Mathematics - grounded in logic?
There is an interesting webpage (https://rationalwiki.org/wiki/Pseudomathematics) that states:
“Pseudo-mathematics is any work, study or activity which claims to be mathematical, but refuses to work within the standards of proof and rigour which mathematics is subject to … Unfortunately for practitioners of pseudo-mathematics, mathematics is an absolute science of black and white - everything is right or wrong. There is no often scope for debate or discussion, as only mathematical proof is relevant.”
If only everything was as black and white as that suggests. The reality is that there are numerous mathematical statements which are widely accepted as correct by the vast majority of mathematicians, in spite of there not actually being a rigorous proof of the statement. The fact is that a mathematician cannot possibly examine every proof, and that means that he often relies on what he perceives to be the commonly accepted belief. In most cases, the commonly accepted belief that there is a rigorous proof of a certain mathematical statement is correct, but that is not always the case.
There are well-documented instances where errors in mathematical proofs have not been discovered for several years after they were first published, including proofs that received widespread attention (see, for example, Kempe’s flawed proof and Tait’s flawed proof). And these were not proofs where one language makes statements about another language, and where there is a possibility of language confusion.
What is so curious about mathematicians is that you would expect that if someone points out that a proof is not sufficiently rigorous, or that it includes an inadmissible step, the mathematician would be grateful that that was brought to his attention. But what actually happens? When it is pointed out that certain proofs, such as various proofs of incompleteness, including Gödel’s original proof and many other proofs of incompleteness (see The flaw in Gödel’s incompleteness proof: here and Analysis of incompleteness proofs: here) are not actually as rigorous as a mathematical proof should be, mathematicians refuse to accept the facts. When it is pointed out that such a proof includes an unproven assumption instead of a rigorous logical argument, or when it is pointed out that a step in the proof relies on a step that is mathematically inadmissible, they seem to take it as an affront, as an insult to their self-perception as bastions of absolute correctness.
For more on why the use of term ‘mathematical proof’ can mean something quite different to a perfectly rigorous logically reasoned argument, see the page Mathematical Proof.
Pseudo-logic and pseudo-mathematics
So, we know that there must be proofs that are not in fact rigorous mathematical proofs, but which, for one reason or another, have been accepted as correct, in spite of failing to meet the criteria for a rigorous mathematical proof. Surely this is not acceptable? Preconceived notions about the correctness of proofs should be set aside, and proofs should be subjected to a logical critical analysis. If a proof fails the test that requires it to be a rigorous mathematical proof, then its result should not be accepted. This rooting out of pseudo-proofs should be just one step in an inevitable, if slow, step towards a more complete understanding of how language, logic and mathematics are interrelated. Mathematics and logic cannot be fully understood unless the role of language is taken in account. Confusions such as those that appear in Gödel’s proof of incompleteness (see The flaw in Gödel’s proof: here) and various other proofs of incompleteness (see Analysis of Incompleteness Proofs: here) masquerade as genuine logic and genuine mathematics, whereas in reality they are the antithesis of real logic and real mathematics.
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