Is that all there is?
Moufang Loop
membership? Seems unlikely some how. But if it is, how come it is so
useful in the sciences?
ML
Virgil
Moufang Loop <Moufan...@btinternet.com> wrote:
No one really knows why, but a lot of people have noticed it.
Even Albert Einstein has commented on the surprising usefulness of
mathematics in physics.
tony
For your second question, it is not the fact that mathematics is useful in
sciences. It is the fact that our physical universe follows the rules of
mathematics. That's why math is so applicable for sciences.
Owen Holden
"Moufang Loop" <Moufan...@btinternet.com> wrote in message
news:3D56CAB4...@btinternet.com...
IMO, The generality of our variables limit their application.
If the objects of the world are included in the range of the individual
variable, as with Frege, Russell, Wittgenstein, Quine, Carnap,
etc. then applied mathematics has sense.
The range in Logicism includes all objects, including the physical ones.
But, in the case of Zermelo, von Neuman, etc. we must presume
that application has meaning. The former way seems to me more
sensible in spite of the apparent advantages of granting only
{ }, {{ }}, {{{ }}}, etc. as values.
Difficulties concerning 'types' is also relevant, imo.
e.g. the number one is the set of unit sets for Frege.
We can say that the unit set of 'Moufang' i.e. {x:x=Moufang},
is a 1, but we cannot say that by the latter method.
The only member of {{ }} is { }.
Owen
Andrew Cady
It would be very easy to construct a consistent mathematics that is
_not_ useful to science. Mathematics is useful because mathemeticians
have chosen to explore the aspects of set membership which are useful.
--
(let ((let '`(let ((let ',let)) ,let))) `(let ((let ',let)) ,let))
Nico Benschop
>
> Moufang Loop <Moufan...@btinternet.com> writes:
>
> > Is mathematics _just_ the theory of the empty set
> > and the relation of membership?
> > Seems unlikely some how.
> > But if it is, how come it is so useful in the sciences?
>
> It would be very easy to construct a consistent mathematics that
> is _not_ useful to science. Mathematics is useful because
> mathemeticians have chosen to explore the aspects of set membership
> which are useful.
Re 'theory of the empty set':
C.L.Siegel in a letter to L.J.Mordell (1964):
" I am afraid that mathematics will perish before the end of
this century if the present trend of senseless abstraction -
as I call it : theory of the empty set - cannot be blocked."
http://home.iae.nl/users/benschop/search.htm
Re 'usefulness of math':
Indeed, I think Maslow applies here - to the effect of:
"If your only tool is a hammer,
you tend to see everything connection in terms of nails."
I guess math is so applicable, because it fits the properties
we are looking for in a desired world model (predictable).
http://home.iae.nl/users/benschop/math-use.htm
-- NB
William Elliot
Yup, from the empty set comes the natural numbers and
"all things are numbers" said some ancient Greek philosopher.
So low and behold! From nothing comes all.
Now quantum physics tell you that also.
The probablity of the universe from nothing is zero.
By quantum uncertainty, this isn't exactly zero but fluctuates.
So eventually in an eternity of nothingness with probablity near to 1 is
the universe. Ya, just multiply an infinitesmial over a long long time.
But what the heck, we know that also.
Once upon a time was no me, but now is all everywhere.
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William Elliot
> C.L.Siegel in a letter to L.J.Mordell (1964):
> " I am afraid that mathematics will perish before the end of
> this century if the present trend of senseless abstraction -
> as I call it : theory of the empty set - cannot be blocked."
> http://home.iae.nl/users/benschop/search.htm
>
by the cultural revolution of the capitlist owners of the world
will bring all of us unto bandruptcy and then into demise within
this century as they continue to systematically destroy all life
support systems to eagerly fill their overflowing coffers.
Larry Hammick
Nico Benschop:
> http://home.iae.nl/users/benschop/search.htm
> http://home.iae.nl/users/benschop/math-use.htm
Nifty pages, Nico.
This formalization of mathematics in terms of sets and ordered pairs is
motivated by mathematical needs, not physical ones, but even so it looks
pretty arbitrary. I guess mathematicians on another planet might see
mathematics differently, although their physical universe is the same as
ours.
Larry
Nico Benschop
>
> Nico Benschop:
> > http://home.iae.nl/users/benschop/search.htm
> > http://home.iae.nl/users/benschop/math-use.htm
>
> Nifty pages, Nico.
Thanks, Larry. A good word is worth a lot.
BTW: Can I visit your home pages?
> This formalization of mathematics in terms of sets and ordered pairs
> is motivated by mathematical needs, not physical ones, but even so
> it looks pretty arbitrary. I guess mathematicians on another planet
> might see mathematics differently, although their physical universe
> is the same as ours. -- Larry
There was a time, not so long ago, that causality (the basis of our
desired world view: 'repeatable' and 'predictable' as main source
for our science & mathematics) had a not neglected counter part:
'teleology' (IIRC) meaning: the future pulls, vs. the more common
'the past pushes' (deterministically). I think it was Aristotle...
That view (apart from some religious and ideological single-focus
adapts) has died unfortunately, since the question: "where do your
ideas come from?" is seldom asked, let alone written about.
Such more balanced approach (taking both seriously, instead of only
one) might yield another kind of 'mathematics', wouldn't it?
Nico Benschop
Content-Transfer-Encoding: 7bit
William Elliot wrote:
>
> > Re 'theory of the empty set':
> > C.L.Siegel in a letter to L.J.Mordell (1964):
> > " I am afraid that mathematics will perish before the end of
> > this century if the present trend of senseless abstraction -
> > as I call it : theory of the empty set - cannot be blocked."
> > http://home.iae.nl/users/benschop/search.htm
>
> The thoughless thought of the New World Order brought to us by
> the cultural revolution of the capitlist owners of the world [*]
> will bring all of us unto bandruptcy and then into demise within
> this century as they continue to systematically destroy all life
> support systems to eagerly fill their overflowing coffers.
Re[*]: The communist world was/is not much different, really.
Nor was the world just after the French revolution (Napoleon &c)
Sad, but true.
The main behavioural modes seem to be (the same as at birth):
Greed + Ignorance (about the consequences).
Next question: what to do about it...
(or just let human nature take its course, till it hits the
wall).
Herman Rubin
Larry Hammick <larryh...@telus.net> wrote:
In a sense it is arbitrary; the idea is to use a minimal
set of basic concepts and assumptions to produce something
adequate. There are many ways of doing this to get the
same "theory", and which is to be preferred is arbitrary.
Most set theory approaches start with the element
relation as primitive, with some assumptions about it,
so sets, and sometimes individuals, are the building
blocks. The "definition" of ordered pair is quite
arbitrary and not of any consequence except for proving
existence theorems; the one used by Quine is far more
complicated, but it works equally well, and in his
model, the usual one does not work.
An equivalent basis starts with the primitive notion
of the application of a function, which functions and
arguments as the "building blocks". In this, the idea
of a set or a class is defined.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
Shmuel (Seymour J.) Metz
at 08:34 PM, Moufang Loop <Moufan...@btinternet.com> said:
>Is mathematics _just_ the theory of the empty set and the relation of
>membership?
That's one approach. It's not the only approach. Even when you take
that approach, what you're talking about is the choice of tools rather
than the choice of subject matter. It's like limiting a painter to
specific pigments; that won't determine his subject matter or style.
>But if it is, how come it is so useful in the sciences?
That is a deep question, given that some of the useful areas were
considered to be of no practical use. Consider the fate of poor Hardy.
We don't know why, although some have claimed that it is because God
is a Mathematician.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
Atid/2, Team OS/2, Team PL/I
Any unsolicited commercial junk E-mail will be subject to legal
action. I reserve the right to publicly post or ridicule any
abusive E-mail.
I mangled my E-mail address to foil automated spammers; reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spam...@library.lspace.org
Shmuel (Seymour J.) Metz
at 12:52 AM, Andrew Cady <an...@jerkface.net> said:
>It would be very easy to construct a consistent mathematics that is
>_not_ useful to science.
Not as easy as you think.
Shmuel (Seymour J.) Metz
In <k2H59.90308$wh1....@news01.bloor.is.net.cable.rogers.com>, on
08/12/2002
at 04:44 AM, "Owen Holden" <oori...@yahoo.com> said:
>e.g. the number one is the set of unit sets for Frege.
That approach runs into problems.
John
Some time back, a set theorist wrote this:
> I think the empty set is used just for reasons of elegance and
> minimalism. ( To draw an analogy, the same reason why physicists try
> to have a model of the origin of the universe in which only
> "nothingness" is assumed... pace big-bang or similar theories)
> However, starting with nothing but the empty set is extremely tedious
> and therefore, many books start with a proper class of "urelements",
> objects which are not sets or classes and which have no members.
Regards,
John
W. Dale Hall
Virgil wrote:
You may be thinking of Eugene Wigner, who wrote "The Unreasonable
Effectiveness of Mathematics in the Natural Sciences," in Communications
in Pure and Applied Mathematics, vol. 13, No. I (February 1960).
If not, that's fine, of course. I was unaware of any quotation from
Einstein expressing any sort of amazement at how well mathematics works.
Dale
Nico Benschop
>
> Virgil wrote:
>
> > Moufang Loop <Moufan...@btinternet.com> wrote:
> >> Is mathematics _just_ the theory of the empty set and the
> >> relation of membership? Seems unlikely some how.
> >
> > No one really knows why, but a lot of people have noticed it.
> >
> > Even Albert Einstein has commented on the surprising usefulness
>
> You may be thinking of Eugene Wigner, who wrote "The Unreasonable
> Effectiveness of Mathematics in the Natural Sciences," in
> Communications in Pure and Applied Mathematics, vol. 13, No. I
> (February 1960).
> If not, that's fine, of course. I was unaware of any quotation
> mathematics works. -- Dale
[*] See some earlier discussion (Chris Hillman sci.math 23july98) in:
http://home.iae.nl/users/benschop/math-use.htm -- NB
Nico Benschop
>
> Moufang Loop <Moufan...@btinternet.com> said:
>
> > Is mathematics _just_ the theory of the empty set and the
> > relation of membership?
>
> That's one approach. It's not the only approach. Even when you take
> that approach, what you're talking about is the choice of tools
> rather than the choice of subject matter. It's like limiting a
> painter to specific pigments; that won't determine his subject
> matter or style.
>
> >But if it is, how come it is so useful in the sciences?
>
> That is a deep question, given that some of the useful areas were
> considered to be of no practical use. Consider the fate of poor
> Hardy. We don't know why, although
> some have claimed that it is because God is a Mathematician.
Yeah, those were mathematicians & physicists
(re: Maslow's hammer theory: - rough quote -
"If all you know, as tool, is a hammer,
then you naturally see all connections in terms of nails."
NB: "The God of mathematicians is a Mathematician,
and His Queen is Number Theory." ...