I have a couple more answers that I really like to the question "What do you believe is true even though you cannot prove it ?" given by edge.org to its contributors.
Ian McEwan (well said):
What I believe but cannot prove is that no part of my consciousness will survive my death. I exclude the fact that I will linger, fadingly, in the thoughts of others, or that aspects of my consciousness will survive in writing, or in the positioning of a planted tree or a dent in my old car. I suspect that many contributors to Edge will take this premise as a given—true but not significant. However, it divides the world crucially, and much damage has been done to thought as well as to persons, by those who are certain that there is a life, a better, more important life, elsewhere. That this span is brief, that consciousness is an accidental gift of blind processes, makes our existence all the more precious and our responsibilities for it all the more profound.
Charles Simonyi (his metaphor using cryptography made me laugh out loud even though it suffers a little from oversimplification):
I believe that we are writing software the wrong way. There are sound evolutionary reasons for why we are doing what we are doing—that we can call the "programming the problem in a computer language" paradigm, but the incredible success of Moore's law blinded us to being stuck in what is probably an evolutionary backwater
....
Leonard Susskind (funny):
Conversation With a Slow Student
Student: Hi Prof. I've got a problem. I decided to do a little probability experiment—you know, coin flipping—and check some of the stuff you taught us. But it didn't work.
Professor: Well I'm glad to hear that you're interested. What did you do?
.....
Freeman Dyson (clever jewel):
Since I am a mathematician, I give a precise answer to this question. Thanks to Kurt Gödel, we know that there are true mathematical statements that cannot be proved. But I want a little more than this. I want a statement that is true, unprovable, and simple enough to be understood by people who are not mathematicians. Here it is.
Numbers that are exact powers of two are 2, 4, 8, 16, 32, 64, 128 and so on. Numbers that are exact powers of five are 5, 25, 125, 625 and so on. Given any number such as 131072 (which happens to be a power of two), the reverse of it is 270131, with the same digits taken in the opposite order. Now my statement is: it never happens that the reverse of a power of two is a power of five.
The digits in a big power of two seem to occur in a random way without any regular pattern. If it ever happened that the reverse of a power of two was a power of five, this would be an unlikely accident, and the chance of it happening grows rapidly smaller as the numbers grow bigger. If we assume that the digits occur at random, then the chance of the accident happening for any power of two greater than a billion is less than one in a billion. It is easy to check that it does not happen for powers of two smaller than a billion. So the chance that it ever happens at all is less than one in a billion. That is why I believe the statement is true.
But the assumption that digits in a big power of two occur at random also implies that the statement is unprovable. Any proof of the statement would have to be based on some non-random property of the digits. The assumption of randomness means that the statement is true just because the odds are in its favor. It cannot be proved because there is no deep mathematical reason why it has to be true. (Note for experts: this argument does not work if we use powers of three instead of powers of five. In that case the statement is easy to prove because the reverse of a number divisible by three is also divisible by three. Divisibility by three happens to be a non-random property of the digits).
It is easy to find other examples of statements that are likely to be true but unprovable. The essential trick is to find an infinite sequence of events, each of which might happen by accident, but with a small total probability for even one of them happening. Then the statement that none of the events ever happens is probably true but cannot be proved.
