One of the things I love about mathematics is that some of the most baffling problems can be expressed in a few simple words that almost anybody can understand. I've been thinking about Goldbach's Conjecture, which is, basically, that every even number greater than 2 may be expressed as the sum of two prime numbers (a prime number has no factor but itself and 1. 2, 3, 5, 7, and 11 are the first 5 prime numbers; 1 is no longer considered a member of the set. It is easy to show that the set of prime numbers is infinite. Suppose it were not. Then make the number X the product of all of the prime numbers in the set. X + 1 would not be divisible by any member in the set. Then either X + 1 must be prime, or it must be the product of some primes not in the set. Therefore the set is not finite.)
For smallish numbers, you can see that the Conjecture holds: 1000 = 997 + 3. You might say, "Well, then it's obvious, because the prime numbers are not that far apart, so all you'd have to do would be to add 3 or 5 or 7 or something, and that will do." Well, not so fast. To get 1006, you have to go all the way to 983 + 23. And the greatest distance between any prime number and the next prime number, both numbers less than X, grows arbitrarily large as X grows large, though not as fast as X grows. So, for example, there's a largest gap of 14 between the primes less than 130: 113, 127. Eventually that largest gap can be as big as a million, though the numbers in question would be staggeringly huge.
The conjecture has not been proved, but a very powerful almost-there has been proved: all even numbers are the sum of a prime number and either another prime number or a "semiprime," that is, a number that is the product of only two primes (35 is a semiprime, as it is the product of two primes, 5 x 7). Semiprimes are a big deal in cryptography, I'm told.
I suppose that Goldbach's Conjecture will eventually be proved, but there are probably a lot of conjectures that will turn out to be undemonstrable. That too fascinates me. A conjecture may be true or false. It has to be one or the other. It cannot be first one, then the other. It is either / or. But it may be impossible even theoretically to demonstrate it.
Whether there is another universe besides ours may be conditionally undemonstrable: given that we are creatures in one universe, we can have no information from another universe. But to be theoretically undemonstrable would mean that under no conditions could any creature demonstrate its truth. Yet the truth would remain, regardless of whether it could be demonstrated ...