On Proof

 

https://barryhisblog.blogspot.com/p/on-proof.html

In discussions about science, people will say that you cannot talk about the "proof" of a scientific theory. This often comes up in discussions between evolution refusers and regular people. People from both sides of the discussion will say, "The concept of proof belongs in mathematics and in measuring the strength of alcoholic beverages, but not in science."

Words can often have multiple meanings.

1) Axiomatic proof.

These proofs are provided by correctly formed arguments in axiomatic systems. Mathematics and logic are given as examples of these systems. The proofs are presented by combining agreed axioms using agreed procedures and rules of combination. It's rather like a game. Consider chess. You play according to a set of fixed, agreed rules. In one important sense, these systems establish what is true only in terms of the rules themselves - only if you accept the rules. "If I accept these rules, then so-and-so inevitably follows". If you move a bishop from a black square to a white one, you are violating the rules. You are not playing the game. In a trivial sense, you have proven that a bishop cannot change his spots, as it were. It is interesting to note that in any axiomatic system that is complex enough to be self-referential, it has been proven, according to its own rules, that there must exist statements that cannot be proven either true or false. These are the famous incompleteness theorems of Kurt Gödel. https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

I live not far from Beaumont-de-Lomagne, in France, the birthplace of Pierre de Fermat, famous for his "Last Theorem" which claimed that the Pythagorean formula for any right-triangle, aⁿ + bⁿ = cⁿ where n=2, has no equivalent whole number solutions where n>2. He wrote in the margin of one of his books that he had a marvellous proof of his claim, but there wasn't enough space there to write it out. For 300 years, generations of mathematicians racked their brains trying to prove him right or wrong. Eventually, Andrew Wiles offered a proof. It was pored over by other mathematicians to check that he had made no mistake - that he had "played by the rules". They found that he hadn't. He had made an error. So what did he do? He retired to his study, and one year later, he had fixed his work. Fermat was vindicated, and Wiles was celebrated. But we will never know if Fermat had a proof based on the mathematics known at the time. Wiles used much more advanced mathematical techniques which would have been unknown to Fermat.
https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem

There is a novel speculating whether Fermat really did have a proof and what that proof might have been, but it doesn't offer one. https://www.amazon.com/Last-Theorem-Arthur-C-Clarke/dp/0345470230/ref=tmm_pap_swatch_0?_encoding=UTF8&qid=&sr=    

2) Proof of alcohol

From https://vinepair.com/spirits-101/what-is-proof/

"According to legend, the concept of “proof” comes from soldiers in the British Royal Navy, who (back in the 18th century) had to douse their gunpowder in rum as a test of its potency. If the wet gunpowder still ignited, it was “proof” the alcohol content was high enough, 57% ABV. If it didn’t ignite, well, you probably had some angry—and armed—British soldiers on your hands. (Another legend has it that rum needed to be at proof so that if a barrel broke on the ship, it wouldn’t render all that precious gunpowder useless.)"

There are other accounts, which you can find at the link, but the principle is the same. This may at first glance sound flippant, but there is a serious point to consider here. Proof spirit is determined by a test. It is not the same type of proof as axiomatic proof, but it is a form of proof.

3) Legal proof

From https://www.law.cornell.edu/wex/beyond_a_reasonable_doubt

"Beyond a reasonable doubt is the legal burden of proof required to affirm a conviction in a criminal case. In a criminal case, the prosecution bears the burden of proving that the defendant is guilty beyond all reasonable doubt. This means that the prosecution must convince the jury that there is no other reasonable explanation that can come from the evidence presented at trial. In other words, the jury must be virtually certain of the defendant’s guilt in order to render a guilty verdict."

 

4) Scientific proof?

You can see that 1) differs from 2) and 3). Axiomatic proofs, like theorems in mathematics, are the unavoidable, inevitable results of playing the game by the rules. 2) and 3) are based on evidence and reason. Doesn't that apply in science too? Can we establish things as being true beyond reasonable doubt? Can we prove them in that sense of the word? I think we can, and we shouldn't be shy of saying so. It has been 163 years, at the time of writing this, that Darwin's Origin Of Species was published. There have been plenty of doubts expressed about evolution, and they are still being expressed. But none of them are reasonable doubts.

Sir Fred Hoyle, often held up as a poster-boy for creationism, despite his other nutty ideas wrote, "Darwin’s theory, which is now accepted without dissent, is the cornerstone of modern biology. Our own links with the simplest forms of microbial life are well-nigh proven.”
–Fred Hoyle and Chandra Wickramasinghe, Lifecloud: The Origin of Life in the Universe (1978), p.15-16 https://barryhisblog.blogspot.com/p/fred-hoyle.html



A word about "assumptions". When you say something is the result of a theorem, it is not an assumption. It is true within your axiomatic system. No argument can be had over the result of a correctly formed theorem. The axioms themselves are the assumptions, and they are not necessarily true in some external sense. They are just the rules of the 'game'. But when you call a scientific conclusion an "assumption", I regard that as an insult. Scientific conclusions are arrived at by applying reason to the evidence. What assumptions are made in science? Only that there are no malicious little gods falsifying the evidence so as to mislead us and that we can normally trust our own faculties. As Galileo put it, "I do not feel obliged to believe that the same God who has endowed us with senses, reason and intellect has intended us to forego their use and by some other means to give us knowledge which we can attain by them."