To have a source of information is to have access to something, and thus that there is at least the possibility of using that information to make reliable judgments about it. The reason sources of information should be important to us is twofold. First there is obviously a strong connection between the ability to have knowledge about things and to have information about them, as it seems impossible to have knowledge without having information (which is not to preclude the possibility there may be other factors that turn that information into what we would consider knowledge, although that is a matter for another time). Thus the ability to identify our sources of information seems like an important component in any evaluation of the way we reason, because knowledge that doesn’t stem from them in some way isn’t really knowledge about anything at all. Secondly it also seems that having sources of information is what allows us to meaningfully refer to things. Specifically it would seem that we can only successfully make reference to something by pointing at it though our information (by identifying it as the cause of that information). Without the ability to pin down what we are talking about in this way our words become completely general (they could be about anything that has the right relationships), and thus describe a pure formalism. And this has important epistemological implications as well, because what a pure formalism can tell us, and how we evaluate them, is different from claims that are about some subject matter.
This brings us to the central problem, identifying the sources of information available to us. Obviously we can’t “prove” what we do and don’t have information about. To prove something, in the strict sense of a deduction from simpler principles, presupposes certain sources of information, sources that justify the premises of the proof and the claim that the inferences involved are valid inferences to make when the domain under consideration is sources of information. As with many other places in epistemology what we must do is consider the claim that we have a particular source of information as “evidence”, a claim that does not itself need justification to be used as the basis for further claims. However, like all evidence, this claim itself must be falsifiable, it must be possible to reject a proposed source of information. And that is how we go about seeing what sources of information we have, we can hypothesize that whatever we like is a source of information, but we must then critically evaluate that source. The way to perform that evaluation is by testing the consistency of a proposed source of information, the assumption being that if a source of information is consistent then it is likely that there is something causing it to be consistent, and that this something, whatever it is, is what the information is about. Because if this information is not in fact about something, but rather comes from a void (or a random process of some kind) then the probability that it will remain consistent as more and more information is drawn out diminishes, and thus in the ideal infinitely long run such a way of evaluating our sources of information will leave us only with the those that are actually sources.
Let us consider then our senses as a source of information. Certainly our senses seem consistent. Obviously we sense different things at different times, but the changes seem to follow predictable patterns, and thus we are fully capable of interpreting our senses as revealing information about an external world which is generally consistent over time, allowing for changes in that world. And there is a further consistency between what we sense and the reports of other people (remember the reports of other people are further evidence of consistency, they don’t prove anything, and so we are not unjustifiably assuming that other people are real). But, of course, our senses are not perfectly consistent in this way, as is demonstrated by phenomena such as optical illusions. And this might seem to invalidate our senses as a source of information by our own criteria. But, when it comes to these “inconsistencies” we possess explanations of them, both in the sense that we can predict and thus compensate for them, and in the sense of knowing reasons why they occur. Of course these explanations are themselves justified by our senses, but this doesn’t invalidate them. What it shows is that our senses really are consistent, despite what might seem like some apparent inconsistencies in them, it’s just that the way in which they are consistent is more complicated than we originally supposed. A real inconsistency in our senses would have to be something that we couldn’t explain and thus was essentially unpredictable, and if they existed they would throw all the information we derive from our senses into question, because there would be a real possibility that any information we derive from them is really a product of one of these unpredictable deviations. It would probably drive us mad.
Similarly, we seem to have information about our own mind. As with our senses, our access to our own mind doesn’t reveal the same internal state to us at all times, but the internal state revealed in this way seems to change in generally predictable ways. Of course this might seem to presuppose the accuracy of our own memories, possibly illegitimately since our ability to recollect things is part of our minds. But, again, it is important to point out that what we are looking for here is only consistency, and not a justification. Our only goal is to examine our memories, and our mind in general, for consistency, not to prove that they are accurate. Obviously when it comes to our own minds we can’t compare our observations of them to those of other people (since they don’t seem to have access to our mind), but this doesn’t mean that this source of information is deficient in some way, it simply means that we have fewer opportunities to detect inconsistencies. I would also like to note here that even though we can consider ourselves to have information about our own minds this doesn’t mean that the mind is as the information seems to portray it, just as the external world isn’t necessarily as the senses portray it. The natural interpretation of the senses is as revealing an external world that is essentially continuous, but we know that it is actually made of discreet atoms. Likewise just because our access to our own minds is naturally interpreted in terms of qualia or “feels” doesn’t mean that such things actually exist, it just means that they reflect something that does exist.
Now let’s turn to a more interesting and controversial case, the possibility that we have access to information about mathematical objects or some mathematical realm through mathematical intuition of some kind. Certainly there are domains in which it seems that our mathematical intuitions are relatively consistent both internally and with the mathematically intuitions of most other people, namely logic, arithmetic, and possibly geometry. However, inconsistencies with these intuitions are certainly possible, and sometimes even mathematically profitable. For example, it is possible to construct internally consistent logics that defy our mathematical intuitions about truth and to use them productively. More significantly there are a large number of cases in which our mathematical intuition is essentially silent, where both of two mutually exclusive possibilities are consistent with everything that is mathematically intuitive to us, such as the continuum hypothesis or the axiom of choice (although the two are not independent of each other). And, on these issues, it is even possible for the mathematical intuition of different people to pull them in opposite directions. Now such inconsistencies do not themselves necessarily refute mathematical intuition as a source of information. The real problem is that no justified explanation can be given for these inconsistencies, as we could with the apparent inconsistencies of the senses. Certainly we can hypothesize reasons, but none of them are backed up by anything besides the necessity of coming up with some explanation (nothing in our source of information about the mathematical world, or about our minds, or the external world, has anything to say about our connection to the hypothesized mathematical world or its limitations). This refutes mathematical intuition as a source of information, unless insights from some other source of information can validate it and explain away the apparent problems. (Of course it doesn’t rule out the possibility that mathematical intuition is a source of information about something subjective, but if it is construed as such it is only an addition to the information we have about our own minds, specifically it would be information about the psychological sources of mathematical intuition.)
What the case of mathematical intuitions brings up is that we need some way to distinguish between genuine sources of information and instincts. Because instincts may be apparently consistent among people, but that doesn’t mean they provide information about anything except what was of survival value in the past. For example, we could imagine that all people are born with the belief that “X is Y”, but such a universal belief does not reveal any information to us about this “X”. The most obvious distinction between the two seems that a source of information can always divulge genuinely new facts, but instinct (and thus intuition) seems limited to a finite number of facts or general rules. Thus I would explain our mathematical intuition as a kind of useful instinct that leads us to believe in certain general rules about inference, addition, and so on. But it is obviously not a source of information, because when we get beyond those general rules, to things that are independent of them, our mathematical intuition is silent or inconsistent.
As a final note I would like to say a few words about how taking the claim that we have a certain source of information as evidence integrates with my previous assertion that we should treat particular pieces of information derived from such sources themselves as evidence. It might seem that saying we have a particular source of information would be to entail the truth of the information derived from it, thus making treating those specific claims as evidence redundant. But this is not actually the case. Saying that we have a particular source of information is not to make a claim not about what is and isn’t the case, but about reference; it is to say that there is something that the information from that source reaches out to. And just because certain claims are produced by a source of information doesn’t mean that they are correct. As was pointed out in the case of our senses the claims produced may occasionally contradict each other, at least under their natural interpretations, and further investigation is needed to determine which are in error and why. And thus claims stemming from this source of information may indeed turn out to be false (those false claims still convey information, just not the information they were originally thought to).