Fake News and Fake Mathematics
10 Feb 2017, last updated 22 Sept 2020
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, it can be seen at Gödel’s original incompleteness paper. An English translation of the paper is entitled “On Formally Undecidable Propositions of Principia Mathematica and Related Systems”, viewable online at Gödel’s Incompleteness paper - English translation:.) 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, 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.
And in this mathematical world, the same happened with another paper that Gödel submitted, and since it had a similar intuitive assertion, it was also rejected. Eventually, in this hypothetical world, someone else actually did a rigorous assessment of Gödel’s work, and proved that Gödel’s intuitive assumption stated precisely the opposite of what was actually the case (and in our actual world, this in fact happened about 50 years after Gödel first wrote his paper - for details of this see Gödel’s Intuitive Error No.2).
As the years rolled on in this mathematical world, large numbers of people still attempted to prove the notion of incompleteness that Gödel had tried to prove, but 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
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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