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There are several words I find myself reluctant to use these days, because they get used beyond their meanings. One of them is “proof”, which gets used for situations where “demonstrate” would be a lot more accurate. Lots of fields of study don’t really use the concept, but that’s lost on politicians and many journalists, who demand, for example, “scientific proof.” That’s not available.

One can disprove things easily in science, since all you need is to demonstrate a counter-example. But no matter how sure you are of something, and it’s possible to have a great deal of confidence in e.g., the second law of thermodynamics, you can’t “prove” it in the way you can with mathematics.
Mathematical proof is solid, within its limits. Those are broad enough to be useful, but they aren’t absolute. Gödel demonstrated that.

Legal proof is something else again. It seems a bit like money, something that we collectively treat as being real, because it’s useful. But it has a lot to do with what you can convince a court about, and courts are made up of humans. My sole experience with legal proof is acting as a juror, in a case where all the evidence was from an accomplice. English law allows a conviction on that, but “proof beyond reasonable doubt” has to be applied very strictly. And when the prosecution are reminding the jury of that, you take it seriously.

I make my living in a specialised field of computer programming, where the hard jobs come down to trying to decide if the bug is in the program, the compiler, or the hardware. Any of those types can do a good job of masquerading as another type, even without edge cases. Those have included the hardware working as designed, but the discovery of a general misunderstanding as to the effects of that design, the revelation of which means that the hardware is a lot less useful than the manufacturer thinks. In time off from that I try to help keep a network of several hundred computers running. It is complex enough that while everything in it is deterministic, the emergent behaviour has “elements of randomness tantamount to wilful perversity”. I make sure to tell new people the story of N-rays, as a cautionary tale.

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