1

I am a new researcher in kinetic theory of gases, and I always see that the proofs are either formal or rigorous.

Could anyone please clarify to me these descriptions?

Thank you.

insipidintegrator
  • 6,642
Marrie
  • 85
  • 2
    Why do you consider them as alternatives? – Mauro ALLEGRANZA Jul 26 '22 at 12:15
  • What's the question? Explain what formal means? – FShrike Jul 26 '22 at 12:16
  • 1
    I think most people would say that "formal" and "rigorous" were synonyms. "informal" or "heuristic" would mean a common sense argument which is intended to persuade, though it is not a substitute for a proper proof. – lulu Jul 26 '22 at 12:19
  • "formal" in the sense of Formalization can be a way to achieve rigor, but we can have a rigorous proof that uses usual "mathematical jargon" (a mix of natural language and symbols). – Mauro ALLEGRANZA Jul 26 '22 at 12:22
  • As written by FShrike, the question will only make sense if the word formal is defined. For example, do you mean formalizing proof in the language of set theory? If yes, mathematics will be hard to follow by most of the mathematicians! – mathcounterexamples.net Jul 26 '22 at 12:26
  • 1
    Being a mathematician, for me, It seems quite hard to differentiate a proof from formal or rigorous. I have never done formal or rigorous proof using set theory axioms, which is the only rigorous math for me. It seems quite personal point of view. I would say that usually math is enough in its proof, but not rigorous. – R. W. Prado Jul 26 '22 at 12:34
  • As far I can say to anyone, formalizing intuitions, like "formal" vs "rigorous" proofs, is commonly criticized approach to do in mathematics. If someone asks you to make it, even if you can formally and rigorously do it, please, do not make it. There will always someone with a slight different point of view, which will put doubts in your ideas, whether it is good or not. – R. W. Prado Jul 26 '22 at 12:45
  • Adding to others comments, let me say, that in this way any word in any question need additional definition and this can be infinite. Short answer - formal, exact, rigorous are same for me in mathematics. It's some type of world view. Ideology. – zkutch Jul 26 '22 at 12:47
  • 5
    Context matters! Language is funny because sometimes, 'formal' and 'rigorous' are synonymous, e.g 'we can prove this theorem formally/rigorously using...', while sometimes it's completely the opposite, e.g 'let us formally manipulate this equation', meaning a manipulation without much regard for rules, just to get a sense of what's going on. In the first case, the synonymous usage is because we're using 'formal' in the sense of formal vs informal, whereas in the second case, we're using 'formal' in the sense of the form (i.e appearance) of something. – peek-a-boo Jul 26 '22 at 12:59
  • This is a good question because there is most definitely a difference between a rigorous proof and a formal proof. – John Douma Jul 26 '22 at 13:37
  • I suggest the down-voters try to search for this. They will get answers that are as incorrect as the comments here. Formal proofs came about in the early 20th century when mathematicians were wondering if they could feed axioms and inference rules to a machine and have the machine prove theorems. See Entscheidungsproblem. – John Douma Jul 26 '22 at 14:01

1 Answers1

2

Rigorous proofs are proofs whose logic is undeniable. These should be understood by someone with training in mathematics but not necessarily by a layman. These are what you usually see in textbooks.

A formal proof is verifiable through a procedure that can be verified by anyone. These proofs should be mechanical in nature. You may have seen these proofs in high school geometry.

They are multi-line statements that are justified by axioms, (the things you assume are true), and inference rules. The inference rules are what allow you to use the conclusions from the previous lines to conclude other facts. The list of statements is finite and ends with the proposition you are trying to prove.

John Douma
  • 11,426
  • 2
  • 23
  • 24