Saturday, 9 September 2023

Is mathematics universal? An argument from sudoku

 I have a horrible feeling that a long time ago I believed, and perhaps even tried ti persuade others, that mathematics is different from other forms of human knowledge and endeavour.  The works of Shakespeare, Beethoven and Rembrandt, for example, are contingent on human attributes, such as our language and emtions and our senses of sight and hearing.  Other species, however evolved, would be unlikely to appreciate these works as we do.  And our science is contingent on the way the universe happens to work: creatures in our universe might appreciate our ideas, but aliens in a completely different universe with different physical laws would not.

But mathematics, I once may have thought, is different.  Mathematical truths, like the facts that there are infinitely many primes or exactly 26 sporadic finite simple groups, are (it would seem) true universally and don't depend on the way humans have evolved or on the physical laws that happen to hold in our universe.  So it makes sense to send the pattern 2, 3, 5, 7, 11, 13, 17, 19, ... as a signal to outer space as a message to potential aliens (although whether it is wise to do so is a different question - our own relationships with other creatures don't suggest that engagement with more powerful species is likely to end well.)

I link to think I was always a little uneasy about this arrogant claim and that as I have grown older and perhaps wiser,  I am increasingly aware that mathematics is a cultural construct.  But what prompted this current ramble is, of all things, an absolutely beautiful sudoku presented on the Cracking the Cryptic Youtube channel - x'clusion by Florian Wortmann.  The break-in (which I didn't see for myself, though I should have) is the most wonderful I have seen.  (Spoiler alert: the rest of this parenthesis uses white text -  to appreciate it you probably need to know a couple of sudoku theorems.)

But would sudoku-solvers from an alien species appreciate it, or does it depend on the structure of the human brain? I can imagine aliens with a different brain structure, with much larger memory.  Such a species could hold all possible sudoku grids in their working memory, and solving a sudoku for them would be quickly achieved by finding the one grid compatible with the puzzle by a brute force search - not using the fascinating logic which our brains require us to apply.

I think this example suggests that mathematics is not as universal as I might once have thought, and that aliens whose brain happened to be structured differently might well have no interest in our mathematics.

(In thinking about this I have also been influenced by this video about algorithms, which shows that a hypothetical computer with a huge amount of fast-access memory could solve by brute force problems far faster than a more traditional conventional computer - the two examples given being solution of a scrambled Rubik's cube, and breaking the Double-DEC encryption system whose 112-bit key might wrongly be assumed to be immune to brute-force attack in a reasonable time.

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