## Posts Tagged ‘**Mathematical Games**’

## “Nature is pleased with simplicity”*…

As Clare Booth Luce once said, sometimes “simplicity is the ultimate sophistication”…

… The uniformity of the cosmic microwave background (CMB) tells us that, at its birth, ‘the Universe has turned out to be stunningly simple,’ as Neil Turok, director emeritus of the Perimeter Institute for Theoretical Physics in Ontario, Canada, put it at a public lecture in 2015. ‘[W]e don’t understand how nature got away with it,’ he added. A few decades after Penzias and Wilson’s discovery, NASA’s Cosmic Background Explorer satellite measured faint ripples in the CMB, with variations in radiation intensity of less than one part in 100,000. That’s a lot less than the variation in whiteness you’d see in the cleanest, whitest sheet of paper you’ve ever seen.

Wind forward 13.8 billion years, and, with its trillions of galaxies and zillions of stars and planets, the Universe is far from simple. On at least one planet, it has even managed to generate a multitude of life forms capable of comprehending both the complexity of our Universe and the puzzle of its simple origins. Yet, despite being so rich in complexity, some of these life forms, particularly those we now call scientists, retain a fondness for that defining characteristic of our primitive Universe: simplicity.

The Franciscan friar William of Occam (1285-1347) wasn’t the first to express a preference for simplicity, though he’s most associated with its implications for reason. The principle known as Occam’s Razor insists that, given several accounts of a problem, we should choose the simplest. The razor ‘shaves off’ unnecessary explanations, and is often expressed in the form ‘entities should not be multiplied beyond necessity’. So, if you pass a house and hear barking and purring, then you should think a dog and a cat are the family pets, rather than a dog, a cat and a rabbit. Of course, a bunny might also be enjoying the family’s hospitality, but the existing data provides no support for the more complex model. Occam’s Razor says that we should keep models, theories or explanations simple until proven otherwise – in this case, perhaps until sighting a fluffy tail through the window.

Seven hundred years ago, William of Occam used his razor to dismantle medieval science or metaphysics. In subsequent centuries, the great scientists of the early modern era used it to forge modern science. The mathematician Claudius Ptolemy’s (

c100-170 CE) system for calculating the motions of the planets, based on the idea that the Earth was at the centre, was a theory of byzantine complexity. So, when Copernicus (1473-1543) was confronted by it, he searched for a solution that ‘could be solved with fewer and much simpler constructions’. The solution he discovered – or rediscovered, as it had been proposed in ancient Greece by Aristarchus of Samos, but then dismissed by Aristotle – was of course the solar system, in which the planets orbit around the Sun. Yet, in Copernicus’s hands, it was no more accurate than Ptolemy’s geocentric system. Copernicus’s only argument in favour of heliocentricity was that it was simpler.Nearly all the great scientists who followed Copernicus retained Occam’s preference for simple solutions. In the 1500s, Leonardo da Vinci insisted that human ingenuity ‘will never devise any [solutions] more beautiful, nor more simple, nor more to the purpose than Nature does’. A century or so later, his countryman Galileo claimed that ‘facts which at first seem improbable will, even on scant explanation, drop the cloak which has hidden them and stand forth in naked and simple beauty.’ Isaac Newton noted in his

Principia(1687) that ‘we are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances’; while in the 20th century Einstein is said to have advised that ‘Everything should be made as simple as possible, but not simpler.’ In a Universe seemingly so saturated with complexity, what work does simplicity do for us?Part of the answer is that simplicity is the defining feature of science. Alchemists were great experimenters, astrologers can do maths, and philosophers are great at logic. But only science insists on simplicity…

Just why do simpler laws work so well? The statistical approach known as Bayesian inference, after the English statistician Thomas Bayes (1702-61), can help explain simplicity’s power. Bayesian inference allows us to update our degree of belief in an explanation, theory or model based on its ability to predict data. To grasp this, imagine you have a friend who has two dice. The first is a simple six-sided cube, and the second is more complex, with 60 sides that can throw 60 different numbers. Suppose your friend throws one of the dice in secret and calls out a number, say 5. She asks you to guess which dice was thrown. Like astronomical data that either the geocentric or heliocentric system could account for, the number 5 could have been thrown by either dice. Are they equally likely? Bayesian inference says no, because it weights alternative models – the six- vs the 60-sided dice – according to the likelihood that they would have generated the data. There is a one-in-six chance of a six-sided dice throwing a 5, whereas only a one-in-60 chance of the 60-sided dice throwing a 5. Comparing likelihoods, then, the six-sided dice is 10 times more likely to be the source of the data than the 60-sided dice.

Simple scientific laws are preferred, then, because, if they fit or fully explain the data, they’re more likely to be the source of it.

…

In my latest book, I propose a radical, if speculative, solution for why the Universe might in fact be as simple as it’s possible to be. Its starting point is the remarkable theory of cosmological natural selection (CNS) proposed by the physicist Lee Smolin. CNS proposes that, just like living creatures, universes have evolved through a cosmological process, analogous to natural selection.

Smolin came up with CNS as a potential solution to what’s called the fine-tuning problem: how the fundamental constants and parameters, such as the masses of the fundamental particles or the charge of an electron, got to be the precise values needed for the creation of matter, stars, planets and life. CNS first notes the apparent symmetry between the Big Bang, in which stars and particles were spewed out of a dimensionless point at the birth of our Universe, and the Big Crunch, the scenario for the end of our Universe when a supermassive black hole swallows up stars and particles before vanishing back into a dimensionless point. This symmetry has led many cosmologists to propose that black holes in our Universe might be the ‘other side’ of Big Bangs of other universes, expanding elsewhere. In this scenario, time did not begin at the Big Bang, but continues backwards through to the death of its parent universe in a Big Crunch, through to its birth from a black hole, and so on, stretching backward in time, potentially into infinity. Not only that but, since our region of the Universe is filled with an estimated 100 billion supermassive black holes, Smolin proposes that each is the progenitor of one of 100 billion universes that have descended from our own.

The model Smolin proposed includes a kind of universal self-replication process, with black holes acting as reproductive cells. The next ingredient is heredity. Smolin proposes that each offspring universe inherits almost the same fundamental constants of its parent. The ‘almost’ is there because Smolin suggests that, in a process analogous to mutation, their values are tweaked as they pass through a black hole, so baby universes become slightly different from their parent. Lastly, he imagines a kind of cosmological ecosystem in which universes compete for matter and energy. Gradually, over a great many cosmological generations, the multiverse of universes would become dominated by the fittest and most fecund universes, through their possession of those rare values of the fundamental constants that maximise black holes, and thereby generate the maximum number of descendant universes.

Smolin’s CNS theory explains why our Universe is finely tuned to make many black holes, but it does not account for why it is simple. I have my own explanation of this, though Smolin himself is not convinced. First, I point out that natural selection carries its own Occam’s Razor that removes redundant biological features through the inevitability of mutations. While most mutations are harmless, those that impair vital functions are normally removed from the gene pool because the individuals carrying them leave fewer descendants. This process of ‘purifying selection’, as it’s known, maintains our genes, and the functions they encode, in good shape.

However, if an essential function becomes redundant, perhaps by a change of environment, then purifying selection no longer works. For example, by standing upright, our ancestors lifted their noses off the ground, so their sense of smell became less important. This means that mutations could afford to accumulate in the newly dispensable genes, until the functions they encoded were lost. For us, hundreds of smell genes accumulated mutations, so that we lost the ability to detect hundreds of odours that we no longer need to smell. This inevitable process of mutational pruning of inessential functions provides a kind of evolutionary Occam’s Razor that removes superfluous biological complexity.

Perhaps a similar process of purifying selection operates in cosmological natural selection to keep things simple…

It’s unclear whether the kind of multiverse envisaged by Smolin’s theory is finite or infinite. If infinite, then the simplest universe capable of forming black holes will be infinitely more abundant than the next simplest universe. If instead the supply of universes is finite, then we have a similar situation to biological evolution on Earth. Universes will compete for available resources – matter and energy – and the simplest that convert more of their mass into black holes will leave the most descendants. For both scenarios, if we ask which universe we are most likely to inhabit, it will be the simplest, as they are the most abundant. When inhabitants of these universes peer into the heavens to discover their cosmic microwave background and perceive its incredible smoothness, they, like Turok, will remain baffled at how their universe has managed to do so much from such a ‘stunningly simple’ beginning.

The cosmological razor idea has one further startling implication. It suggests that the fundamental law of the Universe is not quantum mechanics, or general relativity or even the laws of mathematics. It is the law of natural selection discovered by Darwin and Wallace. As the philosopher Daniel Dennett insisted, it is ‘The single best idea anyone has ever had.’ It might also be the simplest idea that any universe has ever had.

Does the existence of a multiverse hold the key for why nature’s laws seem so simple? “Why simplicity works,” from JohnJoe McFadden (@johnjoemcfadden)

* “Nature does nothing in vain when less will serve; for Nature is pleased with simplicity and affects not the pomp of superfluous causes.” – Isaac Newton, *The Mathematical Principles of Natural Philosophy*

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**As we emphasize the essential,** we might spare a thought for Martin Gardner; he died on this date in 2010. Though not an academic, nor ever a formal student of math or science, he wrote widely and prolifically on both subjects in such popular books as *The Ambidextrous Universe* and *The Relativity Explosion *and as the “Mathematical Games” columnist for *Scientific American*. Indeed, his elegant– and understandable– puzzles delighted professional and amateur readers alike, and helped inspire a generation of young mathematicians.

Gardner’s interests were wide; in addition to the math and science that were his power alley, he studied and wrote on topics that included magic, philosophy, religion, and literature (c.f., especially his work on Lewis Carroll– including the delightful* Annotated Alice*— and on G.K. Chesterton). And he was a fierce debunker of pseudoscience: a founding member of CSICOP, and contributor of a monthly column (“Notes of a Fringe Watcher,” from 1983 to 2002) in *Skeptical Inquirer*, that organization’s monthly magazine.

## “We’re supposed to keep evolving. Evolution did not end with us growing opposable thumbs”*…

The CEO of Enron – now in prison – happily applied ‘selfish gene’ logic to his human capital, thus creating a self-fulfilling prophecy. Assuming that the human species is driven purely by greed and fear, Jeffrey Skilling produced employees driven by the same motives. Enron imploded under the mean-spirited weight of his policies, offering a preview of what was in store for the world economy as a whole…

Frans de Waal on the flaws in the “competition-is-good-for-you” logic: “How Bad Biology is Killing the Economy.”

###

**As we concentrate on cooperation,** we might spare a thought for Martin Gardner; he died on this date in 2010. Though not an academic, nor ever a formal student of math or science, he wrote widely and prolifically on both subjects in such popular books as *The Ambidextrous Universe* and *The Relativity Explosion *and as the “Mathematical Games” columnist for *Scientific American*. * *Indeed, his elegant– and understandable– puzzles delighted professional and amateur readers alike, and helped inspire a generation of young mathematicians.

Gardner’s interests were wide; in addition to the math and science that were his power alley, he studied and wrote on topics that included magic, philosophy, religion, and literature (c.f., especially his work on Lewis Carroll– including the delightful* Annotated Alice*— and on G.K. Chesterton). And he was a fierce debunker of pseudoscience: a founding member of CSICOP, and contributor of a monthly column (“Notes of a Fringe Watcher,” from 1983 to 2002) in *Skeptical Inquirer*, that organization’s monthly magazine.

## “Everyone’s quick to blame the alien”*…

In August of 1977, volunteer astronomer Jerry Ehman reviewed readings from Ohio State’s Big Ear Radio Observatory (that’s a scan, above, of Ehman’s notations on the print-out he was assessing)…

He was sitting in his kitchen when he spotted a pattern that a couple of physicists had theorized 18 years earlier would signify alien chatter, according to Michael Brooks, the author of

13 Things That Don’t Make Sense. The printout read 6EQUJ5, a human way of cataloging the 72-second burst of sound registering at a frequency of 1420 MHz. The significance? E.T. may have phoned our home long before Spielberg set otherworldly hearts aglow with his 1982 film…

Scientists have rules out terrestrial sources– the signal came from “out there.” So there are two possibilities: It was an actual alien communication, or Ehman stumbled across a previously undiscovered natural astrophysical phenomenon. And as H. Paul Shuch (an engineer, radio astronomer, and executive director emeritus of the SETI League) observes, “either one would be worthy of a Nobel Prize, if only we knew which.”

Read the whole story at “The ‘Wow’ Signal. or That Time Jerry Ehman May Have Heard From Aliens.”

* Aeschylus, *The Suppliant Maidens*

###

**As we phone home,** we might spare a thought for Martin Gardner; he died on this date in 2010. Though not an academic, nor ever a formal student of math or science, he wrote widely and prolifically on both subjects in such popular books as *The Ambidextrous Universe* and *The Relativity Explosion *and as the “Mathematical Games” columnist for *Scientific American*. * *Indeed, his elegant– and understandable– puzzles delighted professional and amateur readers alike, and helped inspire a generation of young mathematicians.

Gardner’s interests were wide; in addition to the math and science that were his power alley, he studied and wrote on topics that included magic, philosophy, religion, and literature (c.f., especially his work on Lewis Carroll– including the delightful* Annotated Alice*— and on G.K. Chesterton). And he was a fierce debunker of pseudoscience: a founding member of CSICOP, and contributor of a monthly column (“Notes of a Fringe Watcher,” from 1983 to 2002) in *Skeptical Inquirer*, that organization’s monthly magazine.

## Fun with numbers!…

Gary Foshee, a collector and designer of puzzles from Issaquah near Seattle walked to the lectern to present his talk. It consisted of the following three sentences: “I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?”

The event was the Gathering for Gardner [see

here], a convention held every two years in Atlanta, Georgia, uniting mathematicians, magicians and puzzle enthusiasts. The audience was silent as they pondered the question.“The first thing you think is ‘What has Tuesday got to do with it?'” said Foshee, deadpan. “Well, it has everything to do with it.” And then he stepped down from the stage.

Read the full story of the conclave– held in honor of the remarkable Martin Gardner, who passed away last year, and in the spirit of his legendary “Mathematical Games” column in *Scientific American*— in * New Scientist*… and find the answer to Gary’s puzzle there– or after the smiling professor below.

“I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?”… readers may hear a Bayesian echo of **the Monty Hall Problem** on which (R)D has mused before:

The first thing to remember about probability questions is that everyone finds them mind-bending, even mathematicians. The next step is to try to answer a similar but simpler question so that we can isolate what the question is really asking.

So, consider this preliminary question: “I have two children. One of them is a boy. What is the probability I have two boys?”

This is a much easier question: The way Foshee meant it is, of all the families with one boy and exactly one other child, what proportion of those families have two boys?

To answer the question you need to first look at all the equally likely combinations of two children it is possible to have: BG, GB, BB or GG. The question states that one child is a boy. So we can eliminate the GG, leaving us with just three options: BG, GB and BB. One out of these three scenarios is BB, so the probability of the two boys is 1/3.

Now we can repeat this technique for the original question. Let’s list the equally likely possibilities of children, together with the days of the week they are born in. Let’s call a boy born on a Tuesday a BTu. Our possible situations are:

* When the first child is a BTu and the second is a girl born on any day of the week: there are seven different possibilities.

* When the first child is a girl born on any day of the week and the second is a BTu: again, there are seven different possibilities.

* When the first child is a BTu and the second is a boy born on any day of the week: again there are seven different possibilities.

* Finally, there is the situation in which the first child is a boy born on any day of the week and the second child is a BTu – and this is where it gets interesting.There are seven different possibilities here too, but one of them – when both boys are born on a Tuesday – has already been counted when we considered the first to be a BTu and the second on any day of the week. So, since we are counting equally likely possibilities, we can only find an extra six possibilities here.

Summing up the totals, there are 7 + 7 + 7 + 6 = 27 different equally likely combinations of children with specified gender and birth day, and 13 of these combinations are two boys. So the answer is 13/27, which is very different from 1/3.

It seems remarkable that the probability of having two boys changes from 1/3 to 13/27 when the birth day of one boy is stated – yet it does, and it’s quite a generous difference at that. In fact, if you repeat the question but specify a trait rarer than 1/7 (the chance of being born on a Tuesday), the closer the probability will approach 1/2.

[See **UPDATE**, below]

**As we remember, with Laplace, that “the theory of probabilities is at bottom nothing but common sense reduced to calculus,”** we might ask ourselves what the odds are that on this date in 1964 the World’s Largest Cheese would be manufactured for display in the Wisconsin Pavilion at the 1964-65 World’s Fair. The 14 1/2′ x 6 1/2′ x 5 1/2′, 17-ton cheddar original– the product of 170,000 quarts of milk from 16,000 cows– was cut and eaten in 1965; but a replica was created and put on display near Neillsville, Wisconsin… next to **Chatty Belle**, the World’s Largest Talking Cow.

The replica on display (*source*)

**UPDATE:** reader Jeff Jordan writes with a critique of the reasoning used above to solve Gary Foshee’s puzzle:

For some reason, mathematicians and non-mathematicians alike develop blind

spots about probability problems when they think they already know the

answer, and are trying to convince others of its correctness. While I agree

with most of your analysis, it has one such blind spot. I’m going move

through a progression of variations on another famous conundrum, trying to

isolate these blind spots and eventually get the point you overlooked.Bertrand’s Box Paradox: Three identical boxes each have two coins inside:

one has two gold coins, one has two silver coins, and one has a silver coin

and a gold coin. You open one and pull out a coin at random, without seeing

the other. It is gold. What is the probability the other coin is the same

kind?A first approach is to say there were three possible boxes you could pick,

but the information you have rules one out. That leaves two that are still

possible. Since you were equally likely to pick either one before picking a

coin, the probability that this box is GG is 1/2. A second approach is that

there were six coins that were equally likely, and three were gold. But two

of them would have come out of the GG box. Since all three were equally

likely, the probability that this box is GG is 2/3.This appears to be a true paradox because the “same” theoretical approach –

counting equally likely cases – gives different answers. The resolution of

that paradox – and the first blind spot – is that this is an incorrect

theoretical approach to solving the problem. You never want to merely count

cases, you want to sum the probabilities that each case would produce the

observed result. Counting only works when each case that remains possible

has the same chance of producing the observed result. That is true when you

count the coins, but not when you count the boxes. The probability of

producing a gold coin from the GG box is 1, from the SS box is 0, and from

the GS box is 1/2. The correct answer is 1/(1+0+1/2)=2/3. (A second blind

spot is that you don’t “throw out” the impossible cases, you assign them a

probability of zero. That may seem like a trivial distinction, but it helps

to understand what probabilities other than 1 or 0 mean.)This problem is mathematically equivalent to the original Monty Hall

Problem: You pick Door #1 hoping for the prize, but before opening it the

host opens Door #3 to show that it is empty. Given the chance, what is the

probability you win by switching to door #2? Let D1, D2, and D3 represent

where the prize is. Assuming the host won’t open your door, and knows where

the prize is so he always opens an empty door, then the probability D2 would

produce the observed result is 1, that D3 would is 0, and that D1 is …

well, let’s say it is 1/2. Just like before, the probability D2 now has the

prize is 1/(1+0+1/2)=2/3.Why did I waffle about the value of P(D1)? There was a physical difference

with the boxes that produced the explicit result P(GS)=1/2. But here the

difference is logical (based on the location of the prize) and implicit. Do

we really know the host would choose randomly? In fact, if the host always

opens Door #3 if he can, then P(D1)=1 and the answer is 1/(1+0+1)=1/2. Or if

he always opens Door #2 if he can, P(D1)=0 and the answer is 1/(1+0+0)=1.

But if we observe that the host opened Door #2 and assume those same biases,

the results reverse.To answer the question, we must assume a value for P(D1). Assuming anything

other than P(D1)=1/2 implies a bias on the part of the host, and a different

answer if he opens Door #2. So all we can assume is P(D1)=1/2, and the

answer is again 2/3. That is also the answer if we average the results over

many games with the same host (and a consistent bias, whatever it is). The

answer most “experts” give is really that average, and it is a blind spot

that they are not using all the information they have in the individual

case.We can make the Box Paradox equivalent to this one by making the random

selection implicit. Someone looks in the chosen box, and picks out a gold

coin. The probability is 2/3 that there is another gold coin if that person

picks randomly, 1/2 if that person always prefers a gold coin, and 1 if that

person always prefers a silver one. Without knowing the preference, we can

only assume this person is unbiased and answer 2/3. Over many experiments,

it will also average out to 2/3 regardless of the bias. And this person

doesn’t even have to show the coin. If we assume he is truthful (and we can

only assume that), the answers are the same if he just says “One coin is

gold.”Finally, make a few minor changes to the Box Paradox. Change “silver” to

“bronze.” Let the coins be minted in different years, so that the year

embossed on them is never the same for any two. Add a fourth box so that one

box has an older bronze coin with a younger gold coin, and one has a younger

bronze coin with an older gold coin. Now we can call the boxes BB, BG, GB,

and GG based on this ordering. When our someone says “One coin is bronze,”

we can only assume he is unbiased in picking what kind of coin to name, and

the best answer is 1/(1+1/2+1/2+0)=1/2. If there is a bias, it could be

1/(1+1+1+0)=1/3 or 1/(1+0+0+0)=1, but we can’t assume that. Gee, this sounds

oddly familiar, except for the answer. :)The answer to all of Gary Foshee’s questions is 1/2. His blind spot is that

he doesn’t define events, he counts cases. An event a set of outcomes, not

an outcome itself. The sample space is the set of all possible outcomes. An

event X must be defined by some property such that every outcome in X has

that property, *and* every outcome with the property is in X. The event he

should use as a condition is not “this family includes a boy (born on a

Tuesday)”, it is “The father of this family chooses to tell you one of up to

two facts in the form ‘my family includes a [gender] (born on a [day]).'”

Since most fathers of two will have two different facts of that form to

choose from, Gary Foshee should have assigned a probability to each, not

merely counted the families that fit the description. The answer is then

(1+12P)/(1+26P), where P is the probability he would tell us “one is a boy

born on a Tuesday” when only one of his two children fit that description.

The only value we can assume for P is 1/2, making the answer

(1+6)/(1+13)=1/2. Not P=1 and (1+12)/(1+26)=13/27.And the blind spot that almost all experts share, is that this means the

answer to most expressions of the simpler Two Child Problem is also 1/2. It

can be different, but only if the problem statement makes two or three

points explicit:1) Whatever process led to your knowledge of one child’s gender had access

to both children’s genders (and days of birth).

2) That process was predisposed to mention boys over girls (and Tuesdays

over any other day).

3) That process would never mention facts about both children.When Gary Foshee tells you about one of his kids, #2 is not satisfied. He

probably had a choice of two facts to tell you, and we can’t assume he was

biased towards “boy born on Tuesday.” Just like Monty Hall’s being able to

choose two doors changes the answer from 1/2 to 2/3, Gary Foshee’s being

able to choose two facts changes the answer from 13/27 to 1/2. It is only

13/27 if he was forced to mention that fact, which is why that answer is

unintuitive.

Other readers are invited to contribute their thoughts.

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