Re: Cantorian pseudomathematics

MoeBlee wrote: 
Jean-Claude Arbaut wrote:

MoeBlee wrote:
And to start the theory. It has not come out of nowhere :-)


I don't know what you mean.

I answered to "Inutituion is used in conceptualizing things and in figuring out ways to prove things.". What I mean is set theory was not born just by the great thought of Cantor. There ware thousands of years
of mathematics behind, which help greatly, and which even where necessary for the theory to be imagined.

A formal system is not only a formal system; it has an history, and it seems preferable to take it into account, at least to understand why *this* theory and not another.

The only disargeement is about which axioms are to be chosen at the
beginning.


There are some criteria we can use. There are people who disagree about
some of these points, but in general:

The axioms have to be consistent.

As to a consistent recursive axiomatization that proves all the truths
of arithmetic, the incompletenss theorem tells us that we can't have
that. So we have to do the best we can. So we want a formalization that
gives us at least familiar arithmetic and that formalizes analysis and
that can formalize geometry. I think we should also be able to use the
formalization, at a level up, for meta-theory.


Reasonnable.

Economy is helpful. We'd like to keep the primitives and axioms to a
managable list. (Z set theory has an infinite number of axioms, but all
expressible in one schema.)


Still reasonnable.

We'd like the axioms to be as straightforward if possible. Axioms that
can be regarded informally as making pretty basic assertions are
preferred.


Highly recommended

Some people strongly prefer, or demand, that the axioms express or
prove only the mathematics we set out to prove and not statements about
things like uncountable sets that are considered by some people to be
unwanted artifacts.

Yes, strongly prefered. Two reasons for that: artifacts are not a priori necessary if the theory can still prove all we want to (however there may always be artifacts), and with fewer artifacts, there are also fewer risks that the theory be inconsistant.

However, if one demands that this criterion be met,
then one has to settle for compromising other criteria or show a
formalization that does not compromise the other criteria.

That's the interesting question - the one I really would like to know the answer of.

Some people don't mind that the formalization engenders unexpected
areas of study such as that of transfinite sets, and some people will
even add axioms to enrich the  investigations of the deductive strength
of the axioms and the conceptualizations that are thus suggested.


Crazy kiddies :-)

Some people strongly prefer, or demand, that the formalization have
fidelity as an  analgoue to the physical world or to some other concept
of a paradigmatic state-of-observable-affairs.


Again, strongly prefered.

Again, if this criterion
is demanded, then one has has to settle for compromising other criteria
or show a formalization that does not compromise the other criteria.


Everything has a cost...

But mathematicians do not have to commit to just a single
formalization. Indeed, much of mathematics is the study of comparing
formalizations.

Not really "much of mathematics", or I really don't know mathematics well. Err, I don't pretend to know them *very* well either.

It would be a misconception to think, for example, that
study of set theory commits one to dogmatically holding to set theory
or to any particular system (the many versions of Z set theory, NBG,
Morse-Kelly, type theory, second order logic, NF, et al).

No, not at all. In fact, they are probably the most innocent :-) Other mathematicians use set theory in their everyday work, and some of them even try to get interested in the implications, that's pretty good.
I don't accuse particularly mathematicians. In fact, I'm interested in mathematics mainly for fun, not to make people follow my views.

Morevover,
most naive critiques (such as found on the Internet) of formalism and
set theory are ill-premised and ignorant of what formalism even is and
of the range of thought that formalism allows.

I... think I have *some* knowledge about that. Just some. Anyway, my critiques are not mathematical - as far as I know, nobody has found a valid mathematical critique. May I dare to say mine are more "philosophical" ?

Moreover, these naive
critiques are oblivious to other perspectives such as consequentialism
and structuralism (I mean 'structuralism' in the mathematical sense,
not in the sense of the linguistic or literary schools of thought). And
many of the naive critiques blatantly conflate formalism and realism.


And for that matter, nothing really suggest set theory. 


What suggests set theory is that notions of property, class, and
relation are so fundamental, even before the advent of set theory per
se, to logic and mathematics. Also, what suggests set theory is that it
seems that from just the two primitive notions of identity and
membership we can express virtually all of mathematics


That is appealing, I agree. Consequences are not all as appealing.

(and even
identity can be formulated in terms of membership). And what suggests
set theory is its success. It gets the job done, as in the sense I just
mentioned of so economically formalizing mathematics.

One may hope the same success without the "artifacts". This is a reason to search.

I mean, a geometric definition of real numbers would be as acceptable -
introducing an axiom for continuity.



Fine. Anyone is welcome to give rigorous and conistent formalizations.
But since you've mentioned this, it seems more natural (at least to me)
to discuss geometry in terms of sets than vice versa.

I can understand that. Construction of geometry from real numbers is very beautiful.

Lines and planes
are sets of points. Points are elements in lines and plances. Figures
are sets of points. Points are elements in figures. We don't even need
'point', just 'is a member of', and we define everything else.

That looks natural nowadays. When Cantor proved that a square has as many points as a line, that didn't look as natural to mathematicians.
It's probably a matter of habit.

This is the way I think set theory can be criticized: not by finding an
hypothetic contradiction, but by proposing other mathematics. Why should
all Earth stay on one little (though powerfull) set of axioms ? ;-)

It's not the fault of set theory that people prefer it.

It's true. One could claim thatit's the fault of education, but I would not dare: I know perfectly well the traps behind such a claim, and they have been observed with the emergence of the theory of evolution in school, in some countries. Therefore, I won't make a big critique on that matter: I just noticed that during a large part of math education (until 2nd year of undergraduate studies), set are not much useful to
introduce concepts. They are mainly useful to prepare students for topology and measure theory.

If there's a
better system that is also so MUCH better that it is worth the
"migration costs" (as software engineers like to say),

It's very likely the migration cost would be very low: actually, apart from some branches especially set-centered (topology for example), mathematics don't differ so much from pre-Cantor maths. I discovered that by reading books just preceding Cantor's work, and those just following. Jordan's analysis course greatly resemble the equivalent we have nowadays (it made already some use of sets, in 1909).

then people will
switch. And there is a lot of work in category theory and other
systems.

Set theory didn't get success immediately either. And nobody his asked to switch.

It would be a misconception to think that people have to
choose a formalization like they choose a religious denomination or a
political party.  That seems ridiculous to me. On the contrary,
mathematicians study the relations, the similiarities and differences,
among systems.

While mathematics can cope with multiple theories, I doubt mathematicians could. Few try to get interested in today's alternatives, for example, and nobody can expect that. Therefore, yes, it is like a religion or a political party, at least when it concerns everyday work.

Obviously, the usual check applies: make sure any other theory is not
too far from reality.


A mathematical theory that doesn' formalize the mathematics needed for
scientists and engineers doesn't have to be used by engineers.


I wouldn't be used, anyway.

A theory
that causes scientists and engineers to come to incorrect predictions
and to build bridges that collapse will not be used by scientists and
engineers.

And, honnestly, I don't think it would be a great idea for mathematicians to use it. That would be a waste of work. Just see how engineers use mathematics, sometimes years after the concepts has been introduced. It would be a pain to loose that. But amazingly, they never used anything closely related to set theory, or I'm not aware.

Meanwhile, set theory axiomatizes the mathmatics that is
used by science and engineering and we don't hear too much about
bridges collapsing because an engineer was misled by theorems about
transfinite sets.


True.

You can dispose of infinite sets.

Of course you can. But at what cost? 

That's what I'd like to know, honnestly.

What formalization does one
substitute for formalizations that have infinite sets? What theorems
can be derived from these formalizations? And how straightfoward are
the formalizations? That is, how much more complicated are the formulas
of the alternative formulations?

Certainly not too complicated. Euler's work is mostly independant of any formalism, for example (as long as the formalism is strong enough to express this work). It seems to me that most parts of analysis follows from the construction of real numbers, which in turn doesn't need the full set theory. As I said, a geometric approach may be enough.

I admit that, in principle, there
should be no finite upper bound on how complicated an acceptable
formulation can be. But one cannot have unreasonable expectations that
mathematicians will embrace what is not intellectually feasible.


Certainly.

Since any choice is a priori usable, it won't hurt
if it's based as much as possible on intuition. For example, actual
infinity *could* be discarded: it's just an hypothesis,


In a sense, it is just an hypothesis. But it's not even meant to be an
hypothesis for empirical testing. It is an abstraction from which other
abstractions can be derived as certain of the derived abstractions
"code" the mathematics used for science.The "coding" is given by the
structural relations among the abstract constructs, not by a
literal-minded indexing of observable objects and events with
mathematical symbols.

But, again, it won't hurt if one can make such a relation between real objects and mathematical ones (as close as possible, needless to say).

The structure of the relations is what is needed
to be captured, especially since that gives the very generality that
mathematics needs to provide the sciences.


Formalization gives the objectivity that all proofs are, at least in
principle, mechanically checkable. For me, that is a tenet that IS a
doctrine. If proofs do not admit of objective verification (which can
be as physical as looking at markings on a *** of paper or as
abstract as Turing computability) then we must be content with the
situation in which one man's handwaving is another man's direct
perception of the self-evident.

If I understand what you been by relations and structure, I'm not sure I'll agree that's the most important in mathematics. I don't like to see mathematics as a game between relations. It's beautifull and ugly at the same time, and too far from reality for my taste.

Moreover, the formulas can be formally interpreted through the methods
of models. And the formulas can be informally understood to express
abstract relations - abstract structural relations that "code" the
essential properties of things like numbers and shapes that the
sciences rely upon. If one proposes that the formalization should
generate ONLY those formulas that have a literal scientific
interpretation, then fine, but, at a certain point one is tempted to
invoke another time honored heuristic principle of the sciences: PUOSU.


PUO-what ?

MoeBlee

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