FOM: defining ``mathematics''

Vladimir Sazonov sazonov at
Fri Mar 10 17:34:15 EST 2000

Dear Professor Insall,

I am very sorry for so late answering!

I am not sure that this too lengthy discussion (I am trying to reply
all your questions and to clarify my position) may be interesting
to all FOMers. Thus, if there will be no explicitly expressed interest
it would be probably better to continue this privately, if you want.
Your opinion? Now I reply to FOM as your posting also was to FOM.

You wrote:

> This is the (belated) fourth part of my reply to professor Sazonov.


> > 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?),
> Just ``the truth''.  For example, one problem of mathematics is that of
> deciding whether PA is consistent or inconsistent.  The current type of
> activity seems to have no hope of showing that PA is consistent, and seems
> unlikely to show that it is inconsistent.  The theorems that have been
> proved in this regard are ``true'' about PA.  They are mathematical, having
> been proved by mathematical logicians.

I do not know what is `Just ``the truth''' in contrast with the truth
in the real world. By the way, the fact that a theorem is proved
in a formal system is the fact of the real world about proofs.
Therefore consistency statements also have in principle quite
realistic counterparts in our world. We can believe that they
are true (in our world) because of some intuition related with PA,
which is some (probably not uniquely possible) extrapolation of a
simple intuition on small numbers (sets of pebbles).

Thus, I see here some intuition, but no absolute mathematical truth.
However, we should remember that intuition is not absolutely reliable
thing. If you want a truth, use Tarski semantics (which is strictly
speaking a technical notion, very intuitive and plausible one) in
the formal framework of ZFC. No problem! ZFC itself has some
well-known intuitive informal background. That is quite enough.

> >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.
> I completely agree with this discussion of formalisms.  However, I still
> object to the reduction of mathematics to (the study of) formalisms.

But what is lost?

Formalism, as I understand it and as I described it in my postings,
is not against of meaning, intuition (and related illusions of
existence of mathematical objects). It just gives them a right place.

> > "Reality beyond the marks on the page" lies in pragmatics, not in the
> > subject matter of mathematics.
> Why not?  The early geometers presumably carried out actual geometric
> constructions physically as well as mentally.  I expect that, to them, the
> reality of the geometric objects they studied was quite important to their
> discoveries and understandings in geometry.

For any mathematician his intuition and illusions are quite important.
But this does not mean that illusions become real even when mathematician
relies on them. Physical geometric constructions are real, but they
belong to physics rather than to mathematics.

> Also, I feel certain that a
> formal system can be, and probably has been, developed to deal with (at
> least certain aspects of) pragmatics, and so the study of pragmatics becomes
> a part of mathematics, according to your definition in terms of formal
> systems.

Yes, when dealing with formal systems devoted to some reality
we inevitably will think on pragmatics. In this sense pragmatics
is a part of *activity* of any science.

If you actually mean some formal theory of pragmatic like a formal
theory of natural numbers then such a formalism, as any other, also
should belong to subject matter of mathematics. Of course, when
working with such a formalism mathematicians will take into account
corresponding intuition behind of it.

But pragmatic in general is probably a subject matter of philosophy
or may be of philosophical logic, but not of mathematics.

Besides considering pragmatic, mathematicians may, e.g., make some
measurements, physical experiments related with their mathematical
research. Will you include this also in mathematics? Let us better
restrict ourselves to formal systems (together with corresponding
intuition and not forgetting why we are investigating them). That
is more than enough for mathematics.

> >
> >
> > 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'.''
> >
> >
> > I think, without a correct philosophy of mathematics it is impossible
> > to define what it is, and vice versa.
> You may be right, but what do you mean by a ``correct philosophy'' of
> mathematics?

I am not a philosopher, but rather a theoretical computer scientist.
I have no intention to give a general definition of a correct
philosophy. For me a philosophy of mathematics which puts illusions
into its foundation is unacceptable. I prefer a realistic (in the
normal sense of this word!) explanation of the nature of mathematics.
I see quite real things in mathematics - formal systems. In that or
other form they always were present and I cannot imagine
mathematics without (explicit or implicit or partial) formalisms.
These formalisms seem to me rather solid ground and starting point
for understanding its nature. Other approaches which I ever seen
appear to me unclear.

This does not mean that, given a formalism like ZFC or Heyting
Arithmetic with Formal Church Thesis and Markov's Principle,
or Lambda Calculus or anything else, we cannot have corresponding
intuitive views on their semantics or ontological status. Say, we could
discuss in which sense Axiom of Choice or Continuum Hypothesis or may
be Anti-Foundation Axiom of P.Azcel may be considered as true or
plausible, reasonable (the latter axiom is actually more interesting
for me; e.g. we could discuss relation of AFA to querying World-Wide
Web). But, of course, not whether they are "really" true! What does
it mean this "really"? There are so many possibilities in mathematics
for various intuitions and for relations to our reality!

> Then, even once you have the notion of ``correct philosophy''
> down, what you have said here (vice versa) makes it impossible, in practical
> terms, to obtain either a definition of mathematics or a ``correct
> philosophy'' of mathematics, since your claim is that each requires the
> other.  (BTW, if your claim is correct, then this claim itself is part of
> the ``correct philosophy'' of mathematics, is it not?)

I do not see here any serious contradiction. Or should we go into
fruitless discussion like what was the first egg or chicken?
I think, you over-complicate the story.

I say that I do not believe in objective existence of Platonic
world, etc. and suggest a kind of a clear and meaningful formalistic
view on mathematics (probably not so original) which is seemingly free
of illusions and quite realistic and which completely includes what
we already have in contemporary mathematics, except some strange
beliefs in an ABSOLUTE. If you want to have a feeling of something
*relatively* (99%) absolute, just work in ZFC. But in principle, you
can work in *arbitrary* formalism whose relation to ZFC and in general
to any ABSOLUTE may be problematic.

> > 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.
> > ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
> > ++++++++++
> >
> > ++++++++++++
> >
> >
> >
> > Why a definition of mathematics should be "mathematical itself"?
> Let me revise my comment to the following:  I guess this might make a good
> Webster's type of definition, but it is hardly rigourous.

Rigorous = mathematical. I repeat, that I cannot give mathematical
definition of mathematics.

Do you really not understand the definition or you are only
unsatisfied with its form (circularity, as you say, etc.)?

Why not concentrate on the content instead of a form?

> My reason for desiring a rigourous definition is that others (non-rigourous
> ones) are not definitions at all.
> >
> > 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.
> Here is a ``Webster-type'' definition of engineering:  1.  The application
> of scientific principles to practical ends as the design, construction, and
> operation of efficient and economical structures, equipment, and systems.
> 2. The profession of or work performed by an engineer.
> Now, to study engineering formally with this definition, one must find the
> definitions of the other words used in the definition, etc.  Of course, in a
> Webster's dictionary, there will eventually be a definition cycle, in which
> one must know the meaning of some term t in order to understand the
> definition given for t.  These terms are essentially ``undefined'', even
> though the dictionary gives a definition for them.  Of course, one may take
> all such terms as undefined in the formalist sense, so that they may be
> applied to anything one wishes to apply them to.

By the way, this is a caricature on the formalism.
I think, Hilbert just exaggerated to clarify only some
particular point. The goal of formalism is to formalize
*something*. But this "something", unlike the corresponding
formalism may be not sufficiently clear to be put into the
foundation of mathematics.

Note, that when I wrote that mathematics is a kind of engineering
it was only an auxiliary explanation of how and why mathematics
creates formalisms, that these formalisms should be useful and work
*like* engineering devices. This topic could be put aside or
postponed for further comments on formal systems. The main point
is that mathematics is a science on formal systems or on
formalization of anything what could be formalized. Mathematics
creates and investigates (meaningful) formal systems.

What is circular here?

> My experience with natural
> language dictionaries is that almost every term I find in such a dictionary
> is  a member of such a definition cycle, and so cannot be made rigourous.
> As such, it might be understandable in natural language, but cannot be
> effectively studied formally.

Why do you need to study a definition of mathematics formally?
Please, let you just read the definition and discuss it contents.

> (It can certainly be studied informally,
> however, making the study of natural language a viable subject.)  If it is
> your purpose to only give an ``informal definition'' of mathematics, you may
> or may not have succeeded, but what have you gained over Webster, and
> others?:
> Mathematics: ``The study of number, form, arrangement, and associated
> relationships, using rigourously defined literal, numerical, and operational
> symbols.''

For the Webster it is probably OK. Some important key words are
mentioned in a suitable order to give some impression.

But what is the nature of mathematics? How does it work?
What is its main point? Number? Form? Symbol? Rigor?
Formal systems include (at least implicitly) essentially what
is necessary and sufficient for doing mathematics. For those who
doubts whether Chess game also belongs to Mathematics let us
recall that formal system should formalize something and should
(like an engineering device) make our thought stronger (as some
arithmetical calculation rules from the beginning school).

> > Mathematics deals with formal systems ("devices" of a special kind).
> > This is the main point.
> By ``devices'', do you mean certain idealized, or actual physical objects?

Both. But usually we write our formal rules just on a sheet of paper.
Or use our physical brains or computers.

> If you accept the ``existence'' of ideal ``devices'', then you are a
> Platonist.

I accept ANY kind of ideal objects. But I do not pretend to consider
them outside of any formalism, just in a "vacuum". That is why I am
not a Platonist.

> If you do not, then you must produce, physically, every such
> device (formal system) that you claim exists.  Thus you are a
> constructivist, finitist, etc.  How do you even use FOL at all, in this
> case?

Usually by writing corresponding symbols on a paper.

By the way, I am not constructivist, finitist, ultrafinitist, etc.,
because, in principle, I accept any reasonable (interesting,
meaningful, with any kind of meaning) formalism with or without
the law of excluded middle, etc.

> Previously, you appeared to question the existence of the number
> 10^10^10, presumably because it is so enormous as to be physically
> unattainable.

Not only by this reason. 1 also does not exists as we already
discussed. However, one pebble exists. But of course 10^10^10
of pebbles do not exist. Nevertheless, when working in PA,
I can *imagine* existence of 1 and 10^10^10, etc. However, you
seems believe in existence of this number just in a "vacuum".

> Now you wish to allow the formal system of FOL to exist,
> although it has formulas of length 10^10^10^10 in it?

Why did you conclude this? FOL is a system of formal rules in a
language defined also by some rules for WFF. If you are able to
write down a formula of such a "length" then it exists *really*.
Please write it if you can!

On the other hand, if you consider FOL in terms of an (imaginary)
ZFC universe or the like then you can formally prove that formulas
of length 10^10^10^10 exist. But this has no relation to the *real*
existence. It is the characteristic feature of Platonists to mix
reality with illusions.

> This makes no sense.
> If the number 10^10^10 does not exist, then neither does a formula of that
> length, for how can an existent formula have a nonexistent length?  This
> destroys the closure property for your formulas under the conjunction,
> disjunction, negation, etc, operations.

Are we discussing real formal systems or imaginary ones
"living" in an imaginary world of ZFC?
Please, show the *real* place for a *real* FOL in the *real*
world where the closure property is destroyed.

> Suppose you attempt to get around
> this problem by developing a new and improved FOL in which abbreviations are
> allowed.  When a formula gets to be a certain size, so that its negation,
> for instance, would have a nonexistent length, the formula is
> ``abbreviated'' by a ``new'' symbol.  However, this cannot be iterated for
> as many as 10^10^10 times, so one must stop before this.  Thus all the
> formulas, including those which are abbreviations, can be constructed in
> less than 10^10^10 time periods of at most 1 in 10^10^10 parts of a second,
> but many cannot be used, because when that much time has passed, there is no
> more time left, since such times are described using numbers larger than
> 10^10^10.  Thus, in one second (10^10^10 iterations of these very short time
> periods), all of useful FOL has been constructed, but now cannot be used,
> because there are no other seconds left in which to apply them.  This is
> paradoxical, to say the least.  You may object that ``time'' should be
> quantized differently.  Please tell me how.  What is the shortest unit of
> time?  Is it not 1 part in n seconds, where n is the ``largest integer'' in
> your formal system which identifies 10^10^10 with some infinite number?  If
> not, why?  This consequence of the formalist position brings us to the
> question of ``truth in the real world'', even when we wish to know what our
> formal system actually is, which we must know, or else, the formal system
> does not exist.

Your questions are of course somehow suitable, but mostly
inappropriately stated. I hope that the latter conclusion may be
clear after my comments above. You could also read my paper

V.Yu.Sazonov, On Feasible Numbers, in: Leivant D., ed.,
Logic and Computational Complexity, LNCS Vol. 960,
Springer, 1995, pp.30--51.

available also via

> >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?).
> Knowing the existence of absolute truth is not the same as knowing what that
> absolute truth is.  Thus claiming to know there is absolute truth, does not
> constitute a claim that one knows what is absolutely true.

Yes, of course! But claiming to know there is absolute truth
is useless and meaningless. Also by having some canonical samples
of absolute truth and a "general direction" for this truth Platonists
thereby restrict mathematics to following these samples and

On the other hand, why do not claim something more realistic and
verifiable/refutable? Strictly speaking, I only say that I do not
believe to those who is claiming to know there is absolute truth
(let me restrict - in mathematics).

> However, the
> claim that there is no absolute truth is formalizable as a refutable
> formula, and as such, is untenable.  For an informal argument, suppose
> person A claims that there is no absolute truth.  Is the claim of person A
> true or false in the real world?  If it is true, then it is absolutely true,
> being true in the real world, and this is contrary to the claim of person A,
> so this claim is false in the real world.  One may claim that the denial of
> absolute truth is here restricted to mathematics, and hence not applicable
> to ``truth in the real world''.  However, the notion of ``truth in the real
> world'', while perhaps not definable, is formalizable, via Tarski's
> definition of satisfaction, and, as such, is a part of mathematics,
> according to your definition.

Sorry, I think that these quasi-logical exercises have no serious relation
to our discussion. You take one isolated phrase "there is no absolute truth"
and make some formal manipulations with it.

The relation of Tarski's definition of satisfaction with the truth
in the real world is not so clear or direct. And I see no reason to
conclude that ``truth in the real world'' is a part of mathematics.

> >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.
> Ah but the claim that the Platonist position is unreasonable amounts to a
> pretention to know what is reasonable or not, or at least to know (some of)
> what is not reasonable.

Why not?

> It seems to me we do not (completely) know either
> way at this point in history.  The only currently clear criterion for
> ``reasonableness'' is the criterion of consistency.  Others, such as
> ``effectiveness'', ``recursiveness'', ``feasibility'', etc., place
> mechanical bounds on the apparently nonmechanical human mind, and so are
> themselves unreasonable limits on human endeavours such as mathematics.

I do not understand you notes on mechanical bounds, etc. and on
their relation to my views.

Let me repeat, a formalism is reasonable if it formalizes
something in a good way (and also if it has some reasonable
relations with other reasonable formalisms so that all of them
together play well and can be included in mathematics).

> > 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.
> Would you say then that it is possible that Cohen's theorem, namely that CH
> is independent from ZFC, is not ``absolutely true''?  If that is the case,
> what kind of truth value does it have?  Do you have in mind a model of
> reality in which this theorem is formalizable, but fails to be true?  I know
> of no way to come up with such a model.  I believe no such model exists,
> either in physical reality or abstractly.  Now, this particular theorem is a
> theorem of mathematics, being a theorem of mathematical logic, and is
> absolutely true.

In general, if a theory T is (believably) consistent then any theorem
of T having finitary physically verifiable interpretation may be
physically believable because possible counterexamples evidently
(actually G"odel and Cohen presented corresponding algorithms)
lead to a contradiction in T. In this sense independence of CH from
ZFC is physically true (believable). I do not see any reason to use
here the words ``absolutely true''.

But why your questions like this are related with concrete objects
(proofs in a formal system)? Why not ask about continuous and nowhere
differentiable function, or any other physically non-interpretable
and non-verifiable theorems/hypotheses (like CH)? Do you want to
force me to agree that there are some true statements in the real
world, like the laws of physic? Of course I agree. But if you will
assert that, e.g.  odd + 1 = even is an absolute *mathematical*
(not only physically verifiable) truth, I will say no. It is just a
consequence of some formal (however natural and reasonable) axioms.
If we would choose different axioms this may be disprovable.
(However, for this concrete goal I do not see a reason to change
Peano axioms. But in principle there may be other more interesting
reasons for that.)

What will you say on geometric "absolute truth":
the sum of angles of any triangle is equal to 2 direct angles?

> >      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.
> >
> I guess I no longer understand your notion of Platonism.  ``Naive
> Platonism'' refers to what, in particular, other than the human intuition?

Yes, Naive Platonism is just a kind of normal human intuition, specifically
that one related with using formal theories based on classical FOL.

> ``Platonism'', as I understand it, contends that there are facts that can be
> discovered and reasoned about using mathematics and its formalisms, and that
> some of these facts are purely mathematical in nature.  It does not pretend
> to know which purported facts are actual facts, i.e., absolutely true (true
> in the real world),

May be in your Platonistic imagined world? But it is not real!

> except for a small fraction of them, such as the
> tautologies, and results of FOL and other demonstrably sound systems of
> reasoning.

How do you know what is demonstrably sound? See, e.g.
my paper mentioned above. It presents in a sense demonstrably
sound ("almost consistent") system FEAS. Will you include it in
your list?

> Some Platonists may claim to know what is absolutely true beyond

> this, but not all.  Remember that the Greek geometers, Platonists
> extraordinaire, considered Euclidean geometry to embody absolute truth about
> the real world, although they considered the real world to be merely an
> approximation to the truths of geometry.

They would be right if they would consider this in an opposite way.

> However, they realized that there
> are undefined terms in any such theoretical development of abstract
> geometry, and so the theory could have other applications than the one
> originally intended.  This does not mean that the theory embodies no
> absolute truth, for there are conclusions it draws which we observe as true
> in the real world. That is, it makes predictions, as any scientific theory,
> which are then observed in the real world.

Here I would agree, if to omit mentioning absolute truth.
It is enough to say that the theory has such and such relation
to the real world.

> This does not mean that
> Euclidean geometry is the best (or only) theory for describing the universe,
> although at the time it was the best available.  However, the fact that
> Bolyai and Lobachevsky, and Gauss and Riemann were able to obtain a
> consistent geometry by denying the fifth postulate of Euclid is observable,
> or, if you prefer, demonstrable, and so is true in the real world.  This
> corresponds to a theorem of mathematics, namely that there are consistent
> geometries which deny the fifth postulate, which is therefore absolutely
> true.

I do not see here anything essentially new in comparison with
examples discussed above.

However, what about the sum of angles in arbitrary triangle?
What is here absolute?

> >
> >
> > 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.
> >
> >
> > Where you discovered this?
> Where have I discovered what?  Your comments have frequently indicated that
> mathematics not only does not know what is absolutely true, but that the
> statement that there is absolute mathematical truth does not even have any
> meaning.

Please, look more carefully on the 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).

Formal systems related with some intuition about something are
*meaningful* formal systems.

But you wrote:

> the formalist
> approach would have us believe that no one even knows what the statement
> means.

We need not have absolute mathematical truth to consider
formal systems having a meaning, e.g. in the real world.

You seems are permanently mixing the real world with you imagined
absolute Platonistic world.

> > 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?
> >
> The question is not well-posed.  If you tell me what topology you want me to
> use,

Let us temporary forget on any topology and even on mathematic and
logic! Just look on the picture and ask himself, what do you see as
a human being having only very naive idea on continuity and
discreteness. One moment you see it is continuous. Then, suddenly,
you see that it is discrete. No optical effect, say, with glasses
or your crystalline lens. Only switching something in your head.
Then, recalling some mathematics and logic, let you ask himself how
it is possible that the same picture is for you both continuous AND
discrete? It is *really* true! Will we *ignore* this real truth and
start to speak on pixels and various topologies or will we start
think more on whether our current logic and mathematics in general
is appropriate for describing this kind of reality? (Recall also
somewhat related to this example heap paradox.) Why should we
enforce some traditional logical or mathematical laws to the real
world, but not vice versa? See for example treatment of the
related informal concept of horizon by P.Vopenka in his Alternative
Set Theory by so called semi-sets. You also can find in my paper
some other, in a sense more radical approach.

> then I can, in theory, determine whether the picture represents a
> continuous function, by checking finitely many sets of pixels, a yes-no
> condition that tells me the answer to my question.  Perhaps you do not mean
> ``continuous'' in the same sense I mean?  Well, aside from saying you are
> abusing the well-defined topological term ``continuous'', I would say what
> you are asking may be ``Do the pixels occupied by the nice picture form a
> connected set or a discrete set?''  Again, I say you have not presented me
> with a well-posed problem.  The topology must be given for you to ask me
> this.  In some topologies, the picture may be connected, and in others, it
> may be discrete, and in still others, it may be neither, but in no topology
> can it be both unless the computer screen contains only one pixel.  (In any
> topology, a connected discrete set is a singleton.)  However, once the
> question is well-posed, in a true-or-false format, rather than as a query,
> it is either absolutely true or absolutely false, although we may be unable
> to determine which.
> >
> > Why should we take the formal dogmas of FOL with its contradiction
> > law (A & ~A => anything) as a rule governing the real world?
> Because it is observed to be so.

Please, observe the above example again.
(By the way, the equivalence of A => B to ~A V B
is also observed? or it is our special technical decision?)

> Where, in the real world, is an actual
> contradiction not abhorred?  When any physical theory has been appropriately
> formalized, so that it is considered correct by scientists, contradictions
> are not accepted, because they are observed to not exist in the real world.

Please, observe the above example again.

If our logic would in some reasonable way not contain contradiction
law (A & ~A => anything) then there would be possibility for a theory
to have theorem A, theorem ~A, but not to be trivial (not everything
would be provable). Such a theory could be therefore considered as
non-contradictory (= non-trivial). Probably it would describe reality
in a sufficiently smooth and correct way. Why not? Because Platonists
abhor this? But this is a dictatorship!

> Consider, for example, the fact that Physicists generally agree, currently
> that a ``Theory of Everything'' has not been found.  Why are they not happy
> with the current situation which describes certain phenomena using
> relativistic mechanics, and explains other phenomena using quantum
> mechanics?  As I understand it, the reason the ``relativity-quantum'' theory
> pair is not considered sufficient is that there are overlaps in the domains
> of the two theories at which the two theories make contradictory
> predictions.  Since the Physicists agree that a contradictory prediction
> cannot be true, they do not consider this theory-pair to be a sufficient
> description of reality.
> > 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.
> >
> I presume that by ``practical limit'', you mean ``after the computer has
> performed as many calculations as is reasonable''.

As it *physically* possible by this computer, by any other existing
computer, by any computer of the next Millennium, etc., etc.

> I do not know what
> ``reasonable'' would mean here.  Consider the recent advances of computation
> devices:  Thousands and millions of iterations are possible now, where only
> tens and hundreds were feasible only twenty years ago.  Would you then say
> that ``the (practical) limit'' of the sequence you gave is dependent then on
> the number of computations one can feasibly perform, and on how long one is
> willing to wait for the computer to decide it cannot improve upon its level
> of accuracy, and upon which computer one uses?

I do not know why do you speak about the accuracy. We discuss only
about the difference between 0 and some numbers > 1.
(Not between 0 and 0.0000001.)

A question aside: what about technical progress in construction
of cars, planes, space ships, etc, which would allow to reach more
and more velocity, up to *infinity* (contrary to Einstein)?

Do you know that the number of electrons in the Physical Universe
is estimated as < 2^1000? What are we talking about? Make experiment
and wait (yourself or also your descendants). Forget temporary even
on elementary school mathematics.

Your parenthetical comment,
> then, is quite important here, for I contend that the notion of ``practical
> limit'' that you seem to have in mind does indeed make no sense at all.

I do not know. Unlike you, I have no Platonistic picture which
would enforce me to think in a "uniquely true way" (as Lenin and
Stalin learned us in USSR). It *depends* on an appropriate formal
theory where we would resolve this question. But it surely should
not be 0 (in a corresponding realistic theory if such a theory
is possible). It should be > 1, if any. But, of course, in
any traditional theory like PA it is provable that the limit is 0,
despite the experiment.

> Moreover, because the technical definition of limit of such a sequence in
> Mathematics involves the equivalent of the idealized situation

Yes, to get something mathematical here we should find an appropriate
idealization/formalization. But which concretely? That is the question.

> of ``letting
> an ideal computer (with no lower bound on its level of accuracy) never stop
> calculating'', and observing how this ideal computer is guaranteed to behave
> (in the particular case of calculating 10/(log log n) for ever larger values
> of n), the only sensible way

Yes, yes, "the only sensible way"! Are your really sure?
This is example how Platonism restricts mathematics.

> to interpret the statement that the given
> sequence has a limit, which is some unique number x, is an interpretation
> that yields (always) x=0.
> >
> >
> > 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''?
> >
> >
> > I never said this.
> >
> I did not claim that you said this.  I only asked if it is a consequence of
> your philosophy about (the lack of) truth in Mathematics.

No, because I do not mix mathematical truth (better to say - provability)
with the truth in the real world. There may be some relation between
them (if we had such a goal in constructing a formalism), but not
the identity.

> >
> > 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,
> >
> >
> > 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.
> >
> I agree that such rules *help* make human thought stronger.  But how are
> these rules *not* serving as mathematical formalization?  You even call them
> ``formal'' rules.

Sorry, I do not understand your question.
These rules are formal and they serve as mathematical formalization.
What is the problem?

> >
> > and so can be interpreted any way one may choose.
> >
> >
> > 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.
> >
> What then do you consider (unseriously, as it were) to be ``the'' absolute
> Platonistic meaning which you claim is nonsense?  If you claim only that
> such an absolute meaning does not exist, then why?  Is it because the
> undefined terms of a formal system can be assigned any meaning whatsoever,
> and thereby yield a consistent system using that atomic assignment?  This is
> not convincing to me, because this is true of any consistent informal system
> as well, and yet there is ``truth in the real world'' which you claimed
> previously to ``respect very much''.  My only (Platonic?) claim is that
> there is an absolute (Mathematical) truth, and that our systems of classical
> reasoning are capable of determining (some parts of) that absolute truth.

When I consider a first-order theory I, as any normal mathematician,
try to imagine the world of abstract objects which this theory
"describes". Simultaneously I am trying to relate this imaginary world
with something in the real world. In general that is all.

Different theories give rise do different imaginary worlds.
Sometimes we can relate these different imaginary worlds
and even to embed in some way one into another. It is a happy
that there exist one very reach and flexible imaginary
world/formalism of ZFC into which seemingly *almost all*
other worlds/formalisms may be naturally embedded. But I see
no reason to believe in existence of one unique mathematical
world into which *absolutely all* these worlds (and may be also
worlds *not based on first-order formalisms*) will be embeddable.
Especially it is doubtful existence of a mathematical world which
is not based on any formalism at all, just existing in the
"vacuum" - AN ABSOLUTE.

I completely realize that these worlds are existing only in
imagination, as an illusion and I cannot rely on the illusion
as on a solid ground. Only formalisms are (sufficiently) solid.
Intuition is something vague. That is why it is unreasonable to
believe that any statement in a first order theory is either
true or false, even despite the formal law A or ~ A of the
underlying logic. However, it is desirable to formulate a theory
in such a way that the majority of its interesting statements
will be eventually decidable (provable or disprovable).

> >
> > 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.
> >
> >
> > 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.
> Would you say then that the meaning of the (formalizable) statement that
> ``CH is independent of ZF'' is outside Mathematics?  I would then disagree.
> Its meaning is very much a part of Mathematics, for the meanings of similar
> statements have been a part of Mathematics for a very long time.  For
> example, the (equivalent of the) statement that ``Euclid's Parallel
> Postulate is independent of the other axioms of euclidean geometry.'' has
> been part of Mathematics since Riemann, Bolyai, Gauss and Lobachevski.  Its
> meaning is clear to the educated reader, and represents absolute
> mathematical truth.

We cannot include in subject matter of mathematics a meaning of all
possible mathematical formalisms. It is sufficiently clear what
is a formalism, but we cannot say in advance anything essential
on all possible meanings. Nevertheless, when we will consider
any formalism in mathematics we will do this together with its
meaning. In that concrete case we will have some and we will pay
a lot of attention to this meaning. It was said that mathematics
deals with meaningful formalisms. And that is all. What do you
need else? To say that mathematics deals with (i) formalisms and
(ii) "meanings" (and relations between (i) and (ii))? Do you
realize what are these meanings in general? Probably instead of these
meanings you, as Platonist, need some world of abstract idealized
objects (ZFC universe or the like). But this way you either will
essentially restrict this very wide and extremely vague class of
meanings or your universe would be also extremely vague so that
it hardly could be considered as a Platonistic World you want.

Why do we need these Platonistic fictions, if everything is
sufficiently clear without them. Each time we will have some
informal meaning, whatever you need, if it is related with
some formalism (already written down or arising step-by-step
simultaneously with this meaning/intuition). That is enough.

> >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.
> >
> It may be that nobody could know *all* intuition which can be formalized in
> principle.  However, certain very simple intuitive concepts, such as the
> notion of a two-element group, have already been formalized satisfactorily.
> It is also clear that certain quite large groups can, in principle, be
> formalized similarly, and the mathematical intuition behind work in those
> groups would then be (physically) realized in such a formalization.  This is
> only one way our formalisms are meaningful, but another is that these
> groups, which may in some cases be large enough so as to make physical
> realization (in a computation system) impractical, are frequently useful in
> the physical sciences, both theoretically and for more concrete
> applications.  If the formalization of such a group had no meaning, that
> situation would be completely incongrous with the fact that those groups are
> concretely applicable.  The question of defining Mathematics does not enter
> in when trying to determine the status of the Platonistic Universe, which
> you deny exists.  For if such a Platonistic Universe exists, its existence
> is independent of our terminology used to describe it, such as
> ``mathematical'' or ``non-mathematical''.  It just ``is'',

It is only that you wish that `It just ``is''' without sufficient
grounds for that.

and we could call
> its elements anything we want to, and describe them using any language we
> wish.  The *actual* relationships between the members of this universe
> exist, and some can be known within the framework of the language we choose
> to use, to a certain extent, but others may be unknown from that ``vantage
> point''.  Moreover, the same relationships that are known from one ``vantage
> point'', using a particular language, may be known or knowable via another
> language, but they are still the *same* relationships, and there is a
> (possibly not ``effective'') translation between the two languages that
> explains how the two languages are actually ``talking about'' the same
> *actual* objects, and the same *actual* relationships between the

It seems you forgot that all these idealized objects always arise
just from some concrete languages (formalisms). By which miracle
they became suddenly objectivised? Say, without a formalism (some
well-known rules for exponentiation) such a "natural number" as
10^10^10 could not arise at all. Everything can be understood in
terms of formalisms and their translations one to another. Sometimes
sufficiently universal formalisms, like ZFC, help to reach more
unity. That is enough. You may concentrate on one formalism ZFC,
try to make it canonical (contemporary state of mathematics allows
this) and imagine that an intended universe for ZFC is your Absolute
Platonic world. But this is just a restriction of mathematics (of
the class of all possible mathematical meanings) as I said above.
This restriction might seem not very serious taking into account
the present situation in mathematics. But even now we cannot say
that 100% of mathematics can be interpreted in ZFC.

On the other hand your reference to `certain very simple intuitive
concepts' means that your Platonistic world should consist of
these separate concepts (objects?). Is "the" ZFC universe one of them
or should your `certain' objects in a sense to exhaust this universe?
However, there is only a comparatively small number of such objects
considered by mathematicians. Anyway, you should imagine existence
of much more of them. Why should I believe that your (my) imagination
will give "in the limit" something crystalline clear. I rather
would think that it is inherently vague. (Nothing strange that in
such theory as ZFC which pretends to describe an ABSOLUTE there
are simply formulated and seemingly hopelessly independent
problems like CH.) However, corresponding formalisms are really
sufficiently (if not crystalline) clear and, as our mind is
at present essentially (self)restricted to these formalisms,
an illusion arises that we have something clear even behind of
these formalisms.

> > 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.)
> >
> >
> > 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?)
> >
> I agree that many people are not very concerned about having absolutely
> rigourous abbreviation mechanisms in place for anything at all (including,
> but not limited to, Mathematics, and Mathematical Logic).  However, I ask
> you, how does this subtlety ``make no value in everyday mathematics''?

Everyday mathematics uses only some hints how to get completely
formal proofs. These very informal hints serve as a kind of
abbreviations. They are not formalized (not fixed) and therefore
in principle this could be a source for non-reliability of
mathematical proofs. But our experience shows that mathematics
is not very sensitive to such subtleties.

In principle we could consider formalisms which are more sensitive
to some kinds of abbreviations which are allowed to use in favor
to others which are not allowed. In this case ignoring this subtlety
could lead to a contradiction, unlike the traditional mathematics.

> seems to me to make a significant difference, especially in our discussions,
> where you are denying the existence of objects I am quite comfortable with,
> merely because they are sufficiently complex as to require millions of years
> for a digital computer to completely unravel their meaning.
> >
> >
> > is considered to be
> > contradictory.  Why should ``philosophers of mathematics'' be so
> > biased?
> >
> >
> > Really, why? Someone would like to have a self-contained
> > Platonistic world. It does not exist anyway.
> How do you know this?  Have you a proof that such an absolute Mathematical
> reality does not exist?

Sorry, how do *you* know that it exists? Have *you* a proof?

How do I (and you) know that some, say, geometrical illusion
(I think you have seen such) is just illusion, but not a picture
of something existing in the real world? How do you distinguish
your dreams from the reality when you are waked up? It is needed
probably a special (religious??) training or may be a drug to lost
feeling of these differences. Of course, mathematics itself is a
kind of drug for many of us (including myself, of course). But
at lest sometimes after dreaming we should wake up.

> >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?
> >
> Platonism will still survive such a blow.  But, as with the ``crisis'' of
> Russel's Paradox in Fregean set theory, the view of what actually exists
> will change.  This does not mean that what actually exists will change, only
> our understanding of what actually exists will change.

Sorry, I do not feel this convincing and even understandable.
Some pursuit a ghost. Your comparison with physics does not
help me.

> This is similar to
> the Physical Sciences, in which there is an *actual* physical reality worthy
> of our study (for practical as well as academic reasons), and there are
> competing theories intending to explain that physical reality.  When the
> Michelson-Morley experiment demonstrated that the predictions of the
> aetehereal theory of light failed in the physical world, a new theory had to
> be developed (because of the resulting contradictions).  But the various
> theories did not determine what is *true* in the physical universe.  On the
> contrary, what is true in the physical universe lead scientists to prefer
> the new theory (relativity) over the old one.

Mathematical theories also are not prescribing what is true
(neither in reality, nor in an abstract world of ideas).
However, as any instrument, each of them gives a specific
way to approaching reality.

Physical theory is devoted to describing the physical world.
Mathematics in this process have very important, but auxiliary
role. It supplies physics with specific instrument of thought.
The role of an instrument - to be efficient and convenient in use.

As I understand physics, ethereal theory of light was a kind
(an analogy) of Platonism. Einstein replaced this fiction by
an experimental (quite realistic) definition of simultaneity
via measurements, etc.

> > 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!
> >
> >
> >
> > 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.
> >
> How is it dangerous?  How is it harmful?

As I said above, it intends to restrict mathematics. That is enough.

How is it a self-deception?  (In
> anticipation of your claim that it is self-deception because there is no
> Platonic Universe, I wish to see a demonstration of this claim.  Your
> continued repetition of the claim that there is no Platonic Universe does
> not convince me.

Your continued repetition on existence of Platonistic Universe also
does not convince me.

As to demonstration - just wake up and look around. Like Einstein
asking `what does it mean to be simultaneous events happening in
different places?', ask `what does it mean this Platonistic World?'.
Let us pay more attention to reality than to our dreams.

> <snip>
> >
> >
> > JoeShipman at
> >     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.
> >
> >
> > It is only illusion of an explanation. It is a declaration
> > but not explanation.
> It is as much an ``explanation'' as the theory of relativity is an
> explanation of certain observed phenomena in the physical universe.  What do
> you require for an ``explanation''?

Yes, explanation is a *theory*, like the theory of relativity.
It should consists of some formalism and supporting experiments.
Again, as I understand physics, actually there was no theory
of ether in this sense. It was a fiction from the very beginning.
The concepts like electrons, etc. are also a kind of fictions.
But they are included in a framework of some physically verifiable
and actually verified formalisms. I do not know how to include
Platonism in the form you are advocating into a (verifiable? in
which sense?) formalism. I see here nothing except a self-deception.

Traditionally, an hypothesis is said to
> ``explain'' an observed phenomenon if the given hypothesis predicts (i.e.
> implies) the observed phenomenon.  Thus, supposing that an actual Platonic
> Universe does exist, one would expect, on quite rational grounds, that the
> Mathematics developed by many different Mathematicians would tend to
> demonstrate the unity (i.e. mutual consistency and interpretability) to
> which Professor Shipman refers.

Yes, there is an observed phenomenon of mutual consistency and
interpretability. A reasonable and simple explanation for that
(involving no fictions) consists in a lot of work done by
mathematicians to relate and unify various theories they created
(especially taking into account that before this unification
there were well-known problems with mathematical rigor in
some of their theories; this makes the value and role of the
resulting unified theory extremely high). It seems that having
this goal we would be able to reach it to some extent. We could
get to one or may be two, three or some more unifying formalisms
for various parts of mathematics. Nothing strange. What is strange
(but explainable) that we have at present essentially only one
such formalism ZFC. Probably this is only a temporary state of
affairs because essentially new interesting formalisms not
reducible to ZFC may be found. But when one such a unifying
formalism have been discovered a strong tendency between
mathematicians arises to work only in such formalisms that are
reducible to ZFC. It becomes a kind of a forced standard (like
some standards in programming practice - a result of big market).
Simultaneously an illusion arises of a unique Platonistic World
of Mathematics. This illusion is just a *consequence* of achieved
(I think, temporary) unification, not vice versa.

> >"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.
> >
> >
> And just how do you propose that one ``realize this fact''?

Essentially in the same way as for ZFC: by translation of
numerous reasonable formalisms of computability into
formalism of TM, by getting corresponding experience
which allows us to predict without a proof for which
kind of formalisms it is possible.

I do not
> consider the Church-Turing Thesis as self-evident.

Yes, it is not self-evident, but it is plausible.
I do not see what is the point and the reason of your objections.

I need no Platonism to Realize Church Turing Thesis.
Do *you* need Platonism for that? Analogously I need
no Platonism to realize that ZFC is sufficiently universal
formal system. Just a lot of examples demonstrate this.

By the way, I see essentially no difference between this
Thesis and another thesis: "epsilon-delta definition
of the continuity notion is a reasonable formalization
of this notion". Analogously - for the notion of topological
space, metric space, integral, holomorphic complex
number functions, etc. Especially holomorphic functions
are analogous - they have several quite different equivalent
definitions confirming that the notion is really very
reasonable one. The main difference with Church Turing
thesis - a very large class of various formalizations
of computability prove to be equivalent (e.g., in PA;
we should fix where; in weaker theories like Bounded
Arithmetic we should be much more careful, say, in
defining what is Turing computability to have a universal
Turing Machine satisfying s-m-n and fixed point theorem!).

If it were self-evident,
> then no philosophy (at all) would, IMHO, be needed to realize it.

Which philosophy? Normal mathematical intuition as in any
branch of mathematics and proving several appropriate theorems.

That is,
> its self-evidence would transcend the Formalist-Platonist debate, due to its
> obvious nature.  (Of course, if it were self-evident, then it would be an
> absolute Mathematical truth, which a Platonist would accept as fact, but the
> Formalist would (presumably) reject as utter nonsense.)
> Regards,
> Matt Insall

Best wishes,

Vladimir Sazonov

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