Mathematical proof

Introduction to Mathematics

Fractals and Life

The First Animals Were Living Fractals

Mathematics and Knowledge

In this episode of the BBC science podcast 'The Infinite Monkey Cage' the panel discusses the nature of mathematical knowledge.

Eugene Wigner

Eugene Wigner (1902 - 1995) was a Hungarian mathematician who won the Nobel Prize for Physics in 1963 for his work on modelling the atomic nucleus.

In 1960 he published a paper (reproduced below) in which he discussed the relationship between mathematics and the natural sciences. It is a philosophical work in which he considers the nature of empirical predictions and the certainty of mathematical knowledge

Wigner the Unreasonable Effectiveness of Mathematics

What is Maths About?

What is Maths About?

Is There Beauty in Mathematics?

Beauty in Mathematics

Martin Gardner

Martin Gardner

Maths and Infinity

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 Heisenberg'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 (1948 - 2017) was 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 was grounded in solid statistics (often drawn from United Nations data), and he developed interesting and innovative methods of displaying his data through which he was able to appeal even to the most hardened statistic-phobes. He was 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 work focused on the developing world, which he showed 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 felt 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.

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

Coincidences