## How could the concept of 'evidence' be defined, and how significant is it?

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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.

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 – 2012-12-08T17:54:22.877

@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 – 2012-12-08T18:54:48.960

Could you, perhaps, offer a more practical approach to the definition of 'evidence'? For example, what is 'evidence' in Mathematics? – James – 2012-12-08T20:38:05.497

@James - Mathematics has proofs, not evidence. – Rex Kerr – 2012-12-08T20:43:55.310

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 – 2012-12-08T20:54:15.227

@James - Well, evidence in mathematics is just probabilities that something is the case. I really don't understand what you're not understanding. Are you familiar with how to calculate probabilities or not? I'm not even sure where to begin. – Rex Kerr – 2012-12-08T21:53:27.680

My point is the following: in Science, if I find an example of a metal that is liquid at room temperature and show that it is a metal and that it is a liquid at room temperature, that is empirical evidence for the fact that there must be at least one metal that is liquid at room temperature. Could the same understanding of 'evidence' be applied to Mathematics? What if I randomly find you an example of an even number above 1,000? Is that not empirical evidence of the existence of even numbers above 1,000? – James – 2012-12-08T22:05:02.807

@James - That is called an existence proof (a constructive one) or a counterexample, depending on how you approached it. "Evidence" may prompt mathematicians to search for a proof or probabilistic argument, but they don't use the term the way you have here (that I have ever seen). – Rex Kerr – 2012-12-08T22:19:48.073

Three points: 1.Could you give me an example of a probabilistic argument used as evidence in a simple scenario? 2. How is a probabilistic argument more an example of 'evidence' than an existence proof? Surely nothing 'new' is discovered in each case? 3. Is the notion of probability analytic or synthetic? – James – 2012-12-08T22:24:11.130

@James - I already gave you an example with dice. I also have no idea what you are asking me to do in point 2; I don't know what you mean by any of "probabilistic argument", "evidence", and "existence proof". If you define them all, maybe I can say something useful. – Rex Kerr – 2012-12-08T22:29:00.970

You yourself mentioned each of those terms in your previous comment! – James – 2012-12-08T22:43:33.773

@James - Indeed, but you are asking for things that seem nonsensical to me, which suggests that you aren't using the same definitions I am. You also don't have enough reputation to chat, but the analytic-synthetic distinction isn't going to help you at all here. (If you take mathematics as analytic, probability is a branch that deals with interfacing with the synthetic, I suppose. But the terminology only serves to confuse a clear understanding of what is going on, IMO.) – Rex Kerr – 2012-12-08T23:06:31.717

You are right: terminology seems to be confusing matters here. Perhaps we should start from the following question: in your opinion, by what steps is knowledge gained in Mathematics (i.e. is there a particular general method for gaining knowledge)? – James – 2012-12-08T23:21:38.293

@James - You assume things, and state your assumptions, and then prove what follows from those assumptions. You may do various other ancillary stuff to inspire you (try out cases on paper, note a pattern on a computer, go on a walk, whatever), but proof of logical consequences of axioms is the core of mathematical knowledge. – Rex Kerr – 2012-12-08T23:42:02.573

Are you implying Mathematics to be a Language, rather than a Science, in the way that it uses man-made rules to avoid the notion of empirical evidence altogether? – James – 2012-12-08T23:53:00.197

Also, are you sure that there have not been discoveries in Mathematics where the mathematician has started by hypothesising a result, and then making the assumptions to be able to prove it? – James – 2012-12-08T23:59:07.730

@James - Mathematics is not a language in any useful (colloquial) sense of the word. Language can be formalized at least in part by using mathematical techniques. One can surely find mathematicians hypothesizing a result and then trying out assumptions to get that result; you can also find them meditating while listening to Pink Floyd, or arguing vociferously with a colleague about something neither of them actually know for sure, or thinking in the shower. These activities are interesting as sociology but do not form the core of the mathematical endeavor. – Rex Kerr – 2012-12-09T00:05:42.780

What about theories which are believed to be true, but have not yet been proved to be true (such as the Riemann hypothesis, for example)? Do you think that any proposition unsupported by reasoning should be analysed by mathematicians, or simply discarded without any opposing reasoning? Or does it depend on the level of authority of the mathematician making the proposition? – James – 2012-12-09T00:36:09.380

@James - Theorems believed to be true (theorem not theory) are worked on until they are proved (or proved false). That is part of the process of doing mathematics: constructing proofs of logical consequences of axioms. As a sociological matter, those problems of interest to more prominent mathematicians tend to get more attention, but that doesn't mean you can assume something is true or false. You can take as an axiom something that is widely believed to be true but not proven (and then you prove that, for example, Goldbach's conjecture implies so-and-so). – Rex Kerr – 2012-12-09T10:45:39.640

What would you say about Science and History, and their relation to 'evidence'? – James – 2012-12-09T12:53:39.230

@James - It feels like I'm just repeating myself, but: Science uses probabilistic reasoning explicitly (c.f. "propagation of error" and "statistical tests"); evidence is something that changes your estimate of the probability of something (either mathematically or intuitively). History usually is only able to do this intuitively. – Rex Kerr – 2012-12-09T13:10:34.387

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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.

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 – 2012-12-11T20:36:39.057

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 – 2012-12-12T21:25:46.370

Interesting, thanks! Sounds like it's worth a look if I get the time. – Rex Kerr – 2012-12-12T21:31:38.893

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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.