The purity of probability?

[cianosheehan]

Is probability as "pure" as we think? Some interesting pages that suggest there are more things going on;

http://www.datadiwan.de/SciMedNet/library/articlesN71+/N71time_Dressler.htm

http://paranormal.se/topic/pk-experiment_med_djur.html

http://demo.cs.brandeis.edu/pr/ee/summary.html

[gamble4pro]

You have to define "pure" for a precise answer.
Probability is just a measurement. It measures events, but it not provides any certainty about them. It has his own rigorousness, because is mathematically defined, but also its own relativity. Different information taken into account (i.e. different probability fields) generates different probability results.
The concept of probability also involves a lot of philosophy. It leads to the concept of "possibility", which is a zero degree phylosophical category.

[Lexander]

It should say something about probability theory that generally everything is developed from the Kolmogorov Axioms (which say nothing about how you actually determine the value of P(A)). Even then, there are debates over whether the Axiom of Countable Additivity should be replaced by the Axiom of Finite Additivity. From where I am standing, even if the math is nice and clean, the application gets messy.

[PairTheBoard]

I mentioned this point in the Sklansky Philosophy Thread in which he asks if God could find 3 integers such that the sum of the cubes of two of them equals the cube of the third - something "Mathematics" has proved impossible. As Lexander and Gamble allude to above, Mathematics is not a homogenous science. There are different versions of mathematics with results in one version not accepted in others. Goodel's Theorum showed that Mathematics will never be "complete". There will always be additional axioms possible which allow consistent mathematics to be done with expanded results.

Probability is just a special field of Mathematics. Some say it's not even necessary, that it's just part of Real Analysis with it's own specialized vocabulary. Mathematics combined with Scientific Models can provide great tools for predicting experimental results. But as far as telling us what reality really IS it amounts to little more than a long poem with a specialized structure.

PairTheBoard

[cianosheehan]

Interesting posts guys,
I dont know much about maths, science and probability, but I find this subject very very interesting. Do you hold the opinion that "mathematics" is a way of explaining something that already is, rather than that which is and that which has come from what has been, is a way of representing what "mathematics" is? Or are the two equal?

[PairTheBoard]

I don't understand your question.

PairTheBoard

[Wozza]

I think that Cian is asking a chicken and egg question. Is mathematics our best effort of describing reality or is reality a result of inate mathematical functions?

Another way of asking the question would be to ask if mathematics existed before anyting came into existence.

[Siegmund]

Those links don't seem to question the "purity of probability" - but point out situations where the standard set of assumptions that most the common probabilistic tools are based on might not apply.

Closely related is your other question, about whether mathematics Just Is, or is a way of explaining what we see. I am solidly of the view that mathematical structures have always been and will always be, independent of physical reality and whether any sentient beings are around to think about it. Almost nothing mathematical is stated in the form "this is true"; it is all carefully arranged: "if these first few facts are true, then so are these others." People are sloppy about listing their assumptions. And especially in probability and statistics, people are sloppy about making sure the assumptions are satisfied before they apply the tools to their data.

Re the Sklanksy question about whether God can find 3 integers such than x&3+y^3=z^3, that's as pointless a question as the old "can god create an immovable object?" question. No, not without changing the meaning of integer, +, ^, =, or 3, he can't; just as anything that doesn't move when a force is applied to it is something other than what we mean by 'object'.

If you allow tampering with the assumptions, it becomes easy to do, of course: for instance, assume that 0=12 (that is, define "=" to mean "congruent modulo 12" and 1^3+11^3=2^3+10^3=3^3+9^3=4^3+8^3=5&3+7^3=0^3.

All sorts of fun apparent paradoxes can arise if you slip a distribution whose moments are undefinded or a set that's not measurable into the standard probability tools. Says nothing about the tools' validity, only about how careful the user is.