Kolmogorov Axioms and Gauss dictum
Han de Bruijn
http://en.wikipedia.org/wiki/Probability_axioms
Let (O,F,P) be a probability space with sample space O, event space F
and probability measure P; let E be an event (E in F).
The Kolmogorov axioms for (O,F,P) are:
(1) P(E) >= 0 for all E in F
(2) P(O) = 1
(3) P(E1 \/ E2) \/ .. \/ En) = Sum(k=1..n) P(Ek)
provided that P(Ei /\ Ej) = 0 for all (i =/= j)
Now let O = {1,2, ... ,n} be an Initial Segment of the Naturals.
P(Ek) = 1/n > 0 for k = 1..n ; these are the elementary events.
For example: a number (k in O) equals 1, a number (k in O) equals 5.
Let F = {E1,E2, .. ,En} : elementary event space.
P(E1 \/ E2 \/ E3 \/ .. \/ En) = n.1/n = 1 .
We conclude that (O,F,P) is a Probability Space. All axioms are OK.
Let events be Ee = element in O is even, Eo = element in O is odd.
Hence the event space is F = {Ee,Eo} . Built with elementary events,
then P(Ee) = [(n-1)/2]/n if n = odd, = [n/2]/n if n = even
P(Eo) = [(n+1)/2]/n if n = odd, = [n/2]/n if n = even
In any case P(Ee) + P(Eo) = 1 = P(O) .
I protest against the use of infinite magnitude as something completed
which is never permissible in mathematics. Infinity is merely a way of
speaking, the true meaning being a limit which certain ratios approach
indefinitely close, while others are permitted to increase without
restriction. ( C.F. Gauss [in a letter to Schumacher, 12 July 1831] )
Now take the limit of {1,2, ... ,n} for n->oo . We replace E by dE.
(1) P(dE) = 0 : probability of an "indefinitely close" to zero chance.
According to Gauss dictum, this zero probability is not completed.
(2) P(O) = 1 : probability of a natural being - trivially - a natural
Because the zero probabilities are not completed, their infinite
sum (not completed as well) can still be equal to one (analogous
to an integral as the limit of a Riemann sum).
P(Ee) = lim(n->oo) [(n-1)/2]/n OR [n/2]/n = 1/2 : probability evens
P(Eo) = lim(n->oo) [(n+1)/2]/n OR [n/2]/n = 1/2 : probability odds
Remember P(dE) = 0 . Yet P(O) = 1 . All axioms are still fulfilled,
because an infinite magnitude as something completed is not permitted
in mathematics.
Han de Bruijn
LudovicoVan
> an infinite magnitude as something completed is not permitted
> in mathematics.
Why not? Why a mathematician should refrain from using mathematical
reasoning and logic and providing a formalization of "an infinite
magnitude as something completed", whatever that might end up being as
a mathematical object.
My main point here is that mathematics (and logic) is indeed _the_
priviledged place where purely theoretical constructs get conceived
and perused. Improperly: the referent of mathematics is "ideals",
while (as a case in point) physics is "just another matter".
That said, I would agree that the problem of the feasibility of any
mathematics to applications is the only criterion for judging the
"value" of a mathematical effort. But no naive reductionism should be
allowed here either, because we just cannot know in advance where
anything might eventually lead, plus it is almost a theoretical
certainty that "we need to fail in order to progress" in some deep
fundamental sense (e.g. boundary analysis).
-LV
David C. Ullrich
<umu...@gmail.com> wrote:
>
>http://en.wikipedia.org/wiki/Probability_axioms
>
>Let (O,F,P) be a probability space with sample space O, event space F
>and probability measure P; let E be an event (E in F).
>
>The Kolmogorov axioms for (O,F,P) are:
>
>(1) P(E) >= 0 for all E in F
>
>(2) P(O) = 1
>
>(3) P(E1 \/ E2) \/ .. \/ En) = Sum(k=1..n) P(Ek)
> provided that P(Ei /\ Ej) = 0 for all (i =/= j)
No, these are not the Kolmogorov axioms.
>Now let O = {1,2, ... ,n} be an Initial Segment of the Naturals.
>
>P(Ek) = 1/n > 0 for k = 1..n ; these are the elementary events.
>
>For example: a number (k in O) equals 1, a number (k in O) equals 5.
>
>Let F = {E1,E2, .. ,En} : elementary event space.
>
>P(E1 \/ E2 \/ E3 \/ .. \/ En) = n.1/n = 1 .
>
>We conclude that (O,F,P) is a Probability Space. All axioms are OK.
>
>Let events be Ee = element in O is even, Eo = element in O is odd.
>
>Hence the event space is F = {Ee,Eo} . Built with elementary events,
>
>then P(Ee) = [(n-1)/2]/n if n = odd, = [n/2]/n if n = even
> P(Eo) = [(n+1)/2]/n if n = odd, = [n/2]/n if n = even
>
>In any case P(Ee) + P(Eo) = 1 = P(O) .
>
>I protest against the use of infinite magnitude as something completed
>which is never permissible in mathematics. Infinity is merely a way of
>speaking, the true meaning being a limit which certain ratios approach
>indefinitely close, while others are permitted to increase without
>restriction. ( C.F. Gauss [in a letter to Schumacher, 12 July 1831] )
>
>Now take the limit of {1,2, ... ,n} for n->oo . We replace E by dE.
I have no idea what you mean by this, since you haven't said what
E is.
>(1) P(dE) = 0 : probability of an "indefinitely close" to zero chance.
> According to Gauss dictum, this zero probability is not completed.
>
>(2) P(O) = 1 : probability of a natural being - trivially - a natural
> Because the zero probabilities are not completed, their infinite
> sum (not completed as well) can still be equal to one (analogous
> to an integral as the limit of a Riemann sum).
>
>P(Ee) = lim(n->oo) [(n-1)/2]/n OR [n/2]/n = 1/2 : probability evens
>P(Eo) = lim(n->oo) [(n+1)/2]/n OR [n/2]/n = 1/2 : probability odds
>
>Remember P(dE) = 0 . Yet P(O) = 1 . All axioms are still fulfilled,
>because an infinite magnitude as something completed is not permitted
>in mathematics.
Actually infinite magnitides _are_ allowed in mathematics.
You and Gauss have not been authorized to tell us what
is and what is not allowed.
To save a little time:
"What? You're claiming you understand these things better than
Gauss did?"
"Yes. Let's assume for the sake of atgument that he was the
greatest mathematician in history. It's still true that he was
working almost two centuries ago - it would be a big surprise
if some of his _opinions_ (as opposed to things that he
actually proved) were not at present generally regarded
as simply mistaken."
Now, if you feel like it for some reason, you're free to
investigate the question of what mathematics you can
do with only finite sets. You're not going to be able to
do much probability.
>Han de Bruijn
Han de Bruijn
> On Thu, 6 Jan 2011 06:54:43 -0800 (PST), Han de Bruijn
>
>
> >http://en.wikipedia.org/wiki/Probability_axioms
>
> >Let (O,F,P) be a probability space with sample space O, event space F
> >and probability measure P; let E be an event (E in F).
>
> >The Kolmogorov axioms for (O,F,P) are:
>
> >(1) P(E) >= 0 for all E in F
>
> >(2) P(O) = 1
>
> >(3) P(E1 \/ E2) \/ .. \/ En) = Sum(k=1..n) P(Ek)
> > provided that P(Ei /\ Ej) = 0 for all (i =/= j)
>
> No, these are not the Kolmogorov axioms.
Might have missed some details, yes. Read the abovementioned wiki page
for filling in these details.
> >Now let O = {1,2, ... ,n} be an Initial Segment of the Naturals.
>
> >P(Ek) = 1/n > 0 for k = 1..n ; these are the elementary events.
>
> >For example: a number (k in O) equals 1, a number (k in O) equals 5.
>
> >Let F = {E1,E2, .. ,En} : elementary event space.
>
> >P(E1 \/ E2 \/ E3 \/ .. \/ En) = n.1/n = 1 .
>
> >We conclude that (O,F,P) is a Probability Space. All axioms are OK.
>
> >Let events be Ee = element in O is even, Eo = element in O is odd.
>
> >Hence the event space is F = {Ee,Eo} . Built with elementary events,
>
> >then P(Ee) = [(n-1)/2]/n if n = odd, = [n/2]/n if n = even
> > P(Eo) = [(n+1)/2]/n if n = odd, = [n/2]/n if n = even
>
> >In any case P(Ee) + P(Eo) = 1 = P(O) .
>
> >I protest against the use of infinite magnitude as something completed
> >which is never permissible in mathematics. Infinity is merely a way of
> >speaking, the true meaning being a limit which certain ratios approach
> >indefinitely close, while others are permitted to increase without
> >restriction. ( C.F. Gauss [in a letter to Schumacher, 12 July 1831] )
>
> >Now take the limit of {1,2, ... ,n} for n->oo . We replace E by dE.
>
> I have no idea what you mean by this, since you haven't said what
> E is.
I _have_ said that E is an element of event space F.
Remember that Gauss was a great contributor to probability theory as
well. What much probability exactly am I not going to be able to do?
Han de Bruijn
Marshall
>
> What much probability exactly am I not going to be able to do?
So many possible responses, I can't pick one.
Marshall
Aatu Koskensilta
But you just did. What exactly with much probability you were able to
do instead of not being with much probability not able to do. Se on pala
kakkua puhua englantia, se on kuppini teet�. Puhua englantia kanssanne
olisi pallo!
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Han de Bruijn
> Marshall <marshall.spi...@gmail.com> writes:
> > On Jan 10, 12:23 am, Han de Bruijn <umum...@gmail.com> wrote:
>
> >> What much probability exactly am I not going to be able to do?
>
> > So many possible responses, I can't pick one.
>
> But you just did. What exactly with much probability you were able to
> do instead of not being with much probability not able to do. Se on pala
> kakkua puhua englantia, se on kuppini teet . Puhua englantia kanssanne
> olisi pallo!
Command not found.
Han de Bruijn
David C. Ullrich
Actually you didn't say that - you're confused about free versus
bound variables. But never mind that - supposing that you had
said that, it doesn't matter, because I realized that dE was supposed
to be an event.
_Which_ event? Below you say that P(dE) = o while
P(Eo) = 1/2 and P(O) = 1. So evidently dE is an event
other than Eo and O.
(i) Reallly? What did he contribute to probability theory?
(ii) He cannnot have contributed anything to rigorous
probability theory, since that didn't exist until Kolmogorov.
(iii) Supposing you're right, so what? That has no bearing
whatever on the question of whether he's authorized to
tell me what's allowed in mathematics.
>What much probability exactly am I not going to be able to do?
Giggle. Pick up a book on probability. A book on real mathematical
probability theory, based on measure theory as above. Try
reproving the results. Let us know how far you get.
>Han de Bruijn
Han de Bruijn
> On Mon, 10 Jan 2011 00:23:56 -0800 (PST), Han de Bruijn
>
Evidently, yes.
The _Gauss_ distribution?
> (ii) He cannnot have contributed anything to rigorous
> probability theory, since that didn't exist until Kolmogorov.
_Too_ rigorous perhaps?
> (iii) Supposing you're right, so what? That has no bearing
> whatever on the question of whether he's authorized to
> tell me what's allowed in mathematics.
>
> >What much probability exactly am I not going to be able to do?
>
> Giggle. Pick up a book on probability. A book on real mathematical
> probability theory, based on measure theory as above. Try
> reproving the results. Let us know how far you get.
Oh yeah, pick up a book ..
I've had the illusion that you could come up with a simple _example_
where my setup would not work.
Han de Bruijn
Tonico
He's talking in tongues....Suomi's, apparently.
Tonio
David C. Ullrich
So you're not going to tell me what you do mean by dE? Very strange.
Yeah, I realized later.
>> (ii) He cannnot have contributed anything to rigorous
>> probability theory, since that didn't exist until Kolmogorov.
>
>_Too_ rigorous perhaps?
>
>> (iii) Supposing you're right, so what? That has no bearing
>> whatever on the question of whether he's authorized to
>> tell me what's allowed in mathematics.
>>
>> >What much probability exactly am I not going to be able to do?
>>
>> Giggle. Pick up a book on probability. A book on real mathematical
>> probability theory, based on measure theory as above. Try
>> reproving the results. Let us know how far you get.
>
>Oh yeah, pick up a book ..
>
>I've had the illusion that you could come up with a simple _example_
>where my setup would not work.
Ok. Show me a proof of the Law of Large Numbers using your "setup".
>Han de Bruijn
Han de Bruijn
> On Mon, 10 Jan 2011 06:52:28 -0800 (PST), Han de Bruijn
>
Sorry. I wrote:
Now take the limit of {1,2, ... ,n} for n->oo . We replace E by dE.
This must be:
Now take the limit of {1,2, ... ,n} for n->oo . We replace Ek by dE.
Where Ek are defined as above (elementary events at initial segment of
Naturals). Note that all dE (limits of Ek for n->oo) are the same.
Nope. Because I cannot see there is something lost with my setup.
Han de Bruijn
master1729
> I protest against the use of infinite magnitude as
> something completed
> which is never permissible in mathematics. Infinity
> is merely a way of
> speaking, the true meaning being a limit which
> certain ratios approach
> indefinitely close, while others are permitted to
> increase without
> restriction. ( C.F. Gauss [in a letter to Schumacher,
> 12 July 1831] )
>
Gauss is not cantor for sure :D
guess Gauss is a crank to cantorians :p
no infinite ordinals for Gauss.
thanks !
ive been looking for that quote some time.
i knew Gauss said it , but i forgot the exact letter.
fits well with me.
have it known : Gauss + Kronecker vs cantor
oh well , according to cantorians , Gauss or Kronecker dont matter.
cantor is like Chuck Norris.
nobody can beat him.
* roundhouse kick *
master1729
> On Jan 10, 3:43 pm, David C. Ullrich
> <ullr...@math.okstate.edu> wrote:
> > On Mon, 10 Jan 2011 00:23:56 -0800 (PST), Han de
> Bruijn
> >
> > >> Now, if you feel like it for some reason, you're
> free to
> > >> investigate the question of what mathematics you
> can
> > >> do with only finite sets. You're not going to be
> able to
> > >> do much probability.
> >
> > >Remember that Gauss was a great contributor to
> probability theory as
> > >well.
> >
> > (i) Reallly? What did he contribute to probability
> theory?
>
> The _Gauss_ distribution?
LMFAO
>
> >
> > Giggle.
giggle x 2
tommy1729
quasi
wrote:
>cantor is like Chuck Norris.
>
>nobody can beat him.
Moreover, he had Bruce Lee as backup (Hilbert).
quasi
master1729
Im a fan of Tony Jaa ( Ramanujan )
:)
tommy1729