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Berry’s paradox

August 29, 2012

At the moment I’m reading ‘The Information‘ by James Gleick. I’m only halfway through and it’s a pretty good read so far, though a lot more verbose than Chaos (his first book). It’s sort of ironic that a book about information would have a lower information density than a book about chaos.

When Gleick talks about Russell and famous Set Theory paradoxes, he briefly touches upon the Berry paradox [page 179-180].

It has to do with counting the syllables needed to specify each integer. Generally, of course, the larger the number the more syllables are required. In English, the smallest integer requiring two syllables is se·ven. The smallest requiring three syllables is e·le·ven. The number 121 seems to require six syllables (“one·hun·dred·twen·ty·one”), but actually four will do the job, with some cleverness: “e·le·ven·squared”. Still, even with cleverness, there are only a finite number of possible syllables and therefore a finite number of names, and, as Russell put it, “Hence the names of some integers must consist of at least nineteen syllables, and among these there must be a least. Hence the least integer not nameable in fewer than nineteen syllables must denote a definite integer.”[…] Now comes the paradox. This phrase, the least integer not nameable in fewer than nineteen syllables, contains only eighteen syllables. So the least integer not nameable in fewer than nineteen syllables has just been named in fewer than nineteen syllables. [syllable notation mine]

It took me a while to figure out what was meant by this being a paradox rather than just a linguistic trick. In fact my girlfriend —who is a linguist— still thinks it is a trick, so maybe I’m confused about this still. My understanding is that it is a paradox because of the word “not”. If there would be an integer that cannot be described in less than 19 syllables, then seemingly it can be described in merely 18 after all. Therefore there cannot be a smallest integer which cannot be described in fewer than 19 syllables, which in turn means that all the integers between zero and infinity can be described using the permutations of a finite number of syllables in a limited length sequence, which is clearly bunk. Thus; paradox.

The Wikipedia page claims the resolution is due to ambiguous language, which is pretty much what girlfriend has been trying to tell me as well. I’m clearly missing something yet as I don’t understand how that solves anything.

Biology (especially evolutionary Biology) and Cosmology have always been interests of mine. My level of understanding in either field is probably best described as “blundering amateur”. I am not able to parse —let alone make use of— equations such as one would find in Relativity, Quantum Mechanics, Thermodynamics or Game Theory. I am however a firm believer in the notion that pretty much everything can be explained* through regular language without recourse to mathematics.

I am at the same time disgusted by the sensationalist approach of many documentaries made these days by the likes of Discovery Channel & National Geographic. I switch over almost immediately whenever I’m confronted with a gravelly voice prophesying destruction on a scale hitherto unimagined by scientists ‘when we return after these messages’. Fuck you Discovery Channel for ruining science by removing that which is best about it; knowledge. Even the quality of BBC documentaries has plummeted to lamentable depths over the past decade, which is clearly the most glaring sign of the upcoming apocalypse we could ever hope to get.

At least there is an ever growing number of individuals and small groups who are making quality stuff and distributing it on YouTube and the like. Potholer, Sixty Symbols and Ozmoroid are just a few examples of people who make some excellent content that is free for all. We also see more and more universities recording lectures and putting them on social media for all to see.

Last week I stumbled upon a lecture series by Sean Carroll (from CalTech) about Dark Matter and Dark Energy, two topics about which I knew preciously little. These lectures (24 × half an hour each) are unfortunately not free —they are in fact rather pricey— but they are excellent. Carroll is by far the most intelligible lecturer I know on topics as complicated and unintuitive as Big Bang Cosmology, The Standard Model and Dark Matter/Energy. I have a feeling he pulled as few punches as humanly possible and despite the very high information content, Carroll remains calm and composed. The script is well written (except for a few rather lame jokes) and seems exhaustive in terms of both factual and historic content. I especially liked his treatment of the WMAP data of the Cosmic Background Radiation. It is quite shocking how much information there is embedded in this one image.

Wilkinson Microwave Anisotropy Probe data

Carroll explains what we know for sure to be true, what we think might be true, what we suspect could be true, what we know couldn’t possibly be true, what we’ve yet to know and how we came to know these things. I highly recommend this production for those who wish to learn more about the ‘Dark Side of the Universe‘.

* if one wishes to teach rather than merely explain, I concede that mathematics is often unavoidable.

About three months ago we attended a tasting menu based on Slovak wines at a great local restaurant. Every dish (out of a total of 8 courses) was served with two wines, one from the Vino Nichta winery and the other by Ostrožovič. By and large we liked Ostrožovič better and I decided to contact the winery to see if they would ship to us directly. The following is an —only somewhat— abbreviated transcript of the discussion that spanned a sum total of four emails.

David: “I was wondering if it would be possible to order wines directly from you. We’re interested in the following wines […], roughly 10~20 bottles of each.”

Veronika: “If you mail us the delivery address we can have it shipped in one day.”

David: “That’s great, if you can give me a price estimate for purchase+shipping we can finalize the order.”

Veronika: “We just shipped you the boxes, they should arrive tomorrow. Cash on delivery; xxx,xx €.”

David: “Erm, right…”

Actually I didn’t say that last part, I just thought it. I don’t know whether this aggressive sales strategy is due to an incomplete grasp of polite English by Veronika or whether she’s just efficient at what she does, but the upshot is our pantry is now stocked with a solid 72 bottles of quality wine (to wit: Furmint Sur-Lie, Furmint Solaris, Lipovina Natur, Muškát Žltý and Muškát Saturnia).

Add to this a small batch we bought at the local wine tasting specialist three days ago —located inside the Catholic church bell-tower on the main square, reachable via a narrow staircase I would not enjoy navigating while inebriated— and I think we’re all set for a year and a half, maybe even two. Note to self; do not, repeat not, go on a drinking binge.