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I am interested in the following epistemological problem.

Usually, Probability is seen as a mathematical theory that we use to describe the physical world. Precisely, following the Scientific Method we have observed phenomena which are "random" (like the tossing of a coin), and we have axiomatized their behaviour via measure spaces, measures, sigma-algebras... and other mathematical concepts (see Kolmogorov's axioms of Probability). In this way, Probability is not different from any other physical theory, like Electromagnetism, Relativity, Quantum Mechanics... etc. So we can say that Probability is a product of the Scientific Method.

(Of course, Probability is also an interesting mathematical theory by itself - again like Electromagnetism, Relativity, Quantum Mechanics... - but that is another story which does not concern my question.)

However, it seems to me that most of the Scientific Method is actually rooted in Probability! Indeed, when we test any physical theory, we are somehow discarding the possibility that really improbable coincidences can happen - and how can we justify that if not by Probability? I try to make it clear by an example: Let say that we want to test the photoelettric effect. Then, without going into details, we set up a device and we check if electrons are emitted in a certain way when the light hits a material. In principle, it could be that there is no causal relation between turning on of the light and the emission of the electrons, in fact this is (part of) what are we testing. So it is possible that the electrons are emitted when we turn on the light just by a pure coincidence, even if we repeat the experiment one million times. In order to exclude such an amazing coincide, it seems to me that we implicitly relies on Probability.

So is Probability a product of the Scientific Method, or is a part of it?

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    "In this way, Probability is not different from any other physical theory" I don't think that this statement is true and I think that you might need to rethink this topic. Mechanics (Newtonian, quantum) use probability, but that doesn't mean that probability is a physical theory in the same way that the ones you listed are. Mechanics also use analysis and algebra, but that doesn't mean that either of those are "physical theories". I think that this confusion might be the impetus behind your question. – Not_Here Jun 23 '17 at 22:02
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    Probability theory is a branch of mathematics that is applied to physical theories, just like analysis and algebra are, and it is not itself a physical theory. You can talk about probability distributions as purely abstract mathematical objects, you don't need to invoke physics or chemistry at all. Probability is used by scientists but it itself is not a physical theory in the same way that electromagnetism is. – Not_Here Jun 23 '17 at 22:03
  • @Not_Here OK, I got what you mean, and somehow I addressed what you wrote in the paragraph "Of course, Probability is also an interesting mathematical theory...". Maybe I went to far saying that "Probability is a physical theory...", anyway the problem if Probability is implicitly used in the Scientific Method still stand. – Ron Artigan Jun 24 '17 at 8:27
  • probable came before the scientific method, probability came after. – Mozibur Ullah Jun 25 '17 at 0:49
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    probability is a mathematical device - or method - for explaining observations in empirical reality. It is not comparable to the scientific method. – Swami Vishwananda Jun 27 '17 at 7:55
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You say "probability is seen as a mathematical theory that we use to describe the physical world". I would say rather that probability and statistics are tools we use to help us test scientific hypotheses. To understand this let's go back a few steps and ask why we use probabilities.

Philosophers commonly distinguish between physical (or metaphysical) possibilities and epistemic possibilities. The first are concerned with what events happen or might happen and these are directly related to properties of the world. The latter are concerned with what we might reasonably believe to be true given our current knowledge. The dual of physical possibility is determinism; the dual of epistemic possibility is certainty. Probability can be thought of as a quantitative measure on possibility - a degree of possibility. Physical probability might then be thought of as the amount of possibility of some event happening, while epistemic probability represents the degree of credibility of some proposition given our available information.

To see that these are quite different, recall that probability theory was invented in the 17th century at a time when deterministic newtonian mechanics was believed to hold true of the whole universe. The fact that there are no physical possibilities in newtonian mechanics does not preclude us describing our state of information probabilistically. People playing a game of 'chance' can still talk of probabilities. We can even coherently bet on events in the past: for example, if there was a football match last week and neither you nor I know the result, we could have a bet on the outcome, even though this outcome is already physically determined. The odds that we agree upon reflect our assessment of the epistemic probability of the outcome given our current information.

If there are genuinely indeterministic 'stochastic' processes in the physical universe then there are physical possibilities as well as epistemic ones. Quantum mechanics is often understood to be indeterministic, because of the way the Born rule interprets the wave function probabilistically, though it is worth pointing out that some physicists do not accept the Born rule and consider QM to be a deterministic description of a multiverse. But whether or not there are physical possibilities, there are always epistemic possibilities: there are always propositions of which we are nearly certain, others less so, and others hardly at all, depending on what information is available to us.

In ordinary everyday usage we use epistemic probabilities all the time, though usually without much rigour. We may judge that it is highly probable that the defendent is guilty given that his fingerprints were on the murder weapon, the victim's blood is on his shirt, a security camera shows him fleeing the scene, and witnesses heard him threatening to kill the victim. We may judge it highly improbable that Johnny really did his homework considering that he has used the "a dog ate it" excuse many times, his clothes show evidence of him playing football, and there are no dogs in the neighbourhood.

The harder question to answer is, in the context of scientific method, is it helpful to use the concept of epistemic probability to describe how credible some scientific statements are versus others? Here we get disagreement. At one extreme we have Popper who in "Logic of Scientific Discovery" maintains that we cannot speak of the probability of a theory or hypothesis being correct - that a theory would always have probability zero, though in his later writings he allows that we can reason probabilistically about hypotheses, but such reasoning is still deductive and not inductive. At the other end we have Edwin Jaynes who in his book "Probability Theory: the Logic of Science" argues that we can understand scientific inference as an application of Bayesian probability theory. I suspect most scientific practice lies somewhere in between. We use probabilitistic and statistical methods to help test hypotheses, but we do not usually speak of the probability of a theory being true.

Your question, "is probability a product of the scientific method, or is it a part of it?" suggests a false dichotomy. Probability is a tool that has evolved to help us address problems in scientific inference. As such it is both the product of scientific progress and a part of it.

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To answer the questions, a distinction between Probability (a natural phenomena), and Probability Theory (a human invention/discovery), is necessary.
As a natural phenomenon, Probability comes/existed way before the Scientific Method (before humans existed).
The Probability Theory, on the other hand, is a "recent" human invention/discovery which helps humans understand probabilistic processes, in conjunction with the scientific method. Therefore, Probability Theory is a part/tool of the Scientific Method.

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The 'Scientific Method' does not exist. How science is done varies across disciplines, across history, and across continents. But processes with specific aspects hold the ability to convince humans.

One of those aspects is repeatability, and another is the ability to predict things that would otherwise seem unlikely. Earlier theories of science like Bacon's emphasized the former. A more modern approach like Popper's realizes that repeatability itself is an unlikely event, and puts all of the explanatory power of science in the hands of probability theory. Fischer (Student) had already formalized this into the mechanics of hypothesis testing.

But later philosophers of science disagree, starting with Kuhn. They point out that a theory has to have an accessible mental model and an intuitive appeal in order to be spread and used. The ultimate skeptic here is Feyerabend, who points out that we can go back to the fact that science has always been about convincing people and say that is all we need -- a transferrable mental model with an intuitive appeal and some other psychological aspect that makes it seem reliable.

On the other hand, probability theory is mathematics. Something is not true in math because it happens to be repeatable or to predict reality. It is true because of its internal structure and the power of its basic concepts. Mathematical theories are meant to be assembled out of 'pure' thoughts that we cannot reject fully once we have them. From a neo-Intutionist point of view, mathematics itself is the science of exploring the ability of our shared intuitions to combine properly. It is the oldest branch of psychology, in that it studies ideas themselves. It tries to determine which sets of ideas work together and do not lose cohesion in combination.

In that way, in the way that what mathematics produces is not results but ways of combining thoughts to get new results, probability theory is a scientific fact. But the theory itself is a product of the mind, validated by the mathematical process, it is not a part of reality, it is a way of thinking. We trust it because it arises out of a process that we developed to consistently give us new ways of thinking that we can use well.

So even to the degree probability theory is a part of science, it comes well along in the history of science. And even though it is part of the predictive mechanisms that we currently use to validate scientific thought, it has never been part of the process itself until very recently, and it may well not be a necessary part.

Neither side of this complex insight seems to hold water.

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