Can you ever prove the future will be like the past?

No, not in the strict sense of proof, and that's the problem of induction. Every time you expect the future to resemble the past, the sun to rise, bread to nourish, a chair to hold you, you're reasoning from what you've observed to what you haven't. The Scottish philosopher David Hume pointed out in 1739 that no logical argument can guarantee that leap. Logic can show that a triangle has three angles; it can't show that tomorrow will follow yesterday's rules.

The obvious reply is "but induction has always worked before". Hume's sting is that this reply uses induction to defend induction. It assumes the very thing in question, so it goes round in a circle.

Bertrand Russell made the danger vivid with a chicken. Every day of its life, the farmer feeds it, and every day its confidence grows. Then one day the farmer wrings its neck instead. The swans of Europe tell the same story: for centuries "black swan" was a London figure of speech for something impossible, because every swan anyone had recorded was white. In 1697, Dutch explorers led by Willem de Vlamingh saw black swans in Western Australia.

Hume didn't conclude that we should stop expecting things. He said we can't help it: we form expectations by custom or habit, and without them we'd know nothing beyond what's in front of us. The puzzle is why that habit deserves to be called rational. The philosopher C. D. Broad called induction "the glory of science and the scandal of philosophy", and the scandal has never been fully cleared up.