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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?

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 – 2012-12-08T18:54:48.960Could 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 acounterexample, 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 – 2012-12-08T22:19:48.073Three 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 allhere. (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 – 2012-12-08T23:06:31.717You 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 (

theoremnottheory) 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 – 2012-12-09T10:45:39.640What 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