FOM: defining ``mathematics''
Vladimir Sazonov
sazonov at informatik.uni-siegen.de
Sat Jan 8 16:22:42 EST 2000
The following are my comments on postings of Samuel Buss,
Stephen G Simpson, Karlis Podnieks, Harvey Friedman,
Jan Mycielski, Matt Insall, Vaughan Pratt and Joe Shipman
related with formalism and defining what is mathematics.
Cf. also my previous posting.
Unfortunately, it is not easy to explain the formalist
point of view as I understand it. (However, it seems to me
so simple and clear!) Professor Mycielski is right that
this term has a bad history. Therefore a lot of
misunderstanding arise. But I like this term as short and
linguistically most appropriate one and still hope it can
be introduced again in a positive context to denote a
"sober" and philosophically correct view on mathematics.
It could be objected that this term have been occupied and a
new one should be used (say, rationalism - as Mycielski
suggests; it seems to me that our points of view are
very analogous, except the terminology). But I do not feel
that "formalism"
***as it is usually treated***
really denotes anything reasonable what deserves to have a
name. What it usually denotes is rather some philosophical
curiosity, nonsense. We should recover or reconstruct its
*proper* meaning.
I am sorry that this posting proves to be so long.
Probably it would be better to present, instead,
a concise description of this point of view. However,
it is actually not the first my attempt to do this.
Anyway, some objections would arise just because the
term is "occupied". So, I am trying here to reply to
these objections related to using this term or to clarify
some points where there is no serious controversy but
the danger of misunderstanding may arise.
Stephen G Simpson <simpson at math.psu.edu>
Date: Tue, 21 Dec 1999 19:07:24 -0500 (EST)
wrote:
Buss hangs his definition of mathematics on a methodological
issue: rigor.
MY COMMENT:
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.
SIMPSON:
But this doesn't seem to fit with the history of the
subject.
Our current standard of mathematical rigor evolved only in
the 19th and 20th centuries. Would Buss claim that there was no
serious mathematics in the 17th and 18th centuries, prior to that
evolution?
MY COMMENT:
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.
As to another suggestion of Prof. Simpson to consider
``quantity'' as subject matter of mathematics, cf. below.
"Karlis Podnieks" <podnieks at cclu.lv>
Fri, 24 Dec 1999 09:50:50 +0200
wrote:
Thus, Hegel defines quality as "first, direct definiteness", and
quantity - as "definiteness that has become indifferent to the
Existing" (sorry, I'm not very strong neither in German, nor in
English). For me, "definiteness that has become indifferent to
the Existing" is just the notion of a model that has become
distracted from its "original", and hence, can be investigated
independently of this "original". In other words, this is the
notion of self-contained models (models that can be used by
robots).
I am fascinated by this equivalence of quantity and
self-contained models since the beginning of 1970s when I read
Hegel for the first time. I'm glad to see outstanding people
approaching this position from various angles...
Could we agree on defining mathematics as the science of
self-contained models? Let us call it Hegel's definition, not
Podnieks's. Could this help?
MY COMMENT:
I like very much this clarification by Podnieks of what
is quantity. But what are these "self-contained models
(models that can be used by robots)"? I think, these are
exactly formal systems. Thus, we have a chain of
equalities
quantity = ... = formal systems
with the most nontrivial philosophical step done by
Hegel. It seems I also remember (rather vaguely) that step
from a course of Marxist philosophy when I was a student.
I am very grateful to Podnieks for recalling and for
explaining it so clearly.
Thus, we can say that mathematics is the science on
quantity or, essentially equivalently, the science on
self-contained models (for "robots"!!!) or, equivalently,
the science on formal systems.
As I do not feel myself as a philosopher, I prefer
the last more "operational" version in terms of formal
systems. It seems to me simpler and sufficiently general
and adequate. Also, how could we infer from the definition
in terms of quantity the main feature of mathematics - its
rigor? Only by the above chain reducing quantity to formalism.
This seems somewhat complicated. How many people, even
professional philosophers, will realize these equivalences
without additional explanation? The first opinion would be
that it is about numbers existing objectively. Again the
danger of Platonism.
However, I am also fascinated by this equivalence.
Harvey Friedman <friedman at math.ohio-state.edu>
Date: Tue, 28 Dec 1999 13:05:02 -0500
wrote replying to Buss:
> 4. One of the distinguishing features of mathematics is the use of
> proof and of mathematically rigorous reasoning. Especially
noteworthy
> is the fact that mathematicians will nearly always agree on
whether a
> given asserted theorem has been correctly proved. When there
are
> agreements on whether a proof is correct, mathematicians try to
> resolve their disagreement by breaking their argument into
smaller
> steps, down to the level of first-order logic if necessary (but
> not down to the level of set theory.)
I would not characterize the way mathematicians try to resolve their
disagreements over correctness of proofs in the way that you are. The
process mathematicians most commonly use has nothing whatsoever to do
with
first order logic, or no more than any other mathematics has to do with
first order logic. It is the following professional and mathematical
process:
1) first demand that there be a manuscript available;
2) try to reconstruct the proof, asking questions if necessary;
Assuming this fails,
3) try to give another proof;
Assuming this fails and one is suspicious,
4) give counterexamples to lemmas and/or other assertions in the
manuscript; and/or
5) prove that certain definitions are ill defined for various reasons.
In other words, the usual way to show that a proof Pi is incorrect is to
engage in normal mathematical activity without regard to first order
logic
or ZFC. There is nothing formal about this process, and so it makes no
sense to me to compare the roles of set theory and first-order logic in
this process.
MY COMMENT:
Here I ignore comparison of FOL with set theory. Only on rigor.
The point of Buss here is formal character of mathematical
proofs. I agree with this completely and do not see the
real reason for the above objections.
I think here is some mixture of various aspects of mathematical
activity. Who likes to read the formal (or semi-formal) proof
written by somebody else, whose way of writing is probably not
the best one, say for the given reader? It is essentially the
same as reading a program written by other programmer. As a
creating person any mathematician prefers to guess himself
how the prove was (or may be) discovered. This is much more
useful and interesting than just to read and check even
absolutely formal proof. But eventually, the ideal result
should be anyway a formal(izable) proof, i.e. a construction made
according to some (sufficiently) formal rules. Moreover, this
process may be not very conscious. Mathematicians "know" their
formal rules of reasoning just by training (like swimming or
driving bicycle or car). We do many things mechanically
(formally) without thinking very much how this is happening.
The goal of formalisms in mathematics consists not in forcing
mathematicians to write everything absolutely formally but,
rather to organize, mechanize, accelerate thought and to make
it precise.
Say, one of the goal of formalization consists in the possibility
to write proofs in the manner like:
"... then apply modus ponens five times more ..."
instead of real applying it five times. If mathematics would not
be based on formal rules it would be impossible to make such
"accelerations of thought". We essentially explain one to another
how it is possible in principle to write a formal proof. Of course
it does not always look like the above artificial example. But
I believe that any mathematical proof (whichever informal it looks
like) can be most reasonably treated in analogous way. Of course
we may and should use additionally various images, appealing to
the intuition, to the real life, etc. The final (ideal) result is
essentially the same - formal(izable) proof. Eventually, such a
proof may be checked only in accordance with the formal rules of
a given formalism. And this is actually confirmed by
FRIEDMAN:
Buss continues:
>Lasting disagreements
> on whether a proof is correct are rare; on the contrary, they
can
> usually be resolved to *everyone's* satisfaction.
> I believe that this fact is due directly to the fact that
> mathematicians are reasoning with methods that are *formalizable
> in principle* in a first-order system. This makes the notion
> of mathematical rigor a robust and objective concept.
I don't think this explains why things are so often fully resolved. For
instance, it is likely that one can decide *in principle* whether a
comet
will destroy human life on this planet within 200 million years. Just
wait
200 million years and see. However, that is too impractical a method,
and
it doesn't prevent people from having sharply different views about
this.
On the other hand, I have no doubt that ZFC and first order logic would
play a role in any sufficiently long dispute about the correctness of an
argument if the process I outlined above were to continue to fail over
and
over again, even after the "proof" was cut up into manageable pieces and
examined independently. But it would come into play only as a last
resort
and only when it can be applied to tiny subproofs.
MY COMMENT:
Of course as "as a last resort"! And it is the main point
that mathematics is based on existence of this "last resort".
As to formalizability *in principle*, there may be two main versions:
(1) A mathematician argues that the proof (with such and such
abbreviations, if he pays attention to this important subtlety)
may be really written, say, in a book of a reasonable size.
(2) He/she even do not try to be sufficiently precise in this sense.
Only relies on the ordinary mathematical practice of writing/checking
proofs. So that the length of the resulting formal proof even cannot be
reasonably estimated.
(By the way, we could doubt that we have really a rigorous proof
in the last case. However, usually these doubts may be resolved.)
This only shows that "absolute" mathematical rigor (if any) in
real mathematical practice is yet not achieved even nowadays.
I mean even not writing proofs "absolutely" formally, but only
giving a guarantee for each concrete (usual semi-formal) proof
that it is possible.
Also I am not sure that an *absolute* mathematical rigor is
achievable in mathematical practice. Only some short proofs
may be (even only in principle, in the sense of (1)) completely
written in a formalism. We probably always will have possibility
for a progress in mathematical rigor.
Actually we do not know what will happen if we will try to be more
rigorous (formal) than now. Let us, say, fix some specific formal
rules of abbreviation in FOL, but not other. (Say, we could allow
abbreviations only of formulas, but not terms.) Then we will get
one specific version ZFC' of ZFC (or PA' of PA). Is it provable
*in principle* (in the sense (1) above) the same class of theorems
in ZFC' and in ZFC'' (for other choice of abbreviating mechanisms)?
For the case of ZFC or PA such a program (like the Mizar project
already discussed in FOM) would rather discover some partial
abbreviating mechanisms related to the contemporary mathematical
practice and allow to understand better their role. It seems more
essential considering instead of ZFC or PA much more week theories
(probably based on a new version of first-order logic) related with
the (in a sense more subtle and new) idea of feasibility. (1) above
leads to a new version of consistency of a formal system and
therefore to new possibilities for formalizability. In mathematics
the need of a stronger rigor arises when it is evoked by some new
ideas to be formalized (like the idea of an infinitely small
quantity) which cannot be formalized by the old methods and old
standards of rigor.
Jan Mycielski <jmyciel at euclid.Colorado.EDU>
Date: Wed, 29 Dec 1999 13:00:08 -0700 (MST)
wrote:
Indeed if we regard the problem of defining mathematics as a
problem of natural science (that is mathematics is viewed as a physical
process just like other physical processes), then the answer is:
mathematics is the process of developing ZFC, i.e., the process of
introducing definitions and proving theorems in ZFC.
[As every theory of a real physical phenomenon this definition
is
not complete. Indeed we ignore here the rare phenomenon of addition and
uses of new fundamental axioms beyond ZFC (e.g. large cardinal
axioms).
<snip>
]
My prefered formalism (for ZFC) is not first-order logic, but
logic without quantifiers but with Hilbert's epsilon symbols. In this
formal language quantifiers can be defined as abbreviations. This has
the
advantage that the statements in such a language do not refer to any
universes. So this does not suggest any existence of any Platonic (not
individually imagined) objects.
MY COMMENT:
Thus, at least some extensions of ZFC should be also considered
as describing what is (contemporary?) mathematics. Why to consider
only extensions of ZFC? 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?
MYCIELSKI:
Also this point of view shows that there is no qualitative
(ontological) difference between ZFC and PRA. (Integers such as 10^10^10
and sets such as a well-ordering of the continuum seem equally
imaginary,
i.e., without any intended outer physical interpretation.)
MY COMMENT:
Completely agree! This is confirmed also by a very nice results of
Professor Mycielski from 80th on "isomorphism" of any formal theory
(say, ZFC) with some its locally consistent version (s.t. each
finite set of axioms has a finite model, probably unrealistically
large one). Thus, say, infinite cardinals may be treated as large
finite objects! I consider this result as having a great philosophical
value for f.o.m.
MYCIELSKI:
I have not seen in the literature any clear exposition of the
philosophy stated above. All Platonists reject it. Their definition of
mathematics (a description of a Platonic universe independent from
humanity) assumes more but it does not seem to explain more. Hence it is
inferior. [Of course the Platonic definition puts mathematics in the
realm
of science, while the ZFC definition puts it in the realm of art in as
much as it is independent of any intended physical meaning. This may
have
some negative political implications for mathematicians, but it seems to
me that truth is more important.]
MY COMMENT:
With my additions presented above this kind of philosophy could be
called *formalism* (however, Professor Mycielski wrote that he do
not like this term; this seems only terminological disagreement) or
*formal engineering* (cf. also my previous posting to FOM).
Really, was formalism described by anybody as a consistent point
of view on mathematics, not as an shameful brand or a kind of
soul-less bureaucracy? (By the way, bureaucracy organizes the life
of a society. This seems not so bad, and, moreover, inevitable.)
I would say that Platonists pretend to "put mathematics in the
realm of science" without any real grounds for this. Science
about our vague illusions? But mathematics considered as science
on such very concrete and quite useful things ("devices",
"mechanisms") which are formal systems seems to be not only just
the art.
"Matt Insall" <montez at rollanet.org>
Date: Thu, 30 Dec 1999 18:44:01 -0800
wrote in reply to my posting on the definition of mathematics
given by Buss:
Sam Buss wrote:
> "Mathematics is the study of objects and constructions, or of aspects
of
> objects and constructions, which are capable of being fully and
completely
> defined. A defining characteristic of mathematics is that once
mathematical
> objects are sufficiently well-specified then mathematical reasoning
can be
> carried out with a robust and objective standard of rigor."
I am afraid that these "fully and completely defined objects" would lead
us imperceptible again and again back to Platonism. Also what does it
mean "capable of being fully and completely defined"? I think that this
or other way it is inevitable to explain all of these in terms of
appropriate
formal systems.
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
++++++++++
I'm not sure, professor Sazonov, why you should be ``afraid'' of being
lead
back to Platonism. Are you convinced that our mathematics has no
reality
beyond the marks on the page, as I understand the pure formalist
position to
be?
MY REPLY:
As I wrote in the cited posting, I understand mathematics as
a kind of formal engineering and of course as a science investigating
formal systems. Formal systems are very useful things. Their goal
*in general* is not finding a "mathematical" truth (what does it
mean?), but rather to serve as levers, accelerators for human thought.
Formalisms organize and govern our intuition and thought. E.g., how
would we efficiently multiply natural numbers without formal rules of
multiplication of decimal numbers learned at school? Of course
formalisms we are considering in mathematics usually have
some relation to the real world (or to some other formal systems
having some relation to the real world, etc). When we write a formal
symbol "5" we think, say, on a set of five pebbles. When we multiply
numbers according to the mentioned formal rules we expect that the
input and output of this process will correspond to such and such real
experiment with pebbles. Otherwise we hardly would be interested in
these formal rules of multiplication. Mathematical formalism should be
helpful for human thought.
"Reality beyond the marks on the page" lies in pragmatics, not in the
subject matter of mathematics.
MATT INSALL:
The question was ``What is
mathematics?'', not ``What is your philosophical belief about what parts
of
mathematics possess some metaphysical property such as `existence beyond
marks on the page'.''
MY REPLY:
I think, without a correct philosophy of mathematics it is impossible
to define what it is, and vice versa.
MATT INSALL [Sun, 2 Jan 2000 07:10:11 -0800 ] citing my posting:
<snip>
Mathematics is a kind of *formal engineering*, that is engineering
of (or by means of, or in terms of) formal systems serving as
"mechanical devices" accelerating and making powerful the human
thought and intuition (about anything - abstract or real objects or
whatever we could imagine and discuss).
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
++++++++++++
I like this ``definition'' also, as far as it goes. The problem I see
in
all this is that the ``definitions'' all seem to appeal to terms not
previously defined. Thus I would not really call this a definition,
because
too much of the description is undefined. In particular, how does one
define ``engineering'', or ``human thought and intuition''? I guess
this
might make a good Webster's type of definition, but it is hardly
mathematical itself.
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
++++++++++++
MY REPLY:
Why a definition of mathematics should be "mathematical itself"?
As to definition of ``engineering'' and ``human thought and
intuition'' I do not think that I could or need do anything
better than Webster. Any of us knows sufficiently well what
it is. (Say, it is enough to know that engineering is creating
any useful devices.) I do not understand why it bothers you.
Mathematics deals with formal systems ("devices" of a special kind).
This is the main point. Are these formal systems meaningless or
useless? In principle they can, but mathematicians, like engineers,
prefer to do something reasonable, rational. This is the only
(pragmatic) restriction on the class of formal systems considered.
Platonists pretend to know on existence of an ABSOLUTE TRUTH (but I
do not believe them!) and restrict the formalisms of mathematics to
those which are true (what does it mean?). Formalist (or rationalist,
as Prof. Mycielski probably would say) position consists in considering
ANY REASONABLE formal system. There is no pretension here on knowing
what is reasonable or not. Only our experience could help to judge
this in each concrete situation.
MATT INSALL continues citing my posting:
<snip>
Finally, I would like to stress that mathematics actually deals
nothing with truth. (Truth about what? Again Platonism?) Of course
we use the words "true", "false" in mathematics very often.
But this is only related with some specific technical features of
FOL. This technical using of "truth" may be *somewhat* related
with the truth in real world. Say, we can imitate or approximate
the real truth. This relation is extremely important for possible
applications. But we cannot say that we discover a proper
"mathematical truth", unlike provability. This formalist point of
view is not related with rejection of intuition behind formal
systems. But the intuition in general is extremely intimate thing
and cannot pretend to be objective. Also intuition is *changing*
simultaneously with its formalization. (Say, recall continuous
and nowhere differentiable functions.) Instead of saying that
a formal system is true it is much more faithful to say that it is
useful or applicable, etc. Some other formalism may be more
useful. There is nothing here on absolute truth.
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
++++++++++++
Okay, so if we do not deal with truth, then what would you say is the
``truth in the real world'' of the following statement: ``If f is a
continuous function defined on the real numbers, then f has the
intermediate
value property.'' I submit that as mathematicians, we do, and should,
care
about the ``truth in the real world'' of such a statement.
MY REPLY:
Yes, of course! But this is ``truth in the real world'' (which is
related rather with pragmatic and possible applications of
corresponding mathematical theory), not the ``absolute mathematical
Platonist truth''.
When I say that "mathematics actually deals nothing with truth"
I mean that *ideally* its goal is not finding a truth, whether it
is
(i) mathematical truth (let me ask again, what does this truth
mean?), except, of course, the facts of provability in a formal
system or the like, or
(ii) a truth on the real world (which I respect very much).
The latter truth is the goal of *other* sciences like physics,
biology, etc. mathematics can/should only supply them with some
machinery (formalisms) which will help them. Occasionally,
during working on a formal system, mathematicians can find some
real truth in a "real" science for which this formalism was
especially created or in which it is applied. But this is not the
goal of mathematics *in general*. This is only a (very important)
witness that the formalism considered is good. Of course, the real
situation with concrete mathematical practice is not so pure. (Say,
recall Newton!) But it seems it is unreasonable to heap up everything,
especially in a short definition. However, recall this definition:
Mathematics deals with formal systems making powerful the human
thought and intuition (about anything - abstract or real objects or
whatever we could imagine and discuss).
"Thought and intuition about anything"! Which else truth and meaning
do you need? However they are moved from the subject matter of
mathematics, they are here, very close. Nothing is lost, except of
the nonsense Platonism as a philosophy. Even the naive Platonism of
working mathematicians can be used freely.
MATT INSALL:
The problem I
see is that it is either true or false, but not both, but the formalist
approach would have us believe that no one even knows what the statement
means.
MY REPLY:
Where you discovered this? Look more carefully on the definition
above.
As to "either true or false, but not both", I recall my
favorite example which I already presented in FOM. Look on
a nice picture in a computer display.
Is it continuous OR discrete,
or both continuous AND discrete?
Why should we take the formal dogmas of FOL with its contradiction
law (A & ~A => anything) as a rule governing the real world?
Relations of mathematical formalisms with the truth in real world
may be more complicated.
Another (related) example: It is formally provable (and Platonists
will say - mathematically true) the fact
limit of the sequence 10/(log log n) = 0
with log base 2 logarithm. But it is definitely false in our world!
Calculate this sequence by a real computer, step-by-step.
The practical limit (if it makes a sense at all) should be > 1.
MATT INSALL:
If this were correct about such statements as this, then do we, as
human beings (not, per se, as mathematicians in particular) know what
anything means? In fact, would you say, professor Sazonov, that there
is no
such thing as ``truth in the real world''?
MY REPLY:
I never said this.
MATT INSALL:
For if it is because we
formalize Mathematics that we lose meaning, is it not the case that even
the
very statements we make about the ``real world'' are formalizations, of
a
sort,
MY REPLY:
Not every statements can serve as *mathematical* formalization.
We should have formal *rules* (not only symbols), like
(x+y)z=xz+yz or (uv)'=u'v+uv', A=>B,A/B, etc. (of course, assuming
that they have such and such intuitive meaning) which allow to deduce or
calculate *mechanically* and therefore make stronger human thought.
MATT INSALL:
and so can be interpreted any way one may choose.
MY REPLY:
As to mathematical formalisms, it depends on our intentions.
Arithmetical variables are usually interpreted, say, as finite
sets of pebbles. Variables of the first order logic may have
any imaginary interpretation.
I am not against a meaning of symbols and intuition.
I only cannot consider seriously "the" absolute Platonistic meaning
because, by my opinion, it is nonsense.
MATT INSALL:
After all, whether
we are doing mathematics or not, we are only putting marks on the page.
Thus, even the statement that `` mathematics actually deals nothing
with
truth'' has no meaning outside the virtual marks on the virtual page on
my
computer monitor. When you restrict mathematics to the tenets of pure
formalism, everything must be so restricted.
MY REPLY:
Formalism, as I understand it, (unlike Platonism) can only extend
mathematics. Formalism does not reject meaning. It allows ANY kind
of meaning. It only gives it somewhat different role. Mathematics
considers meaningful formalisms. But the meaning of these formalisms
is in a sense outside of mathematics, even if concrete mathematicians
pay a lot of attention to it. Nobody could know what kind of
intuition can be formalized in principle. Any attempt of describing
a Platonist world including ANY potential mathematical intuition
seems to me extremely non-serious. *In general* we can only mention
the fact that our formalisms are meaningful. And this seems to me
quite enough in the definition of mathematics.
MATT INSALL citing my posting:
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
++++++++++++
By the way, as an example of useful and meaningful formal system
I recall *contradictory* Cantorian set theory. (What if in ZFC or
even in PA a contradiction also will be found? This seems
would be a great discovery for the philosophy of mathematics!)
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
++++++++++++
I think this would be a disaster. It is bothersome enough that
``Cantorian'' set theory (I think you actually mean Fregean set theory.
Cantor's approach was decidedly NOT formalistic.)
MY REPLY:
Read "formal" as "sufficiently formal" or "sufficiently rigorous".
It is completely applicable to Cantor. Even contemporary
logicians not always pay enough attention to using
abbreviation mechanisms when proving in FOL. They are
also not absolutely rigorous in this sense. Usually this
subtlety (like Choice Axiom for many of working mathematicians)
makes no value in everyday mathematics. (But who knows?)
MATT INSALL:
is considered to be
contradictory. Why should ``philosophers of mathematics'' be so
biased?
MY REPLY:
Really, why? Someone would like to have a self-contained
Platonistic world. It does not exist anyway. Other one
(a formalist) would like to have a *sufficiently* reliable and
reasonable formalism. It is happy that ZFC is so reliable (until now).
But *let us only imagine* that suddenly a contradiction will be
found in ZFC and even in PA or PRA or even in the exponential
arithmetic. (As I know, Edward Nelson even tried to find a
contradiction based on using exponentiation; I also believe
that this operation is rather problematic from the point of
view of f.o.m.) Formalist view on mathematics will still exist.
(Cantorian or Fregean set theory can serve us even as
contradictory one. It actually serves to the most of mathematicians
who use set theory without knowing the precise formulation
of ZFC.) What about Platonism?
Vaughan Pratt <pratt at CS.Stanford.EDU>
Date: Thu, 30 Dec 1999 18:11:30 -0800
wrote replying to the posting of Mycielski:
>My prefered formalism (for ZFC) is not first-order logic, but
>logic without quantifiers but with Hilbert's epsilon symbols. In this
>formal language quantifiers can be defined as abbreviations. This has
the
>advantage that the statements in such a language do not refer to any
>universes. So this does not suggest any existence of any Platonic (not
>individually imagined) objects.
When I do mathematics, regardless of what might be happening in my brain
cells, I feel as though I am working in a world of mathematical objects.
The perception of a Platonic universe is very strong for me,
independently
of its reality or lack thereof. I'd find it hard if not impossible
to prove things if I had to work in a framework expressly designed to
eliminate that perception!
MY COMMENT:
I also cannot imagine a mathematician who is working in this strange
manner, i.e. without any intuition behind the formalism considered.
Karlis Podnieks has a Web page with his book where he argues
that the *naive* Platonism of working mathematicians is normal
an useful thing with which I completely agree. We can use *any* kind
of intuition when working with a formal system if this intuition
really helps. However, Platonism *as a philosophy* seems to me very
dangerous and harmful for mathematics. The philosophy should not
be based on a self-deception.
I think the advantage of a language not referring to any universes
mentioned by Mycielski consists in its "honesty". No problem: we can
introduce quantifiers as abbreviations and use the naive Platonism.
But in this case we will better understand the role of this Platonism
as something *secondary*. Probably Mycielski have in mind also some
other technical and philosophical advantages of Hilbert's epsilon
symbol. It would be interesting to know.
JoeShipman at aol.com
Date: Fri, 31 Dec 1999 10:23:56 EST
wrote:
In a message dated 12/30/99 5:32:40 PM Eastern Standard Time,
jmyciel at euclid.Colorado.EDU writes:
<snip>
<<Their definition of
mathematics (a description of a Platonic universe independent from
humanity) assumes more but it does not seem to explain more. Hence it
is
inferior.>>
On the contrary, it explains the unity (mutual consistency and
interpretability) of almost all the mathematics developed by thousands
of
mathematicians over the centuries.
MY COMMENT:
It is only illusion of an explanation. It is a declaration
but not explanation. "Mutual consistency and interpretability"
is very much alike to Church-Turing Thesis: any (reasonable)
notion of computability may be reduced to Turing Machines. It
is just a sufficient (sufficient for the present mathematics,
probably not for the future mathematics) flexibility and
expressibility of a language/theory considered. As for the
case of Church-Turing Thesis no Platonism (as a philosophy)
is needed to realize this fact.
Vladimir Sazonov
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