Re: Cantorian pseudomathematics

Jean-Claude Arbaut wrote:
> 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.

I very much agree.

> > 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.

The part about risks seems to be right, at least as common sense. I
don't know how it would play though if we tried to give the statement a
more precise formulation.

> > 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.

It stands to reason that if you splurge on some criteria then you'll
have to skimp on other criteria.

> > 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...

Right, unless someone can show us a best of both worlds formulation.
I'm not aware that any set theorist wouldn't be interested in such an
achievment.

> > 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.

I don't claim that there is a great proportion of mathematical study
that is about comparision of formalizations, but there is much of it.
One could spend a lifetime and still not get caught up to the field.

> > 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" ?

Just to be clear, I wasn't claiming that your critiques are among those
I was referring to.

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

Not to you, I understand. But, as I mentioned, they are to others.
Moreover, though I am not prepared to flesh out this subject, I should
mention that there are proponents of transfinite theory who point to
results in transfinite theory that illumine the more quotidian studies
also.

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

The search is always on. And results are always coming in.

> 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.

I don't know about mathematicians in general, but the sense I get from
perusing the dialgogues of those in mathematical logic and set theory
is that generally logicians and set theorist are not so doctrinaire as
to regard a particular formalization or axiomatization as if it were a
religious or political doctrine.

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

I want to learn more about alternatives too. There's plenty to study
about them. Unfortunately, at this point I'm still trying to get up to
speed on some very basic mathematics.

> 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.

Even after the assumption of a complete ordered field, you still need
to talk about all kinds of set operations on them, and plain calculus
talks about functions that have infinite domains almost continually
(pun accidental).

> 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.

I don't see it as a game. But, at least for me (and for plenty of other
people), I understand mathematical concepts structurally. I see, for
example, that the von Neumann construction captures the Peano
structure. Actually, mathematically literally in this case. The von
Neumann naturals are a Peano stucture. And I see how the Peano axioms
capture the essential structure of plain ol' counting numbers. The von
Neumann naturals look funny with those nested curly braces, but that is
an artifact I can live with, since I'm interested in the axiomatic
development, the rigor, and what is captured structurally, which is
deeper than a naive literalism that demands that natural numbers are
not "really" nested within their successors.

> > 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 ?

You understand that I'm not talking about you (you're not demanding the
abandonment of set theory for no alternative nor making ridiculous
claims).

MoeBlee

.

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