I am quite the fanboi for the YT channel Numberphile [Tony on Euler / Fermat] which has not-too-long pieces explaining quirky bits of the Mathoverse. One of the reg'lar contributors is Tony Padilla, Associate Director of the Nottingham Centre of Gravity, UK. He featured on Sean Carroll's Mindscapes podcast because of his 2022 book Fantastic Numbers and Where to Find Them: A Cosmic Quest from Zero to Infinity. I'm defo the fanboi for Mindscapes although I usually skip it when Carroll invites a fellow cosmologist to geek out on black holes and the end of time. Padilla is a bit different because although he knows as much as anyone alive today about gravity, he doesn't go at it with alienating gravitas . . . he's more chirpy and engaging. On the podcast, the book was mentioned, so I reserved a copy online for delivery at my branch library. It takes forever for the inter-county library system to move a book "available" in Wicklow to "behind Clodagh's desk" in our library. Cripes, I could have walked to Wicklow and back several times in the lag-time.
Fantastic Numbers is as much fun as it can be when Graham's Number and TREE(3) are far too big to fit in my head. Indeed Padilla riffs on about how any attempt to internalize either of those very large numbers would result in black hole head death even if each memorized / read digit was smaller than the smallest small imaginable. Can't find time to struggle through 300 pages of higher math? There's a 5 point exec summary at NextBigIdeaClub.
Now here's the thing. How does me-the-punter rate such a book when so much of it is teetering on the edge of my comprehension? I mean, I might want a correct exec summary of Graham's number so I can wow my pals down the pub. In a similar position wrt dictionaries, I applied the Fodor' Guide test. This is the idea that, IF you find something sketchy in a part of the book about which you do have some knowledge, THEN you may need to crank up the crap-detector for the bits where you seek new information to cram into your head. Careful! black hole head death awaits those who overdo the knowledge acquisition schtick.
On p.280, there is some 'random' data to help show how Cantor proved that the infinite set of natural numbers did not include all ordered [1st, 2nd, 3rd] numbers . . . and so there was a larger infinity than, like, infinity. Well, I looked sideways at those 85 significant figures and asked "are they random? they don't look random; there are a lot of lucky-7s and only a single "1". Most people are lousy at writing a string of "random numbers": they shy away from including repeats, for a start. 9% of the numbers between 1 and 100 are dupes 11 22 33 44 etc. But there are no such cases in this gang of 84 pairs.
The correct tool for assessing whether bin counts are the same is the Chi.Sq or even χ² test. And I tallied up the count for each of the 10 different digits expecting ~10% of the total [8.5] for each each. Whoa! not even close. It is vanishingly unlikely [p < 0.0005] that these 5 decimals were randomly generated. I first concluded [because 2 + 2 = 22] that there was a coded message on p.280 of this book. Because the dataset is impoverished in 0 1 and 2 ((essential for a [01-26] = [A-Z])) we have 12,08 = L,H in one register and 23,24,08,06,08 = W,X,H,F,H in the other. So prolly no code here. Just some human pretending (not too well) to be a random number generator.
This is why it is important to run job applications through a spell-checker. If your letter shows that you are careless about such details when it is easy to be careful; it might make HR think that you'll be careless with the cash-register. That's a long old way from supersymmetry and the cosmological constant but has more impact on your health, happiness and employability.
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