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Continuous and discrete uniform distributions of N
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Ross A. Finlayson
Dec 20, 2012, 1:12:38 AM12/20/12
to
In the consideration of what a uniform probability distribution over
the natural integers would be, we can begin as modeling it with real
functions, basically as the Dirac delta defined not just at zero, but
at each natural integer, scaled or divided by the number of integers,
with commensurate width, preserving the properties of a p.d.f. that
it's everywhere greater than or equal to zero and less than or equal
to one, and that its area or integral evaluates to one.
Then, where Dirac's delta is described as a spike to infinity at zero
with unit area, then to have a spike to one at each natural integer,
basically dividing the spike fairly among the natural integers, the
consideration then is in as to whether that still would have area one,
besides that each of the points would have f(x)=1. (Of course this is
all modeled standardly with real functions.) Yet, it has area two
(exactly because the integral of EF = 1).
Another notion of what would be the nearest analog to a uniform
probability distribution over the entire set of naturals would be for
each to have probability 1/omega, that as a discrete instead of
continuous distribution, the sum over them equals one. Here then
there's a consideration that there is a continuous distribution, of N,
because a p.d.f. exists and a p.d.f. (or CDF) defines a continuous
distribution. Then there's also a consideration that there's a
discrete distribution, of N, defined as one iota for each.
EF: continuous uniform distribution of N
(EF + REF)/2: continuous uniform distribution of N
f(x)=iota: discrete uniform distribution of N
Regards,
Ross Finlayson
FredJeffries
Dec 21, 2012, 11:23:18 AM12/21/12
to
On Dec 19, 10:12 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com>
wrote:
You still ain't never caught a rabbit.
Dirac delta, infinitesimals, irrational numbers, transfinite
ordinals, ... are legitimate not because they have been rigorously
defined but because they can be used to SOLVE PROBLEMS. See, for
instance, Jesper Lützen's "The Prehistory of the Theory of
Distributions"
http://books.google.com/books?id=pC7vAAAAMAAJ
No one has ever anywhere actually used the concept of a uniform
distributions on N to solve any problem.
You can't even show how to use it to calculate the area of a triangle.
>
> Regards,
>
> Ross Finlayson
wrote:
Dirac delta, infinitesimals, irrational numbers, transfinite
ordinals, ... are legitimate not because they have been rigorously
defined but because they can be used to SOLVE PROBLEMS. See, for
instance, Jesper Lützen's "The Prehistory of the Theory of
Distributions"
http://books.google.com/books?id=pC7vAAAAMAAJ
No one has ever anywhere actually used the concept of a uniform
distributions on N to solve any problem.
You can't even show how to use it to calculate the area of a triangle.
>
> Regards,
>
> Ross Finlayson
Bill Taylor
Dec 21, 2012, 11:41:44 PM12/21/12
to
On Dec 22, 5:23 am, FredJeffries <fredjeffr...@gmail.com> wrote:
> Dirac delta, infinitesimals, irrational numbers, transfinite ordinals,
> ... are legitimate not because they have been rigorously defined
Yes, that is PRECISELY why they are legitimate.
> Dirac delta, infinitesimals, irrational numbers, transfinite ordinals,
> ... are legitimate not because they have been rigorously defined
> No one has ever anywhere actually used the concept of a uniform
> distributions on N to solve any problem.
that two randomly chosen naturals will be co-prime, for example.
And many others of that type.
-- Blunderbuss Bill
** Dogma is a bitch! (pun intended)
porky_...@my-deja.com
Dec 22, 2012, 11:14:10 AM12/22/12
to
On Friday, December 21, 2012 11:41:44 PM UTC-5, Bill Taylor wrote:
> On Dec 22, 5:23 am, FredJeffries <fredjeffr...@gmail.com> wrote: > Dirac delta, infinitesimals, irrational numbers, transfinite ordinals, > ... are legitimate not because they have been rigorously defined Yes, that is PRECISELY why they are legitimate. > No one has ever anywhere actually used the concept of a uniform > distributions on N to solve any problem. Sure they have. You can use it to calculate the probability that two randomly chosen naturals will be co-prime, for example. And many others of that type. -- Blunderbuss Bill ** Dogma is a bitch! (pun intended)
So, you're saying there *exists* the uniform distribution of positive integers (or natural numbers if you wish). Well, well, well, would you please then enlighten the unwashed masses like myself and tell us that's the probability of selecting an arbitrary positive integers?
> On Dec 22, 5:23 am, FredJeffries <fredjeffr...@gmail.com> wrote: > Dirac delta, infinitesimals, irrational numbers, transfinite ordinals, > ... are legitimate not because they have been rigorously defined Yes, that is PRECISELY why they are legitimate. > No one has ever anywhere actually used the concept of a uniform > distributions on N to solve any problem. Sure they have. You can use it to calculate the probability that two randomly chosen naturals will be co-prime, for example. And many others of that type. -- Blunderbuss Bill ** Dogma is a bitch! (pun intended)
Regards,
PPJr.
Ross A. Finlayson
Dec 22, 2012, 2:26:24 PM12/22/12
to
On Dec 22, 8:14 am, "porky_pig...@my-deja.com" <porky_pig...@my-
How much time you got?
Obviously enough while we model Dirac's delta with real functions, as
it is rigorous in the framework we have established for standard real
analysis, there isn't the extant framework for a rigorous treatment of
a non-standard probability with a uniform distribution over the
naturals. So, where the goal is to not invent a brand new standard
with mutually consistent results, the distribution is built from the
standard as described above, modeling the asymptotics of real
functions. Then, where there's instead the notion that these things
exist and are concrete, instead there are currently "alternative"
foundations, that are true as they reflect the true character of these
objects.
As above, on the one hand there are the simple discrete uniform
distributions of 0 through n, or conveniently 1 through n, or 0
through n-1, the probability of each value is 1/n. For n = N, and the
set is an ordinal, 1/N is not a standard real value, but it would be
somewhere between zero and one. (And yes, I know that lim_n->oo 1/n =
0.) Then, each integer has an infinitesimal probability that is a
constant, their sum is one. EF is the CDF of that.
Then, another notion is that a continuous probability distribution,
and its CDF, would have that said CDF ranges from zero to one, for the
domain of the naturals, and to be uniform, that the difference between
any two consecutive values, naturally ordered by the domain, is a
constant, as it is. EF is that.
Then, another would have that a probability function for a continuous
distribution, would have area one, and 1 >= p(n) >= 0 for each value.
EF is that.
That is about, on the one hand, seeing the values of EF as constant
monotone increasing, it is the CDF, as constant monotone non-
increasing, it is the p.d.f. EF is that.
Here, constant monotone is a stronger condition than monotone.
Then, as well this distribution would have that its CDF (or CDFs,
where there are distributions for each of the continuous and discrete)
would, among other things, be uniformly dense in values of its range
throughout [0,1]. This has various consequences for results of
density of series in the reals, and as well to reconcile with standard
modern mathematics, would benefit from the establishment of true
foundations, for these things as they would exist. There would be a
relevant construction with these properties. EF is that.
Nothing to see here? Move along.
Regards,
Ross Finlayson
Obviously enough while we model Dirac's delta with real functions, as
it is rigorous in the framework we have established for standard real
analysis, there isn't the extant framework for a rigorous treatment of
a non-standard probability with a uniform distribution over the
naturals. So, where the goal is to not invent a brand new standard
with mutually consistent results, the distribution is built from the
standard as described above, modeling the asymptotics of real
functions. Then, where there's instead the notion that these things
exist and are concrete, instead there are currently "alternative"
foundations, that are true as they reflect the true character of these
objects.
As above, on the one hand there are the simple discrete uniform
distributions of 0 through n, or conveniently 1 through n, or 0
through n-1, the probability of each value is 1/n. For n = N, and the
set is an ordinal, 1/N is not a standard real value, but it would be
somewhere between zero and one. (And yes, I know that lim_n->oo 1/n =
0.) Then, each integer has an infinitesimal probability that is a
constant, their sum is one. EF is the CDF of that.
Then, another notion is that a continuous probability distribution,
and its CDF, would have that said CDF ranges from zero to one, for the
domain of the naturals, and to be uniform, that the difference between
any two consecutive values, naturally ordered by the domain, is a
constant, as it is. EF is that.
Then, another would have that a probability function for a continuous
distribution, would have area one, and 1 >= p(n) >= 0 for each value.
EF is that.
That is about, on the one hand, seeing the values of EF as constant
monotone increasing, it is the CDF, as constant monotone non-
increasing, it is the p.d.f. EF is that.
Here, constant monotone is a stronger condition than monotone.
Then, as well this distribution would have that its CDF (or CDFs,
where there are distributions for each of the continuous and discrete)
would, among other things, be uniformly dense in values of its range
throughout [0,1]. This has various consequences for results of
density of series in the reals, and as well to reconcile with standard
modern mathematics, would benefit from the establishment of true
foundations, for these things as they would exist. There would be a
relevant construction with these properties. EF is that.
Nothing to see here? Move along.
Regards,
Ross Finlayson
David Bernier
Dec 22, 2012, 2:58:01 PM12/22/12
to
"amenable group".
The additive groups Z, Z^2, Z^3 and so on can be given
the discrete topology, where each sigleton, say {(1, 3)}
for the group Z^2, is a closed set.
Then as I recall, a theorem says some type of abelian groups are
amenable.
I don't remember the type condition for abelian groups.
cf.:
http://en.wikipedia.org/wiki/Amenable_group
Of course, N with addition isn't a group. It's a ? semigroup?
(yes), and also a monoid if by N one means the thing that has
zero in it:
http://en.wikipedia.org/wiki/Monoid
Amenability means a sensible, consistent with translations,
averaging procedure exists e.g. for bounded functions on
the group.
For the probabilistic GCD is 1 argument for "random"
elements (m, n) in N^* x N^* , I don't know if it can be
recast using the amenability of Z x Z ...
gcd of (0, 0) : I know "modulo theory" is linked to
ideals in rings. Anyway, since everything non-zero
divides zero, it's dubious about sensibly defining
gcd of (0, 0).
But then, (0, 0) is just 1 element of ZxZ, so
coprimeness yes/no of (0,0) should be irrelevant
to an argument based on amenability.
I don't know what that argument might look like,
assuming it exists, i.e. that I'm on some sort of
"right track" ...
dave
David Bernier
Butch Malahide
Dec 22, 2012, 4:12:11 PM12/22/12
to
On Dec 22, 10:14 am, "porky_pig...@my-deja.com" <porky_pig...@my-
Zero, of course. Needless to say, the probability measure will not be
countably additive. Finite additivity is good enough for many
purposes, and in this case it will have to do.
http://www.brunodefinetti.it/Bibliografia/StatimplicationsFA.pdf
countably additive. Finite additivity is good enough for many
purposes, and in this case it will have to do.
http://www.brunodefinetti.it/Bibliografia/StatimplicationsFA.pdf
FredJeffries
Dec 24, 2012, 2:43:50 PM12/24/12
to
FredJeffries
Dec 24, 2012, 2:47:32 PM12/24/12
to
On Dec 22, 11:26 am, "Ross A. Finlayson" <ross.finlay...@gmail.com>
wrote:
>
More gibberish.
Still no calculation of any meaningful result.
wrote:
>
More gibberish.
Still no calculation of any meaningful result.
Butch Malahide
Dec 24, 2012, 4:56:10 PM12/24/12
to
duckduckgo gave me too many hits to do more than glance at them:
http://en.wikipedia.org/wiki/Coprime_integers
http://mathworld.wolfram.com/RelativelyPrime.html
http://primes.utm.edu/notes/relprime.html
http://www.cs.umb.edu/~offner/files/an_num_th.pdf
http:/http://mathoverflow.net/questions/97041/what-is-the-probability-
that-two-numbers-are-relatively-prime/
www.physics.harvard.edu/academics/undergrad/probweek/sol44.pdf
http://www.mathreference.com/lc-z,cop.html
http://www.mathreference.com/lc-z,cop.html
http://math.stackexchange.com/questions/64498/probability-that-two-random-numbers-are-coprime
http://fermatslasttheorem.blogspot.com/2006/09/odds-of-two-integers-being-relatively.html
Virgil
Dec 25, 2012, 12:22:38 AM12/25/12
to
In article
<6e340766-2e08-4966...@10g2000yqo.googlegroups.com>,
existence in the standard real number system of e an infinitesimal
non-zero lambda smaller than any positive standard real but itself
positive, and an infinite cardinality, card(|N), of the infinite set of
naturals such that lambda * card(|N) = 1.
Which cannot occur within the essentially unique (up to isomorphism of
complete ordered Archimedian fields) standard real number system.
--
<6e340766-2e08-4966...@10g2000yqo.googlegroups.com>,
Butch Malahide <fred....@gmail.com> wrote:
> On Dec 21, 8:41�pm, Bill Taylor <wfc.tay...@gmail.com> wrote:
>
> > On Dec 22, 5:23�am, FredJeffries <fredjeffr...@gmail.com> wrote:
>
> > > No one has ever anywhere actually used the concept of a uniform
> > > distributions on N to solve any problem.
>
> > Sure they have. �You can use it to calculate the probability
> > that two randomly chosen naturals will be co-prime, for example.
> > And many others of that type.
You cannot do it using the standard reals because it would require the
> On Dec 21, 8:41�pm, Bill Taylor <wfc.tay...@gmail.com> wrote:
>
> > On Dec 22, 5:23�am, FredJeffries <fredjeffr...@gmail.com> wrote:
>
> > > No one has ever anywhere actually used the concept of a uniform
> > > distributions on N to solve any problem.
>
> > Sure they have. �You can use it to calculate the probability
> > that two randomly chosen naturals will be co-prime, for example.
> > And many others of that type.
existence in the standard real number system of e an infinitesimal
non-zero lambda smaller than any positive standard real but itself
positive, and an infinite cardinality, card(|N), of the infinite set of
naturals such that lambda * card(|N) = 1.
Which cannot occur within the essentially unique (up to isomorphism of
complete ordered Archimedian fields) standard real number system.
--
Butch Malahide
Dec 25, 2012, 12:53:31 AM12/25/12
to
On Dec 24, 11:22 pm, Virgil <vir...@ligriv.com> wrote:
> In article
> <6e340766-2e08-4966-b497-0c4d83497...@10g2000yqo.googlegroups.com>,
(as opposed to finitely additive) measure. It seems kind of arbitrary
to say that probability has to be countably additive but does not have
to be, say, aleph_2-additive.
> Which cannot occur within the essentially unique (up to isomorphism of
> complete ordered Archimedian fields) standard real number system.
"Archimedean" is redundant; a complete ordered field is necessarily
Archimedean.
> In article
> <6e340766-2e08-4966-b497-0c4d83497...@10g2000yqo.googlegroups.com>,
> Butch Malahide <fred.gal...@gmail.com> wrote:
>
> > On Dec 21, 8:41 pm, Bill Taylor <wfc.tay...@gmail.com> wrote:
>
> > > On Dec 22, 5:23 am, FredJeffries <fredjeffr...@gmail.com> wrote:
>
> > > > No one has ever anywhere actually used the concept of a uniform
> > > > distributions on N to solve any problem.
>
> > > Sure they have. You can use it to calculate the probability
> > > that two randomly chosen naturals will be co-prime, for example.
> > > And many others of that type.
>
> You cannot do it using the standard reals because it would require the
> existence in the standard real number system of e an infinitesimal
> non-zero lambda smaller than any positive standard real but itself
> positive, and an infinite cardinality, card(|N), of the infinite set of
> naturals such that lambda * card(|N) = 1.
You are tacitly assuming that probability must be a countably additive
>
> > On Dec 21, 8:41 pm, Bill Taylor <wfc.tay...@gmail.com> wrote:
>
> > > On Dec 22, 5:23 am, FredJeffries <fredjeffr...@gmail.com> wrote:
>
> > > > No one has ever anywhere actually used the concept of a uniform
> > > > distributions on N to solve any problem.
>
> > > Sure they have. You can use it to calculate the probability
> > > that two randomly chosen naturals will be co-prime, for example.
> > > And many others of that type.
>
> You cannot do it using the standard reals because it would require the
> existence in the standard real number system of e an infinitesimal
> non-zero lambda smaller than any positive standard real but itself
> positive, and an infinite cardinality, card(|N), of the infinite set of
> naturals such that lambda * card(|N) = 1.
(as opposed to finitely additive) measure. It seems kind of arbitrary
to say that probability has to be countably additive but does not have
to be, say, aleph_2-additive.
> Which cannot occur within the essentially unique (up to isomorphism of
> complete ordered Archimedian fields) standard real number system.
Archimedean.
Virgil
Dec 25, 2012, 1:48:27 AM12/25/12
to
In article
<14be47c2-c95a-45b7...@c16g2000yqi.googlegroups.com>,
probabilities over ANY set of sets making up a partition of the space be
equal to one.
If you want to impose some non-standard definition, that is your
prerogative, but you must not expect all others to accede to it.
<14be47c2-c95a-45b7...@c16g2000yqi.googlegroups.com>,
Butch Malahide <fred....@gmail.com> wrote:
> On Dec 24, 11:22�pm, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <6e340766-2e08-4966-b497-0c4d83497...@10g2000yqo.googlegroups.com>,
> > �Butch Malahide <fred.gal...@gmail.com> wrote:
> >
> > > On Dec 21, 8:41�pm, Bill Taylor <wfc.tay...@gmail.com> wrote:
> >
> > > > On Dec 22, 5:23�am, FredJeffries <fredjeffr...@gmail.com> wrote:
> >
> > > > > No one has ever anywhere actually used the concept of a uniform
> > > > > distributions on N to solve any problem.
> >
> > > > Sure they have. �You can use it to calculate the probability
> > > > that two randomly chosen naturals will be co-prime, for example.
> > > > And many others of that type.
> >
> > You cannot do it using the standard reals because it would require the
> > existence in the standard real number system of e an infinitesimal
> > non-zero lambda smaller than any positive standard real but itself
> > positive, and an infinite cardinality, card(|N), of the infinite set of
> > naturals such that lambda * card(|N) = 1.
>
> You are tacitly assuming that probability must be a countably additive
> (as opposed to finitely additive) measure. It seems kind of arbitrary
> to say that probability has to be countably additive but does not have
> to be, say, aleph_2-additive.
The standard definition of probability requires that the sum of the
> On Dec 24, 11:22�pm, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <6e340766-2e08-4966-b497-0c4d83497...@10g2000yqo.googlegroups.com>,
> > �Butch Malahide <fred.gal...@gmail.com> wrote:
> >
> > > On Dec 21, 8:41�pm, Bill Taylor <wfc.tay...@gmail.com> wrote:
> >
> > > > On Dec 22, 5:23�am, FredJeffries <fredjeffr...@gmail.com> wrote:
> >
> > > > > No one has ever anywhere actually used the concept of a uniform
> > > > > distributions on N to solve any problem.
> >
> > > > Sure they have. �You can use it to calculate the probability
> > > > that two randomly chosen naturals will be co-prime, for example.
> > > > And many others of that type.
> >
> > You cannot do it using the standard reals because it would require the
> > existence in the standard real number system of e an infinitesimal
> > non-zero lambda smaller than any positive standard real but itself
> > positive, and an infinite cardinality, card(|N), of the infinite set of
> > naturals such that lambda * card(|N) = 1.
>
> You are tacitly assuming that probability must be a countably additive
> (as opposed to finitely additive) measure. It seems kind of arbitrary
> to say that probability has to be countably additive but does not have
> to be, say, aleph_2-additive.
probabilities over ANY set of sets making up a partition of the space be
equal to one.
If you want to impose some non-standard definition, that is your
prerogative, but you must not expect all others to accede to it.
>
> > Which cannot occur within the essentially unique (up to isomorphism of
> > complete ordered Archimedian fields) standard real number system.
>
> "Archimedean" is redundant; a complete ordered field is necessarily
> Archimedean.
--
> > Which cannot occur within the essentially unique (up to isomorphism of
> > complete ordered Archimedian fields) standard real number system.
>
> "Archimedean" is redundant; a complete ordered field is necessarily
> Archimedean.
Butch Malahide
Dec 25, 2012, 2:26:40 AM12/25/12
to
On Dec 25, 12:48 am, Virgil <vir...@ligriv.com> wrote:
>
> The standard definition of probability requires that the sum of the
> probabilities over ANY set of sets making up a partition of the space be
> equal to one.
How does that work out in the case of the partition of the space into
>
> The standard definition of probability requires that the sum of the
> probabilities over ANY set of sets making up a partition of the space be
> equal to one.
its one-element subsets? Are you saying that the standard definition
requires every probability distribution to be discrete?
Virgil
Dec 25, 2012, 4:12:12 AM12/25/12
to
In article
<f9fb82de-a071-42f1...@c16g2000yqi.googlegroups.com>,
to your attention.
> Are you saying that the standard definition
> requires every probability distribution to be discrete?
Not at all! Finite spaces cause no problems and for a space of
uncountable cardinality, one can have the probability of every countable
set equal to zero and still have meaningful probability. It is only
spaces of countable cardinality that cause this sort of trouble if one
wishes sets of equal cardinality all to have the same probability.
--
<f9fb82de-a071-42f1...@c16g2000yqi.googlegroups.com>,
Butch Malahide <fred....@gmail.com> wrote:
> On Dec 25, 12:48�am, Virgil <vir...@ligriv.com> wrote:
> >
> > The standard definition of probability requires that the sum of the
> > probabilities over ANY set of sets making up a partition of the space be
> > equal to one.
>
> How does that work out in the case of the partition of the space into
> its one-element subsets?
If the space is N, it doesn't, which is teh point I was trying to bring
> On Dec 25, 12:48�am, Virgil <vir...@ligriv.com> wrote:
> >
> > The standard definition of probability requires that the sum of the
> > probabilities over ANY set of sets making up a partition of the space be
> > equal to one.
>
> How does that work out in the case of the partition of the space into
> its one-element subsets?
to your attention.
> Are you saying that the standard definition
> requires every probability distribution to be discrete?
uncountable cardinality, one can have the probability of every countable
set equal to zero and still have meaningful probability. It is only
spaces of countable cardinality that cause this sort of trouble if one
wishes sets of equal cardinality all to have the same probability.
--
Butch Malahide
Dec 25, 2012, 4:57:37 AM12/25/12
to
> <f9fb82de-a071-42f1-b393-a3e689c24...@c16g2000yqi.googlegroups.com>,
with Lebesgue measure, your "standard requirement" fails in the case
of the partition into one-element sets.
> > Are you saying that the standard definition
> > requires every probability distribution to be discrete?
>
> Not at all! Finite spaces cause no problems and for a space of
> uncountable cardinality, one can have the probability of every countable
> set equal to zero and still have meaningful probability. It is only
> spaces of countable cardinality that cause this sort of trouble if one
> wishes sets of equal cardinality all to have the same probability.
Did anyone say anything about wishing sets of equal cardinality all to
have the same probability? Is that what the OP was talking about? I
didn't try to read his post, as it seemed kind of obscure. Of course
it's not possible, in an infinite sample space, for sets of the same
cardinality all to have the same probability. (I'm assuming the axiom
of choice, so I don't have to worry about Dedekind-finite infinite
sets.)
> Butch Malahide <fred.gal...@gmail.com> wrote:
>
> > On Dec 25, 12:48 am, Virgil <vir...@ligriv.com> wrote:
>
> > > The standard definition of probability requires that the sum of the
> > > probabilities over ANY set of sets making up a partition of the space be
> > > equal to one.
>
> > How does that work out in the case of the partition of the space into
> > its one-element subsets?
>
> If the space is N, it doesn't, which is teh point I was trying to bring
> to your attention.
You said "The standard definition of probability requires that the sum
>
> > On Dec 25, 12:48 am, Virgil <vir...@ligriv.com> wrote:
>
> > > The standard definition of probability requires that the sum of the
> > > probabilities over ANY set of sets making up a partition of the space be
> > > equal to one.
>
> > How does that work out in the case of the partition of the space into
> > its one-element subsets?
>
> If the space is N, it doesn't, which is teh point I was trying to bring
> to your attention.
of the probabilities over ANY set of sets making up a partition of the
space be equal to one." If the probability space is the unit interval
with Lebesgue measure, your "standard requirement" fails in the case
of the partition into one-element sets.
> > Are you saying that the standard definition
> > requires every probability distribution to be discrete?
>
> Not at all! Finite spaces cause no problems and for a space of
> uncountable cardinality, one can have the probability of every countable
> set equal to zero and still have meaningful probability. It is only
> spaces of countable cardinality that cause this sort of trouble if one
> wishes sets of equal cardinality all to have the same probability.
have the same probability? Is that what the OP was talking about? I
didn't try to read his post, as it seemed kind of obscure. Of course
it's not possible, in an infinite sample space, for sets of the same
cardinality all to have the same probability. (I'm assuming the axiom
of choice, so I don't have to worry about Dedekind-finite infinite
sets.)
Shmuel Metz
Dec 25, 2012, 4:46:04 AM12/25/12
to
In <virgil-523D87....@BIGNEWS.USENETMONSTER.COM>, on
12/24/2012
finite additivity is enough. In a more general context, the standard
definition uses countable additivity.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spam...@library.lspace.org
12/24/2012
at 11:48 PM, Virgil <vir...@ligriv.com> said:
>The standard definition of probability requires that the sum of the
>probabilities over ANY set of sets making up a partition of the space
>be equal to one.
For purposes of proving that there is no uniform distribution over N,
>The standard definition of probability requires that the sum of the
>probabilities over ANY set of sets making up a partition of the space
>be equal to one.
finite additivity is enough. In a more general context, the standard
definition uses countable additivity.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spam...@library.lspace.org
Butch Malahide
Dec 25, 2012, 3:08:42 PM12/25/12
to
On Dec 25, 3:46 am, Shmuel (Seymour J.) Metz
<spamt...@library.lspace.org.invalid> wrote:
> In <virgil-523D87.23482724122...@BIGNEWS.USENETMONSTER.COM>, on
that none exists?
<spamt...@library.lspace.org.invalid> wrote:
> In <virgil-523D87.23482724122...@BIGNEWS.USENETMONSTER.COM>, on
> 12/24/2012
> at 11:48 PM, Virgil <vir...@ligriv.com> said:
>
> >The standard definition of probability requires that the sum of the
> >probabilities over ANY set of sets making up a partition of the space
> >be equal to one.
>
> For purposes of proving that there is no uniform distribution over N,
> finite additivity is enough. In a more general context, the standard
> definition uses countable additivity.
How do you define "uniform distribution over N" and how do you prove
> at 11:48 PM, Virgil <vir...@ligriv.com> said:
>
> >The standard definition of probability requires that the sum of the
> >probabilities over ANY set of sets making up a partition of the space
> >be equal to one.
>
> For purposes of proving that there is no uniform distribution over N,
> finite additivity is enough. In a more general context, the standard
> definition uses countable additivity.
that none exists?
gus gassmann
Dec 26, 2012, 6:50:24 AM12/26/12
to
On 25/12/2012 5:57 AM, Butch Malahide wrote:
> Did anyone say anything about wishing sets of equal cardinality all to
> have the same probability?
The subject line says "uniform distributions". What can that mean OTHER
> Did anyone say anything about wishing sets of equal cardinality all to
> have the same probability?
than "sets of equal cardinality have equal measure"?
> Is that what the OP was talking about? I
> didn't try to read his post, as it seemed kind of obscure. Of course
> it's not possible, in an infinite sample space, for sets of the same
> cardinality all to have the same probability.
Butch Malahide
Dec 26, 2012, 12:30:29 PM12/26/12
to
On Dec 26, 5:50 am, gus gassmann <g...@nospam.com> wrote:
> On 25/12/2012 5:57 AM, Butch Malahide wrote:
>
> > Did anyone say anything about wishing sets of equal cardinality all to
> > have the same probability?
>
> The subject line says "uniform distributions". What can that mean OTHER
> than "sets of equal cardinality have equal measure"?
In the case of the continuous uniform distribution on the interval
> On 25/12/2012 5:57 AM, Butch Malahide wrote:
>
> > Did anyone say anything about wishing sets of equal cardinality all to
> > have the same probability?
>
> The subject line says "uniform distributions". What can that mean OTHER
> than "sets of equal cardinality have equal measure"?
[0,1], it means that "intervals of the same length have equal
measure." All intervals have the same cardinality, but their measures
vary.
I'm not sure what it means for a (finitely additive) measure on N.
Perhaps, that all points have equal measure (which would have to be
zero)? Or that sets of the same natural density should have equal
measures? Maybe it should be translation-invariant?
Ross A. Finlayson
Dec 27, 2012, 12:17:28 AM12/27/12
to
distribution of the natural integers would have the properties that
the probability of each value of the sample space and here support set
would be equal.
Then as these sum to one then this value would be infinitesimal and 1/
omega and yes we all know that's not a standard real number, or
rather: not a member of the Archimedean complete ordered field (and 1/
omega is an element of non-Archimedean extra-complete ordered fields
like the surreal numbers, which include all the real numbers as
described, for example, by Ehrlich or Conway.)
Here then as well I would actually described that instead they are
elements of the continuum best represented by the ring of iota-values,
with rather-restricted transfer principle, in that the elements of
continuum as iota-values and elements of the continuum as complete
ordered field are the same thing, but they have different
representations and obviously their rules of formation and
manipulation are different. Their properties as magnitudes hold.
Then, with regards to the probability of elements from U(n) or U_c(n)
and U_d(n) for the "continuous" and "discrete" uniform probability
distributions of natural integers being in particular subsets of the
natural integers, then as above how the U(r) or U(x) would be
represented by measure, here the notion would be as to density, and
from du Bois-Reymond, then to cardinality only for the finite, which
is then basically simply of multiplicity and only coincidentally
cardinality. P(n even | n e N) = 1/2, "the probability of n being
even given n is from the naturals is one half", P(n e {1,2}) = P(n e
{3, 4}), "the probability of n being 1 or 2 is the same as n being 3
or 4", where here it is implicit that the probability is regular and
uniform.
This is simply founded with treating the probabilities as opaque
quantities with additivity, simply underdefined, as to satisfying the
general sense of rigor. Of course, that is merely apologetics, where
the true inquiry is to the actual nature of these quantities.
So, the probability of any given natural being selected at uniform
random from all the natural integers is obviously: the same as that
of any other. It is that first, then as to how to divide the unity of
the sum of those probabilities equally among them is in then to
developments in foundations of real numbers as to support the
reasonable expectations thereof. And, there is longstanding tradition
in the discussion since antiquity as to ruminations on the nature of
these things. Yet, since Cauchy and Weierstrass, and Cantor, many of
those lines of inquiry as posed by our greatest thinkers do not have
their historical place in the context, of the infinitely divisible and
infinitely divided, for the potentially and complete infinity.
And: the only way to line up the points of the line is in a line, and
in the line. Draw the line. EF starts that.
In earlier discussions on this as well I described a framework for
constructing U(n) in ZFC. As well, as above it's described as modeled
by real functions. Again that's to appease the modern, for the avenue
to the real, and concrete, and of the continuum, and, in the
continuum, as divisible, and as divided, as composed.
Then it's a reasonable notion to consider the infinitesimals as the
unity as divisible and the unity as divided in a similar manner as to
consideration of the infinity as the unity potentially and the unity
completely.
Regards,
Ross Finlayson
Ross A. Finlayson
Dec 27, 2012, 9:18:59 PM12/27/12
to
On Dec 26, 9:17 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com>
wrote:
One of the properties of the standard real numbers is that are
iteratively re-divisible infinitely. Basically we know that for any
standard real quantity, it can be halved then halved again ad
infinitum, that multiplied by the series of the negative powers of
two, the limit evaluates to zero as standard real quantity.
Then, another consideration is as to the division or rather partition
of the unit, one, to infinitely many equal parts, that sum back to
one, or that multiplied by the number of partitions, the part is equal
to one. Then, this is the notion that there's a difference between
dividing the continuum, and having divided the continuum.
We're familiar with the fundamental theorems of calculus and
integration, and the definition of the sequence as Cauchy as having a
limit. As well, Leibniz' notation survives: the large S for
summation and d for differentials, that the differential ratio is as
to the free variable, of an infinitesimal rate of change of it, non-
zero and for no finite change. Then, there's at least present the
founder's intent and intuition that the notation reflect the summation
of infinitely many infinitesimal quantities, to retrieve a finite
quantity, and here one that exactly and perfectly corresponds with our
geometrical methods for the computation of area, between real-valued
curves and axes of the coordinate plane or about the origin in general
methods of exhaustion for the computation of area. Then, it's
certainly not unreasonable nor unprecedented to consider that the real
unit of the continuum is composed of points and those of a uniform
partitioning or division _to the points_, besides the inexhaustibility
of division or partition of finite quantities, _to non-points_.
dividing: regions -> regions
divided: regions -> points
Then, where there's the consideration that points have zero width,
that they can't comprise a region with non-zero width, then that's in
as to where only an assemblage of points, and infinitely many,
comprise exactly a region of width one, individua only of the
continuum, where as points they are our familiar points of Euclid and
Hardy, and of Hardy as real numbers.
Then, for this uniform distribution of naturals, or for a putative
mapping from natural integers to the unit interval of reals, then
there is that these infinitesimal quantities, or iota-values, among
real values of the continuum as the continuum includes all magnitudes
and elements of the "linear continuum" of elements that satisfy
trichotomy, density, and continuity between zero and one, are defined
together as a completed thing, that the very definition is as to the
composition of the unit line segment, then via deduction the nature of
the points, instead of definition of the points as elements of a field
from the integers, then ordered field, then complete ordered field.
Plainly there is a consideration that either of these notions
establish trichotomy, density, and continuity of these elements,
between zero and one.
Then, in as to where EF this function is simply representative of
these notions of the continuum as its range, it's of some mathematical
interest. Courtesy its simple properties and resulting available
theorems, given its construction of primary nature in theory of
numbers and geometry, it's in fact a basic building block for a wide
range of theorems, and in particular, an entire class of applicable
theorems, due its properties inaccessible (or otherwise not known to
be accessible) to the standard.
So, then there's a consideration that for the pure or applied, a
putative uniform distribution of the naturals is of interest, and for
that in asymptotics there are established reasonable notions of
density to match those of measure for transfer between the discrete
and continuous. Also, that's applied in number theory as for example
theorems on prime numbers, and as well in systems of for example
random needles dropped on the line and in alternatives to attenuation
of the central limit theorem, such a thing is of more than marginal
utility and placement.
Then for EF, the function, the domain of which is the naturals and
range is [0,1], that in the number-theoretic results on uncountability
of the reals it is as no other function, as well it satisfies being a
CDF, of the naturals. Quite remarkably, it would also be a p.d.f. of
a continuous distribution as its integral evaluates to one, meeting
first the properties of a probability function, before considerations
of its standardness, in its regularity. This complements 1/omega
being a discrete distribution, where the sum of it over the omega-many
terms of the naturals is again: one. Now, where in ZFC it was shown
that there would be a set that would construct a uniform probability
distribution over the elements of the set of naturals, then it remains
that there would be uniform probability distribution over the
naturals.
EF is that.
Regards,
Ross Finlayson
wrote:
iteratively re-divisible infinitely. Basically we know that for any
standard real quantity, it can be halved then halved again ad
infinitum, that multiplied by the series of the negative powers of
two, the limit evaluates to zero as standard real quantity.
Then, another consideration is as to the division or rather partition
of the unit, one, to infinitely many equal parts, that sum back to
one, or that multiplied by the number of partitions, the part is equal
to one. Then, this is the notion that there's a difference between
dividing the continuum, and having divided the continuum.
We're familiar with the fundamental theorems of calculus and
integration, and the definition of the sequence as Cauchy as having a
limit. As well, Leibniz' notation survives: the large S for
summation and d for differentials, that the differential ratio is as
to the free variable, of an infinitesimal rate of change of it, non-
zero and for no finite change. Then, there's at least present the
founder's intent and intuition that the notation reflect the summation
of infinitely many infinitesimal quantities, to retrieve a finite
quantity, and here one that exactly and perfectly corresponds with our
geometrical methods for the computation of area, between real-valued
curves and axes of the coordinate plane or about the origin in general
methods of exhaustion for the computation of area. Then, it's
certainly not unreasonable nor unprecedented to consider that the real
unit of the continuum is composed of points and those of a uniform
partitioning or division _to the points_, besides the inexhaustibility
of division or partition of finite quantities, _to non-points_.
dividing: regions -> regions
divided: regions -> points
Then, where there's the consideration that points have zero width,
that they can't comprise a region with non-zero width, then that's in
as to where only an assemblage of points, and infinitely many,
comprise exactly a region of width one, individua only of the
continuum, where as points they are our familiar points of Euclid and
Hardy, and of Hardy as real numbers.
Then, for this uniform distribution of naturals, or for a putative
mapping from natural integers to the unit interval of reals, then
there is that these infinitesimal quantities, or iota-values, among
real values of the continuum as the continuum includes all magnitudes
and elements of the "linear continuum" of elements that satisfy
trichotomy, density, and continuity between zero and one, are defined
together as a completed thing, that the very definition is as to the
composition of the unit line segment, then via deduction the nature of
the points, instead of definition of the points as elements of a field
from the integers, then ordered field, then complete ordered field.
Plainly there is a consideration that either of these notions
establish trichotomy, density, and continuity of these elements,
between zero and one.
Then, in as to where EF this function is simply representative of
these notions of the continuum as its range, it's of some mathematical
interest. Courtesy its simple properties and resulting available
theorems, given its construction of primary nature in theory of
numbers and geometry, it's in fact a basic building block for a wide
range of theorems, and in particular, an entire class of applicable
theorems, due its properties inaccessible (or otherwise not known to
be accessible) to the standard.
So, then there's a consideration that for the pure or applied, a
putative uniform distribution of the naturals is of interest, and for
that in asymptotics there are established reasonable notions of
density to match those of measure for transfer between the discrete
and continuous. Also, that's applied in number theory as for example
theorems on prime numbers, and as well in systems of for example
random needles dropped on the line and in alternatives to attenuation
of the central limit theorem, such a thing is of more than marginal
utility and placement.
Then for EF, the function, the domain of which is the naturals and
range is [0,1], that in the number-theoretic results on uncountability
of the reals it is as no other function, as well it satisfies being a
CDF, of the naturals. Quite remarkably, it would also be a p.d.f. of
a continuous distribution as its integral evaluates to one, meeting
first the properties of a probability function, before considerations
of its standardness, in its regularity. This complements 1/omega
being a discrete distribution, where the sum of it over the omega-many
terms of the naturals is again: one. Now, where in ZFC it was shown
that there would be a set that would construct a uniform probability
distribution over the elements of the set of naturals, then it remains
that there would be uniform probability distribution over the
naturals.
EF is that.
Regards,
Ross Finlayson
Virgil
Dec 28, 2012, 12:09:34 AM12/28/12
to
In article
<0ef10195-22b2-4847...@pd8g2000pbc.googlegroups.com>,
"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:
> EF is that.
Your EF is EFfing crazy.
--
<0ef10195-22b2-4847...@pd8g2000pbc.googlegroups.com>,
"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:
> EF is that.
Your EF is EFfing crazy.
--
Ross A. Finlayson
Dec 28, 2012, 2:15:39 AM12/28/12
to
> <0ef10195-22b2-4847-a64a-baa35e3d0...@pd8g2000pbc.googlegroups.com>,
that you're willfully ignorant of what EF is, but EF: is what it is.
EF: it is what it is.
The continuum: it is what it is.
Mathematics: it is what it is.
Regards,
Ross Finlayson
> "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:
>
> > EF is that.
>
> Your EF is EFfing crazy.
> --
Now, Hancher, I'll accept that you're quite familiar with crazy, and
>
> > EF is that.
>
> Your EF is EFfing crazy.
> --
that you're willfully ignorant of what EF is, but EF: is what it is.
EF: it is what it is.
The continuum: it is what it is.
Mathematics: it is what it is.
Regards,
Ross Finlayson
Virgil
Dec 28, 2012, 4:12:52 AM12/28/12
to
In article
<a5a8214e-204a-49f4...@uc4g2000pbc.googlegroups.com>,
reads much of sci.math is.
--
<a5a8214e-204a-49f4...@uc4g2000pbc.googlegroups.com>,
> Now, Hancher, I'll accept that you're quite familiar with crazy
With sterling examples of it like you and WM, and others, everyone who
reads much of sci.math is.
--
Ross A. Finlayson
Dec 28, 2012, 1:54:58 PM12/28/12
to
> <a5a8214e-204a-49f4-8bbe-50b960b2c...@uc4g2000pbc.googlegroups.com>,
values from our study of the same principles. Some derive value from
the very closedness of things, that there are right and wrong answers
to our most fundamental questions. Some derive value from the very
openness of things, that there may be right and wrong answers to all
our questions. There are various motivations for its study and
application. Yet, it is largely so that given strong fundamental
principles, there are available general results, and mathematics is
for that those with totally different thinking, can share in truths,
given the principles of mathematics.
Clemens' law of conservation of truth: "For each Great Truth, there's
an equal and opposite Great Truth." Apollo: "Everything in
moderation, including moderation." There are a variety of fallacies,
ad hominem here, or your appeal to the righteousness of the mob, that
have as place in mathematics exhibits, not course.
monomania, check (determination, reliance on mathematical truth and
proof)
megalomania, check (belief in self)
paranoia, reasonable (typical unknowns, and there are many)
delusions, not so many (typical unknowns)
depressive, no just relaxed (yawn)
manic, from time to time (called high-energy, on-task, Olympian)
hypochondriac (not so much)
hyperchondria (not so much)
psychotic (meh)
sociopath, no
neurotic (not so much)
Now I'm not a doctor and I don't play one on TV (MCAT top), but in
terms of promoting the discovery of foundations of mathematics,
including but not limited to those that are are discovered and well-
covered, to me "different" doesn't mean "crazy", and here "infinite"
and "counter-intuitive" don't mean the same thing. And, you can find
someone who'll call you anything, and a quack who'll give you pills
for it, and maybe they know of the schoolyard ribbing of rubber, and
glue. You can trust anyone, to throw you in a ditch for a penny. And
some, you can trust.
For some mathematicians, mathematics is a diamond and a single flaw
would ruin the entire thing. For others, the existence of the flaw is
the only way to cleave the diamond and cut the diamond from its rough
aggregate to its perfect shape. So, when I throw light on the
diamond, it is to see the flaws, not ignore them.
Then, when it comes to spending enough time working on infinities in
mathematics, as one put it, "thinking about infinity makes people
nuts". I think that's not necessarily so, instead that there are
simply not so many people with the capacity and time to entertain the
notions of the infinite for its cerebral beauty, and the mathematics
of it for satisfaction of the mathematical conscience. Though, there
are confounding results, that then for the purposes of establishing
our rigor in mathematics, see requirements for limits in the
discussion, to then work up that limited subset of all what may be
true, to build a walled garden wherein all is true, consistent within
its walls. Then, I'm among those that would have that there is a
Universe, and there are mathematics of it, that tearing down the walls
is not to let in the darkness, but the light.
So, if you think you're surrounded by crazy people, you're probably
right. So, let us maintain decorum in our interminable discussions on
interminability, toward progress, as you and Muckenheim joust each
other keep in mind that if you're doctors of mathematics that a
certain collegial courtesy is apropos, as it is anyways, know that I
find myself quite in control of my faculties and don't so much care
what you find of yours or think of mine, and that EF is a compelling
construct that stands for its own and in the historical context,
particularly as a touchstone in the foundations. I'm for the
conscientious, and conscious, mathematician.
EF: it is what it is.
EF: CDF, p.d.f.
Infinity: topic of our greatest thinkers.
Ad hominem attacks: purview of the playground bully.
Take your ball and go home. Everybody's got one.
So, are you wading in a morass of incompetents, or, alternatively,
engaging in the highest levels of mathematical discourse?
Good luck with that.
Then, for your attempt to divert the course from mathematical
discussions, and there are hundreds of readers who comtemplate these
writings, we return to the notion of _what would be_ the drawing of
the line, _what would be_ the uniform distribution of the naturals
here in the continuous and discrete, _what would be_ progress in
mathematical foundations, and _what it is_.
EF: it is what it is.
What you think of EF: is what it is.
Regards,
Ross Finlayson
> "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:
>
> > Now, Hancher, I'll accept that you're quite familiar with crazy
>
> With sterling examples of it like you and WM, and others, everyone who
> reads much of sci.math is.
> --
There are varieties of mathematicians, and they derive different
>
> > Now, Hancher, I'll accept that you're quite familiar with crazy
>
> With sterling examples of it like you and WM, and others, everyone who
> reads much of sci.math is.
> --
values from our study of the same principles. Some derive value from
the very closedness of things, that there are right and wrong answers
to our most fundamental questions. Some derive value from the very
openness of things, that there may be right and wrong answers to all
our questions. There are various motivations for its study and
application. Yet, it is largely so that given strong fundamental
principles, there are available general results, and mathematics is
for that those with totally different thinking, can share in truths,
given the principles of mathematics.
Clemens' law of conservation of truth: "For each Great Truth, there's
an equal and opposite Great Truth." Apollo: "Everything in
moderation, including moderation." There are a variety of fallacies,
ad hominem here, or your appeal to the righteousness of the mob, that
have as place in mathematics exhibits, not course.
monomania, check (determination, reliance on mathematical truth and
proof)
megalomania, check (belief in self)
paranoia, reasonable (typical unknowns, and there are many)
delusions, not so many (typical unknowns)
depressive, no just relaxed (yawn)
manic, from time to time (called high-energy, on-task, Olympian)
hypochondriac (not so much)
hyperchondria (not so much)
psychotic (meh)
sociopath, no
neurotic (not so much)
Now I'm not a doctor and I don't play one on TV (MCAT top), but in
terms of promoting the discovery of foundations of mathematics,
including but not limited to those that are are discovered and well-
covered, to me "different" doesn't mean "crazy", and here "infinite"
and "counter-intuitive" don't mean the same thing. And, you can find
someone who'll call you anything, and a quack who'll give you pills
for it, and maybe they know of the schoolyard ribbing of rubber, and
glue. You can trust anyone, to throw you in a ditch for a penny. And
some, you can trust.
For some mathematicians, mathematics is a diamond and a single flaw
would ruin the entire thing. For others, the existence of the flaw is
the only way to cleave the diamond and cut the diamond from its rough
aggregate to its perfect shape. So, when I throw light on the
diamond, it is to see the flaws, not ignore them.
Then, when it comes to spending enough time working on infinities in
mathematics, as one put it, "thinking about infinity makes people
nuts". I think that's not necessarily so, instead that there are
simply not so many people with the capacity and time to entertain the
notions of the infinite for its cerebral beauty, and the mathematics
of it for satisfaction of the mathematical conscience. Though, there
are confounding results, that then for the purposes of establishing
our rigor in mathematics, see requirements for limits in the
discussion, to then work up that limited subset of all what may be
true, to build a walled garden wherein all is true, consistent within
its walls. Then, I'm among those that would have that there is a
Universe, and there are mathematics of it, that tearing down the walls
is not to let in the darkness, but the light.
So, if you think you're surrounded by crazy people, you're probably
right. So, let us maintain decorum in our interminable discussions on
interminability, toward progress, as you and Muckenheim joust each
other keep in mind that if you're doctors of mathematics that a
certain collegial courtesy is apropos, as it is anyways, know that I
find myself quite in control of my faculties and don't so much care
what you find of yours or think of mine, and that EF is a compelling
construct that stands for its own and in the historical context,
particularly as a touchstone in the foundations. I'm for the
conscientious, and conscious, mathematician.
EF: it is what it is.
Infinity: topic of our greatest thinkers.
Ad hominem attacks: purview of the playground bully.
Take your ball and go home. Everybody's got one.
So, are you wading in a morass of incompetents, or, alternatively,
engaging in the highest levels of mathematical discourse?
Good luck with that.
Then, for your attempt to divert the course from mathematical
discussions, and there are hundreds of readers who comtemplate these
writings, we return to the notion of _what would be_ the drawing of
the line, _what would be_ the uniform distribution of the naturals
here in the continuous and discrete, _what would be_ progress in
mathematical foundations, and _what it is_.
EF: it is what it is.
Regards,
Ross Finlayson
Virgil
Dec 28, 2012, 8:09:41 PM12/28/12
to
In article
<4a00241c-1c9f-4149...@s6g2000pby.googlegroups.com>,
--
<4a00241c-1c9f-4149...@s6g2000pby.googlegroups.com>,
> There are varieties of mathematicians, and they derive different
> values from our study of the same principles. Some derive value from
> the very closedness of things, that there are right and wrong answers
> to our most fundamental questions. Some derive value from the very
> openness of things, that there may be right and wrong answers to all
> our questions. There are various motivations for its study and
> application. Yet, it is largely so that given strong fundamental
> principles, there are available general results, and mathematics is
> for that those with totally different thinking, can share in truths,
> given the principles of mathematics.
Ross can convey less in more words than almost anyone else.
> values from our study of the same principles. Some derive value from
> the very closedness of things, that there are right and wrong answers
> to our most fundamental questions. Some derive value from the very
> openness of things, that there may be right and wrong answers to all
> our questions. There are various motivations for its study and
> application. Yet, it is largely so that given strong fundamental
> principles, there are available general results, and mathematics is
> for that those with totally different thinking, can share in truths,
> given the principles of mathematics.
--
Ross A. Finlayson
Dec 28, 2012, 9:45:12 PM12/28/12
to
> <4a00241c-1c9f-4149-9c8e-73624b420...@s6g2000pby.googlegroups.com>,
progress, extra the standard, in the mathematics. Goedel proves this
using standard, modern mathematics: any finitely axiomatized theory
strong enough to represent natural arithmetic has true statements
about the objects of the theory that aren't provable, derivable, true
from the axioms of the theory: incompleteness. While that may be
so, it doesn't necessarily conclude that Presburger Arithmetic of
addition on integers is incomplete, as it is shown complete, and it
doesn't necessarily show that an axiomless system of natural
deduction, is incomplete. Yet, it surely gives the interested
mathematician the wherewithal to conclude that there are particular
true features of the objects of the domain of discourse, relevant to
the structure, that aren't those of our standard, modernly.
Then, there are as well considerations that traditional avenues of
inquiry, into the infinitesimal and infinite, have been closed off
from our selection of axioms with basically set theory's definition of
an inductive set or infinity, combined with the restriction of
comprehension of the axiom of regularity.
A. o. Infinity: defines a constant in the language of an infinite
inductive set
A. o. Regularity: restricts naive set-building from sets with
irregular transitive closures, a.k.a. Foundation
The other axioms reflect reasonably intuitive notions of the
composition of sets, as defined by their elements, that nothing
prevents elements from association.
Now, we know that there are various considerations these days, for
example of Aczel's reknowned anti-foundational, foundations, where the
anti-well-founded sets, as objects and members of the domain of
discourse, exist. As well, in NF and NFU, or as well NBG with
Classes, where NBG is ZFC with models, there are considerations of the
embracing of the domain of objects that would not be regular set-
theoretic sets: the class of ordinals, the class of sets, and other
structures that via their definition would not leave ZF containing
them, directly consistent. Obviously enough, this group-noun-game of
having classes for sets leaves what was the primary object, the set,
no-longer the primary object, just as type theories in sets, may find
that types are not the primary objects.
set theory <-> model theory <-> class theory
set theories <-> type theory (eg ramified, stratified) <-> category
theory <-> HOL
geometry
number theory
theory
With theory then and in number theory, a primary area of application
is that of real analysis, and continuum analysis, and here in as to
dividing the continuum, and having divided the continuum. Having
divided the continuum equally, is as to where the infinity of integers
is the continuum to the unit, as the unit is to continuum to the iota-
value, or from Newton: fluents and fluxions: this is a notion extra
the standard's theorems, but from the same principles.
So, there are traditional avenues of inquiry, into the infinite and
infinitesimal, as transcribed from antiquity, and revolutionaries like
Galileo, Wallis, Newton, Leibniz, Euler, Gauss, du Bois-Reymond,
Cantor, that are not within our modern mathematics, which as
curriculum is presented to as wide or wider an audience than ever
before. Then, where it is important for society to have mathematics
lead physics, we have seen in the course of the technological age that
where once mathematics lead physics and the capacity of experiment by
generations, technology has reached in as to where it is not the
capacity for construction of experiment, but the body of mathematics,
that lacks in the explanation of data. Our measurements reveal dark
matter, Avogadro's number grows, the farther we look the bigger it is,
the closer the smaller. We know there are truisms beyond the
classical, and the modern classical, the relativistic. There actually
are differences in things in the macro and micro scales from our
mesoscale, there actually is the anthropocentric, or simply in as to
our place in scale. Yet, there is still a reasonable expectation that
there is order in the Universe, and that then mathematics is and will
remain our best framework for physics, which otherwise would overthrow
not just dogma but science.
So, what are these truly revolutionary ideas in mathematics that will
truly reveal avenues for progression in our science? One might think
they would arise from the continued course in the foundations as we
know them of modern mathematics. Yet, in a hundred years, there isn't
a direct application resulting from that course. Measure theory is
built on the countable. The development of methods in physics is as
to the algebras, to reduce symbolic complexity and that is a right
course, and ever-more-complicated deformations of Euclidean geometry
to bend the methods to the results, and that is not clearly the right
course.
Then, the nature of the continuum is the matter of our discussion.
What is the continuum? We know that our tools of real analysis,
founded on countable additivity in measure theory, give us results
matching those of geometry and experience in the meso-scale. As well
we know there are true features of these objects of discourse not yet
resolved in our modern theory. Then, for what may be directions for
progress in continuum analysis, there is obviously the broad vista of
refinements and developments in the standard, but also there's a
reasonable consideration that the alternative avenues don't end where
we've left them, nor do they necessarily lead off to the weeds or
where the standard could not maintain its track.
So, there is found this simple construction with its plainly
reasonable features, and as well, surprising features, that the
integral of the function, EF, is one, besides that in the fundamental
results of modern mathematics, _that it isn't shown uncountable by the
cohort of results otherwise establishing uncountability of the
reals_. Then, where its features are well-modeled in what are
standard mathematics, as modeled by real functions, in real analysis,
and as having its existence inferred from the existence of a
reasonable notion of uniformity or regularity of the naturals, in ZFC
and number theory, then this gives the reasonable justification that
those abandoned directions are not without course, and not without a
reconciliation, that our edifice of modern mathematics precludes their
existence.
Then, with the other interwined, if not inseparable theoretical
studies, there are notions for set theory, and theory, that an
axiomless system of natural deduction provides a foundation for
results as applied. As well, in the course of the investigation of
these matters, simple axioms of a spiral-space-filling curve of a
natural continuum founds a geometry: from points and space before
points and lines. Then, the demands of the conscientious
mathematician of at once adhering to rigor, and our established
consistency in results, and as well acknowledging and even inviting
those results as would supercede what as modern mathematics precedes
progress, has that those results encompassing what came before, extend
the sphere of knowledge, and find room for developments that may truly
be innovative in discovery, beyond the refinement and specialization
of methods, to their completion, and placement of axiomatics in the
axiomatized.
And, EF is part of that.
So, the notion of a uniform distribution over the naturals, and the
very notion of a constant rate of change over the naturals, seems to
be built into the simple monotone progression of the naturals. One
might find that counting the integers leads to the first being counted
more, and conversely that for any there are more to be counted and
that the more there are, the more there are. While an inductive set
may truly be primitive in our theory, as well in frameworks where it
represents the meter of change, a copy or structural reflection of it,
would be high above. The numerical continuum, first as infinitely
many individua, then as integers, then as a basis for functions from
integers, then for example as reals, is a way to approach that the
numbers aren't defined, instead derived. So, the notion of a uniform
distribution follows from first principles.
Then, the structure of a uniform distribution of naturals, here for
continuous distributions and discrete distributions U_c and U_d, or
U^bar and U^dots, would first see that they are distributions. As
functions defined on the naturals: EF = U_c and 1/omega = U_d. Then
it's quite remarkable that EF, as p.d.f., is its own CDF. The
structure is for our theory of probability. Now, it is well known
that probability doesn't necessarily admit such a function, as
standard real analysis doesn't necessarily admit such a function,
though of course it's readily modeled by real functions, and other
functions with known utility are so constructed (re Dirac's unit
impulse function). Then while non-standard (and not to collide with
the statistical sense of standard as having unit variance), while non-
standard, these functions are at least modeled by the standard. And,
where a corresponding framework for structure of the continuum as non-
standard (here as super-Archimedean ordered field and as well ring of
iota-values) establishes the basis (and in the sense of the vector
basis) of the unit as the range of this function, then as well where
probability is founded on measure theory and real analysis, the
structure of a uniform distribution of the naturals is as well at
least partially evident.
Then the use of a uniform distribution of the naturals, has that it
would correspond with notions of density in the integers of number
theory, where cardinality is mute to it, and about how then
conditional and joint probabilities would be definable in terms of
this simple function, correlating expectations from number theory and
density of integers with statistical expectations. Then as well,
where there are surprising features of the functions that are each of
a continuous and a discrete distribution of the elements of the
support space, these might lead to applications and then, a use of a
uniform distribution of the naturals.
So, with a rationale and a justification for the development of these
ideas in probability, and the corresponding framework of continuum
analysis expanded with renewed investigation of avenues once blocked
for a need to reconcile with our standard, modern mathematics, then,
that is of general interest to many. And, with the fact that re-
exploration and discovery of the good mathematics that great thinkers
see in the true nature of the continuum, may well be a most fruitful
avenue for discovery of novel mathematical structures for physics,
where transfinite cardinals haven't panned out in applications and are
shoehorned into countable additivity for useful measure theory, then
the rationale and justification for the notion, structure, and use of
the uniform probability distribution, and its corresponding
mathematical framework and foundations, is a course for better
understanding of the continuum.
And EF: on the line, in the line, the line: starts that: the
continuum of the line.
Regards,
Ross Finlayson
> "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:
>
> > There are varieties of mathematicians, and they derive different
> > values from our study of the same principles. Some derive value from
> > the very closedness of things, that there are right and wrong answers
> > to our most fundamental questions. Some derive value from the very
> > openness of things, that there may be right and wrong answers to all
> > our questions. There are various motivations for its study and
> > application. Yet, it is largely so that given strong fundamental
> > principles, there are available general results, and mathematics is
> > for that those with totally different thinking, can share in truths,
> > given the principles of mathematics.
>
> Ross can convey less in more words than almost anyone else.
> --
Then, there's a consideration that there certainly is room for
>
> > There are varieties of mathematicians, and they derive different
> > values from our study of the same principles. Some derive value from
> > the very closedness of things, that there are right and wrong answers
> > to our most fundamental questions. Some derive value from the very
> > openness of things, that there may be right and wrong answers to all
> > our questions. There are various motivations for its study and
> > application. Yet, it is largely so that given strong fundamental
> > principles, there are available general results, and mathematics is
> > for that those with totally different thinking, can share in truths,
> > given the principles of mathematics.
>
> Ross can convey less in more words than almost anyone else.
> --
progress, extra the standard, in the mathematics. Goedel proves this
using standard, modern mathematics: any finitely axiomatized theory
strong enough to represent natural arithmetic has true statements
about the objects of the theory that aren't provable, derivable, true
from the axioms of the theory: incompleteness. While that may be
so, it doesn't necessarily conclude that Presburger Arithmetic of
addition on integers is incomplete, as it is shown complete, and it
doesn't necessarily show that an axiomless system of natural
deduction, is incomplete. Yet, it surely gives the interested
mathematician the wherewithal to conclude that there are particular
true features of the objects of the domain of discourse, relevant to
the structure, that aren't those of our standard, modernly.
Then, there are as well considerations that traditional avenues of
inquiry, into the infinitesimal and infinite, have been closed off
from our selection of axioms with basically set theory's definition of
an inductive set or infinity, combined with the restriction of
comprehension of the axiom of regularity.
A. o. Infinity: defines a constant in the language of an infinite
inductive set
A. o. Regularity: restricts naive set-building from sets with
irregular transitive closures, a.k.a. Foundation
The other axioms reflect reasonably intuitive notions of the
composition of sets, as defined by their elements, that nothing
prevents elements from association.
Now, we know that there are various considerations these days, for
example of Aczel's reknowned anti-foundational, foundations, where the
anti-well-founded sets, as objects and members of the domain of
discourse, exist. As well, in NF and NFU, or as well NBG with
Classes, where NBG is ZFC with models, there are considerations of the
embracing of the domain of objects that would not be regular set-
theoretic sets: the class of ordinals, the class of sets, and other
structures that via their definition would not leave ZF containing
them, directly consistent. Obviously enough, this group-noun-game of
having classes for sets leaves what was the primary object, the set,
no-longer the primary object, just as type theories in sets, may find
that types are not the primary objects.
set theory <-> model theory <-> class theory
set theories <-> type theory (eg ramified, stratified) <-> category
theory <-> HOL
geometry
number theory
theory
With theory then and in number theory, a primary area of application
is that of real analysis, and continuum analysis, and here in as to
dividing the continuum, and having divided the continuum. Having
divided the continuum equally, is as to where the infinity of integers
is the continuum to the unit, as the unit is to continuum to the iota-
value, or from Newton: fluents and fluxions: this is a notion extra
the standard's theorems, but from the same principles.
So, there are traditional avenues of inquiry, into the infinite and
infinitesimal, as transcribed from antiquity, and revolutionaries like
Galileo, Wallis, Newton, Leibniz, Euler, Gauss, du Bois-Reymond,
Cantor, that are not within our modern mathematics, which as
curriculum is presented to as wide or wider an audience than ever
before. Then, where it is important for society to have mathematics
lead physics, we have seen in the course of the technological age that
where once mathematics lead physics and the capacity of experiment by
generations, technology has reached in as to where it is not the
capacity for construction of experiment, but the body of mathematics,
that lacks in the explanation of data. Our measurements reveal dark
matter, Avogadro's number grows, the farther we look the bigger it is,
the closer the smaller. We know there are truisms beyond the
classical, and the modern classical, the relativistic. There actually
are differences in things in the macro and micro scales from our
mesoscale, there actually is the anthropocentric, or simply in as to
our place in scale. Yet, there is still a reasonable expectation that
there is order in the Universe, and that then mathematics is and will
remain our best framework for physics, which otherwise would overthrow
not just dogma but science.
So, what are these truly revolutionary ideas in mathematics that will
truly reveal avenues for progression in our science? One might think
they would arise from the continued course in the foundations as we
know them of modern mathematics. Yet, in a hundred years, there isn't
a direct application resulting from that course. Measure theory is
built on the countable. The development of methods in physics is as
to the algebras, to reduce symbolic complexity and that is a right
course, and ever-more-complicated deformations of Euclidean geometry
to bend the methods to the results, and that is not clearly the right
course.
Then, the nature of the continuum is the matter of our discussion.
What is the continuum? We know that our tools of real analysis,
founded on countable additivity in measure theory, give us results
matching those of geometry and experience in the meso-scale. As well
we know there are true features of these objects of discourse not yet
resolved in our modern theory. Then, for what may be directions for
progress in continuum analysis, there is obviously the broad vista of
refinements and developments in the standard, but also there's a
reasonable consideration that the alternative avenues don't end where
we've left them, nor do they necessarily lead off to the weeds or
where the standard could not maintain its track.
So, there is found this simple construction with its plainly
reasonable features, and as well, surprising features, that the
integral of the function, EF, is one, besides that in the fundamental
results of modern mathematics, _that it isn't shown uncountable by the
cohort of results otherwise establishing uncountability of the
reals_. Then, where its features are well-modeled in what are
standard mathematics, as modeled by real functions, in real analysis,
and as having its existence inferred from the existence of a
reasonable notion of uniformity or regularity of the naturals, in ZFC
and number theory, then this gives the reasonable justification that
those abandoned directions are not without course, and not without a
reconciliation, that our edifice of modern mathematics precludes their
existence.
Then, with the other interwined, if not inseparable theoretical
studies, there are notions for set theory, and theory, that an
axiomless system of natural deduction provides a foundation for
results as applied. As well, in the course of the investigation of
these matters, simple axioms of a spiral-space-filling curve of a
natural continuum founds a geometry: from points and space before
points and lines. Then, the demands of the conscientious
mathematician of at once adhering to rigor, and our established
consistency in results, and as well acknowledging and even inviting
those results as would supercede what as modern mathematics precedes
progress, has that those results encompassing what came before, extend
the sphere of knowledge, and find room for developments that may truly
be innovative in discovery, beyond the refinement and specialization
of methods, to their completion, and placement of axiomatics in the
axiomatized.
And, EF is part of that.
So, the notion of a uniform distribution over the naturals, and the
very notion of a constant rate of change over the naturals, seems to
be built into the simple monotone progression of the naturals. One
might find that counting the integers leads to the first being counted
more, and conversely that for any there are more to be counted and
that the more there are, the more there are. While an inductive set
may truly be primitive in our theory, as well in frameworks where it
represents the meter of change, a copy or structural reflection of it,
would be high above. The numerical continuum, first as infinitely
many individua, then as integers, then as a basis for functions from
integers, then for example as reals, is a way to approach that the
numbers aren't defined, instead derived. So, the notion of a uniform
distribution follows from first principles.
Then, the structure of a uniform distribution of naturals, here for
continuous distributions and discrete distributions U_c and U_d, or
U^bar and U^dots, would first see that they are distributions. As
functions defined on the naturals: EF = U_c and 1/omega = U_d. Then
it's quite remarkable that EF, as p.d.f., is its own CDF. The
structure is for our theory of probability. Now, it is well known
that probability doesn't necessarily admit such a function, as
standard real analysis doesn't necessarily admit such a function,
though of course it's readily modeled by real functions, and other
functions with known utility are so constructed (re Dirac's unit
impulse function). Then while non-standard (and not to collide with
the statistical sense of standard as having unit variance), while non-
standard, these functions are at least modeled by the standard. And,
where a corresponding framework for structure of the continuum as non-
standard (here as super-Archimedean ordered field and as well ring of
iota-values) establishes the basis (and in the sense of the vector
basis) of the unit as the range of this function, then as well where
probability is founded on measure theory and real analysis, the
structure of a uniform distribution of the naturals is as well at
least partially evident.
Then the use of a uniform distribution of the naturals, has that it
would correspond with notions of density in the integers of number
theory, where cardinality is mute to it, and about how then
conditional and joint probabilities would be definable in terms of
this simple function, correlating expectations from number theory and
density of integers with statistical expectations. Then as well,
where there are surprising features of the functions that are each of
a continuous and a discrete distribution of the elements of the
support space, these might lead to applications and then, a use of a
uniform distribution of the naturals.
So, with a rationale and a justification for the development of these
ideas in probability, and the corresponding framework of continuum
analysis expanded with renewed investigation of avenues once blocked
for a need to reconcile with our standard, modern mathematics, then,
that is of general interest to many. And, with the fact that re-
exploration and discovery of the good mathematics that great thinkers
see in the true nature of the continuum, may well be a most fruitful
avenue for discovery of novel mathematical structures for physics,
where transfinite cardinals haven't panned out in applications and are
shoehorned into countable additivity for useful measure theory, then
the rationale and justification for the notion, structure, and use of
the uniform probability distribution, and its corresponding
mathematical framework and foundations, is a course for better
understanding of the continuum.
And EF: on the line, in the line, the line: starts that: the
continuum of the line.
Regards,
Ross Finlayson
Ross A. Finlayson
Aug 10, 2018, 2:58:22 PM8/10/18
to
or not, there are various considerations about
what these would be, vis-a-vis standard probability
theory, standard real numbers, usual continuous
and discrete probability distributions, and basically
that in standard probability theory that distributions
have unique pdf's, as here that they may not.
Ross A. Finlayson
Aug 15, 2018, 8:54:06 PM8/15/18
to
the pdf is the differentiated CDF,
then it seems like EF is also as of
under differential operators.
Usually we'd think first of the
exponential as its own derivative
and antiderivative.
"The importance of the exponential function
in mathematics and the sciences stems mainly
from its definition as the unique function which
is equal to its derivative and is equal to 1 when x = 0."
-- https://en.wikipedia.org/wiki/Exponential_function#Derivatives_and_differential_equations
Here this EF (and about its own particular bounds)
if, its derivative is itself, and it equals to 1 when x = oo,
(or rather n = oo), then there is some thinking involved
about what it means here in terms of the space of
differential operators.
In this context with "various" pdfs as about a CDF
of the integers at uniform random, then there's
also here another seemingly unique, special, and
surprising feature of this simple-enough construct
about differential operators and x = oo.
Ross A. Finlayson
May 3, 2020, 2:46:34 PM5/3/20
to
the space of differential operators makes for
that this non-standard probability theory
(after infinitesimals or here the equivalency function)
has relevant novel features in statements about
closures in the space of differential operators.
I.e. it's remarkable, the fact.
For extending singularity theory and making for
potential theory, there's lots to be said.
Basically that the (natural/unit) equivalency function
exists, the integral of the equivalency function exists,
and that the definite integral of it exists,
and is 1 not 1/2: is remarkable.
Ross A. Finlayson
Jun 30, 2022, 6:00:04 AM6/30/22
to
Ross A. Finlayson
Jun 30, 2022, 6:10:01 AM6/30/22
to
Ross A. Finlayson
Jun 30, 2022, 6:18:28 AM6/30/22
to
Why don't you get a mathmamtics degree?
Yeah, sorry, I know I all cvorrected my spelling,
fixed my words, killed mathematics, made it my own,
.
Made it my own, it's like my lawn.
Mayd it my own.
Not that you're not welcome to it, ....
Make it your own.
Ross, Finlayson, mathematics: "make it your own".
Killed mathematics!
Ross A. Finlayson
Jun 30, 2022, 6:30:55 AM6/30/22
to
Ross A. Finlayson
Jun 30, 2022, 7:03:58 AM6/30/22
to
Ross A. Finlayson
Jul 25, 2022, 8:44:33 PM7/25/22
to
Or "long live mathematics", really I think part of the point,
is, one of my favorite quotes, is that it takes mathematics.
'I think my favorite quote was "This is letters".'
Of course it's all the same in "Hilbert's Infinite Living Museum,
..., of Mathematics".
Because last we heard from them was more than 100 years ago.
Then, the point is whether they are mute, or
for that matter: come alive.
I found it easiest to bring all the mathematics I know along with me.
Then it's pretty great but under-served,
these are some of my favorite facts in mathematics.
The "continuous and discrete" is about the perfect dichotomy.
That said then to strongly alienate comment,
it's about as well served.
I'm not quite sure about having pinned all this on myself.
But, it's perfectly rational of course - "modern mathematics
lands on Finlayson". there has to be at least one person
on who could be pinned a mathematics.
I don't for example have any other source than myself
"these" facts.
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