Opiniatrety

Interesting fact indeed. We don't really know much about it, although it's been remarked upon in various places (Karttunen "Possible and Must", Kratzer "Modality", ...). Some people (including me) feel that the choice between "p" and "must p" in English is somewhat similar to choices speakers have in languages with fully grammaticized systems of evidential markers. "must p" seems to mark statements whose truth we only have indirect/inferential evidence for.

Thanks for the info, Kai! In languages with fully grammaticized systems of evidential markers, do the markers have other meanings in other contexts?

If we accept that "must p" marks statements whose truth we have only indirect/inferential evidence for, it raises the question of why 'must' would be used as such a marker. I've been toying with a theory of epistemic modals on which 'might' is peculiarly well suited to conversations in which the participants are trying to figure something out. This presupposes the relativistic account of 'might' (which I know you disagree with): on that theory, the person who doesn't know the fact that establishes not-p can enter "might p" into the presuppositions of the conversation, since the proposition might-p is true as she evaluates it. But if her interlocutor does know that not-p, she is in turn obliged to deny "might p"; otherwise she will have let pass a proposition that she knows to be false as she evaluates it.

If 'might' is peculiarly suited to such conversations, we might also expect 'must' to be suited to such conversations. So we'd say 'must p' when we're trying to push along inquiry concerning p or a related matter. And maybe that accounts for why 'must p' suggests that we have indirect/inferential evidence for p: If we had direct evidence, we wouldn't need to use a locution designed to further inquiry along; we could simply say 'p' and end inquiry.

So if you're inquiring into the question, "Is there an F, and if so what is F?" you might say "There must be an F" in order to rule out the alternative "There is no F" and promote inquiry into the question "What is the F?" But if you know what the F is, then you say "There is an F" which suggests that you know what the F is. (Interestingly, saying "There is an F" doesn't trigger the implicature that you don't know what the F is, even though the maxim of Quantity might suggest that if you knew what the F was you shouldn't have offered the weaker existential generalization.)

This is all very speculative, of course, even for those who accept the relativistic semantics for epistemic modals!

This is the difference between constructive and nonconstructive proofs in mathematics.

In certain cases, for the statement "Some F exists that has properties P," you can prove the statement either by writing down an F that has those properties, or you can indirectly prove that such an F exists - i.e. such an F _must_ exist - without actually constructing it.

I'm sorry to say that the example I dimly remember of this is the existence of a function that is everywhere continuous but nowhere differentiable. You can prove this by constructing the function (it's a series sum of sawtooths of increasing frequency and decreasing amplitude), which is cute, but seems like a inconsequential real-analysis braintwister. Or you can prove it _without_ constructing the function - I think the argument has something to do with measures on the space of functions - which proves that such functions _must_ exist, without giving an example, and arguably teaches you something fundamental about the space of functions.
It is a mathematical necessity that F exists, but though you can talk about it, you can't point to it, a state of affairs capable of frustrating both Ludwig Wittgenstein and Reverend Ike.

It was around this time that I decided to give up mathematics for practical stuff like deconstruction and the theory of quantum mechanics.

But there doesn't seem to be a lack of direct access to the fact asserted in these cases.
1. I must be the first person in the room, there
is obviously no one else here.
2. He must be first in line, he is in line and
there is one in front of him.
3. The answer must be 4, the question asks you to
sum 2 and 2.

These seem best taken as replies to (bad) questions like, "are you sure you're the first person in the room?" or "Is he first in line?" or "Is the answer 4?"

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[for some reason I have decided to leave this spam comment up with a slight redirection of the URL--mw]