Mathematician: Unfortunately, limits to knowledge seem to be built into the nature of the universe, and even into logic itself.
Relativity: Einstein’s theory of special relativity implies that no information can travel faster than the speed of light. That means that information from sufficiently recent, sufficiently far away events will not have had the time to propagate to us yet, making detailed knowledge of such events impossible. In physics speak, we say that these events are outside of our “past light cone“, “space-like separated” from us, or just “elsewhere”. As long as new events of this type keep happening, there will always be things about which we do not and cannot know.
Quantum Mechanics: The Heisenberg uncertainty principle states that the uncertainty we have in a particle’s position and the uncertainty we have in the particle’s momentum cannot both be very small at the same time. In particular, the product of these uncertainties is greater than a constant (). This implies a fundamental limit to the knowledge that is possible because we can know accurately or accurately, but not both.
What’s more, the vast majority of physicists agree that quantum mechanics demonstrates the universe is random at a fundamental level. This means that some events, like the time at which an atom will decay, can be predicted only probabilistically. We can say how likely an atom is to decay in a given time interval, but we will never be able to say precisely when the decay will occur, placing another limitation on what knowledge is possible. (Physicist’s note: After the decay you still can’t say when exactly it happened because according to quantum mechanics the exact time doesn’t actually exist!)
Mathematics: Gödel’s first incompleteness theorem states (essentially) that any mathematical system that is able to express elementary arithmetic (and doesn’t contain any contradictions) must contain true arithmetical statements that cannot be proven within that system. Essentially this implies that there will always be true mathematical statements that we cannot prove.
Add to all of these theoretical considerations the enormous (and possibly infinite) number of things that could be known about our physical universe, and the (most definitely) infinite number of true mathematical statements that could be known, and it is clear that there will always be knowledge that is beyond our reach.