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As we accumulate more evidence to support a given hypothesis we have increasing confidence that the hypothesis is 'correct'.

How do we justify this?

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Statistical hypothesis testing is referring to making decisions based on data. This certainly does not constitute any kind of mathematical proof. Hypothesis testing is simply a statistical procedure for testing whether chance is a plausible explanation of an experimental finding. – mjsa Feb 1 '13 at 22:42
Do you ask about statistical inference or the classical "problem of induction"? Before you rephrase you question, search SE.Philosophy about induction. – Ricardo Feb 2 '13 at 10:46

It's sort of circular, as you would define "evidence that supports a hypothesis" as "things which would increase our confidence in the hypothesis."

To elaborate a bit:

  • You could see for example the Sunrise problem. Laplace's Succession rule shows that the more times you see the sun rise in the past, the greater confidence you have that it will rise in the future.
  • More generally, you can apply Bayes' Theorem to update any hypothesis you have. If you know the probability of evidence occurring given your hypothesis, then your posterior belief in the hypothesis' correctness is directly computable from this and your priors
  • VC theory proves that "simpler" hypotheses are more likely to be correct. Postulating a world in which evidence coheres with "universal laws" is (almost?) always simpler.
  • The Law of Large Numbers indicates that repeated experiments will tend towards truth. This is discussed somewhat more here.
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Thanks. I guess I am seeking a "deeper" reason. The theories above describe what we see, but why is the universe that way? Or am I asking a meaningless question? – Schneider Feb 2 '13 at 2:17
Why is the universe what way? – Xodarap Feb 2 '13 at 13:54
Essentially, yes. Is it the fact we think the Laws of Nature are constant in the past and in the future, that allows us to infer from evidence? – Schneider Feb 4 '13 at 16:52
It's always possible to beg the question and define the future to be independent of the past, if that's what you're asking. Whether that's a good idea is another question. – Xodarap Feb 4 '13 at 19:38
I'm asking what fundamental ideas explain why well-evidenced theories allow us to predict the future. I think I have stumbled upon the answers which is the universality & stability of physical laws, combined with cause & effect. – Schneider Feb 4 '13 at 20:18

We justify it by calling it the "scientific method" which as been around since the western age of modernity and is the basis of all modern science. We propose a hypothesis based upon a hunch, a reason, or observation. We then look for additional 'evidence' or facts to justify our hypothesis. If we can predict a future chain of events based on our accumulated evidence we call it a theory. Given enough evidence we may call it a law. Law is a mental shorthand to explain a series of phenomena; but law as an entity does not exist. We use the word to express the regular succession of certain occurrences in the phenomenal world - for example, the Law of Gravitation.

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If I'm understanding the question correctly, the answer is mainly statistical in nature, perhaps under-girded by a few fundamental presuppositions involving the universe being constrained by logic, assumptions of key self-evident mathematical laws, the constancy of natural laws, etc. These, in turn, are based, in part, on our own observations and those of our fellow humans over both our lives and recorded history, causing us to believe they are true based on the aforementioned core statistical response (I'm running primarily under an empirical lens here...).

Primarily, this is a statistical and multi-variable math problem. There are two major questions we want to answer while testing a hypothesis with properly formulated scientific tests (repeatable, falsifiable, etc.)

  1. Can anyone repeatably get the same results if the test inputs are the same
  2. As I vary the inputs, how well have I defined the transfer function to the outputs.

As for #1, the more people that are able independently validate the experimental results under varying conditions, the more confident we become that we have accounted for all the variables involved. Now granted, there may be limits to this, such as having to test it on Earth, at Earth's current location in the universe, for example. This confidence, in it's purest form, is directly related to statistical sampling theory, which tells us that our confidence in the result is based on both the overall accuracy and precision of the results, and the total number of samples. This would include also running gage tests to determine the accuracy/resolution of testing equipment and other similar methods

As for #2, let's say I have a hypothesis that humans always wear pants outdoors, but I only ever run the experiment at 40 deg F. And I conclude, no matter how many times I run the experiement, that they do, indeed, wear long pants. In this case I'm only sampling 1 point of the transfer function between the input of ambient temperature and the output of what people wear. Obviously, I would need to sample more than one input temperature to fully define the transfer function, and the more points that I gather, again based on statistical law regarding the spacing of the input points and the shape of the transfer function, I get higher and higher confidence that I have "fully defined" the transfer function and that future experiments are unlikely to further refine it.

Of course, the jump from Newtonian physics to Relativity is a good example of where we believed a hypothesis/theory/law had been fully defined in Newtonian physics, but Relativity better refined and expanded the transfer functions involved.

There are other ways of increasing confidence in evidence other than repeatability and increasing the number of samples and refining the resolution of the transfer functions, but I believe that covers the bulk of it from a confidence perspective.

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