FOM: defining ``mathematics''
Vladimir Sazonov
sazonov at informatik.uni-siegen.de
Wed Jan 12 17:16:46 EST 2000
Matt Insall wrote:
> > "sober" and philosophically correct view on mathematics.
> >
>
> Please explain what the criteria are that you use to determine what
> constitutes a ``philosophically correct'' view of mathematics.
I did not think very much on corresponding formulations. At least,
the philosophy should not be based on self-deception of which
Platonism is the very bright example.
Professor Mycielski described his views on "acceptable" philosophy
in a recent posting as (A) respecting physical reality and (B) "minimal".
It seems I agree with him. Platonism surely goes outside (B). Its
fictions *pretending to be real* (however, the discussion on fictions
*as fictions* or *illusions* is useful) are absolutely unnecessary to
understand the nature of mathematics. It is enough to say that our
formalisms are aimed to support our intuition and thought about any
kind of reality. Then, if necessary, we could consider more closely
which kind of intuition and imagination is used for some concrete
formalisms or for some class of formalisms. We even could call
some kind of our illusions (specifically those related with FOL) as
*naive* Platonism of (seemingly all) working mathematicians.
But illusions could not serve as a fundament to (undestanding the
nature of) mathemastics. There is much more solid material -
formal systems. Nobody can say in advance which kind of intuition
(and related illusions and idealizations) could be related with all possible
formalisms.
> I think I understand your position. Let me see if I
> can sum it up according to what I have read both above and below this point:
> You consider mathematics to be the science of formalization, so that the
> objects of study of this science are ``formal systems'' which do not, in and
> of themselves, exist. Please let me know if I have misunderstood.
Sorry, I do not understand this " which do not, in and of themselves,
exist". Formal systems? (Or it is some play of words in English?
I am not so fluent in it.) Formal systems (at least in the contemporary
form) are very concrete objects, at least they can be represented
physically as a list of axioms and rules or the like.
> > I think that rigor may be considered as methodological issue rather
> > for other sciences. For mathematics rigor *in the form of various
> > formalisms* constitutes its subject matter.
> >
>
> I would say that this type of rigour constitutes *part* of the subject
> matter of mathematics. It is a significant aspect of the study of
> mathematics, undoubtedly, but is far from being all there is to mathematics.
Not everything what mathematicians are doing should be included into
subject matter of mathematics. Say, mathematician can appeal that
his theory may be applied for calculating the timetable of sun eclipses
and really to calculate this timetable. This does not mean that subject
matter of astronomy should be included into that of mathematics.
Do you mean something essentially different?
Let us will not heap up everything.
> > As I understand, mathematicians always (at least beginning from
> > Euclid) tried to be rigorous. Sometimes it was problematic. Before
> > obtaining rigorous proof of a theorem we usually have some vague
> > idea and then intermediate proofs. The process of finding a
> > sufficiently satisfactory formal construction (including the very
> > formalism where this construction will be done) may occupy even
> > hundreds of years.
>
> I agree with you on this score. But does this not undermine your thesis
> that mathematics is *merely* the study of these formal constructions?
Not "merely", but study of formal systems serving as formalizations of
any kind of reality.
> The
> fact that the ideas exist *prior* to the constructions, and are studied by
> mathematicians, and are considered to be part of mathematics by those
> mathematicians, means their existence is not contingent upon the
> constructions themselves. Thus the ideas are parts of mathematics that
> have existence beyond the formal structures which represent them.
Of course ideas have existence beyond the formal structures.
Mathematicians may spend a lot of time on elaborating these
ideas/illusions/dreams. But to include them into subject matter
of mathematics means to include there essentially everything.
Is mathematics a science about everything? Or mathematics
is science on formal systems which may be applied in principle
to everything?
> Would you really object to the statement that the number 1 exists
> objectively? When you question this, I question what you mean by
> ``objectively''. If you reduce the question to one of language, again you
> beg the question, because translations can be made, once one sufficiently
> understands another's language.
Your question on existence of the number 1 assumes a natural
language in which we discuss this. Yes, there is the word "one"
in English and its vesrion in other natural languages and, moreover,
peoples have more or less the same associations related with
this word. In this sense the number one exists. But do you see as
vague is such description of this number. It is not better than
entities not related to mathematics, like "house", "flame", etc.
Only when you include it in a (sufficiently) formal context
it becomes a (sufficiently satisfactory illusion of) a mathematical
object. Have you "seen" some funny geometrical illusions having
no real counterparts? Is not this effect of essentially the same
psychological nature? Let *us* govern our illusions, but not vice
versa!
So, we have (illusions) of one, two, three, ..., ten, ... .
All normal peoples have essentially the same associations
related with these concrete words. Then a new word arise:
(natural) numbers. Are you sure that even in this case you
have the same association with this word, say, as me? Even
if we both are considering this term in (general)
mathematical context?
Let me give an example outside of mathematics. Many peoples
otside former Soviet Union know the Russian word perestrojka
(denoting a process initiated in SU by Gorbachev). But I think
that only peoples from fSU know this thing "by their skin".
They have very different associations than other peoples.
The only way to put it on sufficiently solid ground, consists in
making some *agreement* on some concrete formal axioms and
rules about natural numbers. A trivial question: is there a biggest
natural namber? Why not? If we agree, just for definiteness or
because of some reasons that "not", are your sure that there could
be no other questions/suggestions/ideas and corresponding axioms?
How do you fix your Platonistic idea of uniquelly existing (let up
to isomorphism) natural numbers? Why I should simply believe
that they exist uniquelly?
On the other hand, you may appeal to ANY kind of intuition
and explain why you choose such and such axioms for natural
numbers. Probably your intuition will be somewhat convinsing
to other peoples and they agree that the axioms arise naturally.
Now we can work with them mathematically (formally).
> One may question whether infinite numbers, such as \aleph_0,
> exist, on philosophical grounds, but one must accept the existence of
> certain very simple numbers, especially certain finite ones.
I understand not very good what do you need. Let us assume I
would say that "certain very simple numbers, especially certain
finite ones" exist. (Actually I have said above something like
this in the more appropriate for me form.) What will you
do with this existence? If you actually mean naive Platonism,
of working mathematicians, then I am also naive Platonist.
Do you need something more? What? Why?
> > Mathematicians "know" their
> > formal rules of reasoning just by training (like swimming or
> > driving bicycle or car).
>
> I would aggre, except that I believe some of the tools of reason are innate.
> In fact, they probably all are innate, but undeveloped, in most people.
Are the rules of addition and multiplication of decimal numbers
or the rule (uv)' = u'v + uv' innate?
> >Why not arbitrary formal systems? Also
> > the last paragraph shows that in principle these formalisms may
> > be based not on only the ordinary first order logic (FOL).
> > In principle we could consider formalisms even not reducible to
> > FOL. Why should we restrict (also future) mathematics to some
> > special formalisms such as ZFC (or its extensions) or to
> > those based on FOL?
>
> Well, I can think of a few reasons to restrict mathematics to systems
> formalizable in FOL:
> 1. FOL is a very natural model of the practice of mathematics.
> 2. FOL is (demonstrably) sound.
> 3. FOL is (demonstrably) complete.
> 4. I feel I understand FOL, at least reasonably so. :-)
> 5. Every argument I have seen presented in a meeting or a research article
> appears to be formalizable in FOL. (This is true even of all the papers on
> ``fuzzy reasoning'', in which claims are made that the fuzziness of the
> reasoning system somehow improves the results obtained because physical
> reality is actually ``fuzzy''.) That is, behind every argument, there seems
> to be an FOL lurking about.
Yes, "seems to be", but who knows? If you agreed that mathematical
rigor is probably not achieved yet then there is a "gap" where some
new kind of formal reasoning (formal systems) not reducible to
FOL could be discovered.
> In fact
> the part of mathematics which you seem to think comprises all of mathematics
> (the study of formal systems) is modeled mainly upon ideal reasoning systems
> (the devices you mention), and was developed long before there were physical
> realities (computing devices, etc.) associated with it. Thus the only
> ``physical'' model for the development of formal systems when they were
> originally designed was an ideal version of the rational human mind. (To
> wit Boolean logic, etc.)
Let me recall "swimming, driving a bicycle or car". These formalisms
existed and were really (however non consciously) used. I even would
say that they were non consciously discovered (in a rough form) by
mathematicians long before contemporary formalisms of FOL vere
explicitly written. (If you learned yourself to swim, this means that
you discovered some way the rules of swimming.)
Vladimir Sazonov
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