Anything for Aleph

Beyond Infinity by Eugenia Cheng

This is a maths book by a woman and it certainly has that feminine touch. Many of her analogies are drawn from her day-to-day life: children climbing stairs, lego bricks, recipes, hiking holidays. Sounds trite, doesn’t it … but no! I’ve read dozens of books about infinity and this is right up there among the best.

I’ll start again! The thing I hate when I read “introductory” texts in maths or science is that I can’t help wondering where they draw the line between plagiarism and the work in question. I’ve read the same stuff over and over again, written by diverse writers and their repetitive dishing out of the same old stuff makes me feel I’ve been conned. If I had a cent for every author who told me how Gauss, as a child, calculated the sum of the numbers from 1 to 100, I’d have more than €50.50 to my name. (In that connection, don’t miss the very different approach taken by Brian Hayes in Foolproof). The thing about Eugenia Cheng‘s book is that, although she covers familiar ground for me, I don’t think anyone has done it with quite such clarity before. A pleasure to read and I feel I learned a lot of things I thought I knew before.

For instance, I now understand that:  ∞ + 1 ≠ 1 + ∞

I was so taken with this book that I looked the author up and found this presentation she made on UTube. Definitely a lady to pay attention to.

Lies, Damned Lies and Statistics

I want to put down my thoughts on the 50% virus propagation rate now claimed by the Authorities.

First a teentsy weentsy bit of maths: Here’s how propagation works. A group of people have the virus and, on average, each passes it on to ρ more people. If ρ is less than one, we expect the virus to eventually run its course. (I’m assuming there are sufficient uninfected individuals to pass it on to or it will run its course even sooner). The average number of people affected by each initial carrier will eventually be ρ / (1 – ρ). (NB: this formula is only valid when ρ is strictly less that 1).

So, for instance, if ρ = 0.8, one initial carrier will affect 0.8 others, who will affect 0.64, who will affect 0.512, who will … and so on until ultimately a total of 4 people will be affected. And that’s all!  Finito!

So, if ρ = 0.5, as claimed, the total affected by one initial carrier will be one more!

0.5 + 0.25 + 0.125 + … = 1

Here’s a little diagram illustrating how the one person on the left eventually produces one further infected person on the right.

This means that of all the people currently infected, assuming they have not yet had the indelicacy to infect another, they will still only infect one more.

Do you believe the claim that the rate is only 50%? We can think of it an an iceberg where we don’t have all the information but we are making a prediction that in spite of what we don’t know, we don’t expect to have any more that twice as many cases in total as those already in existence. This is a very confident claim, suggesting that the pandemic is virtually over.

Baye’s Theorem

Medical testing is much in the news. I thought I’d do a little note to remove the mystique around the so-called false positives and false negatives which seem to be widely misunderstood.

A false positive is a test result which says you have the condition but actually don’t. It causes anguish but is arguably less dangerous than a false negative which says you don’t have the condition when actually you do.

So if you test for some condition and you get a positive result, just how worried should you be? Imagine the test is known to produce false positives 5% of the time and false negatives 10% of the time. The condition is know to affect 2% of the population.

When 100,000 people are tested,  the test will report as follows:

So the test proves very reliable for eliminating people without the condition but usually further more detailed testing is required to home in on those who have it.

Of course the test returns only positive or negative. It is the effectiveness of the test that provides a warning about how accurate the result is.

Simpson’s Paradox

With all the talk of anti-viral drug research in the news, I thought I’d amuse you with a little drug research paradox attributed to briton Edward Hugh Simpson, but noticed earlier by scotsman George Udny Yule.

Two drugs are being compared. In a trial in Hospital A, the results favour Drug 1 as follows:
In a further trial in Hospital B, the results the results also favour Drug 1 as follows:

Somebody has the idea of combining the two tables. Now it is Drug 2 that heads the effectiveness stakes!

As Mark Twain noted: There are lies, damned lies and statistics.

Absurd Science

Review: The Quantum Astrologer’s Handbook by Michael Brooks

Jerome Cardano was a 16th Century mathematician who, among other things, was the first to look at probability theory and make sense of imaginary numbers. He oscillated from poverty to riches and back, was fêted in royal courts across Europe and, like Galileo, fell foul of the Inquisition. He defended but failed to save his simple son from execution for murder. He has been much maligned in any previous history I’ve read concerning him.

This book is written by a Quantum Physicist, not a Mathematician and it treats him much more sympathetically, warts and all. Brooks is a scientist, but he is perfectly happy to tolerate some of Cardano’s crazier beliefs (e.g. astrology; he wrote a horoscope for Jesus) since he himself defends some rather strange quantum ideas including String Theory.

This book is a very accessible introduction to the craziness of Quantum Theory and how Cardano laid some of its foundations. It requires little or no physics or mathematics knowledge. Its great value for me was how it uncovers how every age has its ideas that seem bizarre in retrospect and how we should show respect for those ideas and wait to see where they lead.

Surreal Mathematician

Review: Genius at Play by Siobhan Roberts

When I’m drowning in the desire to read “something different” but have no idea what the subject might be, I go to Books Upstairs on D’Olier Street.  Last week they didn’t disappoint me when I stumbled upon a biography of one of my great heroes, John Horton Conway, which I hadn’t known existed.

I’ll accept that unless your NQ¹ is fairly high, you have possibly never even heard of JHC but I’m here to tell you that you’re all the poorer for this. He is perhaps the most quirky, playful and creative professor of mathematics to ever to grace the halls of Cambridge and Princeton.

Alert!!!     Don’t switch off

For “mathematics” read “GAMES”

So how would an ordinary mortal come to hear of him?  Well he sprang to prominence when Scientific American’s equally nerd-adored Martin Gardiner published article after article describing his Game of Life (play it here); what Conway calls a zero-player game. The biography reveals that although he revels in the fame it brought, he also bemoans how it has cast a deep shadow over his other accomplishments.

Like Einstein in 1905, he had his annus mirabilis in 1969 inventing ‘Life’, describing the Monster Group (… don’t go there …) and inventing what I think he will eventually be best remembered for: the Surreal Number which I like to call 𝕊.  Don’t get me going on them; I won’t stop.

So why am I bothering to write this up and attract your witty rejoinders?  Because this is laugh-out-loud biography. Conway is a (deliberately) larger than life man, who claims never to have worked a day in his life and he must be, by far, the most entertaining educator ever. It’s so interesting to see also the complete (artistic!) freedom that academia awards to someone like him to “waste” countless hours playing and fooling-around in, no doubt, the confident belief that genius has its own path to follow.

I read this 400-page biography in 3 days; it’s un-put-downable. This is a book about the full expression of creativity and his biographer adopts his quirkiness in her presentation of him. To her credit, she shows him warts and all but at the end, warts aside, I still revere him!

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^{1. Nerd Quotient↩}

‘Tis Brillig

Review: Gödel, Escher, Bach: an Eternal Golden Braid by Douglas R. Hofstadter

Rating: 10/10

If I could take only one book to the proverbial desert island, this would have to be it.

GEB, as it is affectionately known to thousands of fans, is hard to classify.  It is possibly a philosophy book, or maybe a primer on Gödel’s Incompleteness Theorem* or perhaps it’s a plaything. Gödel’s theorem represents to me one of the high intellectual achievements of the 20th century.  It was a devastating hammer-blow to mathematicians; imagine that you’ve given your life’s work to proving, say, Goldberg’s Conjecture.  Gödel says that not everything that is true is provable; so maybe you’ve wasted your life on one such unprovable conjecture!  (The very fine novel: Uncle Petros and Goldbach’s Conjecture by Apostolos Doxiadis imagined just such an outcome and is well worth reading).

Using an intricate structure where each idea is presented first using very witty Plato-like dialogues which draw on the art-work of M. C. Escher and the fugues of J. S. Bach, Hofstadter introduces us to a range of topics from mathematics and meta-mathematics, computing and logic and (most fun for me) recursion, self-reference and self-representation which are found throughout the works of these two creative artists.  After every chapter you’ll find yourself with pen and jotter (or computer) playing about with his ideas.

An example of the delights to be found in GEB are the translations of Lewis Carroll’s Jabberwocky into French and German.

*Gödel’s Incompleteness Theorem states that no consistent system of [mathematics] is capable of proving all truths about the natural numbers. There will always be statements about the natural numbers that are true, but that are unprovable within the system, and statements that are false, but not disprovable within the system. Furthermore the system will not be able to demonstrate its own consistency.