Copyright © James R Meyer 2012 - 2018 https://www.jamesrmeyer.com
Currently we hear a lot about fake news. What we don’t hear much about is fake mathematics. At this point you might be wondering what I mean by fake mathematics.
Fake news might be described as material that is fabricated without any supporting evidence, and which is presented in such a way that naive observers are willing to believe the material without subjecting it to any detailed examination, especially if it concurs with their underlying philosophy.
In a similar vein, fake mathematics might be described as material that is fabricated without any supporting evidence, and which is presented in such a way that naive observers are willing to believe the material without subjecting it to any detailed examination, especially if it concurs with their underlying philosophy.
While we don’t hear much about it, fake mathematics has been prevalent for a great many years. To show that this is the case, we only have to carry out a simple thought experiment. In this thought experiment, we imagine an alternative mathematical world than the one we see today. In our thought experiment, the only proofs accepted by the mathematical community are proofs that have been logically proved, and no proof steps are allowed to be assumed to be correct rather than proven. We now suppose that in this mathematical world (as in our actual world) Gödel submitted his paper on Incompleteness (Footnote: Gödel’s paper was written in German, viewable online Gödel’s original proof in German: here PDF. The English translation of the paper is entitled “On Formally Undecidable Propositions of Principia Mathematica and Related Systems”, viewable online Gödel’s Proof - English translation: here.) to various journals. Unfortunately for Gödel, in this mathematical world, all the reviewers rejected his paper because (as in our actual world (Footnote: Peter Smith, although a staunch advocate of Gödel’s proof, acknowledges this in his paper, Expounding the First Incompleteness Theorem (PDF), that, “Gödel only sketches a proof… The crucial step is just asserted.”)) it failed to prove a crucial step in the proof, and Gödel merely assumed that the crucial step (the Proposition V in his paper) was correct. This was completely unacceptable to the reviewers, and Gödel’s paper was never published in this hypothetical mathematical world.
But, as the years rolled on in this mathematical world, large numbers of people still attempted to prove what Gödel tried to prove, but what he never actually did prove. And all these people either tried to rely on an unproven assumption - just like Gödel did - or else they made basic logical errors. (Footnote: See, for example:
The Flaw in Gödel’s Proof of his Incompleteness Theorem
Paper(PDF): The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem
Analysis of Other Incompleteness Proofs
Common Errors in Incompleteness Proofs
Yet another flawed incompleteness proof) In this alternative mathematical world, such people are ridiculed and are called cranks - because what they are doing strikes against the fundamental ethos of this mathematical world, where the establishment of a logical proof of any claim is of paramount importance.
Now, let us look instead at the mathematical world that we actually inhabit. In our actual mathematical world, such people aren’t called cranks. No, often they are professors and have prestigious positions within our mathematical world. Yes, in our current mathematical world, people that should be called cranks and who should be reprimanded for promoting fake mathematics are accepted and even applauded for what they do. In the actual mathematical world that we inhabit, fake mathematics is sitting alongside normal mathematics, instead of being banished forever from it. Surely this is unacceptable in a community in the 21st century that claims to be based on rationality?
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There is now a paper that deals with the matter of language and the diagonal proof, see On Considerations of Language in the Diagonal Proof.
There is now a new page on a contradiction in Lebesgue measure theory.
I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly).
I have now added a new section to my paper on Russell O’Connor’s claim of a computer verified incompleteness proof. This shows that the flaw in the proof arises from a reliance on definitions that include unacceptable assumptions - assumptions that are not actually checked by the computer code. See also the new page Representability.
For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem
– a page with general links:
– and a page relating specifically to the Gödel mind-machine debate:
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Copyright © James R Meyer 2012 - 2018