Intuition and Bad Mathematics

One would hope that the justice system is one aspect of everyday life based firmly on reason rather than intuition. However, the article below investigates how court judgements may be flawed if they are based on a weak grasp of statistics. Since scientific and forensic evidence has to be presented in terms of statistics, this does lead to a rather worrying state of affairs and means that the interpretation of expert opinion is vitally important in court judgements.

Bad Maths in Court

Nonsense Mathematics

Here's a nice article tweeted by Richard van de Lagemaat this week. It appears that most of us are much more likely to accept the authority of an article if it includes some mathematics (regardless of whether or not these actually make any sense). Its indicative of the awe in which we generally hold professional mathematicians, but perhaps also shows a lack of understanding of maths by the general public. It also makes an interesting companion to the article I posted previously about 'why we mistrust science'.

Maths and Perception

Mobius Strips

A Mobius Strip is a surface obtained by sticking the ends of a band together, giving it a single twist in the process. It was discovered by German mathematician August Ferdinand Mobius and immortalised by some of the work of Dutch artist Maurits Cornelis Escher. Today, it is probably best known as the basis of the ubiquitous recycling symbol.

The shape itself and those obtained by cutting it are somewhat counter-intuitive. For example it appears only to have one surface and one side. If you trace a pencil line across the centre or edge of a Mobius Strip, you find yourself back at the position you started from.

The study of Mobius Strips and shapes like them gave rise to a new branch of mathematics called 'topology', which is concerned with the basic properties of space and connectedness. This developed from discussing questions about simple geometry to the structure of the Universe itself. The shape is also often used in religious analogies when discussing the multifacetedness of God.

Incidentally, I often wondered why those Heath Robinson-style contraptions (which usually seem to be attached to tractors on farms) use a belt twisted into a Mobius shape. Its obvious to me now that by doing so you double the surface area of the belt in contact with the machinery and therefore prolong its life.

The youtube clip below shows some nice bamboozling tricks you can play by creating Mobius Strips in paper and then cutting them up:

Chaos Theory and the Butterfly Effect

Chaos theory is a branch of mathematics which deals with the wildly differing outcomes resulting from small differences in inputs. Its discovery (invention?) is accredited to American meteorologist Edward Lorenz who was trying to come up with a computer program to predict weather patterns in 1961. He discovered that tiny changes can lead to large effects. So tiny, in fact, that this led to the idea that a butterfly flapping its wings in Beijing could have an effect on weather patterns in New York a month later. Chaos Theory resulted in paradigm shifts not only in mathematics, but across the natural sciences. It dealt a blow to the idea that events in nature can be thought of as being deterministic - in much the same way that Heiseberg's Uncertainty Principle did.

A Butterfly in Beijing

Can Eating Chocolate Help You to Win a Nobel Prize?

Here's a nice example that shows that just because something correlates statistically it doesn't necessarily mean that a relationship actually exists. New York cardiologist Dr. Franz Messerli found a correlation between the consumption of chocolate and the number of Nobel Prizes awarded to a country. However, he doesn't claim that one is related to the other.

This shows that even correctly applied statistics can give false conclusions and perhaps that over-reliance on statistical trends can be flawed.

Chocolate Consumption and Nobel Prizes

The Mandelbrot Plot

Game Theory

I've posted about Game Theory (the application of mathematics and reason to optimise decision making) before. It was a term coined by mathematician John von Neumann in 1944 and is now used in fields as diverse as economics, political science, psychology, logic, biology and military stretegy.

I came across a really nice example recently. It is a situation known as the 'prisoner's dilemma' - where two subjects have the choice to cooperate (which is in the best interests of both), or attempt a gamble to win everything for themselves. This is the classic version of the dilemma quoted by Wikipedia:

"Two men are arrested, but the police do not have enough information for a conviction. The police separate the two men, and offer both the same deal: if one testifies against his partner (defects/betrays), and the other remains silent (cooperates with/assists his partner), the betrayer goes free and the one that remains silent gets a one-year sentence. If both remain silent, both are sentenced to only one month in jail on a minor charge. If each 'rats out' the other, each receives a three-month sentence. Each prisoner must choose either to betray or remain silent; the decision of each is kept secret from his partner. What should they do? If it is assumed that each player is only concerned with lessening his own time in jail, the game becomes a non-zero sum game where the two players may either assist or betray the other. The sole concern of the prisoners seems to be increasing his own reward. The interesting symmetry of this problem is that the optimal decision for each is to betray the other, even though they would be better off if they both cooperated."

Here is a clip from the British TV game show 'Golden Balls'. The only twist on the classic version of the situation above is that both contestants are allowed to speak to each other prior to making their decisions. The guy on the right has obviously read about the Prisoner's Dilemma before and has a strategy sorted out in order to optimise his chances of winning the money (or at least not losing it). His partner seems to be left in a state of panic.

 

Bad Statistics

Paraphrasing an article originally published by Ben Goldacre on his Bad Science blog in 2008:

"In 1954 a man called Darrell Huff published a book called How to Lie with Statistics. Chapter one is called 'the sample with built in bias'.

Huff sets up his headline: 'The average Yaleman, Class of 1924, makes $25,111 a year!' said Time magazine. That figure sounded pretty high: Huff chases it, and points out the flaws. How did they find all these people they asked? Who did they miss? Losers tend to drop off the alma mater radar, whereas successful people are in Who’s Who and the College Record. Did this introduce selection bias into the sample? And how did they pose the question? Can that really be salary rather than investment income? Can you trust people when they self-declare their income? Is the figure spuriously precise? And so on.

In the intervening fifty years this book has sold one and a half million copies, it’s the greatest selling stats book of all time (a very tough market) and it remains in print."

Perhaps one and a half million copies is not going to single-handedly change public attitudes towards statistics, however, you might expect statistical reporting to have improved somewhat. This is certainly not the case in the mainstream media where spurious surveys with headline grabbing conclusions are quoted on a daily basis (I will have to try to dig out some examples to support this accusation), making the same mistakes that Huff was fighting against in the '50s. Perhaps the worst thing is that so few journalists are actually prepared to question the data and consider inbuilt bias. I suppose newspapers have effectively passed this responsibility on to the reader.

Glossary of Philosophy

Enlightenment Rationalism Empiricism The Encyclopaedists Analytic synthetic distinction Logic Determinism Kant	Romanticism Rousseau Schelling and Coleridge Hegel Idealism Vs. Materialism Hegel's Dialectics Marx's Dialectical Materialism Schopenhauer & Nietzsche	Analytic Philosophy Bertrand Russell Wittgenstein: Tractatus Wittgenstein: Language Games The Private Language Argument Richard Rorty Timeline	Existentialism Kierkegaard: The solitary wanderer Martin Heidegger Edmund Husserl Sartre: Existence precedes essence Albert Camus
God Proofs of God's Existence: Argument from design Ontological argument Cosmological argument Pascal's Wager Kierkegaard Marx Nietzsche The Death of God	Mind Plato Aristotle Leibniz Epiphenomenalism Dualism Descartes Identity Theory Spinoza Behaviourism Ryle Ghost in the machine	Natural Science Popper Kuhn Feyerabend Science in the post- modern world	Moral philosophy Kant's Categorical Imperative Utilitarianism Nietzsche: The genealogy of morals Sartre and Bad Faith

Using Statistics

Hans Rosling is professor of global health at Sweden's Karolinska Institute. He began his career as a physician, spending many years in Africa tracking a rare paralytic disease (konzo) and discovering its cause: hunger and badly processed cassava. He co-founded Medecins sans Frontiers (Sweden), and has written a respected textbook (Global Health: An Introductory Textbook, Studentlitteratur AB, Sweden, 2006).

His work is grounded in solid statistics (often drawn from United Nations data), and he has developed interesting and innovative methods of displaying his data through which he is able to appeal even to the most hardened statistic-phobes. He is able to clearly show the importance of collecting and understanding real data (in the mathematical sense) in order to understand the current situation and properly plan for the future.

Much of his current work focuses on the developing world, which he shows is no longer worlds away from the west. In this TED talk he shows that the First and Third Worlds are on the same trajectory toward health, prosperity and longer life, and many countries are moving towards this goal twice as quickly as the west once did. He feels the obstacles to true understanding of the situation are merely problems of perception and our preconceived ideas.

Coincidences

Everybody has stories of strange things which have happened to them which seem to be too unlikely to be mere coincidence (in my case going on holiday to Torremolinos and finding myself sitting in a bar next to an old friend). The human brain seems hardwired to remember strange coincidences (and read paranormal explanations into them). Proponents of the supernatural and pseudoscience take advantage (often, but perhaps not always, maliciously) of the fact that most of us are ready to read more into strange events, rather than just accept them as coincidences. I've previously posted about how psychics, especially, play on this.

I often think that in a city of 23 million people, like the one in which I'm living at the moment, you could say that a "chance in a million" occurrence happens to 23 people every day. You only remember the bars in Torremolinos where you end up sitting next to a friend, and remain blissfully ignorant of all of those bars where they just left before you walked in. We are also unaware of factors which increase the likelihood of a coincidental occurrence (perhaps there was an offer on at my local travel agency on trips to Torremolinos that particular weekend), and we don't recognise those things we perceive unconsciously which make particular thoughts pop into our heads.

There are some nice examples of strange coincidences here.

I read an interesting article on the BBC website by a statistician, offering a more mathematical explanation of coincidence.

Coincidences

Logicomix

Logicomix is a graphic novel written by Greek writer Apostolos Doxiadis and theoretical computer scientist Christos Papadimitriou of the University of California, Berkeley. It follows the life of British philosopher and mathematician Bertrand Russell and his attempt to unite mathematics under the umbrella of logic - what became known as "the Foundational Quest", together with his friends Alfred North Whitehead and Ludwig Wittgenstein.

Russell was a quiet, intelligent, introspective man who was haunted by the demons of depression and dissatisfaction with being unable to complete his life's work. Indeed, one of the book’s recurring themes is the connection between logic and madness. Russell himself was surrounded in his family by mental illness, and several of the logicians and philosophers who appear in the story went insane or suffered from mental instability.

Below is the book's glossary which gives the background to a number of the topics that appear in it and its main characters. It gives a nice summary of the work of most of the main figures in logic throughout history:
Logicomix Glossary

The Monty Hall Problem

The Monty Hall Problem is a 'veridical paradox' (something which seems to defy intuition - as with the Achilles and the tortoise paradox). It was invented (discovered?) by Marylin Vos Savant and first published in Parade Magazine in 1990. It is named after US game show host Monty Hall, although other than supplying his name he has nothing to do with the problem.

In the game show you must chose one of three doors. Behind one is the star prize, and behind the other two are (for some reason) goats. You've chosen one of the doors, hoping that the car is behind it, but if you've chosen badly, you get the goat.

Monty then teases you by opening one of the two unselected doors and revealing a goat (he always chooses a losing door), and he offers you the option of switching doors.

The question is: is there an advantage in switching doors? Should you change your choice in order to increase your chances of winning the car.

The intuitive answer is that there is no advantage in changing your mind, since it seems there is now a 50/50 chance that you will win the car. However, it turns out that given this type of choice you should change your mind. I have found a nice program here that proves the point. It is a simulator in which you can run the scenario automatically up to 1000 times, electing either to keep your original choice or change your mind each time.

If this still seems counter-intuitive, this guy explains it much better than I can, in terms of probabilistic outcomes:

Paradoxes

How long is a piece of string?

In a Horizon documentary made for the BBC in 2009, comedian and professional dunce Alan Davies set out to measure a piece of string he bought in a hardware shop. After meeting a series of pompous and condescending experts he decided it's an impossible task. According to the mathematician his string may be of infinite length, while the physicists tell him that until he goes about measuring it, it actually has no length (the observer effect).

Its an interesting exercise which brings together branches of natural sciences, maths and philosophy.

Explaining mathematics

Here is a good example of why the explanation of a mathematical problem is not always as easy as you think it is going to be:

The Axioms of Arithmetic

During the 20th Century, mathematicians decided that they had to properly define the foundations of maths. Work by philosophers such as Bertrand Russell showed that there were inherent contradictions in the subject that might put many important assumptions in doubt. As several theorems proved to be false, there were concerns that many branches of maths were built on houses of cards that may be found to tumble.

To try to resolve this, a list was produced of axioms which can be taken as self-evident. This means they can be assumed as first principles upon which all other theories can be built. However, even these axioms have not been immune to questioning - and, if you need axioms upon which you build the axioms, doesn´t that also mean you need axioms to build the axioms of the axoms....(ad infinitum)? Anyway, these are the seven Axioms of Arithmetic:

1. For any numbers m, n: m + n = n + m
2. For any numbers m, n, k: (m + n) + k = m + (n + k)
3. For any numbers m, n, k: m(n + k) = mn + mk
4. There is a number 0 which has the property that, for any number n: n + 0 = n
5. There is a number 1 which has the property that, for any number n: n x 1 = n
6. For every number n, there is another number k such that: n + k = 0
7. For any numbers m, n, k, if k ≠ 0 and kn = km, then m = n

A truel

From Fermat´s Last Theorem, by Simon Singh,

"A truel is similar to a duel, except there are three participants rather than two. One morning Mr Black, Mr Grey and Mr White decide to resolve a conflict by truelling with pistols until only one of them survives. Mr Black is the worst shot hitting his target on average only one time in three. Mr Grey is a better shot, hitting his target two times out of three. Mr White is the best shot, hitting his target every time. To make the truel fairer Mr Black is allowed to shoot first, followed by Mr Grey (if he is still alive), followed by Mr White (if he is still alive) and round again until only one of them is alive. The question is this: Where should Mr Black aim his first shot?"

If you've not seen this question before, please feel free to drop me a comment with your answer and a short description of why.

This kind of question is part of the branch of maths which has been called Game Theory, a term coined by John von Neumann in 1944. This became fundamental in the study of military strategy, particularly during the Cold War. People of my generation will be familiar with the 1983 film War Games, in which a young Matthew Broderick attempts to teach game theory to a computer which is about to start a global nuclear war, using his Commodore Vic 20.

Fermat's Last Theorem

Pierre de Fermat (1601-1665) was a French mathematician who, together with Isaac Newton and Gottfried Leibnitz, is credited with the invention of calculus (calculations which allow the mathematical description of two factors relative to each other). He also made notable contributions to analytic geometry, probability, and optics. His first love, however, was number theory - the philosophy of mathematics. He was more interested in solving and setting problems, rather than his own fame and fortune.

One of these, "Fermat's Last Theorem" (named as such because it became the last of his theorems requiring proof), states that no three positive integers x, y, and z can satisfy the equation x^n + y^n = z^n for any integer value of n greater than two. Fermat set the challenge for mathematicians to prove the theorem correct. In a scientific sense, it can't be proved, since this would involve testing an infinite series of numbers. However, maths provides the opportunity to prove a theorem without the limitations of perception, and therefore is considered the purest form of truth and knowledge. Despite the simplicity of the equation, for three hundred years, the challenge beat all of the best minds in the world. Tantalisingly, Fermat left some indications when he died that he had come up with the proof himself. In 1995, softly spoken British mathematician Andrew Wiles, a maths professor at Princeton University, gave a talk at a conference at Cambridge in which he claimed to have met Fermat's challenge. It was the culmination of thirty years of work. He'd been fascinated by the probem since he came across it in a library book at the age of ten. The documentary below shows what was involved in him producing his proof, and the nightmare that resulted from it:

The Game of Life

The Game of Life is a computer simulation devised in the 1960's by the British mathematician John Horton Conway. It's a very good example of how a few simple rules can quickly create order out of chaos. The simulation takes place on a 2-dimensional grid divided into cells. Each cell has eight neighbouring cells and can be either "alive" or "dead". The rules which determine it's fate are very simple:

• If a cell has one or no living neighbours, it will die of loneliness.
• If it has too many neighbours - four or more - it will die from overcrowding.
• New cells are "born" whenever an empty square has exactly three living neighbours.

This clip shows a Game of Life Grid in action:

You can create your own grid by going to the website here

In this interview, John Conway gives his thoughts about the Game of Life: