I'm strongly inclined to say yes. Here's an argument. If there's even one technological civilization elsewhere in our unimaginably vast universe, then that civilization must have discovered enough math to produce technology. But we have no reason at all to think that it's a human civilization, given the very different conditions in which it evolved: if it exists, it belongs to a different species from ours. So: If math depends on human consciousness, then we're the only technological civilization in the universe, which seems very unlikely to me.
Here's a second argument. Before human beings came on the scene, did the earth orbit the sun in an ellipse, with the sun at one focus? Surely it did. (Indeed, there's every reason to think that the earth traced an elliptical orbit before any life at all emerged on it.) But "orbiting in an ellipse with the sun at one focus" is a precise mathematical description of the earth's behavior, a description that held true long before consciousness emerged here. Kepler may have discovered that description, but the truth of the description predated him and every other human. So at least one true mathematical description is independent of human consciousness.
Here's a third argument. If the answer to your question is no, then there were zero mathematical truths before human beings came along, in which case there weren't more than zero mathematical truths. But the fact that zero isn't more than zero is a mathematical truth. So there couldn't have been zero mathematical truths. So the answer to your question couldn't be no.
You have described a fascinating phenomenon that I think is remarkably common, though I don't agree that it always happens It certainly happens frequently in my experience. Perhaps we both have very bright colleagues whom we happen to know very well, and can anticipate what they will say! I am delighted to see "the phenomenon" so well described. However, in the form you present it, I think most philosophers and psychologists would say that the question you ask is a psychological one, not a philosophical one, and that no doubt it is amenable to empirical research. Still, it does prompt a philosophical thought or two. I am put in mind of Wittgenstein's observation that 'In philosophy it is not enough to learn in every case what is to be said about a subject, but also how one must speak about it. We are always having to begin by learning the method of tackling it.' Perhaps when you ask a colleague about your problem, you have to decide not just what to say but how to say it, and that is enough for your mind to turn up the answer. This is true in philosophy, but perhaps if the problem is one whose shape is completely obscure to you, it is in effect a philosophical problem for you, at least temporarily. Then you see the way through the confusion to what is actually going on. I wonder what sort of experiments a psychologist might suggest to answer your question, 'What principle could explain the phenomenon?' I am sure there are bad philosophical principles that some people would drag in, such as this: you already knew the answer in a previous life (Plato). I would go very gingerly with answers like that.
Update: An interesting article about one of my computer science colleagues on the subject of cheating in chess and touching on the nature of "intelligence" in chess just appeared in Chess Life magazine; the link is here
Some preliminary thoughts: Racism seems to involve treating a group of persons who share an ethnic identity with derision, disrespect, and partial disadvantage. Accounts of racism today are controversial, but I propose that a comprehensive account of racism should involve both action as well as attitudes. What you write suggests that one reason why racism is inappropriate / unjust / wrong, is that persons cannot help being a certain ethnicity. I suggest, however, that racism would be wrong whether or not one could voluntarily adopt or abandon a race or ethnicity. If I converted to Judaism and, in the eyes of the world I became Jewish, anti-semitism would still be wrong even though I could have remained a Christian. In a related way, I suggest it would still be wrong to discriminate against homosexuals whether or not a person can choose whether to be homosexual or not.
It should be added as a side point that the very category of "race" is vexing. Some think of race as a social construct. If that is true, then (paradoxically) it could turn out that races do not exist as real things / categories, but racists do. This might be analogous to the idea that while it turns out that there are no witches (persons with supernatural powers to cast spells etc) but there have been witch-hunters.
On to intelligence: I suspect that some kinds of preferential treatment of persons based on intelligence would seem like racism. The following examples seem unjust: a policy in which only highly intelligent people have a right not to be tortured, but less intelligent people may be tortured for any reason whatever; a policy in which intelligent people can enslave those less intelligent, etceteras. But sometimes discrimination in which intelligence is a factor seems fair and prudent. Wouldn't you want intelligent persons to be pilots, surgeons, sailers, etc, rather than persons who are not intelligent --here I mean intelligent in the sense of mastering the relevant skills? Presumably, too, for a university to accept students on the basis of intelligence (including the capacity to learn) seems reasonable, right?
But you may be on to a very interesting worry. Some persons may be very vain and assume that they are superior to others on the grounds of some kind of measure of intelligence, when they are utterly inferior when it comes to matters of compassion, caring for others, generosity, courage, humility, poetic and artistic expressiveness, and so on. I suggest that someone we might call intelligent could turn out to be merely clever, but that is different from recognizing that someone is wise.
You asked, "If the physical processes that give rise to thought are rational, how can a human being have an irrational thought?" You might be misinterpreting the claim that "the physical processes...are rational." Presumably what's meant by the claim is that the physical processes can be discovered and understood by rational means, such as empirical investigation and logical reasoning. The claim doesn't mean to attribute rationality to the physical processes themselves: the processes don't literally investigate or reason, either well or badly. So the fact that the physical processes can be discovered and understood rationally doesn't imply that irrational thoughts can't result from those processes. Furthermore, we can rationally investigate the physical causes of irrational thoughts, even if science isn't very far down that road at present. In any case, we should resist the suggestion that the universe sometimes violates the laws of logic: that suggestion is either impossible or not even intelligible.
I'm sympathetic to most of what Professor Heck says, if we consider things from a deontological or even a consequentialist point of view, where the relevant consequences are external to the agent. Fantasy does not violate anyone's rights, and fantasy that never motivates action will not result in actions that harm anyone. But I think there is a plausible way of looking at things that would still find fault with fantasizing about having sex with children, and that would come from the aretaic (or virtue-theoretic) way of thinking, according to which the primary bearer of value is to be found in characteristics of agents. One who indulges in fantasies about sex with children is doing something that both reflects--and also perhaps perpetuates and sustains--a certain trait of character that we might think is not entirely wholesome or admirable. To the extent that we can regard one who indulges in such fantasies as having a trait of character that is improvable, we might also think that some attempt to eliminate or at least diminish the inclination to indulge in such fantasies would result in that person having some improvement in character. It may be that habituation can only go so far, and that virtue theorists (such as Aristotle) overrate the extent to which one can habituate better character traits, but it certainly does seem that a virtue theorist could find the character of someone who tends to indulge in such fantasies at least improvable, and this way of looking at things does, I think, put a different face on this kind of case than what Professor Heck has indicated.
These are great questions. There have been philosophical arguments that suggest that it is impossible to know whether dreams occur while we sleep or are just confabulations we create as or after we awake (call this 'dream skepticism'). These arguments fail once we consider all the evidence and use abductive (best explanation) reasoning. When we wake people during REM, they are likely to report dreams. When we wake them during other phases of sleep, they are unlikely to report (or remember) dreams. When we record neural activity using EEG and now fMRI, we see activity that correlates both with the sorts of experiences reported by the dreamer and with the sorts of activity that correlates with similar waking experiences (fMRI cannot get at all of what you call the "granularity of experiences" but see the link below for an initial attempt). Etc. This body of data could be explained away by a dream skeptic, but that explanation would likely look ad hoc and fail to make predictions as good as the hypothesis that dream experiences are real, in the sense that they occur during sleep and have roughly the content people report (though first-person reports of even waking experience can be inaccurate in lots of ways, and since we cannot report the content of dreams while they occur, the inaccuracies of memory come into play... except that people can make reports of a sort--e.g., intentional eye movements--during lucid dreaming: see http://en.wikipedia.org/wiki/Lucid_dream)
Dream skepticism is one version of the more general problem of other minds--how can we know what other people are experiencing from the first person point of view (or that they are experiencing anything). The sort of 'best explanation' approach described briefly above is, I think, the best way to approach the general problem too. We have lots of reason, including lots of evidence, to think that neural activity correlates with (and is, in some sense, the basis of) conscious mental activity. So, we have lots of reasons to think that creatures with neural activity similar to our own have conscious experiences similar to our own. And this view makes lots of accurate predictions and explains lots of other observations. Etc.
For brain scanning of dreams, see: http://www.wired.com/wiredscience/2013/04/dream-decoder/
For a nice philosophical discussion of dreaming, read Owen Flanagan's Dreaming Souls: http://www.amazon.com/Dreaming-Souls-Evolution-Conscious-Philosophy/dp/0195142357
My answer is a little different from Olilver's. Why do so many scholars and intellectuals think that language is necessary for thought? Answer: Because it really is easier to think about definite rather than indefinite things. But indefinite and formless things also have to be thought about. It takes more of an effort of course to think in a pathfinding sort of a way about something new, and one may or may not be thinking "in" language, whatever that means (muttering to oneself, sub-vocally?) If one is trying to come to an understanding of some hard and new logical or mathematical matter, it may be more like shaping forms in ones mind, and then moving them, and less like chattering in French. If one insists on calling "shaping forms", or whatever the metaphor is "a kind of language", then of course the claim is drained of any content, and with that of any interest. People of say that mathematics is a language, or a "language", something like a language. But it has a function and a status very different from those of a language.
Not an easy question to respond to!
If, by 'thought' you are referring to what persons think about (as in: I am thinking about math), then because it seems as though we thinking persons can think about the world around us and about ourselves and abstract objects (as in mathematics), then the truth of what we think about the world and ourselves and abstracta (e.g. math) will (I suggest) very much depend on how things are. I might think I am an expert in mathematics, but it turns out I am rubbish. The 'reality of [my] thought' that I am a math expert does not depend upon whether it is true or false, but the consequences of its being true or false might be quite significant in reality.
Some philosophers (such as myself) are prepared to go in what must seem like a hopelessly mysterious direction. So, some of us treat the objects of our thinking to be propositions or abstract (non-concrete) states of affairs. For example, i hope to live in a universe in which there are unicorns, while my friend Judy does not. This may be understood as there being an abstract state of affairs:
Human persons living in a universe in which there are unicorns
Which I hope obtains (or is true) and Judy hopes is not true or it does not obtain.
This position (historically linked to Plato) holds that truths and falsehoods about reality should be understood in the obtaining or not obtaining of states of affairs. So, to employ the terms you suggest, I propose that (assuming our universe contains no unicorns, alas), the following state of affairs does not obtain:
Human persons living in a universe in which there are unicorns
And this truth, assuming it is true, is not dependent upon your mind or mind. It does depend on the "external world" in the sense that if a single horned horse showed up in our world with tears that could heal us, then the above state of affairs would obtain (and it would be true we are living in a universe with unicorns, whether or not we believed it or were skeptical).