At the 8th MaxEnt conference in 1998, held in Cambridge UK, Ed Jaynes was the star of the show. His opening lecture has the following abstract: “We show how the character of a scientific theory depends on one’s attitude toward probability. Many circumstances seem mysterious or paradoxical to one who thinks that probabilities are real physical properties existing in Nature. But when we adopt the “Bayesian Inference” viewpoint of Harold Jeffreys, paradoxes often become simple platitudes and we have a more powerful tool for useful calculations. This is illustrated by three examples from widely different fields: diffusion in kinetic theory, the Einstein–Podolsky–Rosen (EPR) paradox in quantum theory [he refers here to Bell’s theorem and Bell’s inequalities], and the second law of thermodynamics in biology.”
Unfortunately Jaynes was completely wrong in believing that John Bell had merely muddled up his conditional probabilities in proving the famous Bell inequalities and deriving the famous Bell theorem. At the conference, astrophysicist Steve Gull presented a three line proof of Bell’s theorem using some well known facts from Fourier analysis. The proof sketch can be found in a scan of four smudged overhead sheets on Gull’s personal webpages at Cambridge University.
Together with Dilara Karakozak I believe I have managed to decode Gull’s proof, https://arxiv.org/abs/2012.00719, though this did require quite some inventiveness. I have given a talk presenting our solution and point out further open problems. I have the feeling progress could be made on interesting generalisations using newer probability inequalities for functions of Rademacher variables.
a <- 1:360
b <- 1:360
AB <- outer(-sign(cos(pi * a / 180)), sign(cos(pi * b / 180)), "*")
d <- outer(a, b, "-")
ABvec <- as.vector(AB)
dvec <- as.vector(d)
out <- aggregate(x = ABvec, by = list(dvec), FUN = mean)
dvals <- out[ , 1]
corrs <- out[ , 2]
plot(dvals, corrs, type = "l")
lines(dvals, -cos(pi * dvals / 180), col = “magenta”)
'data.frame': 719 obs. of 2 variables:
$ Group.1: int -359 -358 -357 -356 -355 -354 -353 -352 -351 -350 …
$ x : num -1 -1 -0.999 -0.998 -0.997 …
You can see the context of this R programming exercise, here:
Take two large wooden disks and colour half of each black, half white. Fix to a wall. There’s a pointer painted on the disks, near the edge, in the middle of the black half of the circumference. Painted on the wall, just above the top of each disk, is another pointer. Around the side of the disk, painted on the wall, equally spaced, are the numbers 1 to 360; 360 on top. Spin each disk. Wait till it stops. The pointers painted in the disks each point to a number alpha, beta between 1 and 360, painted on the wall. The pointers at the top of the disk painted on the wall point either to black, or to white, in the disk. That defines A = +/- 1 and B = +/- 1. You repeat this fairground game many, many times, and average A times B for each value of delta = alpha – beta.
Image: two fairground “wheels of fortune” which have come to rest at A = +1 (black on top), B = -1 (white on top), alpha = 40 (approx), beta = 170 (approx)
The earlier drawn graph is, I believe, piecewise quadratic.
I want to document the more than 20 species of wild mushrooms which I’ve collected and enjoyed eating this year. I will go through my collection of photographs in reverse chronological order. But above, the featured image, taken back in September: Neoboletus luridiformis, the scarletina bolete; in Dutch, heksenboleet (witch’s bolete. Don’t worry. The guy to avoid is the devil’s bolete).
I get my mushroom knowledge from quite a few books and from many websites. In this blog I will just give the English and Dutch wikipedia pages for each species. I highly recommend Google searching the Latin name (though notice – scientific names do change, as science gives us new knowledge) and if your French, German or other favourite language also has a wikipedia page, nature lover’s web pages, forager’s webpages, or whatever, check them out, because ideas of edibility and of how to cook mushrooms which are considered edible varies all over the world. If at some time there was a famine, and the only country people who could survive were those who went out in the forest and found something they could eat, then their fellows who had allergic reactions to those same mushrooms did not survive, and in this way different human populations are adapted to different fungi populations. It’s also very important to consult local knowledge (in the form of local handbooks, local websites) since the dangerous poisonous look-alikes which you must avoid vary in different parts of the world.
Do not eat wild mushrooms raw. You don’t know what is still crawling about in it, and you don’t know what has pooped or pissed on it or munched at it recently. Twenty minutes gentle cooking should destroy anything nasty, and moreover, it breaks down substances which are hard for humans to digest. The rigid structure of mushrooms is made of chitin (which insects use for their external body) and we cannot digest it raw. Some people have allergic reactions to raw chitin.
Appendix: some mushrooms and fungi to be wondered at, but not eaten
1. Paralepista flaccida
Tawny funnel, Roodbruine schijnridderzwam. Grows in my back garden in an unobtrusive spot, fruiting every year in December to January. Yellow-pinkish spore print, lovely smell, nice taste. Also after frying! The combination of aroma/taste/spore-print just does not fit any of the descriptions of this mushroom or those easy to confuse with it which I can find. There is a poisonous lookalike which however is not supposed to taste good, so that’s why I dared to eat this one. It grows close to a Lawson cyprus but there may be other old wood remains underground in the same spot.
Honey fungus, echte honingzwam. These fellows are growing out of the base of majestic beech trees at Palace het Loo. The trees are all being cut down now; excuse: “they’re sick”; true reason: high quality beech wood is very valuable. The trees are hosts to numerous fungi, animals, birds. The managers of the park have been doing their best to kill them off for several decades by blowing their fallen leaves away and driving heavy machinery around. Looks like their evil designs are bearing fruit now.
Shaggy ink cap, Geschubde inktzwam. One of the last ones of the season, very fresh, from a field at the entrance to the Palace park. These guys are so delicious, fried in butter with perhaps lemon juice, and a little salt and pepper, they have a gentle mushroom flavour, they somehow remind me of oysters. And of Autumns in Aarhus, picking them often from the lawns of the university campus.
Slippery jack, bruine ringboleet. This one looks rather slimy and it is said that it needs to be cooked well, it disagrees with some people. It didn’t disagree with me at all, but I must say it did not have much flavour, and does feel a bit slippery in your mouth.
Fly agaric, vliegenzwam. This mushroom contains both poisons and psychoactive substances. However, both are water soluble. One therefore boils these mushrooms lightly for 20 minutes in plenty of lightly salted water with a dash of vinegar, then drain and discard the fluid; then they can be fried in butter and brought up to taste with salt and pepper. They are then actually very tasty, in my opinion.
Another use for them is to soak them in a bowl of water and leave in your kitchen. Flies will come and investigate it, taste some get high (literally and figuratively) and drop dead. The smell is pretty disgusting at this stage.
I understand you can dry them, grind to powder, and make tea. This allegedly destroys the poisons but leaves enough of the psychoactive substances to have interesting effects. I haven’t tried it, since one of the effects is to set your heart racing, and since I have a dangerously irregular hearth rhythm already, I should not experiment with this.
Some people munch a small piece raw, from time to time, while walking in the forests. I have tried that – teaspoon size, desertspoon size even, without noticing anything except that perhaps for a moment everything sparkled more beautifully than usual. Probably that was the placebo effect.
Amanita muscaria is not terribly poisonous. If you cook and eat three or four you will probably throw up after an hour or two and also experience rather unpleasant hallucinations. To be rounded off with diarrhea and generally feeling unwell. You might find yourself getting very large or very small, it depends of course whether you nibble from the right-hand edge of the mushroom or the left-hand edge. You might believe you can fly so it can be dangerous to be in high places on your own. The poisons may damage your liver but being water soluble they are quite efficiently and rapidly excreted from the body, which is a good thing, so eating them just once probably won’t kill you and probably won’t give you permanent damage. Several other Amanita species are deadly poisonous. With poisons which do not dissolve in water and do not leave your body after you’ve eaten them, but instead destroy your liver in a few days. One must learn to recognise those mushrooms very well. In my part of the world: Amanita phalloides – the death cap (groene knolamaniet); Amanita pantherina – the panther cap (panteramaniet). I have seen these two even in the parks and roadside verges in my town, as well as in the forests outside. More rare is Amanita virosa – the destroying angel (kleverige knolamaniet). But I believe I have seen it close to home, too. It is a white mushroom with white gills and consequently many people believe you must never touch a white mushroom with white gills. Consequently, writers of mushroom books themselves generally have the idea that edible white mushroom with white gills, which do exist, do not taste particularly good, either, and so one should not bother with them. Hence they do not explain well how you can tell the difference. We will later (i.e., earlier this year) meet the counterexample to that myth.
Because of the psychoactive effects of Amanita muscaria it is actually presently illegal, in the Netherlands, to be found in possession of more than a very small amount.
7. Sparassis crispa
The cauliflower mushroom, grote sponszwam. One of my favourites. It does have the tendency to envelope leaves and insects in its folds. Before cooking it has a wonderful aroma, almost aromatic, but on frying it seems to lose a lot of flavour.