to discover theta functions?

First, it was a process of discovering of a concrete mathematical 
thing, and the result was formalized (without which we would not 
have anything to discuss)! Could you imagine, a purely intuitive 
thinking, and only afterwards corresponding formalization? 
I can imagine only a mixture of both. Therefore this was not a 
pure mathematical intuition. If it was mathematical, it was not 
pure, but based in some way on mathematical formalisms.  

Poincare was a great mathematician with a good feeling 
of what is a proof or provable, formalizable. He was able 
to not pay so much attention to formal aspects. He did 
this subconsciously. What is the problem? The story 
how it came to theta function may be interesting, as is 
interesting the story of arising mathematics, but let us 
not mix pre-mathematics with mathematics (both in history 
and in the head of a mathematician). They have a strong 
relation one to another, but are not identical. Even if 
some pre-mathematical considerations may play more important 
role, they became mathematical only with formalisation. 
I think, we should be more distinctive to understand better 
the nature of mathematics. 


> > Pure formalism is possible, but mathematically uninteresting.
> > Also, this view needs no Platonism, Intuitionism, etc. as
> > philosophies of Mathematics. The relation between intuition and
> > formalism is extremely delicate thing and should be discussed
> > in each concrete situation separately. In general, we can say
> > only about some acceleration of thought and making it and
> > intuition more powerful, reliable, solid, etc. by formalisms.
> > Also, nothing is asserted on the concrete nature of formalisms
> > (based on First Order Logic or quite arbitrary?). They should
> > be just meaningful, helpful for thought! That is all,
> > in general.
> 
> Also, I suggest, the relation between ordinary languages and
> formalisms is inseparable without distortion.  But then I believe
> that falls in with what you have said.  To this, I add the
> nature of human brains, considered as influences of the
> sort Kant had in mind, though of course without providing
> physiological considerations.

This part is not clear enough to me. Do you mean (agree) 
that the ordinary languages used for representing a formal 
reasoning became somewhat unordinary (adapted to the formalism), 
as any scientific text is different from what we listen in the 
street? 


In conclusion, I would say that Prof. Fisher and me seemingly have 
no essential disagreements, at least in this conversation. I used 
the term "formalist view on mathematics" somewhat in non-traditional 
sense. This could bring some problem in understanding what I mean. 
I think, all misunderstandings are easily resolvable. 


> Gordon Fisher
> Prof Emeritus, Mathematics and Computer Science
> one-time Senior Lecturer in Mathematics, and History &
>              Philosophy of Science, Univ of Otago & Unive
>              of Waikato, New Zealand
> James Madison Univ
> Harrisonburg VA 22801 USA
> gfisher at shentel.net

-- 
Vladimir Sazonov                        V.Sazonov at csc.liv.ac.uk 
Department of Computer Science          tel: (+44) 0151 794-6792
University of Liverpool                 fax: (+44) 0151 794 3715
Liverpool L69 7ZF, U.K.       http://www.csc.liv.ac.uk/~sazonov