The role of infinity in math
Martin Johansen
How is it defined?
Why is it needed?
At what precition does math work?
These are questions which seem to have to accepted answer, what do you
think?
Arturo Magidin
Martin Johansen <mart...@is.online.no> wrote:
>What is the role of infinity in math:
>
>How is it defined?
Depends on the field of math you are working on. It means different
things in calculus than it does in set theory, for example.
>Why is it needed?
Because it is useful. In Calculus, for example, it is a very useful
shorthand for a number of related phenomena, that are important and
that we want to be able to talk about. In set theory, it is the
foundation of a lot of interesting mathematics that could not be done
without the concept of infinity as it is used there. In other areas
(such as functional analysis) it is downright essential.
>At what precition does math work?
That does not seem to be a very clear question. And it will again
depend on what you mean by "precision", and what field of math you are
talking about.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
mag...@math.berkeley.edu
Martin Johansen
"Arturo Magidin" <mag...@math.berkeley.edu> wrote in message
news:c0e8bj$2c9l$1...@agate.berkeley.edu...
> In article <xixWb.3454$rj4....@news2.e.nsc.no>,
> Martin Johansen <mart...@is.online.no> wrote:
> >What is the role of infinity in math:
> >
> >How is it defined?
>
> Depends on the field of math you are working on. It means different
> things in calculus than it does in set theory, for example.
Ok, math related to measurement, calculus, algebra, number theory, etc.
> >Why is it needed?
>
> Because it is useful. In Calculus, for example, it is a very useful
> shorthand for a number of related phenomena, that are important and
> that we want to be able to talk about. In set theory, it is the
> foundation of a lot of interesting mathematics that could not be done
> without the concept of infinity as it is used there. In other areas
> (such as functional analysis) it is downright essential.
It is practical, yes. But why use it instead of a high number, it would not
produce as nice results, but what is it about the concept of infinity that
makes it usefull? (When it comes to math dealing with measurement)
> >At what precition does math work?
>
> That does not seem to be a very clear question. And it will again
> depend on what you mean by "precision", and what field of math you are
> talking about.
Again all measurement work with presition, but what about math in itself?
Arturo Magidin
Martin Johansen <mart...@is.online.no> wrote:
>
>"Arturo Magidin" <mag...@math.berkeley.edu> wrote in message
>news:c0e8bj$2c9l$1...@agate.berkeley.edu...
>> In article <xixWb.3454$rj4....@news2.e.nsc.no>,
>> Martin Johansen <mart...@is.online.no> wrote:
>> >What is the role of infinity in math:
>> >
>> >How is it defined?
>>
>> Depends on the field of math you are working on. It means different
>> things in calculus than it does in set theory, for example.
>
>Ok, math related to measurement, calculus, algebra, number theory, etc.
It still means different things; in algebra, the usual meaning will be
closer to that of set theory: a set is "infinite" if and only if it
can be put into one-to-one bijective correspondence with a proper
set of itself; and finite otherwise. An "infinite dimensional vector
space" is a vector space where any finite set is not a basis, or where
(assuming the Axiom of Choice) every base forms an infinite set. In
"number theory", it will depend: if you are doing analytic number
theory, "infinity" will probably be related to the notion of infinity
from complex analysis (the one point compactification of the complex
plane). In algebraic number theory, "infinity" has to do with the
archimedean norms that can be put on a number field (they are called
the 'places at infinity').
In standard calculus, "infinity" is used as shorthand for certain
things. Saying a function "goes to infinity as x approaches y" means
that for every value N, there exists a positive integer e such that if
0<|x-y|<e, then f(x) is bigger than N. Saying that a function
approaches a value v as the variable "goes to infinity" means that for
every e>0 there is an N>0 such that if x>N, then |f(x)-v|<e. Taking
"limits as n goes to infinity" means one thing for functions and a
slightly different (but closely related) things for sequences. Taking
a sum "from one to infinity" means taking a special kind of limit of a
special sequence associated with the sum...
"Infinity" has a very precise meaning, but what that "precise meaning"
is depends entirely on the context on which you are working.
>> >Why is it needed?
>>
>> Because it is useful. In Calculus, for example, it is a very useful
>> shorthand for a number of related phenomena, that are important and
>> that we want to be able to talk about. In set theory, it is the
>> foundation of a lot of interesting mathematics that could not be done
>> without the concept of infinity as it is used there. In other areas
>> (such as functional analysis) it is downright essential.
>
>It is practical, yes. But why use it instead of a high number, it would not
>produce as nice results, but what is it about the concept of infinity that
>makes it usefull? (When it comes to math dealing with measurement)
When you are measuring things in the real world, you don't use
"infinity." As for very high numbers, it is not the same thing to say
that something gets larger than anything we might possibly name ("goes
to infinity") than to say that it gets larger than a specific large
number we mention.
They do not mean the same thing.
>> >At what precition does math work?
>>
>> That does not seem to be a very clear question. And it will again
>> depend on what you mean by "precision", and what field of math you are
>> talking about.
>
>Again all measurement work with presition, but what about math in itself?
Again, your question is not clear. "Precision", with regards to
measurement, means something that only makes sense when you talk
about measurements. It is something related to the margin of errors
and so on. But what does it mean to ask whether math is "precise"?
Axiomatic mathematics is "precise" in the sense that all non-primitive
notions must be explicitly and very clearly defined if they are to be
used, but is not "precise" in any sense realted to margin of errors.
Your question, as stated, does not make sense. You need to be very
clear about just what you mean by "precision" before it can be
answered, and you need to be clear about what kind of math are you
David C. Ullrich
<mart...@is.online.no> wrote:
>
>[...]
>
>> >At what precition does math work?
>>
>> That does not seem to be a very clear question. And it will again
>> depend on what you mean by "precision", and what field of math you are
>> talking about.
>
>Again all measurement work with presition, but what about math in itself?
Math in itself works to a prisishun of 3 significant digits.
(That doesn't make much sense, does it? No. Well, like
Arturo's been saying, you need to clarify what you mean by
the question.)
************************
David C. Ullrich
Michael N. Christoff
"Martin Johansen" <mart...@is.online.no> wrote in message
news:xixWb.3454$rj4....@news2.e.nsc.no...
> What is the role of infinity in math:
>
As Arturo mentioned, it depends on the branch of math your are in. But for
example, take the set of natural number, or sometimes refered to as the
counting numbers 1,2,3,... etc... How many of these are there? If you tell
me some number M is the largest (hence there would be exactly M counting
numbers), I could always say "but what about M+1?". So there are in fact an
infinite number of counting numbers since no 'largest' counting number
exists.
> How is it defined?
>
Well there are actually levels of infinity. The set of all counting numbers
is what is called a 'countably infinite' set. The link below may be of
help:
http://en.wikipedia.org/wiki/Countable
> Why is it needed?
>
What I wrote above is an example. If we want to talk about the 'size' of
the set of all counting numbers, we need to get into infinities since no
finite set of counting numbers will ever be able to contain _all_ of the
counting numbers.
As mentioned above, another application is found in computer science. One
could talk about the set of all computer programs which is also 'countably'
infinite set. This is because there is a 1 to 1 correspondence between the
counting numbers and each possible computer program.
> At what precition does math work?
>
Despite what your encyclopedia or dictionary may tell you, math is not all
about making measurements. Take theoretical computer science. There are
many branches of math that do not even use numbers at all as their
fundamental objects!
> These are questions which seem to have to accepted answer, what do you
> think?
>
Each branch of math will have its own possibly unique 'accepted answer'.
l8r, Mike N. Christoff
Dave Rusin
> What is the role of infinity in math:
None at all.
Mathematicians (and teachers) would probably do everyone a great
service if they never made reference to "infinity"; leave the
mysticism to someone else (Buzz Lightyear, perhaps?)
Certainly we have great need of some sets which are not finite;
the set of counting numbers comes to mind. Since the sets are not
finite, we call them "in-finite sets", but notice that this is a
negative word: we know what the sets are _not_ (they're not finite)
but we don't waste energy thinking about what they _are_ (that is,
we don't try to grapple with "infinity" per se).
When you have a set that's not finite, and people ask you how
many things you've got, you can say "infinitely many", but they're
still just ordinary things (points, numbers, whatever); there's
just a lot of them -- not a finite number of them at all, so, well,
an "in-finite number" of them.
Mathematicians WILL sometimes use the phrase "infinity" as a sort of
slang to refer to something that takes more words to describe properly,
but that's just between consenting adults. There is really nothing
"infinite" about most of these things. For instance, if you puncture
a balloon, you have some rubber which you can stretch out to give a
model of the (complex) plane; running this backwards, you can roll up
the plane into a sphere, but you have to add back the point of the
puncture. That extra point isn't part of the rubber plane, but it is
part of the sphere. For some reason a more pedestrian name is not
used; instead, it's call the "point at infinity" (or just "infinity").
But it's still just a point.
dave
Chan-Ho Suh
<ru...@vesuvius.math.niu.edu> wrote:
> In article <xixWb.3454$rj4....@news2.e.nsc.no>,
> Martin Johansen <mart...@is.online.no> wrote:
>
> > What is the role of infinity in math:
>
> None at all.
>
> Mathematicians (and teachers) would probably do everyone a great
> service if they never made reference to "infinity"; leave the
> mysticism to someone else (Buzz Lightyear, perhaps?)
>
> Certainly we have great need of some sets which are not finite;
> the set of counting numbers comes to mind. Since the sets are not
> finite, we call them "in-finite sets", but notice that this is a
> negative word: we know what the sets are _not_ (they're not finite)
> but we don't waste energy thinking about what they _are_ (that is,
> we don't try to grapple with "infinity" per se).
>
> When you have a set that's not finite, and people ask you how
> many things you've got, you can say "infinitely many", but they're
> still just ordinary things (points, numbers, whatever); there's
> just a lot of them -- not a finite number of them at all, so, well,
> an "in-finite number" of them.
> [rest snipped]
Dave, I understand part of what you're saying here: that infinite often
means not finite-- but I'm quite puzzled by your overall point.
What if someone asks me how many hyperbolic 3-manifolds there are, and
I reply " Omega to the Omega power". I'm not only saying there are a
not-finite amount of them, I'm saying the collection can be ordered as
a particular order type. And note I'm not doing this to be weird, but
that is actually the way you would count them. First count all the
hyperbolic 3-manifolds with 0 cusps, then those with 1 cusp, then those
with 2 cusps, etc. It would seem I am in fact grappling with what they
are, not just what they are not -- to use your wording.
So I guess I'm not understanding how ordinals fit into your explanation
of why infinity plays no role in math. It seems to me that I am
classifying different types of infinite numbers.
Dan Christensen
> In article <xixWb.3454$rj4....@news2.e.nsc.no>,
> Martin Johansen <mart...@is.online.no> wrote:
> >What is the role of infinity in math:
> >
> >How is it defined?
>
> Depends on the field of math you are working on. It means different
> things in calculus than it does in set theory, for example.
>
> >Why is it needed?
>
> Because it is useful. In Calculus, for example, it is a very useful
> shorthand for a number of related phenomena, that are important and
> that we want to be able to talk about. In set theory, it is the
> foundation of a lot of interesting mathematics that could not be done
> without the concept of infinity as it is used there. In other areas
> (such as functional analysis) it is downright essential.
>
[snip]
Can the notion of infinity really not be dispensed with in
mathematics? If not, can it be restricted to a very small area of
study?
In calculus (and functional analysis?), the formal definitions of
limits, etc. do not make explicit use of the notion of infinity. As
far as I know, it is only used as a convenient shorthand notation
there. In number theory, we say only that each number has a suitably
defined successor. What does "infinity" really get you outside of,
say, the theory of cardinal numbers?
Dan
Visit DC Proof Online at http://www.dcproof.com
Stephen J. Herschkorn
>In article <xixWb.3454$rj4....@news2.e.nsc.no>,
>Martin Johansen <mart...@is.online.no> wrote:
>
>
>
>>What is the role of infinity in math:
>>
>>
>
>None at all.
>
>Mathematicians (and teachers) would probably do everyone a great
>service if they never made reference to "infinity"; leave the
>mysticism to someone else (Buzz Lightyear, perhaps?)
>
[...]
Are Martin and Dave constructivists? From Davis and
Hersh,_The_Mathematical_Experience_:
"The constructivists regard as genuine mathematics only what can be
obtained by a finite construction. The set of real numbers, or any
other infinite set, cannot so be obtained."
Loc. cit.,
"Constructivists are a rare breed, whose status in the mathematical word
sometimes seems to be that of tolerated heretics surrounded by orthodox
members of an established church."
Personally, I am *not* of this "rare breed."
--
Stephen J. Herschkorn hers...@rutcor.rutgers.edu
Torkel Franzen
> "The constructivists regard as genuine mathematics only what can be
> obtained by a finite construction. The set of real numbers, or any
> other infinite set, cannot so be obtained."
This is denounced as pure mysticism by those who want to actually
be able to obtain things in mathematics by a finite construction.
Dave Rusin
>What is the role of infinity in math:
I responded:
>None at all.
In article <A8FWb.6777$I67.2...@news4.srv.hcvlny.cv.net>,
Stephen J. Herschkorn <hers...@rutcor.rutgers.edu> wrote:
>Are Martin and Dave constructivists? From Davis and
>Hersh,_The_Mathematical_Experience_:
>
>"The constructivists regard as genuine mathematics only what can be
>obtained by a finite construction. The set of real numbers, or any
>other infinite set, cannot so be obtained."
I don't think I'm a constructivist (though I don't claim to have an
authoritative delineation of the schools of mathematical thought).
I happily use existence proofs in which the thing whose existence is
claimed is never, in fact, constructed. I merrily work with infinite
sets and other betes noires of the constructivists.
The example given in D&H's quote is probably perfect: I find the
real numbers to be very useful. All of them, individually, and the
whole set of them. It's an infinite set. But inside that set I
find only numbers. I don't find a thing called "infinity". Well,
OK, so infinity is not a real number. What is it, then, and why
should I care about it? (You can argue that it's a complex "number",
as I did in a prior post. But then, it just sits in my bag of complex
numbers next to 0 and i and sqrt(2). It doesn't glow and sparkle
and boggle my mind. It's just a point with a funny name, one whose
arithmetic rules are a little different.) Take a look at the theorems
of mathematics: are any of them about "infinity"? (Not the points
"at infinity" on a curve; they are ordinary points in projective space.
Not limits in calulcus which "equal infinity"; those are limits which
fail to exist but have an easily-described behaviour. Not "infinite
sets"; they're studied as sets, or ordered sets, or something, and
they just happen not to be finite. I just don't think I've ever seen
an article about "infinity" itself.)
One can argue that sets exist in a platonic realm, and the whole
development of set theory is an attempt to capture, in concrete
axioms, the features that we "know" are there. Sometimes real-analysis
courses build on students' prior knowledge of numbers and simply
present the defining axioms for R as a means of capturing the essence
of that prior experience, so that there is a foundation on which to
build proofs. Broadly speaking, this is how most productive mathematics
functions: we have an object in mind, and we set some axioms to mirror
its behaviour, then investigate that model. Well, great. Now what
sort of previously-conceived object is "infinity", that we should
study it?
I'm all in favor of enlivening the minds of school kids who are
sympathetic to mathematics. Physicists can talk about black holes,
and mathematicians can talk about vastly huge sets. Fine. But we
get a lot of gobbledygook in sci.math and elsewhere which would
probably be avoided if we stopped getting the students to think
about "infinity" as a (mathematical) object. That's how I interpreted
the OP's first question; that's why I responded as I did.
dave
Dave Rusin
wrote in article <110220041729426026%s...@math.ucdavis.nospam.edu> :
>
>> When you have a set that's not finite, and people ask you how
>> many things you've got, you can say "infinitely many", but they're
>> still just ordinary things (points, numbers, whatever); there's
>> just a lot of them -- not a finite number of them at all, so, well,
>> an "in-finite number" of them.
>> [rest snipped]
>What if someone asks me how many hyperbolic 3-manifolds there are, and
>I reply " Omega to the Omega power". I'm not only saying there are a
>not-finite amount of them, I'm saying the collection can be ordered as
>a particular order type.
>So I guess I'm not understanding how ordinals fit into your explanation
>of why infinity plays no role in math. It seems to me that I am
>classifying different types of infinite numbers.
That's odd; I wouldn't say you're dealing with numbers at all. You've
got a collection of things which you've classified in big book, and
the table of contents looks like this:
* * * * * * ***... * * * * * ***... * * * *...
[etc]. That is, you've got a pattern in mind which you can think of as
an ordered set, or a kind of tree or something. But it's the manifolds
you're studying, right? Or ordinals [order types] more generally.
But not "infinity".
I think in fact your example shows we're on the same wavelength here.
You know the set of manifolds is not finite, so then you ask, "what is it?"
The answer "infinity" is correct as a synonym for "not finite", but
you know that there are many non-finite things, so you want more
specificity. By contrast the naive questioner who wants to know what
"infinity" is seems to think that it's a single, fixed thing that has
a nature sui generis. There are finite sets and non-finite sets;
many species of well-ordered sets, only the most boring of which are finite;
sets which can be parameterized by the counting numbers, or by the
real numbers, or by some other set. We know something when we know
that these sets are not finite. That's one thing that they all _are not_;
what is the thing that all of these _are_? What is this thing called
"infinity"? I don't think that adds anything useful (besides saying
"not finite") and I think that you know that --- that's why you count
the items in your set differently instead of just saying "we've reached
infinity".
Maybe I mis-interpreted the OP's question. There are plenty of
non-finite things of interest in mathematics: ordinals and cardinals,
sequences and series, non-terminating algorithms, etc., and if
that's what the question was about I apologize. I thought the person
was talking like a movie producer. ("Oh, those mathematicians are
so different from us; they can study 'infinity'!")
dave
Ross A. Finlayson
>> obtained by a finite construction. The set of real numbers, or any
>> other infinite set, cannot so be obtained."
>
>be able to obtain things in mathematics by a finite construction.
There are not only finitely many numbers in mathematics.
The ultrafinitists simply pursue a model of the finite consideration of things infinite.
The limit approximation is a useful tool. It characterizes finitely some aspects of functional relations that imply their tractability in mainline calculist systems.
The circle's not a regular polygon. A regular polygon can only approximate a circle, and not be one. One third can not be the solution of summing finitely many positive integer powers of 1/2, yet it can infinitely.
"No classes in set theory."
Consider the empty set, {}. Consider if that conceptualization is {;seogsergsgsbs; s...}, or that the empty set is the set of all sets, or not.
If the set of all sets that are the set of all sets is the empty set, would not the empty set be the set of all sets?
Ross F.
Darryl L. Pierce,,,
> What is the role of infinity in math:
It's only a supporting cast member. But,it won an Oscar for Best Supporting
Transfinite Number In A Scientific Discipline.
<snip>
> These are questions which seem to have to accepted answer, what do you
> think?
I think.....very dirty thoughts about Rachael Ray.
--
Darryl L. Pierce <mcpi...@myrealbox.com>
Visit the Infobahn Offramp - <http://mypage.org/mcpierce>
"What do you care what other people think, Mr. Feynman?"
Arturo Magidin
Dan Christensen <dch...@netcom.ca> wrote:
>mag...@math.berkeley.edu (Arturo Magidin) wrote in message news:<c0e8bj$2c9l$1...@agate.berkeley.edu>...
>
>> In article <xixWb.3454$rj4....@news2.e.nsc.no>,
>> Martin Johansen <mart...@is.online.no> wrote:
>> >What is the role of infinity in math:
>> >
>> >How is it defined?
>>
>> Depends on the field of math you are working on. It means different
>> things in calculus than it does in set theory, for example.
>>
>> >Why is it needed?
>>
>> Because it is useful. In Calculus, for example, it is a very useful
>> shorthand for a number of related phenomena, that are important and
>> that we want to be able to talk about. In set theory, it is the
>> foundation of a lot of interesting mathematics that could not be done
>> without the concept of infinity as it is used there. In other areas
>> (such as functional analysis) it is downright essential.
>>
>[snip]
>
>Can the notion of infinity really not be dispensed with in
>mathematics?
Depends on what you mean by "notion of infinity". You can certainly
excise the word "infinity" everywhere that it appears in mathematics;
but in most places, you would still need the concept codified by that
word.
>In calculus (and functional analysis?), the formal definitions of
>limits, etc. do not make explicit use of the notion of infinity.
What is "the notion of infinity"? Functional analysis deals mainly
with infinite dimensional vector spaces, so I do not see how you could
get away from it.
>In number theory, we say only that each number has a suitably
>defined successor.
There is more to "number theory" than the natural numbers. There's
analytic number theory, that requires complex analysis.
Craig Feinstein
That is a very controversial question. On the one hand, there is my
disturbed brother-in-law Ben (who posted a few threads and believes
that there are only a finite number of number). He is of the
ultra-finitist philosophy and he is not alone, believe it or not.
Although, most people of that philosophy would not go so far as to do
a computer search for the largest number. See
http://en.wikipedia.org/wiki/Ultraintuitionism
Then there is the intuitionist philosophy which is a little more
mainstream but is still not the majority opinion. Their objection is
not based on whether infinity exists or not, but on whether we can
even say anything about infinity if it did exist. What separates them
from the classic mathematicians is that they reject Aristotle's Law of
the Excluded Middle that the proposition (A or not A) is a tautology,
since one can never know for sure if for each x, (A(x) or not A(x)) is
a tautology when x ranges over an infinite domain. While this may seem
crazy and counter-intuitive, intuitionists might respond that the
theorems that result from this assumption are also counter-intuitive -
i.e., Cantor's aleph null and aleph one. Why should there be many
types of infinities? Infinity is infinity. And also, look at all of
the paradoxes of set theory - Is there a set of all sets, etc. When
one eliminates the Law of the Excluded Middle, one avoids a lot of the
crazy paradoxes associated with infinity.
The classic philosophy (Platonism) says that all of this stuff
(infinity) is in fact real. And just because we cannot perceive it
does not mean that it does not exist. This is the majority opinion,
not necessarily because the majority believe it, but more because it
is the most practical way of thinking of mathematics, since the
conclusions have applications in natural sciences. If we were to go to
what my brother-in-law proposes, replacing differential equations with
difference equations, etc., we would get the same theorems, but they
would be much more difficult to write and describe. The "derivative"
of x^2 would not be 2x but would be 2x+1/M, where M is the largest
number. In practice, this number 1/M would be so small that it
wouldn't even matter, so why write it down on paper? Also, the
Pythagorean theorem would not hold, and it would be much more messy to
describe this relationship. But this way of thinking still avoids
paradoxes.
Craig
Will Twentyman
> What is the role of infinity in math:
For part of your answer, I suggest reading
http://www.kuro5hin.org/story/2003/6/3/95744/71866
>
> How is it defined?
It depends on the context. In calculus, it is usually a concept, not a
value. In complex analysis (as modeled by the Riemann sphere), it is a
number. In set theory, there are multiple distinct infinities
corresponding to various cardinalities.
>
> Why is it needed?
It's easier to write "x -> oo" than "as x grows larger than any
arbitrary value epsilon".
>
> At what precition does math work?
That depends on what the acceptable error is in your statistical
analysis/approximation.
On a more serious note, I'm not sure what you're trying to ask. My
inclination is to so something like "infinite precision" or "absolute
precision".
> These are questions which seem to have to accepted answer, what do you
> think?
Huh? This question doesn't make sense.
--
Will Twentyman
email: wtwentyman at copper dot net
Will Twentyman
> "Arturo Magidin" <mag...@math.berkeley.edu> wrote in message
> news:c0e8bj$2c9l$1...@agate.berkeley.edu...
>
>>In article <xixWb.3454$rj4....@news2.e.nsc.no>,
>>Martin Johansen <mart...@is.online.no> wrote:
>>
>>>What is the role of infinity in math:
>>>
>>>How is it defined?
>>
>>Depends on the field of math you are working on. It means different
>>things in calculus than it does in set theory, for example.
>
>
> Ok, math related to measurement, calculus, algebra, number theory, etc.
real analysis, combinatorics, statistics, differential equations, logic, ...
>
>
>>>Why is it needed?
>>
>>Because it is useful. In Calculus, for example, it is a very useful
>>shorthand for a number of related phenomena, that are important and
>>that we want to be able to talk about. In set theory, it is the
>>foundation of a lot of interesting mathematics that could not be done
>>without the concept of infinity as it is used there. In other areas
>>(such as functional analysis) it is downright essential.
>
>
> It is practical, yes. But why use it instead of a high number, it would not
> produce as nice results, but what is it about the concept of infinity that
> makes it usefull? (When it comes to math dealing with measurement)
Because no matter how large a number you pick, it is possible to create
a function with a behavior as x -> oo that does not exhibit that
behavior prior to your large number.
Also, infinity is the only useful way to start talking about how many
integers there are.
>
>
>>>At what precition does math work?
>>
>>That does not seem to be a very clear question. And it will again
>>depend on what you mean by "precision", and what field of math you are
>>talking about.
>
> Again all measurement work with presition, but what about math in itself?
It depends on the branch and how much error is acceptable. For many
branches, error=0. For others, precision doesn't make sense. What do
you mean by precision in logic?
Ob pet-peeve: please spell "precision" correctly. Arturo corrected you
and you responded without the correction.
Will Twentyman
> In article <xixWb.3454$rj4....@news2.e.nsc.no>,
> Martin Johansen <mart...@is.online.no> wrote:
>
>
>>What is the role of infinity in math:
>
>
> None at all.
How many integers are there?
>
> Mathematicians (and teachers) would probably do everyone a great
> service if they never made reference to "infinity"; leave the
> mysticism to someone else (Buzz Lightyear, perhaps?)
Modern set theory deals a great deal with infinity as a core concept.
>
> Certainly we have great need of some sets which are not finite;
> the set of counting numbers comes to mind. Since the sets are not
> finite, we call them "in-finite sets", but notice that this is a
> negative word: we know what the sets are _not_ (they're not finite)
> but we don't waste energy thinking about what they _are_ (that is,
> we don't try to grapple with "infinity" per se).
>
> When you have a set that's not finite, and people ask you how
> many things you've got, you can say "infinitely many", but they're
> still just ordinary things (points, numbers, whatever); there's
> just a lot of them -- not a finite number of them at all, so, well,
> an "in-finite number" of them.
You are focusing on the individual elements, but when people talk about
how many elements there are in a set, the desire is to get a meaningful
value. In fact, simply saying that it is infinite is not always enough
detail. Thus the cardinals to measure levels of infinity.
>
> Mathematicians WILL sometimes use the phrase "infinity" as a sort of
> slang to refer to something that takes more words to describe properly,
> but that's just between consenting adults. There is really nothing
> "infinite" about most of these things. For instance, if you puncture
> a balloon, you have some rubber which you can stretch out to give a
> model of the (complex) plane; running this backwards, you can roll up
> the plane into a sphere, but you have to add back the point of the
> puncture. That extra point isn't part of the rubber plane, but it is
> part of the sphere. For some reason a more pedestrian name is not
> used; instead, it's call the "point at infinity" (or just "infinity").
> But it's still just a point.
The Riemann sphere has a point on it that *is* infinity. It is a value,
not a concept, in that model of the complex numbers.
Martin Johansen
"Will Twentyman" <wtwen...@read.my.sig> wrote in message
news:402a6...@newsfeed.slurp.net...
> Martin Johansen wrote:
>
> > What is the role of infinity in math:
>
> For part of your answer, I suggest reading
> http://www.kuro5hin.org/story/2003/6/3/95744/71866
>
> >
> > How is it defined?
>
> It depends on the context. In calculus, it is usually a concept, not a
> value. In complex analysis (as modeled by the Riemann sphere), it is a
> number. In set theory, there are multiple distinct infinities
> corresponding to various cardinalities.
>
> >
> > Why is it needed?
>
> It's easier to write "x -> oo" than "as x grows larger than any
> arbitrary value epsilon".
>
> >
> > At what precition does math work?
>
> That depends on what the acceptable error is in your statistical
> analysis/approximation.
>
> On a more serious note, I'm not sure what you're trying to ask. My
> inclination is to so something like "infinite precision" or "absolute
> precision".
Yes this was what I was looking for, thanks. Math does work at infinite
precision.
> > These are questions which seem to have to accepted answer, what do you
> > think?
> Huh? This question doesn't make sense.
You are right, it doesn't. I'll try again:
These are questions which seem not to have an accepted answer, what do you
think?
>
Leonard Blackburn
> Dave Rusin wrote:
>
> >In article <xixWb.3454$rj4....@news2.e.nsc.no>,
> >Martin Johansen <mart...@is.online.no> wrote:
> >
> >
> >
> >>What is the role of infinity in math:
> >>
> >>
> >
> >None at all.
> >
> >Mathematicians (and teachers) would probably do everyone a great
> >service if they never made reference to "infinity"; leave the
> >mysticism to someone else (Buzz Lightyear, perhaps?)
> >
> [...]
>
> Are Martin and Dave constructivists? From Davis and
> Hersh,_The_Mathematical_Experience_:
Not necessarily.
For example, I'm not a constructivist. I use ordinals and cardinals all
the time. I like classical math and set theory, and after reading
D. Rusin's posts I think I am in complete agreement with him.
Leonard Blackburn
Carlos Moreno
> The classic philosophy (Platonism) says that all of this stuff
> (infinity) is in fact real. And just because we cannot perceive it
> does not mean that it does not exist. This is the majority opinion,
> not necessarily because the majority believe it, but more because it
> is the most practical way of thinking of mathematics
What?!!!! (read that as "I'm stunned in disagreement and intrigue")
> would be much more difficult to write and describe. The "derivative"
> of x^2 would not be 2x but would be 2x+1/M, where M is the largest
> number. In practice, this number 1/M would be so small that it
> wouldn't even matter, so why write it down on paper?
Limits and derivatives and in general differential calculus makes
sense and is 100% "tangible" without having to resort to the
existence of such largest number, which can so easily be shown
does not exist. The limit of x^2 as x approaches 0 is *exactly*
0 -- it's not 0 + 1/M. It's not "as close to 0 as we want" -- no,
it is 0, because the definition directly implies so (a definition
that is tangible and expressed in tangible terms, unambiguous, and
very indisputable, I think).
> Pythagorean theorem would not hold, and it would be much more messy to
> describe this relationship. But this way of thinking still avoids
> paradoxes.
How does it avoid the paradox that there is the same number of
real numbers in the interval (0,1) as in the interval (0,2) ??
If you state that there are M real numbers in the interval (0,1),
where M is the largest number that exist, how do you explain
that there is exactly M numbers in the interval (0,2)? (but
there is also exactly M + M numbers... What? M + M is M??
I don't think that's an easy thing to explain, or avoid
paradoxes)
Carlos
--
Michael N. Christoff
"Carlos Moreno" <moreno_at_mo...@xx.xxx> wrote in message
news:OVSWb.4713$5a1.3...@weber.videotron.net...
> Craig Feinstein wrote:
>
> > The classic philosophy (Platonism) says that all of this stuff
> > (infinity) is in fact real. And just because we cannot perceive it
> > does not mean that it does not exist. This is the majority opinion,
> > not necessarily because the majority believe it, but more because it
> > is the most practical way of thinking of mathematics
>
> What?!!!! (read that as "I'm stunned in disagreement and intrigue")
>
> > would be much more difficult to write and describe. The "derivative"
> > of x^2 would not be 2x but would be 2x+1/M, where M is the largest
> > number. In practice, this number 1/M would be so small that it
> > wouldn't even matter, so why write it down on paper?
>
> Limits and derivatives and in general differential calculus makes
> sense and is 100% "tangible" without having to resort to the
> existence of such largest number, which can so easily be shown
> does not exist.
I think you've missed the point.
http://en.wikipedia.org/wiki/Mathematical_constructivism
l8r, Mike N. Christoff
Dave Rusin
Will Twentyman <wtwen...@read.my.sig> wrote:
>How many integers are there?
How would you possibly answer this question?
A. "There's an infinity of them". I would never say that.
B. "There's an infinite number of them." Don't say that; it makes people
think there are infinite (natural) numbers.
C. "There are infinitely many of them", or "There's an infinite set of them."
That's OK. I don't see a bit of difference between saying that and
saying "there's not a finite number of them" (But there is at least one!)
D. "There are aleph-1 of them" or "the set of them is countably infinite".
Look up the definition of "countably infinite" and tell me this isn't
nearly a tautology!
So what's your point? I thought I made it clear that mathematics encounters
infinite sets and limits and all sorts of situations where we might
bandy about the term "infinity", but it's just a shorthand for something
more precise, or else just a synonym for "not finite". Or are you saying
"infinity" is the set of integers? I think that's awfully limiting...
>> Mathematicians (and teachers) would probably do everyone a great
>> service if they never made reference to "infinity"; leave the
>> mysticism to someone else (Buzz Lightyear, perhaps?)
>
>Modern set theory deals a great deal with infinity as a core concept.
I think you mean it deals primarily with infinite sets and compares and
contrasts them. OK. But I think the emphasis is on other structures (such
as the existence of functions of various types between sets, or possible
well-orderings on sets). The fact that most of the sets are infinite is a
given, and not itself the subject of much attention. (Disclaimer: I am not
a set theorist!) Some people do wrestle with the precise question of how
we define the predicate "S is infinite" (usual choice: existence of a
bijection S->T where T is a proper subset of S) but I don't think that's
considered a core issue of modern set theory!
I believe you're reading much too much into the OP's question. This is
a secondary-school student who links mathematics with measurement and
precision. Perhaps someone has whispered about a thing called "infinity"
and he wants to know what it is and why we care. Not infinite sets;
not ordinals and cardinals, just a thing called "infinity", which is
seen as some kind of number (I guess). I don't believe that's a useful
concept in itself.
I've given a talk like "My infinity can beat your infinity!" to high
school students. First thing I do is to ask them to stop thinking about
any such things, and instead to think about infinite _sets_ instead.
Much more mathematically sound and it still seems to intrigue them.
>> When you have a set that's not finite, and people ask you how
>> many things you've got, you can say "infinitely many", but they're
>> still just ordinary things (points, numbers, whatever); there's
>> just a lot of them -- not a finite number of them at all, so, well,
>> an "in-finite number" of them.
>
>You are focusing on the individual elements, but when people talk about
>how many elements there are in a set, the desire is to get a meaningful
>value. In fact, simply saying that it is infinite is not always enough
>detail. Thus the cardinals to measure levels of infinity.
You mean, "Thus the cardinals to represent different answers to 'how many'".
There are finite cardinals, too, and I think you're reinforcing my point:
"infinity", per se, is pretty useless: if you want to answer "how many",
you point to a cardinal. If that cardinal is finite, great. If not, you
can say "not finite". If you want to give more information, you specify
the appropriate cardinal, thus making it clear that the answer "infinity"
is at best a shorthand for "not finite". It's not really a useful concept
in and of itself.
>The Riemann sphere has a point on it that *is* infinity. It is a value,
>not a concept, in that model of the complex numbers.
Um, right. I said it was a point _called_ infinity, but if now you want
to tell me that this is, in fact, the definition of infinity, then I
don't object, exactly; but if infinity is _really_ a part of the
Riemann sphere, then I wonder what the heck you meant by trotting out
infinite sets before? Again I think you're sort of making my point for me:
as much as we use the term "infinity" in mathematics, there really is
no single thing that encompasses
a synonym for "not finite"
the whole panoply of infinite sets
a point on the Riemann sphere
etc.
not to mention the pop-science connections to Big Bangs or whatever.
It's a subtle thing, but careful attention to subtleties can help
newcomers. Use "infinite", the definable adjective, whenever possible
instead of an undefined noun "infinity".
dave
Will Twentyman
As you've probably seen, there tends to be a majority opinion, with a
few dissenters. This is a lot like most other fields. We also get the
occasional crackpot. Nobody tends to get kicked out unless they are
doing something that is logically inconsistent.
Personally, I would have issues with some of the Ultraintuitionist stuff
just because I like mathematical induction. As long as they can make it
self-consistent, they can do what they want, though.
Russell Easterly
"Dave Rusin" <ru...@vesuvius.math.niu.edu> wrote in message
news:c0h1sp$dii$1...@news.math.niu.edu...
> Some people do wrestle with the precise question of how
> we define the predicate "S is infinite" (usual choice: existence of a
> bijection S->T where T is a proper subset of S) but I don't think that's
> considered a core issue of modern set theory!
I came up with a simple definition of an infinite set in another thread.
Let S be a set of natural numbers and let x and y be members of S.
Consider these two statements:
1) ExAy(x>=y)
2) AxEy(x<y)
If statement (1) is true then S is finite.
If statement (2) is true then S is infinite.
Russell
- 2 many 2 count
Will Twentyman
> In article <402a6...@newsfeed.slurp.net>,
> Will Twentyman <wtwen...@read.my.sig> wrote:
>
>
>>How many integers are there?
>
>
> How would you possibly answer this question?
>
> A. "There's an infinity of them". I would never say that.
> B. "There's an infinite number of them." Don't say that; it makes people
> think there are infinite (natural) numbers.
> C. "There are infinitely many of them", or "There's an infinite set of them."
> That's OK. I don't see a bit of difference between saying that and
> saying "there's not a finite number of them" (But there is at least one!)
> D. "There are aleph-1 of them" or "the set of them is countably infinite".
> Look up the definition of "countably infinite" and tell me this isn't
> nearly a tautology!
I'd say C, or "There's aleph-0 of them"
Probably C.
>
> So what's your point? I thought I made it clear that mathematics encounters
> infinite sets and limits and all sorts of situations where we might
> bandy about the term "infinity", but it's just a shorthand for something
> more precise, or else just a synonym for "not finite". Or are you saying
> "infinity" is the set of integers? I think that's awfully limiting...
Probably none. At this point we seem to be splitting hairs of one of
the angels dancing on the head of a pin.
>>>Mathematicians (and teachers) would probably do everyone a great
>>>service if they never made reference to "infinity"; leave the
>>>mysticism to someone else (Buzz Lightyear, perhaps?)
>>
>>Modern set theory deals a great deal with infinity as a core concept.
>
>
> I think you mean it deals primarily with infinite sets and compares and
> contrasts them. OK. But I think the emphasis is on other structures (such
> as the existence of functions of various types between sets, or possible
> well-orderings on sets). The fact that most of the sets are infinite is a
> given, and not itself the subject of much attention. (Disclaimer: I am not
> a set theorist!) Some people do wrestle with the precise question of how
> we define the predicate "S is infinite" (usual choice: existence of a
> bijection S->T where T is a proper subset of S) but I don't think that's
> considered a core issue of modern set theory!
>
> I believe you're reading much too much into the OP's question.
See my reply to the OP. I just found your response odd. Perhaps it's
just me.
It is, IMHO, a concept best used carefully and not introduced too soon.
Unfortunately, it gets tossed around a lot.
Virgil
"Russell Easterly" <logi...@comcast.net> wrote:
But (1) only works for well-ordered sets, and not all sets are
well-ordered, e.g., the set of negative integers satisfies (1) so must
be finite by your reckoning.
How do you tell if an ordered-but-not-well-ordered set is finite?
In fact sets need not be ordered at all. How do you tell whether an
unordered set is finite?
Your definitions are simple enough, but wrong.
Russell Easterly
"Virgil" <vir...@COMCAST.com> wrote in message
news:virgil-76749C....@news.nntpservers.com...
> In article <gYudnS7UboD...@comcast.com>,
> "Russell Easterly" <logi...@comcast.net> wrote:
>
> > "Dave Rusin" <ru...@vesuvius.math.niu.edu> wrote in message
> > news:c0h1sp$dii$1...@news.math.niu.edu...
> > > Some people do wrestle with the precise question of how
> > > we define the predicate "S is infinite" (usual choice: existence of a
> > > bijection S->T where T is a proper subset of S) but I don't think
that's
> > > considered a core issue of modern set theory!
> >
> > I came up with a simple definition of an infinite set in another thread.
> > Let S be a set of natural numbers and let x and y be members of S.
> > Consider these two statements:
> >
> > 1) ExAy(x>=y)
> > 2) AxEy(x<y)
> >
> > If statement (1) is true then S is finite.
> > If statement (2) is true then S is infinite.
>
> But (1) only works for well-ordered sets, and not all sets are
> well-ordered, e.g., the set of negative integers satisfies (1) so must
> be finite by your reckoning.
I said S is a set of natural numbers.
That wouldn't include negative integers.
> How do you tell if an ordered-but-not-well-ordered set is finite?
That would be more complicated.
Any set of natural numbers can be well ordered.
> In fact sets need not be ordered at all. How do you tell whether an
> unordered set is finite?
My method won't tell you.
Can't all finite sets be ordered?
> Your definitions are simple enough, but wrong.
My definition works for sets of natural numbers.
That is all I claimed.
Michael N. Christoff
"Will Twentyman" <wtwen...@read.my.sig> wrote in message
For what its worth, I tend to agree with you. I'm not a mathematician but
Rusin's comments don't seem to jibe with the way I have learned the concept
of infinity. Especially comments that seem to reduce the concept of
'infinite set' to merely 'non-finite set'. But perhaps it is appropriate
for the OP's level of familiarity with math in general.
l8r, Mike N. Christoff
Virgil
"Russell Easterly" <logi...@comcast.net> wrote:
> "Virgil" <vir...@COMCAST.com> wrote in message
> news:virgil-76749C....@news.nntpservers.com...
> > In article <gYudnS7UboD...@comcast.com>,
> > "Russell Easterly" <logi...@comcast.net> wrote:
> >
> > > "Dave Rusin" <ru...@vesuvius.math.niu.edu> wrote in message
> > > news:c0h1sp$dii$1...@news.math.niu.edu...
> > > > Some people do wrestle with the precise question of how
> > > > we define the predicate "S is infinite" (usual choice: existence of a
> > > > bijection S->T where T is a proper subset of S) but I don't think
> that's
> > > > considered a core issue of modern set theory!
> > >
> > > I came up with a simple definition of an infinite set in another thread.
> > > Let S be a set of natural numbers and let x and y be members of S.
> > > Consider these two statements:
> > >
> > > 1) ExAy(x>=y)
> > > 2) AxEy(x<y)
> > >
> > > If statement (1) is true then S is finite.
> > > If statement (2) is true then S is infinite.
> >
> > But (1) only works for well-ordered sets, and not all sets are
> > well-ordered, e.g., the set of negative integers satisfies (1) so must
> > be finite by your reckoning.
>
> I said S is a set of natural numbers.
> That wouldn't include negative integers.
It wouldn't work for negatives either, which is what I said.
>
> > How do you tell if an ordered-but-not-well-ordered set is finite?
>
> That would be more complicated.
> Any set of natural numbers can be well ordered.
>
> > In fact sets need not be ordered at all. How do you tell whether an
> > unordered set is finite?
>
> My method won't tell you.
> Can't all finite sets be ordered?
>
> > Your definitions are simple enough, but wrong.
>
> My definition works for sets of natural numbers.
> That is all I claimed.
By implication you claimed more. You said that you had simple definition
in another thread, then stated this one, without making clear that it
was not your other definition.
And I complained that this one did not work for arbitrary sets.
Robin Chapman
>
> "Will Twentyman" <wtwen...@read.my.sig> wrote in message
> news:402ae...@newsfeed.slurp.net...
>> Dave Rusin wrote:
>>
>> > In article <402a6...@newsfeed.slurp.net>,
>> > Will Twentyman <wtwen...@read.my.sig> wrote:
>> >
>> >
>> >>How many integers are there?
>> >
>> >
>> > How would you possibly answer this question?
>> >
>> > A. "There's an infinity of them". I would never say that.
>> > B. "There's an infinite number of them." Don't say that; it makes
>> > people
>> > think there are infinite (natural) numbers.
>> > C. "There are infinitely many of them", or "There's an infinite set of
> them."
>> > That's OK. I don't see a bit of difference between saying that and
>> > saying "there's not a finite number of them" (But there is at least
> one!)
>> > D. "There are aleph-1 of them" or "the set of them is countably
> infinite".
>> > Look up the definition of "countably infinite" and tell me this
>> > isn't nearly a tautology!
>>
>> I'd say C, or "There's aleph-0 of them"
>>
>> Probably C.
>>
>
> For what its worth, I tend to agree with you. I'm not a mathematician but
> Rusin's comments don't seem to jibe with the way I have learned the
> concept
> of infinity.
As you said: you are not a mathematician.
Dave's comments above nicely epitomize the mathematician's concept of
infinite sets. Note B and the avoidance of the "infinite numbers" beloved
of cranks.
> Especially comments that seem to reduce the concept of
> 'infinite set' to merely 'non-finite set'.
Simple etymology: "infinite" means "not finite".
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
"Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9"
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
Craig Feinstein
>
> > The classic philosophy (Platonism) says that all of this stuff
> > (infinity) is in fact real. And just because we cannot perceive it
> > does not mean that it does not exist. This is the majority opinion,
> > not necessarily because the majority believe it, but more because it
> > is the most practical way of thinking of mathematics
>
> What?!!!! (read that as "I'm stunned in disagreement and intrigue")
combination of formalists and Platonists. When they congregate
together, they only believe in the theorems that they prove because
they follow from axioms that are standardly accepted and assumed to
get rid of paradoxes.
However, whenever they relate to natural scientists, it is then when
they suddenly become Platonists and are the first to proclaim that not
only what they are doing is formally right but also is reality.
The reason that the intuitionist philosophy is not mainstream is not
because it is wrong, but because it would hinder the power of
mathematics to describe the natural world.
Craig
Torkel Franzen
> The reason that the intuitionist philosophy is not mainstream is not
> because it is wrong, but because it would hinder the power of
> mathematics to describe the natural world.
No it wouldn't.
Jesse F. Hughes
> The reason that the intuitionist philosophy is not mainstream is not
> because it is wrong, but because it would hinder the power of
> mathematics to describe the natural world.
Really? And that part about where intuitionism avers that mathematics
and proofs cannot be communicated? That doesn't conflict with any
mathematicians' experiences?
--
"Now I'm informing all of you that the people arguing against me are EVIL,
yes they are real, live EVIL people as mathematics is that important, so
it's important enough for Evil itself to send minions like them."
-- James Harris on Evil's interest in Algebraic Number Theory
Chan-Ho Suh
<ru...@vesuvius.math.niu.edu> wrote:
> Maybe I mis-interpreted the OP's question. There are plenty of
> non-finite things of interest in mathematics: ordinals and cardinals,
> sequences and series, non-terminating algorithms, etc., and if
> that's what the question was about I apologize. I thought the person
> was talking like a movie producer. ("Oh, those mathematicians are
> so different from us; they can study 'infinity'!")
>
That's why I don't bother with answering those kinds of questions.
It's hard to tell if it's a movie producer or not ;-)
I basically read your post without thinking of the context of the
possible "movie producer" mindset of the OP. [BTW, this is Dave
Rusin's terminology; I have only the highest regard for movie
producers] Your answer is undoubtedly appropriate in that context;
unfortunately, many people, including me, misunderstood what you were
trying to say. There are a lot of "consenting adults" -- to use your
phrase from a related post -- here on sci.math!
Shmuel (Seymour J.) Metz
at 10:39 PM, "Martin Johansen" <mart...@is.online.no> said:
>What is the role of infinity in math:
The word is used for a number of very different things.
>How is it defined?
In what context?
>Why is it needed?
Like any other technical term, it allows you to express common phrases
more concisely.
>At what precition does math work?
None.
Mathematics is not the same as Engineering. If you need to do a
calculation, your objective will dictate what precision is
appropriate. Note, however, that there is a branch of Mathematics
called Numerical Analysis that will help you determine what precision
you need for a calculation in a particular context.
In <FDxWb.3460$rj4....@news2.e.nsc.no>, on 02/11/2004
at 11:02 PM, "Martin Johansen" <mart...@is.online.no> said:
>It is practical, yes. But why use it instead of a high number,
Because a high number would be totally irrelevant to the uses to which
Mathematicians put the term "infinity". That's like asking a farmer wy
he feeds his cattle corn instead of pumice; they're not remotely
similar.
>Again all measurement work with presition, but what about math in
>itself?
The concept of precision is inapplicable to most of Mathematics. Even
where it is relevant, the Mathematics has no precision, but rather
proves results about the precision of calculations.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
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Shmuel (Seymour J.) Metz
at 11:23 PM, ru...@vesuvius.math.niu.edu (Dave Rusin) said:
>D. "There are aleph-1 of them" or "the set of them is countably
>infinite".
> Look up the definition of "countably infinite" and tell me this
> isn't nearly a tautology!
It isn't a tautology. In fact, it's false. There are only Aleph-0
integers.
>I don't believe that's a useful concept in itself.
Well, one and two point compactifications are useful, and the term
"infinity" is used productively in Projective Geometry.
>Use "infinite", the definable adjective, whenever possible instead
>of an undefined noun "infinity".
I'd say that they are equally meaningful in context and equally
meaningless out of context.
Ross A. Finlayson
Let's start with a set theoretical universe where the empty set is all of the sets.
Then, there is the set containing the empty set, and the set of the empty set and the set containing the empty set: any combination is a set.
Then there is the set containing those and their subsets in enumeration, there plainly are the sets containing each.
A simple rule assigns each ordinal a set representation.
The set of all of those sets is a set. The set of each of those sets is a set. Reiterate for each as above as if it were the empty set. Many sets are generated.
This doesn't yet have any non-regular sets, as any of these sets.
There are infinitely many of the sets.
Each subset of each set is a member of a set.
None of them is the set of all sets. Which is a mint julep?
By simply constructing from the empty set there is no necessary axiomatization of regularity, infinity, or powerset. What other axioms are unnecessary?
How low can you go?
Why yes, two plus two does equal four.
Ross F.
KRamsay
In article <873c9ev...@phiwumbda.org>, je...@phiwumbda.org (Jesse F.
Hughes) writes:
|Really? And that part about where intuitionism avers that mathematics
|and proofs cannot be communicated? That doesn't conflict with any
|mathematicians' experiences?
I don't know that anyone has ever followed Brouwer that far. But I think
I can see how one might come to some of his pessimistic conclusions
about language. On the one hand, we can see people learning to play
certain kinds of language games, as a kind of skill. He expressed great
misgivings about people adopting such an instrumental attitude toward
language, mathematics, or just about anything. On the other hand, if it
really is supposed to be a matter of carrying a thought from one mind to
another, isn't it somewhat remarkable that we feel such confidence that
the thought that arrives at the other end is really "the same" as the one
that set forth? I'm not at all as pessimistic, but I still find language often
sliding toward either a successful but robot-like technique that succeeds
in conveying basically data, or into poetry that is easily lost in translation.
Keith Ramsay
KRamsay
It depends on context. Many of the uses of the term "infinity"
are best thought of as idiomatic expressions. For example,
"the limit of f(x) as x goes to infinity" should be thought of as
an idiom meaning nothing more or less than what the
definition says that it means.
The uses of the term which are least metaphorical this way
are where "infinity" means "the property of being infinite",
in reference to infinite sets. An infinite set S is simply one
having the property that for any finite subset {s1,...,sn} of
S, there exists an element of S different from s1,...,sn.
|Why is it needed?
To answer that one has to compare how things work now
with how they would work if we didn't consider infinite sets.
That's a little slippery, since it's hard to say exactly when one
has crossed the line into talking about sets. A set of integers
for example corresponds to a property of integers, where two
properties are considered equivalent if the same integers
satisfy them. So talking about the set of primes is pretty much
another way of talking about the property of being prime.
One way to analyze the situation is to consider some standard
mathematical systems and dump out the infinite sets. Often the
resulting system of finite structures turns out to be equivalent to
a system of axioms for arithmetic, since finite structures can be
encoded as integers.
The most stubborn problem with trying to found all of mathematics
on such a foundation is that there are properties of finite structures
that we ordinarily prove indirectly by reasoning that involves
referring to infinite sets along the way. For instance, there's a
result called the Paris-Harrington variant of Ramsey's theorem,
that's impossible to prove in PA, which is a standard system for
axioms for arithmetic, even though PH only talks about finite
structures. The usual proof of PH defines a sequence of infinite
sets of integers, each of which individually could be defined purely
in terms of integers, but which together don't have a definition in
the language of elementary arithmetic.
|At what precition does math work?
This seems like a very strange question. If by "precition" you mean
numerical precision, the answer is "absolute precision". If you mean
"linguistic precision", well, take a look for yourself. The precision of
the language of mathematics tends to be rather high, but in practice
varies somewhat depending on the context. In an informal talk, one
may brush off quite a lot of detail in order to convey the gist of an idea.
In publication one needs to be more careful. Imprecision which serves
no purpose is considered poor. Imprecision which is due to your not
being _able_ to be precise is considered bad.
|These are questions which seem to have to accepted answer, what do you
|think?
People would state things differently. Probably people would give a
diversity of answers why we "need" infinity in mathematics. I don't
know that there's much controversy about how actually to use it, though.
Keith Ramsay
KRamsay
In article <OVSWb.4713$5a1.3...@weber.videotron.net>, Carlos Moreno
<moreno_at_mo...@xx.xxx> writes:
|Craig Feinstein wrote:
|
|> The classic philosophy (Platonism) says that all of this stuff
|> (infinity) is in fact real. And just because we cannot perceive it
|> does not mean that it does not exist. This is the majority opinion,
|> not necessarily because the majority believe it, but more because it
|> is the most practical way of thinking of mathematics
|
|What?!!!! (read that as "I'm stunned in disagreement and intrigue")
It's a little hard to accurately characterize the views of typical
mathematicians, since they're not usually so interested in these
philosophical issues.
"Realism" might be a better term here than "Platonism", since the
latter implies more of a commitment to the existence of a kind of
realm of ideas. Without having any polling data in hand, I would say
it seems rather common for mathematicians to have a realist stance
(a philosopher might call it "naive realism") about the commonplace
objects of study in mathematics. For instance, the existence of
infinitely many twin primes would be commonly considered either
actually true or actually false (almost certainly true), independent of
any means we might have or not have of determining which is the case.
I think it would take something remarkable to cause a serious change
in this situation. To some extent, for people dealing seriously with a
subject matter, naive realism is a natural default position, and only
really gives way to some form of nonrealism once the nonrealism is
able to establish a sufficiently solid basis for itself.
Probably formalism is the strongest candidate for such an usurper.
If anything makes me cautious about claiming that realism is the usual
point of view of mathematicians, it's mainly that I've heard mathematicians
make comments somewhat in favor of formalism. For instance, I
remember one guy at an afternoon tea remarking that we could just as
well choose whichever set of axioms for set theory we wanted, and he
thought assuming all sets of reals were measurable (and sticking to
just the axiom of dependent choice) would be handy.
I think formalism raises enough issues, however, that it hasn't really
succeeded in establishing itself as the default understanding by
mathematicians of what their subject is about. Saying that there is
not just one twin prime conjecture (that there are infinitely many
integers n such that n and n+2 are both prime), but an assortment
depending on which axiom system you mean to use to prove it,
grates against many of our intuitions. I don't see any easy way
for a formalist to avoid this kind of pluralism.
That leaves the mathematical community with a lot of this kind of
squishy realism. Fortunately, not much about the way mathematics
is actually done depends on this kind of philosophical issue, so
there's not much motivation to settle the issue convincingly one way
or the other.
Keith Ramsay
KRamsay
Herschkorn" <hers...@rutcor.rutgers.edu> writes:
|Are Martin and Dave constructivists? From Davis and
|Hersh,_The_Mathematical_Experience_:
|
|obtained by a finite construction. The set of real numbers, or any
|other infinite set, cannot so be obtained."
Davis and Hersh have not tried very hard to understand what is meant
by constructivism. Was L.E.J.Brouwer a constructivist? Yes. Did he
accept the set of real numbers? Yes. Was Errett Bishop a constructivist?
Yes. Did Bishop accept the set of real numbers? Yes. Have a look at
any of their work and you'll soon see they aren't the kind of finitist that
Davis and Hersh mistakenly thought constructivists are.
In article <b671fc3e.0402...@posting.google.com>, cafe...@msn.com
(Craig Feinstein) writes:
|Then there is the intuitionist philosophy which is a little more
|mainstream but is still not the majority opinion. Their objection is
|not based on whether infinity exists or not, but on whether we can
|even say anything about infinity if it did exist.
That's silly.
|What separates them
|from the classic mathematicians is that they reject Aristotle's Law of
|the Excluded Middle that the proposition (A or not A) is a tautology,
Rejecting the law of excluded middle is one of the main differences,
although not the only one.
|since one can never know for sure if for each x, (A(x) or not A(x)) is
|a tautology when x ranges over an infinite domain.
Intuitionists accept assumptions that imply that in some cases
"for each x, A(x) or not A(x)" is _false_. Describing it as a matter
of not knowing whether it's a tautology is a little odd. In constructive
terms, knowing "A or B" means having a method which in principle
will select one of them, with a guarantee that the one selected is
true. Certainly one doesn't generally have a procedure for determining
whether A(x) is true or false.
|While this may seem
|crazy and counter-intuitive, intuitionists might respond that the
|theorems that result from this assumption are also counter-intuitive -
|i.e., Cantor's aleph null and aleph one.
Do you mean "i.e." or do you mean "e.g."? If you really mean "i.e.",
you're indicating that you've just stated all the counterintuitive
theorems that result from the assumption. If you only mean "for
example", that's abbreviated by "e.g.".
|Why should there be many
|types of infinities? Infinity is infinity.
This is not the constructivist point of view.
Brouwer didn't accept all of Cantor's infinities, for other reasons
(for example, not accepting the power set axiom as a general
principle), but he did agree that there were different kinds of
infinity. For instance, that the continuum was a different kind of
infinity than the integers.
I would guess Bishop thought there was a somewhat misplaced
emphasis in the theory of cardinalities, but had no special problem
with the theorem that there isn't an onto mapping from the natural
numbers to the real numbers.
The counterintuitive consequences of the law of excluded middle
have more to do with the claim that mathematical objects can exist
allegedly without there being any way of our finding them.
| And also, look at all of
|the paradoxes of set theory - Is there a set of all sets, etc. When
|one eliminates the Law of the Excluded Middle, one avoids a lot of the
|crazy paradoxes associated with infinity.
Which ones? The existence of a set of all sets still results in a
contradiction even without the law of excluded middle. It would
imply that the Russell "set" R={x : x is not a member of x} exists.
If R is a member of R, then by the definition of R, we get that
R is not a member of R, which is a contradiction. Consequently
R is not a member of R. But then by the definition of R, R must
be a member of R, a contradiction.
I didn't need the law of excluded middle there. I needed that a
proof that a proposition P leads to a contradiction constitutes
a proof of "not P". That's the "canonical" way to prove a negative
statement constructively. I also needed that y is a member of
{x: P(x)} if and only if P(y), which is just what {x: P(x)} means.
So evidently there's no such thing as the Russell "set".
|The classic philosophy (Platonism) says that all of this stuff
|(infinity) is in fact real. And just because we cannot perceive it
|does not mean that it does not exist. This is the majority opinion,
|not necessarily because the majority believe it, but more because it
|is the most practical way of thinking of mathematics,
I would say that in practice it helps you do mathematics if you think
of the objects and relationships involves as being just as real as all
the stuff of daily life that one deals with the rest of the time.
Now, it happens that some of the most philosophically minded
constructivists have been people whose philosophy led them to
treat all realms the same way. I remember one telling me with a
chuckle how it was sometimes claimed that we apply the law of
excluded middle in reasoning about real world, but that people
in fact hardly ever did.
| since the
|conclusions have applications in natural sciences.
Nonconstructive mathematics is applied in natural science, but I'm
not entirely convinced this is helpful overall.
|If we were to go to
|what my brother-in-law proposes, replacing differential equations with
|difference equations, etc., we would get the same theorems, but they
|would be much more difficult to write and describe. The "derivative"
|of x^2 would not be 2x but would be 2x+1/M, where M is the largest
|number. In practice, this number 1/M would be so small that it
|wouldn't even matter, so why write it down on paper? Also, the
|Pythagorean theorem would not hold, and it would be much more messy to
|describe this relationship. But this way of thinking still avoids
|paradoxes.
This is a different story entirely. :-) Inquiring minds want to know whether
you married his sister, or (perhaps more unsettling?) your sister married
him.
Perhaps he'd like to hunt down a paper by Jan Mycielski written in
about 1980 where he describes a system where you deal with real
numbers by pretending that they're ratios of really big numbers. In
a model of his axiom system, there are infinite elements, but in any
given proof in the system, you only use finitely many axioms, and
any finite collection of axioms has a finite model where the elements
are merely large.
Keith Ramsay
Jesse F. Hughes
> In article <873c9ev...@phiwumbda.org>, je...@phiwumbda.org (Jesse F.
> Hughes) writes:
> |Really? And that part about where intuitionism avers that mathematics
> |and proofs cannot be communicated? That doesn't conflict with any
> |mathematicians' experiences?
>
> I don't know that anyone has ever followed Brouwer that far.
I'm sure you know a lot more about the state of modern intuitionism
than I do, so I'll take your word for it. Frankly, it's good to hear
that, since that part of intuitionism is certainly the most dubious to
my ears.
> But I think I can see how one might come to some of his pessimistic
> conclusions about language. On the one hand, we can see people
> learning to play certain kinds of language games, as a kind of
> skill. He expressed great misgivings about people adopting such an
> instrumental attitude toward language, mathematics, or just about
> anything. On the other hand, if it really is supposed to be a matter
> of carrying a thought from one mind to another, isn't it somewhat
> remarkable that we feel such confidence that the thought that
> arrives at the other end is really "the same" as the one that set
> forth? I'm not at all as pessimistic, but I still find language
> often sliding toward either a successful but robot-like technique
> that succeeds in conveying basically data, or into poetry that is
> easily lost in translation.
Personally, I find that the mathematical constructions that I carry
out in my poor addled brain are much more dubious than those that I
express in writing. If I understand Brouwer (likely not), it is the
former that yields greater certainty and the latter is just an
approximation of the former. But I'm not really sure what counts as a
mental construction versus an attempt to communicate the same.
--
Jesse F. Hughes
"Leaving things always seems to fix me,
Running seems to ease my worried mind."
-- Bad Livers, "Honey, I've Found a Brand New Way"
John Savard
<mart...@is.online.no> wrote, in part:
>What is the role of infinity in math:
The role of completed infinity is limited to stuff like axiomatic set
theory, but incomplete infinity pervades mathematics.
>How is it defined?
>
>Why is it needed?
>
>At what precition does math work?
If math were limited to any set precision, it would be mere
computation, and if a higher precision were needed, then we would have
to work out math all over again.
That's why infinity is needed in math, so we get it right the first
time.
John Savard
http://home.ecn.ab.ca/~jsavard/index.html
Torkel Franzen
> The role of completed infinity is limited to stuff like axiomatic set
> theory, but incomplete infinity pervades mathematics.
What's incomplete about the infinity of the real numbers?