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What is evidence, and how much of it means that a proposition is true? Does a partial / total lack of evidence mean that a proposition should be ignored?

Is the concept evidence more important to some subjects than others (for example, Mathematics versus Science)?

It seems to me that evidence is more important in Mathematics than in Science, due to the analytical nature of Mathematics and the experimental nature of Science. But this seems to me to be too much of a generalisation - is there a stronger argument for this? Would you even agree with the claim?

Also, it seems to me that History relies almost entirely on evidence - if there was no evidence, then History would surely be shaped by psychology. The language of the evidence must surely influence the way in which the evidence is interpreted in History, unlike in Mathematics, where there is a strict, 'emotionless' language (full of definitions). Would you agree?

What definition of the concept of 'evidence' would encompass more than one subject area?

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As a Bayesian, evidence for a truth-statement x can be thought of as any observation y (also a truth-statement) such that p(x|y) < p(x|~y) (where '~' means not, p means probability-of, and | means given-that-we-observed). I think this covers every use of the term evidence, though in cases where you cannot conveniently calculate probabilities it's more of an analogy than an exact definition. Note also that evidence can be good or poor (depending on how great the difference is), and you can be confused about what evidence actually means if you do your calculation wrong or have insufficient data to perform a good calculation.

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Could you possibly provide some Science and History examples of such calculations? At the moment, I see that your definition incorporates the notion of probability to reflect an unpredictable world, but I cannot quite see how your definition works in practice. – James Dec 8 '12 at 17:54
@James - It mostly works as an analogy in practice, as I mentioned. You can make up examples: I see the number 4, and I know it was generated by the roll of a die. This makes me favor that the die has six sides (p(4|d6) = 1/6) rather than 100 (p(4|d100) = 1/100), if those are the only two possible options. But it's very difficult to know how to quantify probabilities in most situations; one can still use the same intuition, however. Explaining all the bits of intuition that one might gain goes beyond the scope of this answer, I think. (Learning Bayesian probability is more useful.) – Rex Kerr Dec 8 '12 at 18:54
Could you, perhaps, offer a more practical approach to the definition of 'evidence'? For example, what is 'evidence' in Mathematics? – James Dec 8 '12 at 20:38
@James - Mathematics has proofs, not evidence. – Rex Kerr Dec 8 '12 at 20:43
Are you making a distinction between mathematical reasoning, and using mathematical definitions as evidence? Are you saying that there could NEVER be evidence in Mathematics? What if I randomly find you an example of an even number above 1,000? Is that not evidence? – James Dec 8 '12 at 20:54
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Rex Kerr's assessment is correct. Evidence is a piece of information that increases or decreases confidence in a determination.

The logic in mathematics has the same basis as the logic in philosophy (though it comes across perhaps as far more elegant and useful). For discussion, you can proceed from the law of identity. If we know that a = a, then we can make some basic truth-statements. If a = a and b = b, a cannot equal b. If we can make truth-statements, we can assess our confidence that such a statement is true. Confidence is, in all fields, inextricably connected with evidence.

The difference between philosophy and mathematics is that mathematics never, in any way, departs from that language. It could never disregard the law of identity. If I have four objects and take away two, two remain. Two remain because two remain, and there is no amount of evidence (nor any evidence) that can be brought to bear to show that more than two remain. It is because of this that mathematics is capable of making proofs. Working with quantities, whether real or virtual, will invariably result in the same answers to our questions.

These are not man-made rules, but tautologies. A must be A, or A would not exist and posing the question of some object being itself would never have been made.

A bit of a digression, but it shows that Bayesian probability has a strong basis in logical thought.

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You might want to keep an eye on other contemporary developments of the notion of evidence. For example, according to Tim Williamson (see his Knowledge and its limits), the evidence a person has consists of everything that person knows.

This is a thouroughly non-Bayesian understanding of evidence.

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I'm not familiar with Williamson specifically, but basically every apparently non-Bayesian method I've seen either reduces to Bayesian (though the author may not realize it), is qualitatively similar but quantitatively flawed (which humans are in many psychophysical tests), or just plain doesn't work (either logically/mathematically flawed, or is so unlike what we normally call evidence as to not really be the same thing). Is Williamson none of these? (Bayesian inference in principle ought to be applied to all knowledge that you have.) – Rex Kerr Dec 11 '12 at 20:36
The idea is that the right way to think of evidence is not as building blocks on the way to knowledge, but as the end-product of the process -- knowledge itself. Williamson's theory is not about the dynamics of belief update, for example, but designed to solve certain problems with externalist epistemologies such as his. I thought I'd point to his work in an answer, as he is a very important contemporary epistemologist with non-standard views on evidence; it is useful to show that "evidence" is a term of art, and might be used differently by different theorists. – Schiphol Dec 12 '12 at 21:25
Interesting, thanks! Sounds like it's worth a look if I get the time. – Rex Kerr Dec 12 '12 at 21:31

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