Remarkable mathematical truths

Deductive systems are either incomplete or inconsistent. Meaning

  • Inconsistent: they contain contradictions. Statements can be true and false in the same deductive system.
  • Incomplete: Statements can be found that can not be proven to be true or false.

Gödel proved this for us.

Wittgenstein formulated something similar:

The truth is built of true facts and untrue facts: facts that are not based on a system of observation yet are true anyway. Nevertheless, Wittgenstein seems to disagree with Gödel’s incompleteness theorem. Food for a lasting scientific debate. Anyway, Wittgenstein was looking at language and philosophy, not at mathematics.

Final remarkable mathematical truth for now from Cantor.

Cantor proved that one infinity is not the same as the other infinity. He developed a way to compare infinite sets and describe how infinite sets with different characteristics exist.

As an example, Cantor proved that real numbers are more numerous than the set of natural numbers. While both are infinite. He also invented a way to operate on infinite sets.

Cantor ended up in a mental hospital, which seems to be viewed as as heroic achievement among mathematicians—an opinion I do not share.

I recall reading The Mystery of the Aleph by Amir D. Aczel about Cantor. Unfortunately, I have lost my notes and the book. This book was very accessible, I do recall that.

Dawkins’ embellished account

the selfish gene

In the 30th anniversary edition of ‘The Selfish Gene’ (2006) Richard Dawkins writes a vile but correct comment on Fred Hoyle‘s misrepresentation of Darwinism in an endnote (pp. 277-278). He ends his note:

‘Publishers should correct the misapprehension that a scholar’s distinction in one field implies authority in another. And as long as that misapprehension exists, distinguished scholars should resist the temptation to abuse it’.

Very true, though on the same page, in the note referenced on this page (page 59 of the 30th Anniversary edition), Dawkins almost falls into his own trap, saving himself with one little sentence.

richard dawkins

The text of the note to the main text is so incredibly incorrect that it is quite funny, given he does this on the very same page as his scolding on Hoyle.

In the note Dawkins for some reason wants to explain a theory of consciousness by Daniel Dennett. Dennett himself has tried to explain his ideas in several books. For reasons that remain unclear Dawkins wants to summarize Dennett’s work in this 2 page note.

Dawkins takes two technical ideas from the world of computers to illustrate his ideas: the concept of a virtual machine, and the ‘the distinction between serial and parallel processors’. Dawkins starts out with a completely incorrect explanation of what a virtual machine is. As an example of a virtual machine he mentions the Macintosh User Interface. The Mac is a great machine, but the Macintosh User Interface has very little to do with a virtual machine, and the connection with consciousness remains very unclear. For a correct description of virtual machines, Dawkins could have simply relied on the  Wikipedia article on virtual machines.

douglas r. hofstadter

The story derails entirely when Dawkins turns to his description of ‘serial and parallel processors’. The piece is so totally incorrect that it does not make sense to highlight the individual errors here. Since Dawkins fails to see the distinction between processors and processes, he starts off wrong and makes things worse every sentence. And it’s not like this was rockets science at the time of writing. Parallel processing is known and applied in computing since our own Edsger Dijkstra and others invented concepts like the semaphore and the indivisible instruction.

daniet dennett

More linkages to Dennett’s work and that of his friend Douglas Hofstadter on page 59, where Dawkins discusses self-awareness and rejects ideas of self-awareness because

‘it involves an infinite regress if there is a model of the model, why not a model of the model of the model …?’
The Mind’s I‘ and also ‘Gödel, Escher, Bach – An Eternal Golden Braid‘ deal exactly with these issues.

the minds eye

So can we conclude Dawkins has fallen into the trap of asserting a scholar’s distinction in one field implies authority in another?

As I said, almost. On page 280 Dawkins saves himself, on the edge, with this little remark:

‘The reader is advised to consult Dennett’s own account when it is published, rather than rely on my doubtless imperfect and impressionistic – maybe even embellished – one.’
How true.

godel escher bach

I have never had such fun with academic footnotes.