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Blog Archive

10 Feb 2017    Fake News and Fake Mathematics

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.

 

Picture: Fake mathsIn 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) GödelPicture: Pipe 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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Footnotes:

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Diverse opinions and criticisms are welcome, but messages that are frivolous, irrelevant or devoid of logical basis will be blocked. Difficulties in understanding the site content are usually best addressed by contacting me by e-mail. Note: you will be asked to provide an e-mail address - any address will do, it does not require verification. Your e-mail will only be used to notify you of replies to your comments - it will never be used for any other purpose and will not be displayed. If you cannot see any comments below, see Why isn’t the comment box loading?.

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The Lighter Side

NEWS

Paper on the diagonal proof

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.

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Other recently added pages

The Platonist Rod paradox

 

The Balls in the Urn Paradox

 

How you can tell if someone is a crackpot

 

Platonism’s Logical Blunder

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Lebesgue Measure

There is now a new page on a contradiction in Lebesgue measure theory.

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Easy Footnotes

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).

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O’Connor’s “computer checked” proof

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.

Previous Blog Posts

Descartes’ Platonism

The duplicity of Mark Chu-Carroll

A John Searle Inanity

Man versus Machine

Fake News and Fake Mathematics

Ned Block’s Blockhead

Are we alone in the Universe?

Good Math, Bad Math?

Bishops Dancing with Pixies?

Artificial Intelligence

Cranks and Crackpots

The Chinese Room

Links

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:

Gödel Links

 

– and a page relating specifically to the Gödel mind-machine debate:

Gödel, Minds, and Machines

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