Our civilization depends on our ability to predict results of actions, which we do on the basis of our knowledge of how the universe works. It follows that it is important to understand what we can know, and how we can know that we know it.
I will use the following terms in this discussion:
A "thesis" is a proposition put forward for discussion. It may be true, or false, or its truth value may be undetermined or even indeterminable.
A "hypothesis" is a thesis for which there exists some known means for investigating whether it is true, or (more importantly, as we shall see later), whether it is false.
A "theory" is a hypothesis which has been tested and found to be valid over some domain of situations. (This is the scientific sense of the word.)
A theory may be said to be "persuasively demonstrated" if the domain of situations in which it has been found to be valid is "large". (How big "large" must be is something of a judgment call.)
A theory may be said to be "conclusively demonstrated" if it is persuasively demonstrated, and its domain of validity extends to domains to which it did not obviously apply when initially propounded. In other words, it predicted the existence of a new effect which was subsequently found.
A theory may be said to be "provably correct" if it is based on valid premises and the logical steps by which it is deduced from those premises are also valid. (Most scientific theories are NOT provably correct.)
A "theorem" is a theory which is provably correct. The term is more often used with mathematical or logical deductions than with science, since most scientific theories are not provably correct.
A note: Much of this material has been taken from previous posts in "Belief in the Christian god is absurd" and elsewhere. Extensive editing has been done to make the material flow more logically, further explain some concepts, and fix a minor error.
The first real description of the scientific method is attributed to Francis Bacon, Lord Verulam (1561-1626), who discussed it in his "Novum Organum" (1620). The description of a scientific statement as a provisional statement which is tentatively assumed to apply everywhere is an empirical approach, used widely because it works well. No attempt to provide any sort of theoretical basis for this approach had been made until Popper's refutability thesis, some three centuries later. But scientists didn't particularly care; the empirical approach made sense and worked, and attempting to give it a theoretical basis simply wasn't interesting. Nor was a theoretical basis even possible until recently: the mathematical tools needed to do so (information theory, set theory, and predicate calculus) simply did not exist.
Popper's thesis is that the touchstone of science is refutability: a thesis cannot convey information (in the Shannon sense) unless there exists a way to show that it is false. Popper argued vigorously for this [1, 2], but did not offer a proof that it is valid. However, it IS provable, and we shall presently examine the matter here.
It is true, of course, that people doing real science don't much care about any of this. The empirical approach has worked well for centuries, and we have never had a reason to suppose that it might not continue to do so.
Science uses both inductive and deductive logic. Inductive logic consists of conclusions, from sets of observations apparently having common properties, that there exist underlying rules which explain the supposed common properties. Since a posited property may not in fact exist, inductive conclusions are necessarily tentative. Deductive logic consists of conclusions based on prescribed premises, and conclusions drawn from these by rules of inference. If the premises are valid, and the rules correctly applied, the conclusion will be valid.
A scientific theory arises as follows:
- One constructs a thesis, possibly as an inference from observations (inductive logic), or possibly on the basis of a "What if" question, or often on the observation of a peculiar result for which the cause is not obvious.
- One attempts to devise a test of the thesis by deducing the expected results of applying the thesis to some situation. Such test must be able to show that the thesis is correct, in at least some situations, or that it is NOT correct. (As we shall see, this latter is crucial.) If no such test can be made, the thesis must be abandoned as mere speculation.
- Tests are carried out and the results examined. Whether the thesis is confirmed or refuted, our knowledge of the universe has grown.
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