Infinitesimal Probabilities

[Ross A. Finlayson]

Wenmackers writes an interesting article about the
history and modern developments of probability theory
with regards to usual modern logical and number theories.

https://philpapers.org/archive/WENIP.pdf

Continuity and infinitesimals for an infinite space
have varieties of infinities, infinitesimals, and
continuity (eg field continuity, line continuity,
signal continuity), about what can become modern
theories of measure theory, much about Vitali and
Hausdorff, vis-a-vis, Banach and Tarski.

Pruss writes an interesting article

http://alexanderpruss.com/papers/InfinitesimalsAreTooSmall.pdf

with some consideration of a probability distribution
over the naturals. As noted here before, "sweep" or
the "natural/unit equivalency function" is among objects
of mathematics that have surprising unique properties
for more than one function defining a distribution,
about superclassical probability.

Hyperreals after Robinson are generally alluded to
vis-a-vis the long line of du Bois-Reymond or other
such notions of extending the "complete" ordered field
yet only occupying the same space - such notions as
iota values (or, "constant" infinitesimals after Peano)
should see a revitalized (and, re-Vitali-ized) role,
in neat usual considerations of the objects of mathematics.

Mathematicians and philosophers these days are finding anew
these considerations of the mental formation of these objects,
if mostly from a privileged view from Cantor's "Paradise"
(or, "Hilbert's Cantor's Paradise"), quite fundamentally.

[Mitch Raemsch]

If there is unlimited probability does anything repeat
exactly the same. In QM that is a No.

Mitchell Raemsch

[Ross A. Finlayson]

"Extending the proposed super classical quantum mechanics
to the microscopic domain will at best require a similar framework
to the 'hidden variable' interpretations, which have all the
well-known problems arising from Bell's inequalities."

-- https://aapt.scitation.org/doi/10.1119/1.1424268

"This phase plays such a crucial role due to
the uncertainty principle..." [which is an artifact
of quantization of the continuous].

"This fact forces is to consider electrons or photons
as entities possessing both wave and particle properties.
It is well known that if we settle for _less_ precision,
we can find a limit where we recover classical mechanics."

"The central role played by h in establishing a frontier
between classical and quantum domains is ignored by
Lamb, who implies that h (or hbar) is no more than
a constant for obtaining dimensional consistency."


Mathematics _owes_ physics: mathematics and probability
of the infinitesimal and infinite.

[Mitch Raemsch]

Infinite randomness would be non repeating.

[Ross A. Finlayson]

No, flip a coin infinitely many times.

Notice it builds a real number's,
or rather, "a real number between zero and one"'s,
initial segment, the 0's and 1's of the
fair coin tosses (Bernoulli trials, samples).

Notice that with memory, each sample
not only refines the previous sample,
it starts a new one.

So, a rational (repeating) value,
would infinitely sample the same value,
as so many times as there are numbers,
as that the irrational would get so many
infinitely many different samples, instead.

This is a usual comfort to the thinker that
the rationals and irrationals are each dense
in the reals, then for meeting expectations
(due the character of numbers and most simple
processes of the statistical), besides that
for example the irrationals of the complete
ordered field are uncountable, or: that the
probability that the sample is rational
only goes to zero as the samples go to infinity.

(And the probability that it isn't doesn't arrive.)

These days about Ramsey theory, and, measure theory,
and, Banach-Tarski, and Borel, and combinatorics,
there are more interesting theories about all the
character of the real numbers than rational approximations.

Descriptive set theory leaves topology as a neat usual
toggle, here about Cartesian functions vis-a-vis
the range itself, under mappings, as quite simple.

Parastatistics is about suprising results or
counterexamples in statistics (via probability
theory). These for example are part and parcel
of usual statistical ensembles in super-classical
physics.

[Mitch Raemsch]

The problem with that is that it can't done.
Who is going to flip the coin and where does it end?
IF the opportunities to be different are unlimited
you can have infinite randomness.

In quantum mechanics order never repeats exact...
in an unlimited way

[Basil Jet]

On 07/12/2019 22:48, Mitch Raemsch wrote:
>
> Infinite randomness would be non repeating.

The probability of it repeating would be zero, but if you could
guarantee it wouldn't repeat it wouldn't be random.

--
Basil Jet recently enjoyed listening to
Jon Hassell - 1987 - The Surgeon Of The Nightsky Restores Dead Things By
The Power Of Sound

[Transfinite Numbers]

Did you try looking at P(X =< c) instead of
P(X == c) for a stochastic variable?

https://en.wikipedia.org/wiki/Cumulative_distribution_function

About "Parastatistics", I don't know whether it
is related to paraconsistent logics.

In paraconsistent logics there is no:

~(p & ~p)

But without knowing what is "Parastatistics", we
could maybe develop a probability based

on paraconsistent logics.

[Transfinite Numbers]

Ok, somebody did already:

PARACONSISTENT PROBABILITY THEORY AND PARACONSISTENT BAYESIANISM
Edwin D. MARES - Logique & Analyse 160 (1997), 375-384
http://virthost.vub.ac.be/lnaweb/ojs/index.php/LogiqueEtAnalyse/article/view/1437/1212

[Ross A. Finlayson]

We learned about Bayes (expectations and priors) after
classical probability, then also about Jeffreys, in
courses in statistics and probability. The usual notions
of the probability density and cumulative distribution functions,
and moment generating functions, as identifying distributions,
were presented with the results as about the uniqueness of a
pdf and a CDF and an MGF for a given distribution. This is much
then about the Central Limit Theorem, and then for results in
the standardization about z usually besides such usual
notions of the various statistics as so computed from sample
data, then as about estimators and usually with the goal of
finding the unbiased maximum likelihood estimators.
(A usual notion of model fitting and over families of distributions
by their scale and shape parameters finding relevant distributions
as match data wasn't much discussed in these usual courses
in statistics and probability theory.) Things like least squares,
then about computation of variance and mean for normal
distributions, then ANOVA/ANCOVA/MANCOVA, Student's
and Fisher's and t-tests, these are usual introductory applications
of statistics.


The parastatistics are about the Bose-Einstein and Fermi-Dirac
statistics, then about more parastatistics, as what in various
statistical ensembles of particle physics, where QM is probabilistic,
these are methods as what seem to add up.

https://en.wikipedia.org/wiki/Parastatistics

Paraconsistency in statistical hypotheses would seem to involve
mostly being built into the hypothesis itself, though it's mostly
where in statistics "it isn't not so", or, "it isn't so", not, "it is so".
Simply building in parastatistics as a reasonable notion of
"it could be wrong" seems a bit removed from some theory
with "ex falso nihilum" not "ex falso quodlibet", then besides
where there are simply third alternatives (past incompleteness).

Such notions of infinitesimal probability instead of a sum of
measure zero events being impossible, has in a usual consideration
about the natural integers at uniform random as considered in
one of the essays above: that surprisingly there is more than
one pdf about this distribution.

https://groups.google.com/d/topic/sci.math/8B8hPWpDooo/discussion

That there's a similar arrangement for U [0,1] (the distribution)
may be interesting.

[Transfinite Numbers]

Not to forget the herpes-gibberish-moron
I can copy paste buzzwords distribution.

[Transfinite Numbers]

So Ross A. Finlayson comes out of his dybbuk box,
once a while like santa clause vis-a-vis the

ekpyrotic universe, and sings us the song of his
thousend googled herpes blisters is and is not.

[Ross A. Finlayson]

It might help the reader to decode some of Burse's usage,
Burse's box is a reference to some comments about his style
of argument, Burse's blisters is about the mathematical
infinitesimal: as Cantor referred to them as cholera bacilli,
back in the times when the flu was a regular generational plague,
Burse files my mention of infinitesimals under a virus, while
Burse's amoebae are protozoans about the reals being larger
than the standard reals, a term he coined one day.

"Santa Clause" might be seen as generous,
while the ekpyrotic (vis-a-vis, the ergodic) universe,

https://en.wikipedia.org/wiki/Ekpyrotic_universe

proposes at least an infinite past and future for the universe.
(Or, time goes back forever and space goes on forever.)

I would infer as a reference to how in physics I noted that
the universe's origins are both, at once, Steady State and
Big Bang. (Or, maybe it's a reference to something else).
I.e., a generous and even in-group's reading might find
meaningful usage of the terms, that are otherwise disjoint
or wrong.

That might help the reader, it might not. It might help the
reader at least to point out that Burse is sometimes a
silly person and sometimes informal. (Or, "Burse might
be a genius: there are limits.")

The infinitesimal in probabilities usually refers to a notion
that some events have probability zero because they are
not events (statistical events), others from continuous distributions
have measure zero (but are not empty), where "almost everywhere"
is used to refer to measure one, as the complement of the set of
measure zero events has, in the set of possible events. The point
is that they are not impossible, these events, but without some
means of summing together all the probabilities of the events
and finding that they equal one, the whole of them, the usual
notions that the area or integral of a p.d.f. equals one wouldn't
be satisfied. (We know real analysis uses countable additivity.)

So, I noted that the sweep function a.k.a. EF the natural/unit
equivalency function, f(n/d) = n/d, n->d, d->oo, which is very
simple and very simply modeled by real functions, via inspection
has all these relevant properties to do with functions that
describe distributions as not-a-real-function, has "real infinitesimals".

(Newton and Leibniz and many others would point at this function
and say "ah, that's my function, too". That it's integrable and
has area one instead of 1/2 I found this.)


The domain of this function is the natural integers,
the range of this function is [0,1],
the range is a continuous domain,
and this function in the setting of uncountability results
does _not_ find the resulting model of a continuum uncountable,
uniquely among functions. (Or, "Cantor proves the line is drawn.")

(It was also noted in the referenced thread that like the exponential
function which falls under differential operators, another peculiar
unique feature of this continuity model is that it's its own
derivative and anti-derivative. This might be a fact only relevant
to those familiar with the systems of differential equations.)

Then, the referenced papers as work to formalize notions of
the infinitesimal, here particularly for probability theory,
have that more-or-less they either eventually point at this
prototype of a line-drawing continuous domain, or: don't.
I.e., toward completeness, and a definition of "line continuity"
and another of "signal continuity" besides the day's "Dedekind's
field continuity", which is the only one offered by the usual
curriculum, the Equivalency Function is a central object in mathematics.

[Steve Ellis]

Hello moron!

It took me about 15 seconds to find the first falsehood at that link.

It is the very first sentence:

"Suppose that a dart is thrown, using the unit interval as a target;
then what is the probability of hitting a point?
Clearly this probability cannot be a positive real number."

Actually, the probability of hitting a point is
the positive real number known as 1.0, or unity.

It is certain that the dart will hit a point.

I hope this is clear now.

You're welcome!

[Ross A. Finlayson]

It's fair to be strict and formal,
clearly the generous reading would be
"what is the probability of hitting a [given] point",
as clearly otherwise it would be trivial.

The infinitesimal with real character
would clearly be small enough, as that
only summed all together are they non-zero.
(I.e., these infinitesimals are not too small
as to have no real character.)

We know points have no width but there's a
notion they have sides (on the line, or,
in the line: assigning a technical difference
about "on" and "in"), the field's point is
two-sided (with limits above and below) while
the line (or, ray's) point is only one-sided
(from before).


The hyper-reals are only a _conservative_,
meaning offering no new properties, extension
of the standard real numbers (as formalized by
the complete ordered field or after Cauchy,
Weierstrass, Dedekind, augmenting the ordered
field of rationals with LUB and measure 1.0).

There are other models of infinitesimals than hyper-reals,
or for example sur-reals, and indeed historically
most models of infinitesimals are "constant" or
"standard" infinitesimals (eg after Peano) as what
were forming together the integer, besides just
vanishing quantities.

So, it couldn't much be expected to find results in
hyper-reals not accessible to results in standard reals.
(See section 3.2 of Pruss' paper where that's so noted.)

Instead, some notion of "iota-values" defines continuity
for itself, and formally and strictly enough - it's a
_very_ usual course in the development that the modern
standard ignores.


It's like the idea of adding a duck-typing theorem
to formal systems with class and set comprehensions
separately as so they're the same, about what it
would then be required in disambiguating different models
of continuity as to maintain a consistency.

[Ross A. Finlayson]

("Parameterization" is about free variables,
"Parametrization" is about extensionality of measure.)

[Ross A. Finlayson]

On Sunday, December 8, 2019 at 2:44:06 PM UTC-8, Ross A. Finlayson wrote:

It's similar to the one-sidedness and two-sidedness of
the points, the different sigma algebras that define
the measure, that this line continuity has examples from
either side, while the field continuity is the nesting.

I.e., this line continuity has its own sigma algebra
as defines "measure 1.0" for measure theory.

There's not a transfer principle between these two
different models of continuity so much as a "rather
restricted transfer principle" as about the extensionality
of the continuous domains between the integers, eg
the "non-integer" part from the field or the "post-integer"
part from the line.

A formalism of real character of infinitesimals would
demand this usual apparatus.