Re: English question...
- From: "Brian M. Scott" <b.scott@xxxxxxxxxxx>
- Date: Thu, 19 Jul 2007 00:45:58 -0400
[sci.math dropped]
On Wed, 18 Jul 2007 15:36:41 -0700, Seán O'Leathlóbhair
<jwlawler@xxxxxxxxx> wrote in
<news:1184798201.112400.73100@xxxxxxxxxxxxxxxxxxxxxxxxxxx>
in sci.math,sci.lang:
On Jul 17, 6:57 pm, "Brian M. Scott" <b.sc...@xxxxxxxxxxx> wrote:
On Tue, 17 Jul 2007 03:35:00 -0700, Seán O'Leathlóbhair
<jwlaw...@xxxxxxxxx> wrote in
<news:1184668500.078461.320550@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>
in sci.math,sci.lang:
On 17 Jul, 04:11, "Brian M. Scott" <b.sc...@xxxxxxxxxxx> wrote:
On Mon, 16 Jul 2007 20:57:14 -0000, <jwlaw...@xxxxxxxxx>
wrote in
<news:1184619434.256354.272780@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>
in sci.math,sci.lang:
[...]
How many finite groups without proper non-trivial
subsets? A countable infinity. How many finite groups
of any type? A countable infinity. So, in a sense
there are just as many groups without proper non-
trivial subsets as finite groups of any sort.
In a trivial and rather pointless sense,
Maybe but it is hard to formalise other ways of dealing
with infinite sizes.
I don't agree.
Can you find a maths book or other reference which
formalises infinity in your view of infinity?
Eh? My notions of infinity are the usual ones. I already
gave you one formalization that works here: asymptotic
density. Another way of dealing with infinite sizes, albeit
probably not particularly useful in the present specific
context, is measure theory.
[...]
More important, however, I don't see why on
earth you're fussing about formalizing it. It's bloody
obvious that cyclic groups of prime order are extremely
atypical of groups in general, and that informal observation
is all that's needed to answer the original question about
the use of 'but'.
I don't feel that fuzzy informal intuitions about infinity
have any place in serious maths.
(1) This particular observation isn't an intuition; it may
be difficult to formalize, but it's an obvious fact.
(2) It isn't specifically about infinity; it's about the
range of possible structures of groups.
(3) Have you ever listened to research mathematicians
talking shop? Informal intuitions are *very* important in
developing ideas. Published papers, unfortunately, rarely
show this, partly because the standard style doesn't really
allow for it, and partly because space in journals is very
tight.
Not as I learnt it or taught it anyway.
A very important part of a teacher's job is helping students
develop useful ways to think about complex concepts, and
some informal intuitions are extremely useful. For
instance, there's a useful heuristic that if something is
provable for two widgets, it's quite likely to be provable
for any finite number of widgets; it's very hard -- probably
impossible in any useful way -- to specify the exact
conditions under which this heuristic actually applies, but
the heuristic is still useful, and with experience one comes
to recognize when it's likely to apply.
[...]
The trouble with asymptotic densities is that they
typically depend on the sequence used. If I iterate
through the groups in a different sequence to you I can
get a different answer. In fact anything I want between
0 and 1. I guess that you will argue that your sequnce
is somehow the natural one but I could say that it is
just an arbitrarily selected one that achieves the answer
that you want.
You could, but you'd be claiming that ordering by
cardinality was an arbitrary choice; this would look a bit
silly coming from someone who was arguing for the importance
of cardinality in the first place! In fact it's a natural
order for which a real case could be made -- possibly not
the only natural order, but certainly *a* natural order.
Fair enough, it is a natural order but unless it is the only natural
order then it does not give a unique value to the asymptotic density.
In a few particular cases, you can select an obvious natural order but
I don't think that the notion can be generalised or that two different
mathematicians could be relied upon to always select the same natural
order. Another might order the groups based on their decomposition
into simple groups.
I frankly don't think that any serious mathematician would
dispute the assertion that the groups of prime order are
atypical in the first place!
Anyway, my only point in this area is that intuiiton and
infinity don't go well together.
And I don't really agree. *Untrained* intuition has
trouble; my intuitions, on the other hand, have been pretty
decent for decades. And even untrained intuition has
trouble only because it makes the fundamental error of
failing to realize that there's no reason to think that
infinite sets should behave like finite ones.
My experience of teaching infinite cardinals is that
typical intuitions are poor.
Sure, but typical intuitions *are* untrained. That's why
I'm careful to emphasize from the start that most people's
intuitions are based entirely on experience with finite
sets, and that they can't reasonably expect infinite sets to
behave similarly. (And I've managed to teach some of this
material to students in liberal arts math courses, to
students whose formal mathematical knowledge ends with the
arithmetic of fractions and basic algebra.)
It normally takes quite a
while to convince a student that N, Z, and Q all have the
same cardinality and once you have achieved that, you
surprise them again by showing that R is bigger. I don't
remember any student ever correctly guessing which of
these sets had the same cardinality at the beginning of
the class.
I don't think that I ever had to guess: by the time I was
actually thinking about comparing the sizes of infinite
sets, I knew these results and their proofs. That would
have been by the time I was 15 at the latest.
Are you familiar with the Continuum Hypothesis and its
status?
Of course; I've even been through proofs of both its
consistency with ZF and its independence therefrom (though I
don't really recommend Cohen's little book for the latter:
there are better ways to do it!).
Can you call that intuitive?
Sure: why should the axioms of ZFC determine the value of
the continuum?
Intuitively there are fewer even numbers than all whole
numbers, obviously half as many but, unlike cardinalties,
you cannot use this to prove many things since the even
numbers can be put in one to one correspondance with all
integers. Hilbert's Hotel is a good informal
illustration of the oddities of infinity .
Which I don't find particularly odd: I've worked with such
things for so long that this stuff seems perfectly ordinary,
and natural and not the least bit counter-intuitive.
Then you are rather more experienced that the intended
subject of the original question. It was clearly for
someone fairly new to group theory.
Yes, but I never did find Hilbert's Hotel odd. Amusing,
perhaps, but not odd: that's so obviously simply the way
things work.
[...]
"I agree that "but" reads better but "and" also sounds
acceptable to me."
It's acceptable, but the original is substantially better.
So, we don't differ that much.
Not on this, at any rate.
Can we compromise on "significantly" better?
I don't mind.
"I would agree that these groups are very few and special
but it is results like this that give me this intuition.
Since the student is being asked to prove this result, he
may not yet have this intuition and the more neutral
"and" phrasing may be preferable to avoid biasing his
expectations. Let this result begin to give him his own
intuitions. "
I strongly disagree. The result should be phrased as a
mathematician would phrase it, both to strengthen the
student's intuitions and to teach him how to write
mathematics.
Do you learn better by figuring something out for yourself
or being told it?
I don't think that the issue arises here. To any minimal
extent to which it does, I think that the other issues that
I mentioned supersede it hands down.
Brian
.
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