Re: Countable models of ZFC

On Oct 3, 9:55 am, Rupert <rupertmccal...@xxxxxxxxx> wrote:
On Oct 3, 9:29 am, "R. Srinivasan" <sradh...@xxxxxxxxxx> wrote:
[...]
But personally I would prefer to take George as my graduate student,
if I were a Prof. (fortunately I am not). In my view, guys like George
are more likely to radically challenge the status quo. George is not
afraid to call a spade a spade, and though he will get a lot of stuff
wrong, he *could* eventually hit the jackpot with his approach.
Whereas Rupert, like a lot of other smart guys in this newsgroup, is
(in my view) too deeply into the status quo to radically challenge it.

I'm a little unclear about what you mean by the "status quo".

I quite often have the experience that I state a theorem and then
George starts ranting and raving about it, and using a lot of capital
letters, based on philosophical misgivings. Now, there's nothing wrong
with being interested in philosophy, and having philosophical views
which challenge the "status quo", but it shouldn't interfere with your
ability to understand mathematics. There's no doubt that the theorems
I state are theorems of ZFC (for example) and this could in principle,
be checked by computer. George should try to develop the capacity to
recognize facts like those and to separate them from the philosophical
axes he likes to grind.

Now, when you say I'm not going to challenge the "status quo", do you
mean as a philosopher, or as a mathematician? Most mainstream
mathematicians don't worry about foundational issues at all. They
agree on what counts as rigorous mathematical reasoning (most of them
are perfectly happy with the law of excluded middle, impredicativity,
the axiom of choice, and so forth), and they just get on with doing
mathematics, and once something has been established as a theorem
there usually isn't much controversy about it. People who work in
mathematical logic are more likely to think about foundational issues,
but that doesn't need to affect the mathematical part of your work,
you just specify in what formal theory your results go through.

In the context of philosophy the idea of challenging the "status quo"
makes a lot more sense. But there's also less of a consensus to
challenge in the first place. I quite like Geoffrey Hellman's modal
structuralism, myself, which is a form of mathematical realism which
is quite friendly towards strong set-theoretic reasoning. I'm not sure
how much of a "mainstream view" this is. I have the feeling that
people who are more philosophical than mathematical tend to be a bit
more skeptical about mathematical realism and strong set-theoretic
reasoning than I am. But on the other hand these ideas sit comfortably
with generally accepted mathematical practice today. So perhaps in
this sense I don't pose much of a challenge to the "status quo".

But I am interested in other foundational views and am happy to
consider them with an open mind. It's just that I personally have
never found George's points to be very coherently expressed and I also
think that it's unfortunate that he seems to allow his philosophical
views to get in the way of understanding mathematics.

Aatu Koskensilta wrote:

R. Srinivasan wrote:
But personally I would prefer to take George as my graduate student,
if I were a Prof. (fortunately I am not). In my view, guys like George
are more likely to radically challenge the status quo. George is not
afraid to call a spade a spade, and though he will get a lot of stuff
wrong, he *could* eventually hit the jackpot with his approach.

By shouting at people at random? Challenging stagnant orthodoxy is
swell and good, but it is not very effectively done by accusing people
of lying, erratic capitalisation, obsessing over trivialities and so
on. If George were to formulate some coherent conception of
mathematics -- perhaps he has and is carefully hiding it from the evil
liars infecting the news -- and explained it in understandable terms,
something interesting might possibly come out of it (we might recall
Yesenin-Volpin's ideas, which have recently been studied, from a
classical point of view, in context of non-standard models). His
present behaviour only serves to make him look silly.

I agree that George's style of communication puts off people and I
have also been at the receiving end. He may eventually find that being
more diplomatic and polite will get him further ahead in life.

Whereas Rupert, like a lot of other smart guys in this newsgroup, is
(in my view) too deeply into the status quo to radically challenge it.

An absurd idea. Anyone with an interest in logic would be delighted to
come up with something truly novel, to find some hitherto unknown
problem in currently accepted notions, to devise a new way of
conceiving mathematics etc.

What I meant by "status quo" is the following. Today there is a huge
body of knowledge in almost any technical field and it takes several
years to master even some portion of it and, say, graduate with a
Ph.D. Once you have climbed that mountain you are under pressure to
look out for positions in academics and so your research proposals
must usually be oriented towards the mainstream, at least initially.
Otherwise you may not be able to find suitable positions. It may still
be possible for you to come out with radical proposals that seek to
challenge the established ideas, but only after you have "established"
yourselves for several years, if not decades.

But by then your enthusiasm and incentive to mount any such radical
challenge would have dimmed considerably, if only because you would
have already published several papers yourselves in the same areas
that you may wish to challenge. The very nature of logic and
foundations is such that once you have gotten used to one particular
viewpoint (e.g. classical logic, set theory), you more or less become
entrenched and will not easily open your mind to other viewpoints.

I also suspect that there is a lot of unjustified peer pressure on
young researchers to fall in line with "established" ideas and not
challenge them. There is already a mountain of established knowledge
in, for example, mathematics and physics. It seems to me that smart
young researchers working in foundations/logic today simply cannot
afford to question the basis of such long-standing ideas. These guys
are more or less forced to accept that this knowledge exists and they
have to find a logic/philosophy that somehow justifies it.

Whereas I think that logic and philosophy are prior to everything else
and have to be fixed without taking into account any such obligations.
It is the logic that determines what is mathematics and ;physics, and
not the other way round. However, what has been verified by
experiment, say, in Physics, could have a different justification in
some new logic. SImilarly it is quite possible that some useful
mathematical theorems established using set theory could also possibly
be justified by other means, say in a different theory based on a new
logic. Unfortunately people working from such a viewpoint will not
survive in today's academic world.

Regards, RS

.

Relevant Pages

  • Re: Countable models of ZFC
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  • Re: Countable models of ZFC
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    (sci.logic)