Calculus XOR Probability
Han de Bruijn
http://groups.google.nl/group/sci.math/msg/13795822737a77ca?hl=en&
Then we define an interval as the construction set N = {1,2,3, ... ,n},
which is the one in the poster (except zero). Furthermore we define (a)
as natural valued and the function D(a,n) = '(number of naturals <= n
divisible by fixed natural a) divided by n'. Then D(a,n) = floor(n/a)/n
And therefore:
lim D(a,n) = 1/a = D(a)
n->oo
On the other hand, consider the function E(k,n) = '(number of naturals
<= n which are equal to fixed natural a) divided by n'. Then it's found
that E(a,n) = 1/n . It is possible to develop a Random Number Generator
for a Naturals Construction Set, indeed. As it's implemented in most of
our nowadays programming languages. In Delphi (Pascal), for example, it
sounds like: integer Random(integer). But, on the other hand:
lim E(a,n) = lim 1/n = 0 = E(a)
n->oo n->oo
Meaning that there is NO chance that E(a) will happen in ALL naturals.
Meaning that there does _not_ exist a random number generator on _all_
of the Naturals.
But, at the same time, the Kolgomorov axioms of Probability Theory are
nevertheless satisfied. Because:
oo
lim Sigma E(a,n) = lim n.1/n = 1
n->oo n=0 n->oo
Thus we have the conceptual difficulty that _elementary_ probabilities
are virtually zero, while their infinite sequence still sums up to one.
Strangely enough, such seems to be perfectly acceptable to most of us
if the above sum is interpreted otherwise, namely as a Riemann sum:
lim 1/n.n = 1 : n blocks with height 1 and width 1/n
n->oo
Corresponding with an integral(0..1) 1.dx = 1 . And indeed the blocks
become infinitely thin for n -> oo , but nobody has trouble with this.
Now imagine these blocks rotated over 90 degrees, then we have:
lim n.1/n = 1 : n blocks with height 1/n and width 1
n->oo
It makes _no_ difference for the mathematics of the (Riemann) sum, but
- quite suddenly - a sensible understanding seems to be lacking. While
it is not so difficult after all. We only have to accept the fact that
an infinite sum of zeroes is not zero. It's actually _undefined_ until
we say where the zeroes, and the infinity, come from. In our case, the
zeroes come from 1/n while the infinity comes from n. Consequently the
infinite sum is n.1/n = 1 . Nothing undefined after all. How much is 0
times oo ? The answer is therefore: that depends ...
How can the assumption, that an infinite sum of zeroes may not be zero
be destructive for mathematics? Is it really destructive? And how?
Han de Bruijn
David C. Ullrich
No, the axioms of probability theory are not satisfied if we
define the probability of a set of natural numbers to be its
density, which is what all this appears to be doing. Because
the axioms refer to the actual probabilities, and here for
example
sum_a lim_n E(a,n) = 0.
>Thus we have the conceptual difficulty that _elementary_ probabilities
>are virtually zero, while their infinite sequence still sums up to one.
>Strangely enough, such seems to be perfectly acceptable to most of us
>if the above sum is interpreted otherwise, namely as a Riemann sum:
>
> lim 1/n.n = 1 : n blocks with height 1 and width 1/n
> n->oo
>
>Corresponding with an integral(0..1) 1.dx = 1 . And indeed the blocks
>become infinitely thin for n -> oo , but nobody has trouble with this.
>Now imagine these blocks rotated over 90 degrees, then we have:
>
> lim n.1/n = 1 : n blocks with height 1/n and width 1
> n->oo
>
>It makes _no_ difference for the mathematics of the (Riemann) sum, but
>- quite suddenly - a sensible understanding seems to be lacking. While
>it is not so difficult after all. We only have to accept the fact that
>an infinite sum of zeroes is not zero. It's actually _undefined_ until
>we say where the zeroes, and the infinity, come from. In our case, the
>zeroes come from 1/n while the infinity comes from n. Consequently the
>infinite sum is n.1/n = 1 . Nothing undefined after all. How much is 0
>times oo ? The answer is therefore: that depends ...
>
>How can the assumption, that an infinite sum of zeroes may not be zero
>be destructive for mathematics? Is it really destructive? And how?
It's impossible to answer this question until you state _all_ the
axioms of your system (and then persuade someone to read them).
It follows from standard axioms by definition that the sum of
infinitely many zeroes _is_ zero. So you must be throwing
out some standard axioms. You have not stated what the
axioms _are_.
>Han de Bruijn
************************
David C. Ullrich
Tony Orlow
Han, I am not sure what this has to do with XOR, but I agree entirely with your
point. Given some number n and its reciprocal, as n goes to infinity, their
product remains at 1, and given any 0 and any oo, the product MAY be nonzero.
This is one of my favorite math paradoxes, which has guided a considerable
amount of my thinking. If 0*x=0 and oo*x=oo, then what is 0*oo? It depends on
the infinity and on the zero you're working with, so a precise notion of
infinite and infinitesimal numbers is required to really answer any such
question. Good point. I would say, given a unit infinitesimal i and a unit
infinity oo, i*oo=1.
--
Smiles,
Tony
Tony Orlow
The actual probabilities of what? Of each natural satisfying the requirements
of the set? What axioms refer to the probability of each natural in the entire
set being randomly selected? As the size of the set apporaches oo, that
probability approaches 0, in the limit. But, every natural HAS a chance of
being selected, even if it's infinitesimal and considered a 0% chance, if you
are picking one of them at random. So, the chance is not truly zero, and no
axiom says it is, or if it does, it's wrong. Given that each natural has an
infinitesimal but nonzero chance of being selected, the chance of selecting an
n which satisfies E(a,n) can well be seen to be a sum of all these
infintiesimal chances of choosing each of them.
What standard axioms are thrown out when calculus measures the area under a
curve as the sum of the approximating rectangles as their width approaches 0
and their count approaches oo over a finite range? Han doesn't seem to be doing
anything unreasonable as far as I can see.
--
Smiles,
Tony
Virgil
Tony Orlow <ae...@cornell.edu> wrote:
Given any finite set, the axioms of probability theory say what the
probabilities of random selection from that set are.
And there is nothing in those axioms, or any others, that requires
sum_a lim_n E(a,n) = lim_n sum_a E(a,n)
> Of each natural satisfying the requirements
> of the set? What axioms refer to the probability of each natural in the
> entire
> set being randomly selected?
It is the definition of "randomly selected", rather than any axiom, that
governs what "randomly selected" means.
> >
>
> What standard axioms are thrown out when calculus measures the area
> under a curve as the sum of the approximating rectangles as their
> width approaches 0 and their count approaches oo over a finite range?
The properties of area involved are that if one plane region entirely
contains another, it has an area at least as great as that of the region
contained.
Then if outermeasure >= area-to-be-found >= innermeasure and
outermeasure-innermeasure -> 0 as the rectangles get narrower
then area-to-be-found is the common limit.
At least for Reimann integration.
Han de Bruijn
>
> No, the axioms of probability theory are not satisfied if we
> define the probability of a set of natural numbers to be its
> density, which is what all this appears to be doing. Because
> the axioms refer to the actual probabilities, and here for
> example
>
> sum_a lim_n E(a,n) = 0.
But what if we throw out the limits for the moment being. Then for
_any_ finite set {1,2,3,4,5, ... ,n} :
E(a,n) = 1/n and sum_a E(a,n) = n.1/n = 1 . Right or wrong?
Since this result does not depend on n, how can the limit for n->oo
then nevertheless result in zero = 0 ?
Han de Bruijn
Han de Bruijn
> Han, I am not sure what this has to do with XOR, [ .. snip .. ]
We have _one_ and the same formula, with _two_ interpretations,
one in calculus and one in probability theory.
Calculus:
>
> lim 1/n.n = 1 : n blocks with height 1 and width 1/n
> n->oo
Probability:
>
> lim n.1/n = 1 : n blocks with height 1/n and width 1
> n->oo
Calculus says that the outcome is 1 , while Probability Theory says
that, instead, we must do a dirty trick like this:
lim n.1/n = lim n ( lim 1/a ) = lim 0 = 0
n->oo n->oo a->oo n->oo
Consequently, there is a discrepancy between Calculus and Probability.
Therefore one can accept Calculus or Probability Theory, but not both.
Hence the XOR .
Han de Bruijn
Han de Bruijn
Persuasion not needed. You _have_ already read them. My axioms are a
subset of yours, namely the ones that apply to _finite_ sets (of the
form {1,2,3,4,5, ... ,n} in this case).
> It follows from standard axioms by definition that the sum of
> infinitely many zeroes _is_ zero. So you must be throwing
> out some standard axioms. You have not stated what the
> axioms _are_.
It follows from "my" axioms that the sum of n many 1/n's is 1 and that
it makes no difference how big n is. Now let n->oo and I find it hard
to comprehend why that wouldn't be the same as "for all naturals" and
even harder that the sum quite suddenly becomes 0 instead of 1 .
Han de Bruijn
Han de Bruijn
>
> Given any finite set, the axioms of probability theory say what the
> probabilities of random selection from that set are.
Example. Given the set {1,2,3,4,5, ... ,n}, the probabilities of random
selection from that set are E(a,n) = 1/n .
> And there is nothing in those axioms, or any others, that requires
>
> sum_a lim_n E(a,n) = lim_n sum_a E(a,n)
But for the finite set: sum_a E(a,n) = n.1/n = 1. No matter what n is.
Accomplish that first and then start thinking about limits. I really
don't get what your point is here.
> It is the definition of "randomly selected", rather than any axiom, that
> governs what "randomly selected" means.
Agreed. Just _define_ then: E(n,a) = 1/n . No?
Han de Bruijn
David C. Ullrich
<Han.de...@DTO.TUDelft.NL> wrote:
>David C. Ullrich wrote:
>>
>> No, the axioms of probability theory are not satisfied if we
>> define the probability of a set of natural numbers to be its
>> density, which is what all this appears to be doing. Because
>> the axioms refer to the actual probabilities, and here for
>> example
>>
>> sum_a lim_n E(a,n) = 0.
>
>But what if we throw out the limits for the moment being. Then for
>_any_ finite set {1,2,3,4,5, ... ,n} :
>
>E(a,n) = 1/n and sum_a E(a,n) = n.1/n = 1 . Right or wrong?
Right.
>Since this result does not depend on n, how can the limit for n->oo
>then nevertheless result in zero = 0 ?
The limit of the sum is 1. I didn't say otherwise.
You claimed that this thing satisfies the axioms of probability
theory. It does not. Because the thing that seems like it might
satisfy those axioms _is_ the limit of the density. And the
sum of the limits here is not the limit of the sum.
David C. Ullrich
<Han.de...@DTO.TUDelft.NL> wrote:
>David C. Ullrich wrote:
>
>> On Mon, 13 Mar 2006 12:40:50 +0100, Han de Bruijn
>> <Han.de...@DTO.TUDelft.NL> wrote:
>>>
>>> How can the assumption, that an infinite sum of zeroes may not be zero
>>> be destructive for mathematics? Is it really destructive? And how?
>>
>> It's impossible to answer this question until you state _all_ the
>> axioms of your system (and then persuade someone to read them).
>
>Persuasion not needed. You _have_ already read them. My axioms are a
>subset of yours,
No, you said that you're taking the sum of infinitely many
zeroes to be non-zero. So either your axioms are different
or you're making an error.
> namely the ones that apply to _finite_ sets (of the
>form {1,2,3,4,5, ... ,n} in this case).
>
>> It follows from standard axioms by definition that the sum of
>> infinitely many zeroes _is_ zero. So you must be throwing
>> out some standard axioms. You have not stated what the
>> axioms _are_.
>
>It follows from "my" axioms that the sum of n many 1/n's is 1 and that
>it makes no difference how big n is. Now let n->oo and I find it hard
>to comprehend why that wouldn't be the same as "for all naturals" and
>even harder that the sum quite suddenly becomes 0 instead of 1 .
What you find hard to believe has little to do with mathematical
truth. And I didn't say that "the sum becomes 0 instead of 1."
_Unless_ you're talking about a totally different system of
mathematics, in which case you have a _lot_ of preliminary
explaining to do, the sum of infinitely many zeroes _is_
zero. You can't just claim that you're doing standard
mathematics except that you're going to change this one
fact.
The reason for this phenomenon that you find hard to believe
is that when we're talking about infinite sums the sum of
the limit need not be the limit of the sum. You may not
like that, but that's the way it _is_. (That's the way it
is in standard mathematics - if you're talking about
something else you need to clarify _all_ the rules,
starting from the beginning.)
David C. Ullrich
<Han.de...@DTO.TUDelft.NL> wrote:
>Tony Orlow wrote:
>
>> Han, I am not sure what this has to do with XOR, [ .. snip .. ]
>
>We have _one_ and the same formula, with _two_ interpretations,
>one in calculus and one in probability theory.
>
>Calculus:
>>
>> lim 1/n.n = 1 : n blocks with height 1 and width 1/n
>> n->oo
>
>Probability:
>>
>> lim n.1/n = 1 : n blocks with height 1/n and width 1
>> n->oo
>
>Calculus says that the outcome is 1 , while Probability Theory says
>that, instead, we must do a dirty trick like this:
>
> lim n.1/n = lim n ( lim 1/a ) = lim 0 = 0
> n->oo n->oo a->oo n->oo
Nonsense. Probability theory says no such thing.
>Consequently, there is a discrepancy between Calculus and Probability.
>Therefore one can accept Calculus or Probability Theory, but not both.
>
>Hence the XOR .
>
>Han de Bruijn
************************
David C. Ullrich
Han de Bruijn
Wait a minute! What you saying here is the following.
For N = {1,2,3, ... ,n, ... }, that is: THE Naturals, we have:
sum_a E(a) = 0 because E(a) = 0 hence E is not a probability.
But for _ANY_ finite subset {1,2,3, ... ,n} of the naturals we have:
sum_a E(a,n) = 1 (for all n) while lim_(n->oo) E(a,n) = 0 = E(a)
How comes that a result for _any_ finite subset of the kind does not
fit smoothly into a result for the infinite set of naturals N ?
Han de Bruijn
Han de Bruijn
> On Tue, 14 Mar 2006 10:40:12 +0100, Han de Bruijn
> <Han.de...@DTO.TUDelft.NL> wrote:
>
> No, you said that you're taking the sum of infinitely many
> zeroes to be non-zero. So either your axioms are different
> or you're making an error.
Let's withdraw _that_ statement for the moment being. Informal question:
in your opinion, Mr. Ullrich, do infinitesimal quantities have a chance
in standard mathematics? And if so, what are the pro's and con's?
Han de Bruijn
Han de Bruijn
> On Tue, 14 Mar 2006 10:27:03 +0100, Han de Bruijn
> <Han.de...@DTO.TUDelft.NL> wrote:
>
>>Calculus says that the outcome is 1 , while Probability Theory says
>>that, instead, we must do a dirty trick like this:
>>
>> lim n.1/n = lim n ( lim 1/a ) = lim 0 = 0
>> n->oo n->oo a->oo n->oo
>
> Nonsense.
Agreed.
> Probability theory says no such thing.
I sincerely hope that you are right.
Han de Bruijn
ste...@nomail.com
> Han de Bruijn
Because finite and infinite are very different.
Stephen
Tony Orlow
Oh! That's the XOR. I get it. Hmmm.... Do we have the option of rejecting
both? ;) Just kidding.
Seriously, if probability theory claims that this sum over an infinite domain
must be zero because the individual probabilities are each essentially zero,
without regard to the portion of the entire sample space that the set of such
zero-probability occurrences occupies, then I agree there's a serious mistake
there. Calculus is fine. It acknowledges the notion of the infintiesimal. I
agree, Han.
--
Smiles,
Tony
Virgil
Han de Bruijn <Han.de...@DTO.TUDelft.NL> wrote:
Not unless E(n,a) = E(a,n) for all a and all n.
If you really meant E(a,n) = 1/n for all n, then E(a,oo) = 0 and sum 0 =
0.
Virgil
For the same reason that any finite subset is not infinite.
The jump from finite to infinite is just that, a jump. To expect
non-jump behaviour is to court disappointment.
Virgil
Han de Bruijn <Han.de...@DTO.TUDelft.NL> wrote:
> David C. Ullrich wrote:
>
> > On Tue, 14 Mar 2006 10:27:03 +0100, Han de Bruijn
> > <Han.de...@DTO.TUDelft.NL> wrote:
> >
> >>Calculus says that the outcome is 1 , while Probability Theory says
> >>that, instead, we must do a dirty trick like this:
> >>
> >> lim n.1/n = lim n ( lim 1/a ) = lim 0 = 0
> >> n->oo n->oo a->oo n->oo
> >
> > Nonsense.
>
> Agreed.
>
> > Probability theory says no such thing.
>
> I sincerely hope that you are right.
>
There is nothing in probability theory that requires any sort of
"continuity" in the discontinuous jump from choosing at random from a
finite sample space to choosing at random from an infinite one.
Tony Orlow
If oo is a specific infinity, then 0 is a specific infinitesimal, and Han's
logic holds just fine. Do you really think the limit of n * 1/n is 0 as n->oo?
It's simply false.
--
Smiles,
Tony
Virgil
Tony Orlow <ae...@cornell.edu> wrote:
> Virgil said:
> > In article <53651$44169371$82a1e228$12...@news2.tudelft.nl>,
> > Han de Bruijn <Han.de...@DTO.TUDelft.NL> wrote:
> >
> > > Virgil wrote:
> > > >
> > > > Given any finite set, the axioms of probability theory say what the
> > > > probabilities of random selection from that set are.
> > >
> > > Example. Given the set {1,2,3,4,5, ... ,n}, the probabilities of random
> > > selection from that set are E(a,n) = 1/n .
> > >
> > > > And there is nothing in those axioms, or any others, that requires
> > > >
> > > > sum_a lim_n E(a,n) = lim_n sum_a E(a,n)
> > >
> > > But for the finite set: sum_a E(a,n) = n.1/n = 1. No matter what n is.
> > > Accomplish that first and then start thinking about limits. I really
> > > don't get what your point is here.
> > >
> > > > It is the definition of "randomly selected", rather than any axiom,
> > > > that
> > > > governs what "randomly selected" means.
> > >
> > > Agreed. Just _define_ then: E(n,a) = 1/n . No?
> > >
> > > Han de Bruijn
> >
> > Not unless E(n,a) = E(a,n) for all a and all n.
> >
> > If you really meant E(a,n) = 1/n for all n, then E(a,oo) = 0 and sum 0 =
> > 0.
> >
>
> If oo is a specific infinity
E(a,oo) is merely shorthand for lim_{n -> oo} E(a,n), assuming the
limit exists. So that limit either does not exist at all or is
identically zero.
And just to make things quite clear, lim_{n ->oo} f(n) = L is also an
abbreviation , in this case for the statement
for any epsilon greater than zero,
there are only finitely many naturals n for which
|f(n) - L| >= epsilon
Han de Bruijn
In standard mathematics. In all of the _sciences_, however, the infinite
is nothing else than the idealized very large finite.
Han de Bruijn
Han de Bruijn
> Calculus is fine. It acknowledges the notion of the infintiesimal.
That's what I asked to David Ullrich: is it indeed that infinitesimals
are recognized in standard mathematics? Or is that a forbidden area as
well? I mean, as a physicist, I have no problems with infinitesimals ..
Han de Bruijn
Han de Bruijn
> The jump from finite to infinite is just that, a jump. To expect
> non-jump behaviour is to court disappointment.
Meaning that Probability Theory suffers from kind of Balls in a Vase
paradox. But, of course, mainstream mathematics will deny that it is
a paradox and will continue to find it consistent. That distinguishes
mathematics from science.
Han de Bruijn
Han de Bruijn
> There is nothing in probability theory that requires any sort of
> "continuity" in the discontinuous jump from choosing at random from a
> finite sample space to choosing at random from an infinite one.
I'm only flabbergasted that this is the official doctrine. And that no
mainstream mathematician feels any shame or hesitation to advertize it.
Han de Bruijn
David C. Ullrich
<Han.de...@DTO.TUDelft.NL> wrote:
>David C. Ullrich wrote:
>
>> [...]
>How comes that a result for _any_ finite subset of the kind does not
>fit smoothly into a result for the infinite set of naturals N ?
Because N is not a finite set.
This comes up a lot around here - if you assume that infinite
sets work the same as finite sets you're going to be deriving
a lot of falsehoods, because infinite sets do _not_ work the
same.
For example, if we have a finite sum then the sum of the
limits is equal to the limit of the sum. That's simply not
true for infinite sums. It's hard to see how to answer the
question "why not" - you can't prove that it _does_ work
that way and you've been given an example showing that it
does _not_ work that way.
David C. Ullrich
If you mean "very large finite" you should say so.
"Infinite" does not mean the same thing.
David C. Ullrich
<Han.de...@DTO.TUDelft.NL> wrote:
>Virgil wrote:
>
>> The jump from finite to infinite is just that, a jump. To expect
>> non-jump behaviour is to court disappointment.
>
>Meaning that Probability Theory suffers from kind of Balls in a Vase
>paradox.
It means no such thing. There are things that are not precisely
defined in that "paradox", hence the difference of opinions about
the answer.
>But, of course, mainstream mathematics will deny that it is
>a paradox and will continue to find it consistent. That distinguishes
>mathematics from science.
You have not found an inconsistency in probability theory. You
have found, or rather been shown, an aspect of probability theory
that doesn't work the way you expect it to. There's a big difference.
David C. Ullrich
<Han.de...@DTO.TUDelft.NL> wrote:
>David C. Ullrich wrote:
>
>> On Tue, 14 Mar 2006 10:40:12 +0100, Han de Bruijn
>> <Han.de...@DTO.TUDelft.NL> wrote:
>>
>> No, you said that you're taking the sum of infinitely many
>> zeroes to be non-zero. So either your axioms are different
>> or you're making an error.
>
>Let's withdraw _that_ statement for the moment being. Informal question:
>in your opinion, Mr. Ullrich, do infinitesimal quantities have a chance
>in standard mathematics? And if so, what are the pro's and con's?
The question needs to be more precisely formulated.
(For example, I don't know what it means to say that something
"has a chance in standard mathematics".)
Also I don't see why anyone should care about my _opinion_
on a mathematical question.
ste...@nomail.com
> Han de Bruijn
Then the _sciences_ are misusing the word 'infinite'. If
you do not mean 'infinite', do not say 'infinite'.
Stephen
Han de Bruijn
>>in your opinion, Mr. Ullrich, do infinitesimal quantities have a chance
>>in standard mathematics? And if so, what are the pro's and con's?
>
> The question needs to be more precisely formulated.
> (For example, I don't know what it means to say that something
> "has a chance in standard mathematics".)
OK. _Are_ infinitesimals actually _defined_ in standard mathematics?
And if so, could you please tell me how?
> Also I don't see why anyone should care about my _opinion_
> on a mathematical question.
Aren't you a human being? Why then shouldn't I care about the _opinion_
of somebody who has more knowledge about standard mathematics as I have
(at least that's what I find).
Han de Bruijn
ste...@nomail.com
> Han de Bruijn
Why should anyone hesitate to recognize that 'infinite' and 'finite'
are very different things?
Stephen
Tony Orlow
Sure, Virgil, that's the standard treatment that was finally agreed upon as
being the most rigorous, but that doesn't mean it's the only possible approach.
When you talk about the limit of the probability of picking a particular
element from a set, as the set size approaches oo, that probability approaches
0, and so one can say that the chance of picking any particular number from an
infinite set may be considered to be 0%. The problem is that there is actually
SOME chance of picking any given element of the set, since one of the elements
is to be chosen. So, the chance is not truly zero, but some infinitesimal
value.
There is also the notion of a specific unit oo and a specific unit
infinitesimal, say iota. Then the limit of 1/n as n->oo is not 0, but iota,
this infinitesimal value. When you state that the limit statement is shorthand
for there being only finitely many f(n) which are more different from the limit
L than any particular epsilon, that is because epsilon is a finite value. If
epsilon is instead allowed to take the infinitesimal value iota, then there may
be actually infinitely many f(n) such that |f(n)-L|>epsilon, and yet, epsilon
is not exactly 0. Han is addressing the notion of unit infinity and
infinitesimal. Are you surprised? Have a nice day.
--
Smiles,
Tony
Tony Orlow
Hi Han -
Here's an interesting paper on the subject of Continuity and Infinitesimals,
with some historical background: http://plato.stanford.edu/entries/continuity/
"However useful it may have been in practice, the concept of infinitesimal
could scarcely withstand logical scrutiny. Derided by Berkeley in the 18th
century as “ghosts of departed quantities”, in the 19th century execrated by
Cantor as “cholera-bacilli” infecting mathematics, and in the 20th roundly
condemned by Bertrand Russell as “unnecessary, erroneous, and self-
contradictory”, these useful, but logically dubious entities were believed to
have been finally supplanted in the foundations of analysis by the limit
concept which took rigorous and final form in the latter half of the 19th
century. By the beginning of the 20th century, the concept of infinitesimal had
become, in analysis at least, a virtual “unconcept”."
It seems that infinitesimals were dropped in favor of the more "rigorous"
approach using limits. I still think infinitesimals make sense, as I said in my
response to Virgil just now, in this context, as the chance of a particular
element being randomly selected from an infinite set, as you say. But, they
seem to be largely out of vogue, though resurrected in respectable form in
Robinson's Hyperreals: http://en.wikipedia.org/wiki/Hyperreal_number
--
Smiles,
Tony
Tony Orlow
I've seen some pretty-unkosher games played with infinite sums. Can you give a
concrete example of what you're talking about? What limits are you taking?
--
Smiles,
Tony
Toni Lassila
<Han.de...@DTO.TUDelft.NL> wrote:
>OK. _Are_ infinitesimals actually _defined_ in standard mathematics?
>And if so, could you please tell me how?
Oddly enough, the treatment of calculus by rigorous notion of
infinitesimals is called 'nonstandard analysis', so I guess the answer
would strictly be 'no'.
I know nothing of the details.
Tony Orlow
But, David, what IS the chance of any given element being selected from an
infinite set? In standard probability, it's 0%, indistinguishable from no
chance at all. And yet, each element does have some chance, because one of them
is going to be selected. So, how does one represent that kind of chance
mathematically? Isn't that an infinitesimal probability?
--
Smiles,
Tony
Dave Rusin
"Forbidden"?
Mathematicians are a libertine, almost anarchistic group.
Nothing is really forbidden -- you're welcome to pursue whatever
you like. The only requirement is that you define your terms and
clarify your assumptions, and then proceed logically.
There are indeed perfectly logical and useful notions of "infinitesimals".
Emphasis on the plural here -- e.g. one definition of "infinitesimal" is
the set of nilpotent elements in the ring F[x]/(x^2) where F is
a field. I'll bet that's not the one you use in physics, but what
you DO have in mind I don't quite know.
One can build a theory of the calculus on a logical foundation which
involves another kind of infinitesimal; one can even prove a
transfer theorem which allows us to decide which results from
that theory apply to ordinary calculus over the real field.
Some folks prefer this approach, others start in measure theory, etc.
Different strokes for different folks. We're a "big tent" kind of group,
so that's OK.
Socks with sandals? Not a problem. Unkempt hair? Par for the course.
Unintentionally insults the host's spouse? Happens all the time.
None of these things is forbidden in mathematics. The only thing
that will get you thown out of the club is an insistence on vague
language and a refusal to accept logical deduction from clear
starting principles.
dave
A N Niel
> OK. _Are_ infinitesimals actually _defined_ in standard mathematics?
> And if so, could you please tell me how?
Definition:
An element s of an ordered field is called *infinitesimal* if s>0,
but s < 1/n for all natural numbers n.
In some ordered fields (for example the real number system)
there are no infinitesimals. In others (for example Conway's
surreal numbers) there are.
Tony Orlow
> In article <b4dd4$4417ec3d$82a1e228$16...@news2.tudelft.nl>,
> Han de Bruijn <Han.de...@DTO.TUDelft.NL> wrote:
> >Tony Orlow wrote:
> >
> >> Calculus is fine. It acknowledges the notion of the infintiesimal.
> >
> >That's what I asked to David Ullrich: is it indeed that infinitesimals
> >are recognized in standard mathematics? Or is that a forbidden area as
> >well? I mean, as a physicist, I have no problems with infinitesimals ..
>
> "Forbidden"?
>
> Mathematicians are a libertine, almost anarchistic group.
That's a laugh, coming from the guy who patiently tried to explain how I was
wrong about countably infinite sets, and then demanded that I either accept
that there are an infinite number of finite naturals or be banished to his
killfile, ala the Spanish Inquisition. Very open minded. If someone responds to
this post, maybe he'll see my response.
> Nothing is really forbidden -- you're welcome to pursue whatever
> you like. The only requirement is that you define your terms and
> clarify your assumptions, and then proceed logically.
That didn't seem to be good enough a year ago.
>
> There are indeed perfectly logical and useful notions of "infinitesimals".
> Emphasis on the plural here -- e.g. one definition of "infinitesimal" is
> the set of nilpotent elements in the ring F[x]/(x^2) where F is
> a field. I'll bet that's not the one you use in physics, but what
> you DO have in mind I don't quite know.
>
> One can build a theory of the calculus on a logical foundation which
> involves another kind of infinitesimal; one can even prove a
> transfer theorem which allows us to decide which results from
> that theory apply to ordinary calculus over the real field.
> Some folks prefer this approach, others start in measure theory, etc.
> Different strokes for different folks. We're a "big tent" kind of group,
> so that's OK.
>
> Socks with sandals? Not a problem. Unkempt hair? Par for the course.
> Unintentionally insults the host's spouse? Happens all the time.
> None of these things is forbidden in mathematics. The only thing
> that will get you thown out of the club is an insistence on vague
> language and a refusal to accept logical deduction from clear
> starting principles.
Or, perhaps, starting from different clear principles than the ones that a
particular mathematician has invested their life in, or not accepting blindly
their decrees as to truths which they base on unfounded axioms. Just those
extras things, that's all.
>
> dave
>
>
>
--
Smiles,
Tony
Virgil
Tony Orlow <ae...@cornell.edu> wrote:
Only if that infinitesimal value is allowed to be smaller thanitself.
>
> There is also the notion of a specific unit oo and a specific unit
> infinitesimal, say iota.
Nonsense! There is no "specific unit infinitesimal" anywhere outside
TOmatics. Even in Robinson's non-standard reals, there are other
infinitesimals infinitely smaller than any (non-zero) infinitesimal.
The only possible smallest non-negative in any real number system,
standard or otherwise, is zero.
Virgil
Tony Orlow <ae...@cornell.edu> wrote:
One notes that Robinson's hyperreals, and other non-standard number
systems, (unlike TOmatics) are firmly grounded in standard set theories,
and are based on standard axiomatic systems and logic.
Virgil
Tony Orlow <ae...@cornell.edu> wrote:
Actually standard mathematics is quite kosher, it is TOmatics that is
tref.
Virgil
Tony Orlow <ae...@cornell.edu> wrote:
It is a situation where a "random selection", as standardly defined,
cannot be properly executed.
As soon as one gives up the requirement that each naturalis to have
exactly the same chance of being chosen as its predecessor, the anomaly
vanishes.
David C. Ullrich
wrote:
You could just read the thread...
Define a sequence of sequences by
a(n)_j = 1/n for 1 <= j <= n
a(n)_j = 0 for j > n.
The idea being that each a(n) is a sequence a(n)_1, a(n)_2, ... .
When I say sum, or sum_j, that means the sum for j = 1 to infinity.
When I say lim, or lim_n, I mean the limit as n tends to infinity.
Now for every n we obviously have
sum_j a(n)_j = 1.
Hence
lim_n(sum_j a(n)_j) = 1.
On the other hand for every j we have
lim_n(a(n)_j) = 0,
so
sum_j(lim_n a(n)_j) = 0.
The sum of the limit is not equal to the limit of the sum.
Nothing paradoxical or unkosher about it, that's just the
way it works.
************************
David C. Ullrich
David C. Ullrich
wrote:
>David C. Ullrich said:
>> On Tue, 14 Mar 2006 15:52:34 +0100, Han de Bruijn
>> <Han.de...@DTO.TUDelft.NL> wrote:
>>
>> >David C. Ullrich wrote:
>> >
>> >> On Tue, 14 Mar 2006 10:40:12 +0100, Han de Bruijn
>> >> <Han.de...@DTO.TUDelft.NL> wrote:
>> >>
>> >> No, you said that you're taking the sum of infinitely many
>> >> zeroes to be non-zero. So either your axioms are different
>> >> or you're making an error.
>> >
>> >Let's withdraw _that_ statement for the moment being. Informal question:
>> >in your opinion, Mr. Ullrich, do infinitesimal quantities have a chance
>> >in standard mathematics? And if so, what are the pro's and con's?
>>
>> The question needs to be more precisely formulated.
>> (For example, I don't know what it means to say that something
>> "has a chance in standard mathematics".)
>>
>> Also I don't see why anyone should care about my _opinion_
>> on a mathematical question.
>>
>> >Han de Bruijn
>>
>>
>> ************************
>>
>> David C. Ullrich
>>
>
>But, David, what IS the chance of any given element being selected from an
>infinite set? In standard probability, it's 0%,
Yes.
Well, not necessarily yes, because you're leaving out some
important details. But there will be times when the answer
is yes.
>indistinguishable from no
>chance at all.
Depends on what you mean by "no chance at all".
> And yet, each element does have some chance, because one of them
>is going to be selected.
Depends on what you mean by "has some chance". Yes, for any
given element there is some chance that it will be selected.
For any given element the probability of that happening is 0.
Probability = 0 simply does not mean impossible.
> So, how does one represent that kind of chance
>mathematically? Isn't that an infinitesimal probability?
No.
************************
David C. Ullrich
Virgil
Tony Orlow <ae...@cornell.edu> wrote:
> Dave Rusin said:
> > In article <b4dd4$4417ec3d$82a1e228$16...@news2.tudelft.nl>,
> > Han de Bruijn <Han.de...@DTO.TUDelft.NL> wrote:
> > >Tony Orlow wrote:
> > >
> > >> Calculus is fine. It acknowledges the notion of the infintiesimal.
> > >
> > >That's what I asked to David Ullrich: is it indeed that infinitesimals
> > >are recognized in standard mathematics? Or is that a forbidden area as
> > >well? I mean, as a physicist, I have no problems with infinitesimals ..
> >
> > "Forbidden"?
> >
> > Mathematicians are a libertine, almost anarchistic group.
>
> That's a laugh, coming from the guy who patiently tried to explain how I was
> wrong about countably infinite sets, and then demanded that I either accept
> that there are an infinite number of finite naturals or be banished to his
> killfile, ala the Spanish Inquisition. Very open minded. If someone responds
> to
> this post, maybe he'll see my response.
>
> > Nothing is really forbidden -- you're welcome to pursue whatever
> > you like. The only requirement is that you define your terms and
> > clarify your assumptions, and then proceed logically.
>
> That didn't seem to be good enough a year ago.
TO does not define his terms nor clarify his assumptions to the
satisfactin of anyone but himself, so that his TOmatics is an island
unconnected to the mathematical main.
>
> >
> > There are indeed perfectly logical and useful notions of "infinitesimals".
> > Emphasis on the plural here -- e.g. one definition of "infinitesimal" is
> > the set of nilpotent elements in the ring F[x]/(x^2) where F is
> > a field. I'll bet that's not the one you use in physics, but what
> > you DO have in mind I don't quite know.
> >
> > One can build a theory of the calculus on a logical foundation which
> > involves another kind of infinitesimal; one can even prove a
> > transfer theorem which allows us to decide which results from
> > that theory apply to ordinary calculus over the real field.
> > Some folks prefer this approach, others start in measure theory, etc.
> > Different strokes for different folks. We're a "big tent" kind of group,
> > so that's OK.
> >
> > Socks with sandals? Not a problem. Unkempt hair? Par for the course.
> > Unintentionally insults the host's spouse? Happens all the time.
> > None of these things is forbidden in mathematics. The only thing
> > that will get you thown out of the club is an insistence on vague
> > language and a refusal to accept logical deduction from clear
> > starting principles.
Which is a capsule description of TO's operating principles.
>
> Or, perhaps, starting from different clear principles than the ones that a
> particular mathematician has invested their life in, or not accepting blindly
> their decrees as to truths which they base on unfounded axioms. Just those
> extras things, that's all.
If those "clear principles" are clearly enough stated, they ARE the
axioms, and if they are not clearly enough stated, as in TOmatics, they
are useless.
Clear up your principles, TO, or give it up.
David C. Ullrich
<Han.de...@DTO.TUDelft.NL> wrote:
>David C. Ullrich wrote:
>
>>>in your opinion, Mr. Ullrich, do infinitesimal quantities have a chance
>>>in standard mathematics? And if so, what are the pro's and con's?
>>
>> The question needs to be more precisely formulated.
>> (For example, I don't know what it means to say that something
>> "has a chance in standard mathematics".)
>
>OK. _Are_ infinitesimals actually _defined_ in standard mathematics?
Depends on exactly what you mean. There do exist mathematical
structures that contain things known as infinitesmals. Those
infinitesmals are not real numbers, so in particular they're
not allowed as probabilities in standard probability theory.
(And sorry, but in the situations I'm aware of where infinitesmals
come up, if i_1, i_2, ... are all infinitesmals it never happens
that i_1 + i_2 + ... = 1.)
>And if so, could you please tell me how?
Various ways. My favorite uses the compactness theorem from logic.
>> Also I don't see why anyone should care about my _opinion_
>> on a mathematical question.
>
>Aren't you a human being? Why then shouldn't I care about the _opinion_
>of somebody who has more knowledge about standard mathematics as I have
>(at least that's what I find).
>
>Han de Bruijn
************************
David C. Ullrich
Robert Low
> On Wed, 15 Mar 2006 16:36:19 +0100, Han de Bruijn
Heh. In the branch of standard mathematics called
nonstandard analysis :-)
> (And sorry, but in the situations I'm aware of where infinitesmals
> come up, if i_1, i_2, ... are all infinitesmals it never happens
> that i_1 + i_2 + ... = 1.)
Sure it does (assuming I haven't misunderstood '...'). If
N is a hyperfinite natural, and i_1..i_N = 1/N, i_n = 0
for n>N, then the sum over all i_n (n ranging over
all naturals, finite and hyperfinite) is 1. If you
want all the infinitesimals to be non-zero, then
you can easily fudge this by taking i_n = (1-epsilon)/N
for i=1..N, then making the remainder a geometric sequence
whose sum is epsilon.
Tony Orlow
Stop licking your frogs, Virgil.
>
> >
> > There is also the notion of a specific unit oo and a specific unit
> > infinitesimal, say iota.
>
> Nonsense! There is no "specific unit infinitesimal" anywhere outside
> TOmatics. Even in Robinson's non-standard reals, there are other
> infinitesimals infinitely smaller than any (non-zero) infinitesimal.
>
> The only possible smallest non-negative in any real number system,
> standard or otherwise, is zero.
>
We have discussed this. The Archimedean priciple is not violated by nilpotent
infinitesimals, since the intermediate value it requires is only needed between
distiguishable values. On the standard finite scale, infinitesimally close
values are not distinguishable, and it is not required on that scale that
anything lie between them. On the infinitesimal scale, where infinitesimally
close values CAN be distinguished, there IS always an intermediate value in
that infinitesimal range, and so the Archimedean principle of the continuum is
preserved. Han's point is correct. n * 1/n = 1, even for infinite n. Every
element in a set has a probability of being randomly selected that is the
reciprocal of the size of that set, even if that size is infinite. The
reciprocal of a specific infinity is not absolute zero, but some specific
infinitesimal value, which reflects the fact that every element does have some
chance of being selected.
On Continuity and Infinitesimals, from Stanford University:
http://plato.stanford.edu/entries/continuity/
--
Smiles,
Tony
Tony Orlow
Random selection means exactly that: each possible candidate has an exactly
equal probability of being selected. If there are y elements in the set, then
the chance than any given one will be selected is 1/y. This is grade school
stuff, Virgil. You're being silly with all that flailing.
--
Smiles,
Tony
Tony Orlow
0% probability CAN mean entirely impossible, and in fact, does mean that in any
finite set of possible outcomes. So, how do you distinguish between 0%
probability where is is some chance and 0% probability when there is none? Why
do you say the first is not the same as having some infinitesimal chance,
nonzero but smaller than any finite probability? They are obvious two different
kinds of zero.
--
Smiles,
Tony
Gerry Myerson
ru...@vesuvius.math.niu.edu (Dave Rusin) wrote:
> Mathematicians are a libertine, almost anarchistic group.
> Nothing is really forbidden
No, but dividing by nothing is.
--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
Keith A. Lewis
>0% probability CAN mean entirely impossible, and in fact, does mean that in any
>finite set of possible outcomes. So, how do you distinguish between 0%
>probability where is is some chance and 0% probability when there is none? Why
>do you say the first is not the same as having some infinitesimal chance,
>nonzero but smaller than any finite probability? They are obvious two different
>kinds of zero.
Here's how to distinguish:
In any countable outcome domain, p=0 means impossible.
In a continuum, having p=0 for a discrete point isn't the whole story. You
have to look at the probability density function.
--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.
Dave Rusin
Gerry Myerson <ge...@maths.mq.edi.ai.i2u4email> wrote:
>In article <dv9e0e$h1l$1...@news.math.niu.edu>,
> ru...@vesuvius.math.niu.edu (Dave Rusin) wrote:
>
>> Mathematicians are a libertine, almost anarchistic group.
>> Nothing is really forbidden
>
> No, but dividing by nothing is.
Not forbidden at all; there's a one-element group consisting only of 0
and it's perfectly OK to have nothing divided by nothing equalling nothing.
In fact, it's required. This, I think, is a corollary of the well-known
theorem of Preston:
Nothing from nothing leaves nothing
(You gotta have something, if you wanna be with me.)
dave
Dik T. Winter
Oddly enough Han does not like that either.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Gerry Myerson
ru...@vesuvius.math.niu.edu (Dave Rusin) wrote:
> In article <gerry-1E1227....@sunb.ocs.mq.edu.au>,
> Gerry Myerson <ge...@maths.mq.edi.ai.i2u4email> wrote:
> >In article <dv9e0e$h1l$1...@news.math.niu.edu>,
> > ru...@vesuvius.math.niu.edu (Dave Rusin) wrote:
> >
> >> Mathematicians are a libertine, almost anarchistic group.
> >> Nothing is really forbidden
> >
> > No, but dividing by nothing is.
>
> Not forbidden at all; there's a one-element group consisting only of 0
> and it's perfectly OK to have nothing divided by nothing equalling nothing.
> In fact, it's required.
A beginner's mistake in group theory, confusing the additive group, {0},
with the multiplicative group, {1}.
See also Kristofferson's Lemma; Nothing ain't worth nothing, but
it's free.
Dik T. Winter
> Dave Rusin said:
...
> > "Forbidden"?
> >
> > Mathematicians are a libertine, almost anarchistic group.
>
> That's a laugh, coming from the guy who patiently tried to explain how I was
> wrong about countably infinite sets, and then demanded that I either accept
> that there are an infinite number of finite naturals or be banished to his
> killfile, ala the Spanish Inquisition. Very open minded. If someone
> responds to this post, maybe he'll see my response.
Is it? Using the standard definitions, axioms and whatever, it can be
proven that there are only a countably finite number of naturals.
> > Nothing is really forbidden -- you're welcome to pursue whatever
> > you like. The only requirement is that you define your terms and
> > clarify your assumptions, and then proceed logically.
>
> That didn't seem to be good enough a year ago.
It was good enough. You use in your discourse your own definitions, your
own logic and your own assumption, all trying to prove that mathematics
with the standard definitions, the standard logic and the standard
assumptions is wrong.
Virgil
Tony Orlow <ae...@cornell.edu> wrote:
> Virgil said:
> > In article <MPG.1e8200bd4...@newsstand.cit.cornell.edu>,
> > Tony Orlow <ae...@cornell.edu> wrote:
> > > There is also the notion of a specific unit oo and a specific unit
> > > infinitesimal, say iota.
> >
> > Nonsense! There is no "specific unit infinitesimal" anywhere outside
> > TOmatics. Even in Robinson's non-standard reals, there are other
> > infinitesimals infinitely smaller than any (non-zero) infinitesimal.
> >
> > The only possible smallest non-negative in any real number system,
> > standard or otherwise, is zero.
> >
>
> We have discussed this. The Archimedean priciple is not violated by
> nilpotent infinitesimals, since the intermediate value it requires is
> only needed between distiguishable values.
The Archimedean principle either holds universally or does not hold at
all, since it is a universal statement. There is no hidden switch
allowing it t be turned off when inconvenient.
>
> On Continuity and Infinitesimals, from Stanford University:
> http://plato.stanford.edu/entries/continuity/
This citation by TO in fact refutes all TO's claims!
Virgil
Tony Orlow <ae...@cornell.edu> wrote:
> Virgil said:
> >
> > It is a situation where a "random selection", as standardly defined,
> > cannot be properly executed.
> >
> > As soon as one gives up the requirement that each naturalis to have
> > exactly the same chance of being chosen as its predecessor, the anomaly
> > vanishes.
> >
>
> Random selection means exactly that: each possible candidate has an exactly
> equal probability of being selected. If there are y elements in the set, then
> the chance than any given one will be selected is 1/y. This is grade school
> stuff, Virgil. You're being silly with all that flailing.
What "flailing" does TO object to. I merely said what has often been
said before, that the definition of "selection at random" cannot
properly be applied to the elements in a countable set.
It is quite possible to apply a probability to countable sets, e.g., the
naturals, for which the probability of n, P(n), is a slowly decreasing
function of n, such that sum[P(n)] = 1. But that does not satisfy the
definition of random selection.
Virgil
Tony Orlow <ae...@cornell.edu> wrote:
On a closed real interval, one can define an additive measure function
such that the measure of a subinterval of the interval is the probabilty
that a value from that subinterval will be chosen. Then there will be
non-empty sets with probability zero, but that does not mean that it is
impossible for a member is such a set to be chosen.
David C. Ullrich
wrote:
Well sure, but that's not what I was talking about.
Saying we have a non-archimedean field doesn't mean we're
doing non-standard analysis! My "..." was the perfectly
ordinary "...", indicating the limit, if any, of the
finite partial sums.
************************
David C. Ullrich
David C. Ullrich
wrote:
Not necessarily, but that's correct in the situations you have in
mind, so never mind.
>So, how do you distinguish between 0%
>probability where is is some chance and 0% probability when there is none?
You _don't_ distinguish them _via_ probability.
>Why
>do you say the first is not the same as having some infinitesimal chance,
>nonzero but smaller than any finite probability?
Because that's the way it is.
>They are obvious two different
>kinds of zero.
Nope. Infinitesmals are not zero. And probabilities are not
infinitesmals.
(By definition. Your cracks about Rusin's recent comments
about how we're allowed to get away with anything as long
as our definitions and deductions are in order were way
off - you don't seem to realize that there _is_ such a
thing as a definition in mathematics.)
************************
David C. Ullrich
Toni Lassila
wrote:
>In article <22eg129j22u0g7ggv...@no.spam> tlas...@cc.hut.fi writes:
> > On Wed, 15 Mar 2006 16:36:19 +0100, Han de Bruijn
> > <Han.de...@DTO.TUDelft.NL> wrote:
> >
> > >OK. _Are_ infinitesimals actually _defined_ in standard mathematics?
> > >And if so, could you please tell me how?
> >
> > Oddly enough, the treatment of calculus by rigorous notion of
> > infinitesimals is called 'nonstandard analysis', so I guess the answer
> > would strictly be 'no'.
>
>Oddly enough Han does not like that either.
Perhaps it wasn't formulated by engineers? He seems to share a common
mindset for physicists/engineers that mathematicians are somehow
"enemies" or "annoying formalists" that contribute nothing and just
ruin things.
Tony Orlow
> In article <MPG.1e821881e...@newsstand.cit.cornell.edu> Tony Orlow <ae...@cornell.edu> writes:
> > Dave Rusin said:
> ...
> > > "Forbidden"?
> > >
> > > Mathematicians are a libertine, almost anarchistic group.
> >
> > That's a laugh, coming from the guy who patiently tried to explain how I was
> > wrong about countably infinite sets, and then demanded that I either accept
> > that there are an infinite number of finite naturals or be banished to his
> > killfile, ala the Spanish Inquisition. Very open minded. If someone
> > responds to this post, maybe he'll see my response.
>
> Is it? Using the standard definitions, axioms and whatever, it can be
> proven that there are only a countably finite number of naturals.
What type of proof are you suggesting? None of my proofs of the finiteness of
the set of naturals seem to be acceptable to those that disagree. Any other
approach would be interesting to see.
>
> > > Nothing is really forbidden -- you're welcome to pursue whatever
> > > you like. The only requirement is that you define your terms and
> > > clarify your assumptions, and then proceed logically.
> >
> > That didn't seem to be good enough a year ago.
>
> It was good enough. You use in your discourse your own definitions, your
> own logic and your own assumption, all trying to prove that mathematics
> with the standard definitions, the standard logic and the standard
> assumptions is wrong.
>
Most mathemtics is correct, but the treatment of zeroes and infinities leaves a
lot to be desired, and the conclusions of what we currently have are often
clearly wrong. Some of the assumption sof the standard theory I see as
unfounded, and reject them as causing such absurd conclusions.
--
Smiles,
Tony
Tony Orlow
Well, if the probability distribution is normal, anyways.
>
> >So, how do you distinguish between 0%
> >probability where is is some chance and 0% probability when there is none?
>
> You _don't_ distinguish them _via_ probability.
I didn't ask how you don't distinguish.
>
> >Why
> >do you say the first is not the same as having some infinitesimal chance,
> >nonzero but smaller than any finite probability?
>
> Because that's the way it is.
Oh, well, that explains everything.
>
> >They are obvious two different
> >kinds of zero.
>
> Nope. Infinitesmals are not zero. And probabilities are not
> infinitesmals.
And that further clarifies the matter.
>
> (By definition. Your cracks about Rusin's recent comments
> about how we're allowed to get away with anything as long
> as our definitions and deductions are in order were way
> off - you don't seem to realize that there _is_ such a
> thing as a definition in mathematics.)
You don't seem to understand what a response to a question consists of, so
never mind. My comment about Rusin was about his libertine attitude a year ago,
when he put me on his kill list for not agreeing with him, and then continued
to make comments about me, like throwing trash over the fence he just built.
>
> ************************
>
> David C. Ullrich
>
--
Smiles,
Tony
David C. Ullrich
wrote:
>David C. Ullrich said:
>> [...]
>> >
>> >0% probability CAN mean entirely impossible, and in fact, does mean that in any
>> >finite set of possible outcomes.
>>
>> Not necessarily, but that's correct in the situations you have in
>> mind, so never mind.
>
>Well, if the probability distribution is normal, anyways.
What do you mean by "normal" here? The word has a technical
meaning in probability, but there's no such thing as a
normal distribution with only finitely many outcomes.
>>
>> >So, how do you distinguish between 0%
>> >probability where is is some chance and 0% probability when there is none?
>>
>> You _don't_ distinguish them _via_ probability.
>
>I didn't ask how you don't distinguish.
I understand that - you asked how you do distinguish between
two things. It's impossible to distinguish them using probability.
>> >Why
>> >do you say the first is not the same as having some infinitesimal chance,
>> >nonzero but smaller than any finite probability?
>>
>> Because that's the way it is.
>
>Oh, well, that explains everything.
>
>>
>> >They are obvious two different
>> >kinds of zero.
>>
>> Nope. Infinitesmals are not zero. And probabilities are not
>> infinitesmals.
>
>And that further clarifies the matter.
As I say below, you don't seem to understand what a definition
is. Probabilities are real numbers, by definition. There are
no infinitesmal real numbers; that follows from various definitions.
So there are no infinitesmal probabilities. That _is_ all there
is to it.
>> (By definition. Your cracks about Rusin's recent comments
>> about how we're allowed to get away with anything as long
>> as our definitions and deductions are in order were way
>> off - you don't seem to realize that there _is_ such a
>> thing as a definition in mathematics.)
>
>You don't seem to understand what a response to a question consists of,
That's not the problem. _You_ don't seem to understand that the answer
to "how do I do this" is sometimes "that's impossible".
> so
>never mind. My comment about Rusin was about his libertine attitude a year ago,
>when he put me on his kill list for not agreeing with him, and then continued
>to make comments about me, like throwing trash over the fence he just built.
No. He did _that_?
Hard to believe that anyone would make comments about a perfectly
reasonable guy like you.
>>
>> ************************
>>
>> David C. Ullrich
>>
************************
David C. Ullrich
Dave Rusin
>In article <MPG.1e821881e...@newsstand.cit.cornell.edu> Tony Orlow <ae...@cornell.edu> writes:
>> Dave Rusin said:
>...
>>> "Forbidden"?
>>>
>>> Mathematicians are a libertine, almost anarchistic group.
>>
>> That's a laugh, coming from the guy who patiently tried to explain how I was
>> wrong about countably infinite sets, and then demanded that I either accept
>> that there are an infinite number of finite naturals or be banished to his
>> killfile, ala the Spanish Inquisition. Very open minded. If someone
>> responds to this post, maybe he'll see my response.
>
>Is it? Using the standard definitions, axioms and whatever, it can be
>proven that there are only a countably finite number of naturals.
Um, you mean "countably infinite". But I don't recall that in my
conversations with TO he claimed that there were uncountably many
naturals (except perhaps in a very circuitous way, e.g. by claiming
a falsehood, from which every other falsehood logically follows!)
No, the problem I had with Tony was just the opposite: he would claim
that there are only finitely many natural numbers (he referred to them
as "finite naturals"). I was under the impression he still believed that.
Tony's experience is, actually, precisely covered by what I wrote
about mathematical culture. As I explained to him there is absolutely
nothing wrong with considering other sets of "numbers" besides the
natural numbers. (Some of these sets may indeed by uncountably infinite.)
One may, for example, consider the set of all left-infinite
digit-strings ... a_4 a_3 a_2 a_1 where each a_i is in {0,1,...,9}.
It's a well-defined set, and on this set we can define addition and
multiplication in the grade-school way, and we can verify that the
associative, distributive, and commutative laws hold here, as well as
the existence of zero and additive inverses. (So it's a ring.)
But I also pointed out to him that it's not possible to linearly order
this set in a way consistent with sums and products. Well, that's not
a problem for me: I have no particular interest in this ring, so I
don't care that it can't be ordered. He is welcome to pursue it if it
interests him. That's what I meant by our "libertine" culture.
Unfortunately, Tony was not willing to abide by what I claimed to be
the one immutable law of society here: he was unwilling to state
clearly what it was that we studying and to live by the consequences
of that decision. (For example, the fact that there is no good
ordering in this ring did not prevent him from claiming that
"....3636363 is 1.25 times as large as ....272727 ".) Indeed, he
never really claimed to be studying this particular ring -- it was
just one candidate I proposed to him so I would know what the heck
we were talking about; he neither explicitly accepted this definition
nor ignored it.
>>> Nothing is really forbidden -- you're welcome to pursue whatever
>>> you like. The only requirement is that you define your terms and
>>> clarify your assumptions, and then proceed logically.
I'm sure those of you following his postings have continued to see
this kind of nonsense -- an unyielding attachment to intuition even
when it is provably contradictory. As he suggests above, I have
stopped reading his posts. I had no idea that my personal refusal
to pay attention to someone meant that they were "forbidden" from
discussing the things they want. (Someone should tell the TV networks
that they are forbidden from broadcasting because I don't watch!)
Or perhaps I didn't make it clear enough: in an anarchistic society
no one can tell you to shut up; but no one can tell me to listen to
you either.
I do like the bit about the Spanish Inquisition, though. I had no
idea I had so much power. The rest of you had better watch it!
Remember -- NO ONE EXPECTS THE SPANISH INQUISITION !!1!
dave
Tony Orlow
> On Thu, 16 Mar 2006 10:18:11 -0500, Tony Orlow <ae...@cornell.edu>
> wrote:
>
> >David C. Ullrich said:
> >> [...]
> >> >
> >> >0% probability CAN mean entirely impossible, and in fact, does mean that in any
> >> >finite set of possible outcomes.
> >>
> >> Not necessarily, but that's correct in the situations you have in
> >> mind, so never mind.
> >
> >Well, if the probability distribution is normal, anyways.
>
> What do you mean by "normal" here? The word has a technical
> meaning in probability, but there's no such thing as a
> normal distribution with only finitely many outcomes.
I guess what I meant was if each possibility has an equal probability. I guess
I used the wrong word. Sorry.
>
> >>
> >> >So, how do you distinguish between 0%
> >> >probability where is is some chance and 0% probability when there is none?
> >>
> >> You _don't_ distinguish them _via_ probability.
> >
> >I didn't ask how you don't distinguish.
>
> I understand that - you asked how you do distinguish between
> two things. It's impossible to distinguish them using probability.
Not if you allow specfic infinitesimal probabilities, but I am not going to
waste a lot of time convincing you that Cantor's "bacilli cholera" are
applicable, since you probably already have an unchangeable opinion about that.
>
> >> >Why
> >> >do you say the first is not the same as having some infinitesimal chance,
> >> >nonzero but smaller than any finite probability?
> >>
> >> Because that's the way it is.
> >
> >Oh, well, that explains everything.
> >
> >>
> >> >They are obvious two different
> >> >kinds of zero.
> >>
> >> Nope. Infinitesmals are not zero. And probabilities are not
> >> infinitesmals.
> >
> >And that further clarifies the matter.
>
> As I say below, you don't seem to understand what a definition
> is. Probabilities are real numbers, by definition. There are
> no infinitesmal real numbers; that follows from various definitions.
> So there are no infinitesmal probabilities. That _is_ all there
> is to it.
Are you saying it is impossible or unreasonable to consider any non-standard
approaches to probability? I hope not. To be clear, I am not suggesting that
infinitesimal probabilities are part of standard probability theory. I am
saying they should be, because in the context of Han's proposal, they make
perfect sense.
>
> >> (By definition. Your cracks about Rusin's recent comments
> >> about how we're allowed to get away with anything as long
> >> as our definitions and deductions are in order were way
> >> off - you don't seem to realize that there _is_ such a
> >> thing as a definition in mathematics.)
> >
> >You don't seem to understand what a response to a question consists of,
>
> That's not the problem. _You_ don't seem to understand that the answer
> to "how do I do this" is sometimes "that's impossible".
I understand that, if it's true. Here, I am suggesting a way to do it, and you
are saying what I am suggesting is impossible. it's not. You just don't care to
entertain the notion.
>
> > so
> >never mind. My comment about Rusin was about his libertine attitude a year ago,
> >when he put me on his kill list for not agreeing with him, and then continued
> >to make comments about me, like throwing trash over the fence he just built.
>
> No. He did _that_?
>
> Hard to believe that anyone would make comments about a perfectly
> reasonable guy like you.
No, it's just hard to believe that anyone who behaved like that would try to
put forth the notion that mathemticians are all laid back and "libertine",
expecially around here, where it's obviously not the case. We're struggling
here. Mathemticians aren't saying, "If you want to entertain the notion of
infinitesimal probabilities, it seems to make sense in this context, so have
fun." You, here, now, are simply dismissing the idea and saying it's worthless.
That goes on here all the time. You can't deny that.
Tony Orlow
> In article <Iw77...@cwi.nl>, Dik T. Winter <Dik.W...@cwi.nl> wrote:
> >In article <MPG.1e821881e...@newsstand.cit.cornell.edu> Tony Orlow <ae...@cornell.edu> writes:
> >> Dave Rusin said:
> >...
> >>> "Forbidden"?
> >>>
> >>> Mathematicians are a libertine, almost anarchistic group.
> >>
> >> That's a laugh, coming from the guy who patiently tried to explain how I was
> >> wrong about countably infinite sets, and then demanded that I either accept
> >> that there are an infinite number of finite naturals or be banished to his
> >> killfile, ala the Spanish Inquisition. Very open minded. If someone
> >> responds to this post, maybe he'll see my response.
> >
> >Is it? Using the standard definitions, axioms and whatever, it can be
> >proven that there are only a countably finite number of naturals.
>
> Um, you mean "countably infinite". But I don't recall that in my
> conversations with TO he claimed that there were uncountably many
> naturals (except perhaps in a very circuitous way, e.g. by claiming
> a falsehood, from which every other falsehood logically follows!)
>
> No, the problem I had with Tony was just the opposite: he would claim
> that there are only finitely many natural numbers (he referred to them
> as "finite naturals"). I was under the impression he still believed that.
He still does.
There were a bunch of sidebars Rusin introduced. I wasn't going to be
distracted in every direction, but was trying to address certain things. Here,
I think he means ....363636 is 4/3 as large as ...272727. Given the same
infinite of digits, this is true.
I appreciated David's efforts in discussion, but ultimately he dismissed every
idea as ridiculous, and insisted that I admit I was wrong about the set of
naturals and agree with him or be excommunicated. That was just ridiculous.
I've never kill listed anyone.
>
> >>> Nothing is really forbidden -- you're welcome to pursue whatever
> >>> you like. The only requirement is that you define your terms and
> >>> clarify your assumptions, and then proceed logically.
>
> I'm sure those of you following his postings have continued to see
> this kind of nonsense -- an unyielding attachment to intuition even
> when it is provably contradictory. As he suggests above, I have
> stopped reading his posts. I had no idea that my personal refusal
> to pay attention to someone meant that they were "forbidden" from
> discussing the things they want. (Someone should tell the TV networks
> that they are forbidden from broadcasting because I don't watch!)
>
> Or perhaps I didn't make it clear enough: in an anarchistic society
> no one can tell you to shut up; but no one can tell me to listen to
> you either.
Right, you don't have to click on any of my posts if you don't want, but you
also don't have to declare excommunication and compare me to JSH and whoever
else you have in your kill file, as a penalty for not agreeing with you. That
was puerile.
>
> I do like the bit about the Spanish Inquisition, though. I had no
> idea I had so much power. The rest of you had better watch it!
> Remember -- NO ONE EXPECTS THE SPANISH INQUISITION !!1!
That wasn't powerful. It was pitiful. "Covert or be banished!" Ha! Glad you
enjoy your power, dave.
>
> dave
>
--
Smiles,
Tony
Robert Low
> On Wed, 15 Mar 2006 20:15:21 +0000, Robert Low <mtx...@coventry.ac.uk>
>>David C. Ullrich wrote:
>>>(And sorry, but in the situations I'm aware of where infinitesmals
>>>come up, if i_1, i_2, ... are all infinitesmals it never happens
>>>that i_1 + i_2 + ... = 1.)
>>Sure it does (assuming I haven't misunderstood '...'). If
I was trying to guess (perhaps not the most sensible thing)
what HdB might have been talking about.
> Saying we have a non-archimedean field doesn't mean we're
> doing non-standard analysis! My "..." was the perfectly
> ordinary "...", indicating the limit, if any, of the
> finite partial sums.
OK. In the NSA context, of course, that '...' doesn't
mean anything at all. I guess that if you're just saying
'non-Archimedean field' then it means something, but
a monotone sequence that's bounded above need not have
a limit. And in this case, if it does have a limit, then
it seems reasonably clear that the limit will
still be infinitesimal, since each partial sum
is still less than 1/n, for all n.
Virgil
Tony Orlow <ae...@cornell.edu> wrote:
> Dik T. Winter said:
> > In article <MPG.1e821881e...@newsstand.cit.cornell.edu>
> > Tony Orlow <ae...@cornell.edu> writes:
> > > Dave Rusin said:
> > ...
> > > > "Forbidden"?
> > > >
> > > > Mathematicians are a libertine, almost anarchistic group.
> > >
> > > That's a laugh, coming from the guy who patiently tried to
> > > explain how I was wrong about countably infinite sets, and then
> > > demanded that I either accept that there are an infinite number
> > > of finite naturals or be banished to his killfile, ala the
> > > Spanish Inquisition. Very open minded. If someone responds to
> > > this post, maybe he'll see my response.
> >
> > Is it? Using the standard definitions, axioms and whatever, it can
> > be proven that there are only a countably finite number of
> > naturals.
>
> What type of proof are you suggesting? None of my proofs of the
> finiteness of the set of naturals seem to be acceptable to those that
> disagree. Any other approach would be interesting to see.
That is because none of TO's definitions of finiteness are acceptqble to
standard mathematics, which has its own, well established and , at least
among mathematicians, universally accepted, definition of what
finiteness of a set means.
>
> >
> > > > Nothing is really forbidden -- you're welcome to pursue
> > > > whatever you like. The only requirement is that you define
> > > > your terms and clarify your assumptions, and then proceed
> > > > logically.
> > >
> > > That didn't seem to be good enough a year ago.
> >
> > It was good enough. You use in your discourse your own
> > definitions, your own logic and your own assumption, all trying to
> > prove that mathematics with the standard definitions, the standard
> > logic and the standard assumptions is wrong.
> >
>
> Most mathemtics is correct, but the treatment of zeroes and
> infinities leaves a lot to be desired
Not to those who are using the standard mathematical definitions of zero
and infinite.
> and the conclusions of what we
> currently have are often clearly wrong.
Not to those who are using the standard mthematical definitions of zero
and infinite.
> Some of the assumption sof
> the standard theory I see as unfounded, and reject them as causing
> such absurd conclusions.
TO manages to see in TOmatics what is not there in mathematics and
manages not to see what isthere in mathematics.
Virgil
Tony Orlow <ae...@cornell.edu> wrote:
Wrong! The normal probability distributions are among of those in which
a 0 probability does NOT mean impossible.
>
> >
> > >So, how do you distinguish between 0%
> > >probability where is is some chance and 0% probability when there is none?
> >
> > You _don't_ distinguish them _via_ probability.
>
> I didn't ask how you don't distinguish.
Take a (post calculus college level) course in probability theory and
all will be revealed.
>
> >
> > >Why
> > >do you say the first is not the same as having some infinitesimal chance,
> > >nonzero but smaller than any finite probability?
> >
> > Because that's the way it is.
>
> Oh, well, that explains everything.
Take a (post calculus college level) course in probability theory and
all will be revealed.
>
> >
> > >They are obvious two different
> > >kinds of zero.
> >
> > Nope. Infinitesmals are not zero. And probabilities are not
> > infinitesmals.
>
> And that further clarifies the matter.
Take a (post calculus college level) course in probability theory and
all will be revealed.
>
> >
> > (By definition. Your cracks about Rusin's recent comments
> > about how we're allowed to get away with anything as long
> > as our definitions and deductions are in order were way
> > off - you don't seem to realize that there _is_ such a
> > thing as a definition in mathematics.)
>
> You don't seem to understand what a response to a question consists of, so
> never mind. My comment about Rusin was about his libertine attitude a year
> ago,
> when he put me on his kill list for not agreeing with him, and then continued
> to make comments about me, like throwing trash over the fence he just built.
As he was only returning your trash, whats your problem?
Virgil
ru...@vesuvius.math.niu.edu (Dave Rusin) wrote:
> In article <Iw77...@cwi.nl>, Dik T. Winter <Dik.W...@cwi.nl> wrote:
> >In article <MPG.1e821881e...@newsstand.cit.cornell.edu> Tony Orlow
> ><ae...@cornell.edu> writes:
> >> Dave Rusin said:
> >...
> >>> "Forbidden"?
> >>>
> >>> Mathematicians are a libertine, almost anarchistic group.
> >>
> >> That's a laugh, coming from the guy who patiently tried to explain how I
> >> was
> >> wrong about countably infinite sets, and then demanded that I either
> >> accept
> >> that there are an infinite number of finite naturals or be banished to his
> >> killfile, ala the Spanish Inquisition. Very open minded. If someone
> >> responds to this post, maybe he'll see my response.
> >
> >Is it? Using the standard definitions, axioms and whatever, it can be
> >proven that there are only a countably finite number of naturals.
>
> Um, you mean "countably infinite". But I don't recall that in my
> conversations with TO he claimed that there were uncountably many
> naturals (except perhaps in a very circuitous way, e.g. by claiming
> a falsehood, from which every other falsehood logically follows!)
TO has, on numerous occasions, tried to redefine "finite" and "infintie'
for ets to mean "countable" and "uncountale" respectively, and has
misused "finite' and "infinite" in those senses even more often and has
belabored those who objected to such misuse.
Virgil
Tony Orlow <ae...@cornell.edu> wrote:
> David C. Ullrich said:
> > On Thu, 16 Mar 2006 10:18:11 -0500, Tony Orlow <ae...@cornell.edu>
> > wrote:
> >
> > >David C. Ullrich said:
> > >> [...]
> > >> >
> > >> >0% probability CAN mean entirely impossible, and in fact, does mean
> > >> >that in any
> > >> >finite set of possible outcomes.
> > >>
> > >> Not necessarily, but that's correct in the situations you have in
> > >> mind, so never mind.
> > >
> > >Well, if the probability distribution is normal, anyways.
> >
> > What do you mean by "normal" here? The word has a technical
> > meaning in probability, but there's no such thing as a
> > normal distribution with only finitely many outcomes.
>
> I guess what I meant was if each possibility has an equal probability. I
> guess
> I used the wrong word. Sorry.
One of TO's less endearing qualities is his inability to say what he
means without help.
>
> >
> > >>
> > >> >So, how do you distinguish between 0%
> > >> >probability where is is some chance and 0% probability when there is
> > >> >none?
> > >>
> > >> You _don't_ distinguish them _via_ probability.
> > >
> > >I didn't ask how you don't distinguish.
> >
> > I understand that - you asked how you do distinguish between
> > two things. It's impossible to distinguish them using probability.
>
> Not if you allow specfic infinitesimal probabilities
But mathematical probability theory does NOT allow infinitesimal
probabilities at all, and if it did, could not allow TO's version of
infinitesimals without creating myriads of self-contradictions.
, but I am not going to
> waste a lot of time convincing you that Cantor's "bacilli cholera" are
> applicable, since you probably already have an unchangeable opinion about
> that.
We have mathematical opinions that can be changed by mathematically and
logically sound proofs that they are wrong, but TO has yet to show
himself capable of creating any mathematically or logically sound proof
of any his TOmatic claims or conjectures.
> > As I say below, you don't seem to understand what a definition
> > is. Probabilities are real numbers, by definition. There are
> > no infinitesmal real numbers; that follows from various definitions.
> > So there are no infinitesmal probabilities. That _is_ all there
> > is to it.
>
> Are you saying it is impossible or unreasonable to consider any non-standard
> approaches to probability? I hope not. To be clear, I am not suggesting that
> infinitesimal probabilities are part of standard probability theory. I am
> saying they should be, because in the context of Han's proposal, they make
> perfect sense.
But TO does not understand how infinitesimals must work to be compatible
with the axioms of ordered fields. His version of infintesimals destroy
the ordered field properties of his number system, so that whatever he
would have is unworkable as a number system.
>
> >
> > >> (By definition. Your cracks about Rusin's recent comments
> > >> about how we're allowed to get away with anything as long
> > >> as our definitions and deductions are in order were way
> > >> off - you don't seem to realize that there _is_ such a
> > >> thing as a definition in mathematics.)
> > >
> > >You don't seem to understand what a response to a question consists of,
> >
> > That's not the problem. _You_ don't seem to understand that the answer
> > to "how do I do this" is sometimes "that's impossible".
>
> I understand that, if it's true. Here, I am suggesting a way to do
> it, and you are saying what I am suggesting is impossible. it's not.
> You just don't care to entertain the notion.
The notion that TO's version of infinitesimals is mathematically even
coherent is quite entertaining. But unreal.
In TO's version of infinitesimals there simultaneously is no smallest
possible positive element and there is a smallest possble positive
element. In standard mathematics that is called a self-contradiction,
and is enough to get the system rejected, at least outside of TOmatics.
>
> >
> > > so
> > >never mind. My comment about Rusin was about his libertine attitude a year
> > >ago,
> > >when he put me on his kill list for not agreeing with him, and then
> > >continued
> > >to make comments about me, like throwing trash over the fence he just
> > >built.
> >
> > No. He did _that_?
> >
> > Hard to believe that anyone would make comments about a perfectly
> > reasonable guy like you.
>
> No, it's just hard to believe that anyone who behaved like that would try to
> put forth the notion that mathemticians are all laid back and "libertine",
> expecially around here, where it's obviously not the case.
No matter how "libertine" mathematicians may be about new ideas, and
many of them are quite "libertine", they do not necessarily suffer fools
gladly.
> We're struggling
> here. Mathemticians aren't saying, "If you want to entertain the notion of
> infinitesimal probabilities, it seems to make sense in this context, so have
> fun." You, here, now, are simply dismissing the idea and saying it's
> worthless.
TO's notion of infinitesimals is worthless. While it might be possible
to work up a probability theory based on, say, Robinson's hyperreals,
nobody can work up anything consistent using TO's self-contradictory
infinitesimals.
> That goes on here all the time. You can't deny that.
Dismissing an idea that is as clearly self-contradictory as TO's
formulation of infinitesimals does go on all the time. No one denies
that.
Virgil
Tony Orlow <ae...@cornell.edu> wrote:
There it is again, TO's arrogance in believing, or claiming to believe
that he can redefine words or phrases with stndard mathematical meanings
to mean just what he wants them to mean and then force these definitions
on all mathematicians.
TO is wrong! Mathematicians are not impressed by TO's bluster and
arrogance.
And according to the definitin of fniteness of sets that mathematicians
accept, the set of natural numbers is NOT finite. And no amount of
TO's blustering or arrogance will change that.
Not until TO can define a linear ordering on the set of left-infinite
digits strings, which, as Mr. Rusin pointed our, cannot be done.
>
> I appreciated David's efforts in discussion, but ultimately he dismissed
> every
> idea as ridiculous,
Those ideas of TO's which David rejected were ridiculous, at last to
mathematicians.
> and insisted that I admit I was wrong about the set of
> naturals and agree with him or be excommunicated.
As far as I know David is not in any sort of holy orders which would
allow him to "excommunicate" anyone. Nor did he threaten to.
> That was just ridiculous.
> I've never kill listed anyone.
> > I'm sure those of you following his postings have continued to see
> > this kind of nonsense -- an unyielding attachment to intuition even
> > when it is provably contradictory. As he suggests above, I have
> > stopped reading his posts. I had no idea that my personal refusal
> > to pay attention to someone meant that they were "forbidden" from
> > discussing the things they want. (Someone should tell the TV networks
> > that they are forbidden from broadcasting because I don't watch!)
> >
> > Or perhaps I didn't make it clear enough: in an anarchistic society
> > no one can tell you to shut up; but no one can tell me to listen to
> > you either.
>
> Right, you don't have to click on any of my posts if you don't want, but you
> also don't have to declare excommunication and compare me to JSH and whoever
> else you have in your kill file, as a penalty for not agreeing with you. That
> was puerile.
TO's arrogance is puerile. And will not persuade any mathematician as
much as a single valid proof would. But that is asking for something TO
seems incapable of producing. At least something that TO hs yet to
produce.
W. Dale Hall
Tony Orlow wrote:
... stuff deleted ...
>
> There were a bunch of sidebars Rusin introduced. I wasn't going to be
> distracted in every direction, but was trying to address certain things. Here,
> I think he means ....363636 is 4/3 as large as ...272727. Given the same
> infinite of digits, this is true.
>
Presumably, you mean 'given the same infinity of digits'.
However, the digits have a 1:1 correspondence with the natural numbers.
Therefore (according to you) there are only finitely many digits. In
fact, you should be claiming that there are *fewer* digits than there
are natural numbers.
You really should get your terminology under control.
BTW, it seems like you *still* haven't solved the problem of
"everything is divisible by x"
for *every* natural number x coprime to 10. There's no getting around
this for your version of the integers. I've already shown you how to
do that division, so you can't claim it ain't so.
Shouldn't that be an important issue? After all, what use are your
numbers if you can't even do arithmetic? If your numbers are of no
use (i.e., literally "useless"), then aren't they worthless?
Dale.
Han.de...@dto.tudelft.nl
>
> Why should anyone hesitate to recognize that 'infinite' and 'finite'
> are very different things?
If it comes to non-mathematics, physics for example, then the 'finite'
_becomes_ 'infinite' only by increasing the former. Meaning that there
is _no_ essential difference between the two. 'Infinite' in physics is
a way of expressing that something is finite, but very large. So large
that it doesn't matter anymore how large. The other way around: finite
though very large means: that physicists still _are_ interested in the
size of such a quantity. Therefore the same quantity can be finite in
one theory and infinite in another. And physics is a prototype for any
other science, if we're talking about these matters.
Han de Bruijn
Tony Orlow
>
>
> Tony Orlow wrote:
>
> ... stuff deleted ...
> >
> > There were a bunch of sidebars Rusin introduced. I wasn't going to be
> > distracted in every direction, but was trying to address certain things. Here,
> > I think he means ....363636 is 4/3 as large as ...272727. Given the same
> > infinite of digits, this is true.
> >
>
> Presumably, you mean 'given the same infinity of digits'.
Nice typo catch.
>
> However, the digits have a 1:1 correspondence with the natural numbers.
Not necessarily. If they are both 100 digits long, it's true too.
>
> Therefore (according to you) there are only finitely many digits. In
> fact, you should be claiming that there are *fewer* digits than there
> are natural numbers.
I have no problem with infinite strings of digits. The finiteness of the
natural numbers, which doesn't particularly have any bearing on these adic
numbers, derives from the identity relation between element count and value in
the set, and the restriction of finiteness imposed on the values.
>
> You really should get your terminology under control.
Yeah, thanks.
>
> BTW, it seems like you *still* haven't solved the problem of
>
> "everything is divisible by x"
>
> for *every* natural number x coprime to 10. There's no getting around
> this for your version of the integers. I've already shown you how to
> do that division, so you can't claim it ain't so.
>
> Shouldn't that be an important issue? After all, what use are your
> numbers if you can't even do arithmetic? If your numbers are of no
> use (i.e., literally "useless"), then aren't they worthless?
Hi Dale. Been a long time. So long, it appears, that you don't seem to recall
the lengthy response I gave to your "everything is divisible by 7" argument,
where I fully analyzed what you were doing, and confirmed that, with the 10-
adics, indeed ...110, ...111, ...112, etc are apparently divisible by 7. I
explained why this was so, pointed out that in fact ...1115 is NOT divisible by
7, since there is no multiple of 7 consisting of a string of 1's ending in a 5
as is true for the other 9 digits, and finally, reminded you that my infinite
number system is distinctly different from the adics. They're not "left-
infinite", but "center-infinite". I then applied the division by 7 to my unit
infinity, 1:000...000, and found that, with a six-digit repeating sequence in
decimal 1/7, the remainder of dividing decimal infinity by 7 could be any of
six different values, from 1 through 6, and could not be 0, which makes sense,
since no power of 10 could be evenly divisible by 7. So, I showed that my
numbers work there the adics don't.
So, you can claim that I never solved your "problem", that arithmetic doesn't
work with the T-riffic numbers, and that my ideas are worthless, but that's a
bunch of baloney, and if you have any memory of the response which I describe
and a shred of integrity, then you'll admit that I exposed your tricky
maneuver, which I don't think you invented, and that the objection was vacuous
and irrelevant. Okay?
>
> Dale.
>
--
Smiles,
Tony
Han.de...@dto.tudelft.nl
>
> Hi Han -
>
> Here's an interesting paper on the subject of Continuity and Infinitesimals,
> with some historical background: http://plato.stanford.edu/entries/continuity/
[ ... snip ... ]
> Robinson's Hyperreals: http://en.wikipedia.org/wiki/Hyperreal_number
Quite worthwile references, Tony. Thank you!
Han de Bruijn
ste...@nomail.com
> ste...@nomail.com wrote:
>>
>> Why should anyone hesitate to recognize that 'infinite' and 'finite'
>> are very different things?
> If it comes to non-mathematics, physics for example, then the 'finite'
> _becomes_ 'infinite' only by increasing the former. Meaning that there
> is _no_ essential difference between the two.
But that is not what it means to mathematicians. If you
are going to discuss things with mathematicians, you should
either use mathematical definitions, or make it clear that
you are using a different definition.
In mathematics there is an essential difference between finite
and infinite, which is why mathematicians are not all hesitant
to recognize that finite and infinite are very different things.
> 'Infinite' in physics is
> a way of expressing that something is finite, but very large.
But that is not what 'infinite' means in mathematics.
Why you stubbornly refuse to recognize that fact is
quite puzzling. It seems rather childish to insist
everyone use the physicists definition of 'infinite',
assuming of course that is actually the physicists definition.
I seem to recall the last time you provided a quote
to support your position, the quote actually contradicted
your position.
> So large
> that it doesn't matter anymore how large. The other way around: finite
> though very large means: that physicists still _are_ interested in the
> size of such a quantity. Therefore the same quantity can be finite in
> one theory and infinite in another. And physics is a prototype for any
> other science, if we're talking about these matters.
Who was talking about physics? Was your question about
probability theory supposed to by a physics question?
Anyway, if you want to use "infinite" which literally means
"not finite" to mean "finite, but very large", then go ahead.
Just do not be so foolish to assume that is what everyone
else means by the word, or that is what everyone else should
mean by the word.
Stephen
Han.de...@dto.tudelft.nl
> >
> > OK. _Are_ infinitesimals actually _defined_ in standard mathematics?
> > And if so, could you please tell me how?
>
> An element s of an ordered field is called *infinitesimal* if s>0,
> but s < 1/n for all natural numbers n.
And that's precisely what's happening with the probabilities E(a,n) =
1/n at the naturals construction sets {1,2,3, ... ,n} when n becomes
arbitrary large. Then they are < 1/n for all naturals n but stay > 0 .
So I shall conclude that E(a) at _all_ naturals is an _infinitesimal_,
indeed, as Tony Orlow has suggested all the time.
> In some ordered fields (for example the real number system)
> there are no infinitesimals. In others (for example Conway's
> surreal numbers) there are.
Next question: _can_ Probability Theory be extended to the surreals?
Han de Bruijn
Han.de...@dto.tudelft.nl
>
> Depends on exactly what you mean. There do exist mathematical
> structures that contain things known as infinitesmals. Those
> infinitesmals are not real numbers, so in particular they're
> not allowed as probabilities in standard probability theory.
Because probabilities in standard probability theory are defined to be
real numbers. Right? Would it be possible to extend probability theory
to another number system (surreals, I've heard) so that probabilities
can be infinitesimals then?
Han de Bruijn
W. Dale Hall
Tony Orlow wrote:
> W. Dale Hall said:
... stuff deleted ...
>
>>BTW, it seems like you *still* haven't solved the problem of
>>
>> "everything is divisible by x"
>>
>>for *every* natural number x coprime to 10. There's no getting around
>>this for your version of the integers. I've already shown you how to
>>do that division, so you can't claim it ain't so.
>>
>>Shouldn't that be an important issue? After all, what use are your
>>numbers if you can't even do arithmetic? If your numbers are of no
>>use (i.e., literally "useless"), then aren't they worthless?
>
>
> Hi Dale. Been a long time. So long, it appears, that you don't seem to recall
> the lengthy response I gave to your "everything is divisible by 7" argument,
> where I fully analyzed what you were doing, and confirmed that, with the 10-
> adics, indeed ...110, ...111, ...112, etc are apparently divisible by 7. I
> explained why this was so, pointed out that in fact ...1115 is NOT divisible by
> 7, since there is no multiple of 7 consisting of a string of 1's ending in a 5
> as is true for the other 9 digits, and finally, reminded you that my infinite
Really? No sequence of digits to yield a product of an infinite string
of 1's terminated by a 5?
The "number" ...11115 is equal to
(... 44445) * 7
So much for your "fully analyzed" study. I'll note that it took
under two minutes to locate this answer. Where were you looking
that it escaped you so successfully?
> number system is distinctly different from the adics. They're not "left-
> infinite", but "center-infinite". I then applied the division by 7 to my unit
> infinity, 1:000...000, and found that, with a six-digit repeating sequence in
> decimal 1/7, the remainder of dividing decimal infinity by 7 could be any of
> six different values, from 1 through 6, and could not be 0, which makes sense,
> since no power of 10 could be evenly divisible by 7. So, I showed that my
> numbers work there the adics don't.
>
> So, you can claim that I never solved your "problem", that arithmetic doesn't
> work with the T-riffic numbers, and that my ideas are worthless, but that's a
> bunch of baloney, and if you have any memory of the response which I describe
> and a shred of integrity, then you'll admit that I exposed your tricky
> maneuver, which I don't think you invented, and that the objection was vacuous
> and irrelevant. Okay?
>
I actually don't recall the response you're referring to. I'll check
Google groups to locate it.
I'll note also that I haven't impugned your integrity. I have made no
claim to have "invented" the concept of multiplication, even in this
context. Having exposed my trickery: for one, you haven't shown that
not every number isn't divisible by 7 (or any other number coprime to
10), and whether the objection is vacuous or irrelevant depends on
whether the content of this version of numbers is vacuous or irrelevant.
>
>>Dale.
>>
>
>
ste...@nomail.com
> Han de Bruijn
But why would you want that? According to you, "infinite"
means "finite, but very large". If you are being consistent,
then "infinitesimal" should mean "finite, but very small".
Standard probability theory can handle "finite, but very large"
and "finite, but very small" just fine. You only need
"real" infinitesimals if you are willing to accept "real" infinities.
Stephen
W. Dale Hall
I just looked up one response (from Dec 1 2005), where the numbers
you're dealing with are describe as being left-terminated after an
infinite number of intermediate digits.
The number
0.1111.....11115
is then divisible by 7:
0.1111 ... 11115 = 0.015873015873...4444445 * 7
where the left sequence is the periodic sequence 015873, and the right
sequence is the sequence ...44445.
Not really that difficult. Still divisible by 7.
Dale.
W. Dale Hall
... stuff deleted ...
(I know, I've responded elsewhere, but noted a couple of points
that I had missed. Apologies for whatever redundancy has been
generated, and the concomitant acceleration of the heat death
of the universe.)
>
>>BTW, it seems like you *still* haven't solved the problem of
>>
>> "everything is divisible by x"
>>
>>for *every* natural number x coprime to 10. There's no getting around
>>this for your version of the integers. I've already shown you how to
>>do that division, so you can't claim it ain't so.
>>
>>Shouldn't that be an important issue? After all, what use are your
>>numbers if you can't even do arithmetic? If your numbers are of no
>>use (i.e., literally "useless"), then aren't they worthless?
>
>
> Hi Dale. Been a long time. So long, it appears, that you don't seem to recall
> the lengthy response I gave to your "everything is divisible by 7" argument,
> where I fully analyzed what you were doing, and confirmed that, with the 10-
> adics, indeed ...110, ...111, ...112, etc are apparently divisible by 7. I
> explained why this was so, pointed out that in fact ...1115 is NOT divisible by
> 7, since there is no multiple of 7 consisting of a string of 1's ending in a 5
> as is true for the other 9 digits, and finally, reminded you that my infinite
> number system is distinctly different from the adics. They're not "left-
> infinite", but "center-infinite". I then applied the division by 7 to my unit
> infinity, 1:000...000, and found that, with a six-digit repeating sequence in
> decimal 1/7, the remainder of dividing decimal infinity by 7 could be any of
> six different values, from 1 through 6, and could not be 0, which makes sense,
> since no power of 10 could be evenly divisible by 7. So, I showed that my
> numbers work there the adics don't.
But why do you say that no power of 10 could be evenly divisible by 7?
0:0000....010 = 0:000...28571428571430 * 7
The center-infinite string has its left string all zeros, and its right
string equal to the periodic sequence 285714 terminated on the right by
30.
As for your "unit infinity",
1.000...000,
isn't it equal to
...2857142857143.000...000 * 7 ?
(Here, the string to the left of the period extends periodically,
endlessly repeating the string 285714)
Or does the string to the left of the period not admit (my version
of) infinite strings of digits? Any time you admit infinite strings
of digits, the same operation can be applied, without fail. The
presence of a left-terminating infinite string of digits doesn't
save your arithmetic, as another of this spate of redundant posts
of mine has shown. The jist of my argument is that the left portion
of the string can be divided separately, just as one can produce
the periodic decimal expansion of a standard fraction. The fact that
the left and right strings (on the separate sides of the ellipsis)
do not interact allows one to divide both strings separately: there
is no digit position that admits a carry operation from the right
string to the left string.
Only finite-length strings of digits can fail to admit such division.
>
> So, you can claim that I never solved your "problem", that arithmetic doesn't
> work with the T-riffic numbers, and that my ideas are worthless, but that's a
> bunch of baloney, and if you have any memory of the response which I describe
> and a shred of integrity, then you'll admit that I exposed your tricky
> maneuver, which I don't think you invented, and that the objection was vacuous
> and irrelevant. Okay?
>
I looked up a response that appears to be the one you are referring to;
it was dated December 1, 2005. My latest contribution to that thread had
been on August 19, and while I followed the thread for about a month
thereafter, I didn't hang around for three and a half months while you
got your act together. I found the thread "Infinity" to be unbearably
tedious and contentious, and tired of it. If it is my fault for having
too short of an attention span, then I can accept that.
Your language (in referring to my "tricky maneuver") suggests that
my suggestion (that everything is divisible by 7) is not to be taken
at face value. You may accept or deny anything you please, that is
of no concern to me. However, if you're serious that you have somehow
defined a structure that represents integer arithmetic, you'll need
to show how arithmetic works, and how it agrees with what the rest
of the world sees as arithmetic. I don't see how you've done that.
My point is this: your number system doesn't admit arithmetic. That's
all. If you wish to claim that it is an adequate model of the naturals,
then that's fine, but you'll have to admit that arithmetic doesn't
really exist.
Maybe you disagree. Then show me that I'm wrong. In my adult life, I
have never minded being shown to be wrong when I am, and will not have
any problem with it now. I don't have any ego involvement here.
You got the goods? Then go to it, youngster.
Dale.
Han de Bruijn
>
> None of these things is forbidden in mathematics. The only thing
> that will get you thown out of the club is an insistence on vague
> language and a refusal to accept logical deduction from clear
> starting principles.
Though I have the impression that those "clear starting principles" are
biased towards a set theoretic foundation. But, for me as a physicist,
set theoretic principles are far from clear. I mean, everything depends
on what you _accept_ as clear. Is Heavisides' Operator Calculus "clear"?
I find so, but many mathematicians don't.
Han de Bruijn
Han de Bruijn
> In article <MPG.1e82045c1...@newsstand.cit.cornell.edu>,
> Tony Orlow <ae...@cornell.edu> wrote:
>>
>>But, David, what IS the chance of any given element being selected from an
>>infinite set? In standard probability, it's 0%, indistinguishable from no
>>chance at all. And yet, each element does have some chance, because one of
>>them is going to be selected. So, how does one represent that kind of chance
>>mathematically? Isn't that an infinitesimal probability?
>
> It is a situation where a "random selection", as standardly defined,
> cannot be properly executed.
Yes, but the _accumulation_ of these random selections with a question
like "what is the chance that a natural is divisible by three" results
in one-third, which is therefore an executable random selection.
> As soon as one gives up the requirement that each naturalis to have
> exactly the same chance of being chosen as its predecessor, the anomaly
> vanishes.
True, but off-topic here.
Han de Bruijn
Han de Bruijn
> (And sorry, but in the situations I'm aware of where infinitesmals
> come up, if i_1, i_2, ... are all infinitesmals it never happens
> that i_1 + i_2 + ... = 1.)
Sounds strange to me. The Riemann sum which approximates the integral
int_(0,1) dx is equivalent to the sum of infinitesimals sum_n dx ,
where dx = 1/n , hence = n.1/n = 1 . (Well, iff the infinitesimals in
non-standard analysis are expected to behave like those in physics ..)
Han de Bruijn
Han de Bruijn
> Tony Orlow <ae...@cornell.edu> writes in article <MPG.1e824df87...@newsstand.cit.cornell.edu> dated Wed, 15 Mar 2006 16:15:25 -0500:
>
>>0% probability CAN mean entirely impossible, and in fact, does mean that in any
>>finite set of possible outcomes. So, how do you distinguish between 0%
>>probability where is is some chance and 0% probability when there is none? Why
>>do you say the first is not the same as having some infinitesimal chance,
>>nonzero but smaller than any finite probability? They are obvious two different
>>kinds of zero.
>
> Here's how to distinguish:
>
> In any countable outcome domain, p=0 means impossible.
>
> In a continuum, having p=0 for a discrete point isn't the whole story. You
> have to look at the probability density function.
True. But off-topic. Here we have a discrete and _infinite_ substrate:
the natural numbers. And we have p=0 for a discrete point. There is no
probability density function as with continuous distributions. Now, is
it impossible to pick a natural number at random? Let's do it: 9862972
is such a number.
Han de Bruijn
Han de Bruijn
> In article <22eg129j22u0g7ggv...@no.spam> tlas...@cc.hut.fi writes:
> > On Wed, 15 Mar 2006 16:36:19 +0100, Han de Bruijn
> > <Han.de...@DTO.TUDelft.NL> wrote:
> >
> > >OK. _Are_ infinitesimals actually _defined_ in standard mathematics?
> > >And if so, could you please tell me how?
> >
> > Oddly enough, the treatment of calculus by rigorous notion of
> > infinitesimals is called 'nonstandard analysis', so I guess the answer
> > would strictly be 'no'.
>
> Oddly enough Han does not like that either.
Oddly enough, I might change my mind.
Han de Bruijn
Han de Bruijn
> On a closed real interval, one can define an additive measure function
> such that the measure of a subinterval of the interval is the probabilty
> that a value from that subinterval will be chosen. Then there will be
> non-empty sets with probability zero, but that does not mean that it is
> impossible for a member is such a set to be chosen.
OK. That is Geek to me. Can you please elaborate with sort of example?
Han de Bruijn
Han de Bruijn
> On Wed, 15 Mar 2006 16:15:25 -0500, Tony Orlow <ae...@cornell.edu>
> wrote:
>
>>So, how do you distinguish between 0%
>>probability where is is some chance and 0% probability when there is none?
>
> You _don't_ distinguish them _via_ probability.
What? You _don't_ distinguish "chance" (yes or no) _via_ "probability"?
Now I'm completely at lost! What then is the difference between "chance"
and "probability"?
Han de Bruijn
Han de Bruijn
> Perhaps it wasn't formulated by engineers? He seems to share a common
> mindset for physicists/engineers that mathematicians are somehow
> "enemies" or "annoying formalists" that contribute nothing and just
> ruin things.
Ho, ho, ho. Mathematicians have ideas about the _infinite_ that are NOT
kosher from a scientific point of view. Which doesn't mean that they are
doing "wrong" everything else.
Han de Bruijn
Toni Lassila
What is the distribution function of your random variable? I'll also
settle for the probability-generating function, or even the
characteristic function.
Toni Lassila
Mathematics is (among other things) just a tool for modelling stuff.
If you think some aspect of mathematics doesn't accurately model
whatever it is you're studying, don't use it. There's no need to start
reshaping mathematics (poorly and ignorantly, I might add). You don't
throw away a hammer just because it's not a screwdriver.
Besides, I would argue that the physicists have ideas about the
infinity that are much worse and in some cases simply wrong in a
purely logical sense. But often it does not matter because they don't
need to be right, they just need to be in line with physical reality.
If faulty thought happens to bring them to the correct answer, so be
it.
Mathematicians have no such luxury and actually have to carefully
define things like infinity instead of giving contradictory Orlowian
declarations about what they think "infinity" means.
Han de Bruijn
> Han.de...@dto.tudelft.nl wrote:
>
>>ste...@nomail.com wrote:
>>
>>>Why should anyone hesitate to recognize that 'infinite' and 'finite'
>>>are very different things?
>
>>If it comes to non-mathematics, physics for example, then the 'finite'
>>_becomes_ 'infinite' only by increasing the former. Meaning that there
>>is _no_ essential difference between the two.
>
> But that is not what it means to mathematicians. If you
> are going to discuss things with mathematicians, you should
> either use mathematical definitions, or make it clear that
> you are using a different definition.
I _am_ making clear that we are using a different notion, which is not
(yet) the same as a different "definition".
> In mathematics there is an essential difference between finite
> and infinite, which is why mathematicians are not all hesitant
> to recognize that finite and infinite are very different things.
Yes, that's quite clear, after all those years.
>>'Infinite' in physics is
>>a way of expressing that something is finite, but very large.
>
> But that is not what 'infinite' means in mathematics.
> Why you stubbornly refuse to recognize that fact is
> quite puzzling. It seems rather childish to insist
> everyone use the physicists definition of 'infinite',
> assuming of course that is actually the physicists
> definition.
I _know_ what 'infinite' means in mathematics. I am _not_
stubbornly refusing to recognize that fact. I do _not_ insist
that everyone uses the physicists definition of 'infinite',
which nevertheless is the "definition" in _all_ other sciences
as well.
I'm only making clear, time after time, that the mathematical
definition of 'infinity' is at odds with any scientific content.
If that is OK with you, then why bother? If you find that kind
of disturbing, then better start to think about it.
> Who was talking about physics? Was your question about
> probability theory supposed to by a physics question?
Probability Theory _plays_ a role in physics. Hence my concern.
> Anyway, if you want to use "infinite" which literally means
> "not finite" to mean "finite, but very large", then go ahead.
> Just do not be so foolish to assume that is what everyone
> else means by the word, or that is what everyone else should
> mean by the word.
It's the other way around. This is as silly as trying to forbid the use
of the word "ring" in Lords of the Ring, because mathematics claims the
word for itself. Same for common speech words like "measure" and "ideal"
and "infinity". A far more sensible program would be to figure out where
the mathematical definition of infinity leads to "jumps from the finite
to the infinite" which cannot possibly have any counterpart in empirical
science. There are two possibilities. Either start mathematics all over
again, from i.e. constructive principles. Either accept mathematics as
it is and develop kind of 'AfterMath' to protect science from infinities
that cannot arise from the finite. The pleasant news is that many of the
infinities in mathematics are "good" infinities in science as well. That
makes i.e. calculus so highly useful.
Han de Bruijn
David C. Ullrich
>A N Niel wrote:
>> >
>> > OK. _Are_ infinitesimals actually _defined_ in standard mathematics?
>> > And if so, could you please tell me how?
>>
>> Definition:
>> An element s of an ordered field is called *infinitesimal* if s>0,
>> but s < 1/n for all natural numbers n.
>
>And that's precisely what's happening with the probabilities E(a,n) =
>1/n at the naturals construction sets {1,2,3, ... ,n} when n becomes
>arbitrary large. Then they are < 1/n for all naturals n but stay > 0 .
>So I shall conclude that E(a) at _all_ naturals is an _infinitesimal_,
If so then you're not doing (standard) probability theory.
So you can't apply any of the standard theorems, you need
to start from scratch.
Let us know when you've formulated the _definitions_ and _axioms_
of this theory of yours.
>indeed, as Tony Orlow has suggested all the time.
>
>> In some ordered fields (for example the real number system)
>> there are no infinitesimals. In others (for example Conway's
>> surreal numbers) there are.
>
>Next question: _can_ Probability Theory be extended to the surreals?
>
>Han de Bruijn
************************
David C. Ullrich