Big Ideas on Mathematical Logic

My knowledge of Mathematics has dwindled substantially since I got my degree in 1978 but it is still a topic that interests me and I love the Big Ideas events so it was no surprise that I spent the evening of my birthday considering Big Ideas on Mathematical Logic.

Our guide for the evening was Brendan Larvor from the University of Hertfordshire where he specialises in the history and philosophy of mathematics and science.

Over a couple of hours we had an entertaining and thoughtful discussion on mathematical proof (discussion is the whole point of Big Ideas) during which I tried to take some notes while also trying to work out why I did not quite agree with the hypothesis presented. That is my excuse for why the following notes are unstructured and probably inaccurate.

Ancient Greeks first looked at methods of proof. Mathematics was proved using pictures as the use of mathematical notation did not start until around 1630.

Some pictorial proofs are easy and Brendan used the small panes in the windows behind him to prove the that the numbers in the sequence 1+3+5+7+.. are always square numbers (1+3 = 4 which is 2**2 and 1+3+5 = 9 which is 3**2). In pictures this can be seen by adding one more block to two sides of a square to make a slightly bigger square.

Similarly, it's is visibly obvious that 1/2 + 1/4 + 1/8 + ... = 1 as each additional step, 1/2**n, finishes half way between the previous step and 1. Drawing this on a number line makes it obvious.

The invention of mathematical notation made other proofs possible. It is easy to prove that 1/2 + 1/3 + 1/4 + 1/5 + ... is infinite as the sequence can be grouped into an infinite number of segments each of which adds up to more that 1/2, i.e. 1/3 + 1/4 and 1/5 + 1/6 + 1/7 + 1/8.

Mathematical notation is still not always capable of finding proofs. It is fairly easy to see physically that one knot on a line can pass by another (simply by loosening one of them) but there is no formal proof for this. Of course it may be that a proof is not possible because the question is badly formed, and that in itself would be interesting if we could prove it.

There are various standard ways of proving things. Mathematicians like to find other ways to prove things that have already been proved (we may have a proof for Fermat's Last Theorem but it is not the proof that Fermat had, if he had one) and to find other generic ways of proving things. Proof is very important in Mathematics.

The realm of Logic introduced words into Mathematics and, therefore into mathematical proofs. For example, if A implies B and B implies C then A implies C.

This logic is not generally well understood and people still use arguments like you are a man, men play golf therefore you play golf.

Using words to describe mathematics lets linguists into the game and that was the part of the discussion that I had problems with. Just because linguists have rules it does not mean that it makes sense to extend them to the use of language in mathematics. I think that Mathematics is supreme here and if there are any conflicts then it is the upstart linguists that have got a problem to solve.

Some proofs come without insight or meaning. For example computers have solved the four colour map problem by number crunching all possibilities that does not help us to understand why it is true.

It is often said, and I would tend to agree, that the best solutions are simple and elegant but it this just pretension (quite possibly!) or is there something more right about a simple proof than a complex proof. If Fermat did have a proof for his Last Theorem then it was a lot simpler than the one we have now.

We also touched on the world of proof in other areas where proving something absolutely is much harder. For example, Climate Change is accepted by most scientifically minded people, myself included, but without a parallel world that does not have Man on it it is impossible to say absolutely what the impact of Man is. Similarly economists do not have another global economic system to compare ours with.

Big Ideas on Mathematical Logic was two hours of brain stimulating talk with a group of intelligent and interested people in a room above a pub. It was a happy birthday!

Big Ideas on What is Normal?

One of the attractions of the monthly Bid Ideas discussions, and one of the reasons that I keep going to them, is the variety of topics that are covered. I also like the way that most of them are on something that I know something about so that I can contribute to meaningfully but they are also something that I know sufficiently little about so that I can learn something too.

Writing these blog posts is part of the learning process and is where I can restructure, reprocess and rethink my notes from the evening. These write-ups are never a historical record of the evening and they are not meant to be.

What we mean by "normal" is something that I have thought about at various times over the years, usually prompted by a news story, so I was looking forward to this discussion.

Our thought leader for the evening was Richard Barnett (on the right) who describes himself as a writer, teacher and broadcaster, mostly on the cultural history of science and medicine, and a poet. This is what he said, what others said, what I thought at the time and what I thought (much) later when I wrote this up.

The word normal derives from mathematics and most people will be familiar Normal / Gaussian distribution from school mathematics. This was derived from coin tosses and demonstrated that while a 50/50 split is expected a range of results close to this are also common and, for example, 100/0 splits are also possible these happen very rarely.

The question posed here is where to draw the line (if it can be drawn) between normal/expected results and abnormal/unexpected results.

Other distribution models exist each of which poses the question as to what a normal result is.

The mathematical normal can lead to results that need further explanation. For example, the average (mean) number of children per family is 2.4 but nobody has 2.4 children. Some of this depends on the units of measurement. If we measure people's height to the nearest centimetre then we will find thousands of people of average height but if we measure it to the nearest nanometre then we might find nobody of average height.

Normal has other linguistic usages. It can mean to mean mediocre, conformity etc. Here normal means not special when special is something good that we aspire too. In other circumstances normal is what we want to be, especially when we are comparing ourselves to abnormal people like terrorists or paedophiles.

While the definition of normal is fairly static, what counts as normal changes all the time. It used to be normal to drink and drive and, until recently, it was normal for young men to shave. Normal moves slowly and it is not usually obvious when what was normal has become unusual or abnormal.

Normal becomes even harder to define when the thing being described is not easy to measure. You can count coin tosses, heights of people, pints drunk and beards worn but what is a normal book or a normal face?

The more we talked about normal the more that I thought that normal is not a good word to use, especially when there are often better alternatives, e.g. average/mean (mathematical) or commonplace (mediocre). Similarly, it is better to say "I am not a paedophile" than to say "I have a normal sex life"; the first is clear and unambiguous but the second is loaded with context, assumptions and interpretations, all of which change over time.

It was an interesting talk on the various uses of the word "normal", and the issues around each use, which made me realise how poor a word it is and I resolved to try and curtail my own use of it. It was also another win for the certainty of Mathematics over the vagueness of Language.

Big Ideas on Modernism in Mathematics

Big Ideas is one of my very favourite talking-shops (I mean that in a nice way) amongst some stiff competition so I was delighted to be able to attend April's meeting. I was even more delighted than usual as the topic was "What is Modernism in Mathematics?" and my degree is in Mathematics.

Our guide for the evening was Jeremy Gray, Professor of the History of Mathematics at the Open University, pictured below on the right in the cute mock-Tudor bay window of the upstairs room at The Wheatsheaf in Fitzrovia.

As usual with these things, what follows is a mash-up of what the speaker said, what other people said, what I thought at the time and what I think now when writing it up. It is my summary of the topic based on this meeting but is not a summary of that meeting.

While Mathematics always had been abstract in became more so in the late 19th Century. Previously Mathematics had dealt with understandable thing like numbers and shapes but then mathematicians created new things like Sets and Fields to play with. The Algebra that was developed for numbers (2x+y=5) was extended to these new things.

Numbers themselves became from complicated with the addition of Real Numbers, i.e. numbers that are not fractions.

Euclidean Geometry was challenged and found to be lacking.

Some inherent paradoxes were discovered, e.g. does the set of all sets that do not contain themselves contain itself or not?

Euclidean Geometry starts with some basic axioms that do not need proving because they are true by definition, e.g. it is possible to draw a straight line from any point to any point. First it was shown that Geometry was still consistent if some of these were changed, e.g. parallel lines could meet, and then it was shown that any axiomatic Mathematics has inherent paradoxes.

The outcome of this was multiple domains in Mathematics where everything worked nicely in each one but they were all different and so there was no one single truth.

In trying to understand what a single truth might mean or be we discussed how much of Mathematics would we expect an alien civilisation to share. They would have numbers and shapes but would they also have Sets and Fields? And what could they have that we have not thought of? Could they have solved the one truth problem and, if so, would we be intelligent enough to understand it (you cannot teach a dog French).

The conversations we had spent a lot of time on how this new Mathematics of options was mirrored in the real world (Physics) and in Art.

Quantum Mechanics has shown us that the real world is a lot more complicated than we thought (and more complicated than we yet know). For example, particles are also waves and can be in two places at the same time. Perhaps the real world also has many truths like our Mathematics does.

Art changed around the same time and new rules were created for Impressionism, Cubism, etc. For centuries a portrait only had one rule and that was to be a reasonable likeness then other rules were invented that said that a portrait could be something else. Each school of Art has its own rules and each school is equally valid, there is no one truth for what a portrait should look like now.

It was a very animated, intelligent and thought-provoking discussion and I think we were all a little reluctant to end it after only an hour and a half, though quite a few of us stayed on for a while longer just to make that final comment or ask that final question.

I left understanding more about the multiple truths problem than when I went in (we had touched on this at university) but I think that the Physics and Arts analogies made me more relaxed about it.

Mathematics and comics

Rather a long time ago now I was in love with mathematics enough to study it at university. I've forgotten most of what I learned there, I cannot even understand the language any more, but the fondness remains.

Another passion that developed at university was for comics. So when I saw that Comica were doing a talk on mathematics and comics I just had to go.

The talk was on the new graphic novel Logicomix and was with the writer and artist on the comics side and the only mathematician anybody knows, Marcus du Sautoy, on the other.

What followed was another exhilarating talk. This was partly due to Marcus' questioning which showed his interest in storytelling, just as he did a year ago at the Royal Society. But good conversations need all participants to play their parts well and author Apostolos Doxiadis and artist Alecos Papadatos certainly did this with their erudite explanations and insights.

The talk itself would have made it a great evening but the icing on the cake was to get Marcus du Sautoy to sign one of his books and Apostolos Doxiadis and Alecos Papadatos to sign my newly purchased copy of Logicomix. Another Xmas present in the waiting.

I was also able to catch organiser Paul Gravett to thank him for the excellent Comica season; thanks I am delighted to repeat here.

Tim Harford entertains

I know of Tim Harford not from his column in the FT (which is the basis of his new book) but from the only mainstream programme that covers mathematics, More or Less on Radio 4.

An opportunity to see Tim speak at the LSE at a free public meeting seemed to good to miss, so I didn't.

You might have expected an economist speaking at an economics school to say quite a lot about economics but what we actually got was some pretty slight statistics on the subjects of happiness, dating and food and drink.

There was not much to challenge the grey cells but the talk we did get was well rehearsed, well delivered and entertaining enough to justify the time spent consuming it. If that has got you excited then you can listen to it here.

Telling stories with numbers, telling stories with words

As a mathematician by training, and inclination, who is employed to exploit words, a talk entitled "Telling stories with numbers, telling stories with words" was always going to interesting.

I was also interested in the possible Knowledge Management (KM) connections as story telling has been a hot topic there for some years.

But let's start with the geeky stuff first. The map shown here is taken from my iPod touch. When at home and on the wi-fi network I can use the maps application to find out where I am going and then save that map as a photograph so that I can access it later when offline. Simple, but very useful.

The talk was hosted by the Royal Society and was in the form of a conversation between Mark Haddon (author of The Curious Incident of the Dog in the Night Time) and Marcus du Sautoy (ubiquitous maths guy) on the similarities and differences between their two practices.

I am not sure that I agree with all of their observations but it was interesting to hear how they see things.

The main difference between the two was that story telling with words was seen as a open world where gaps are left deliberately for the reader's imagination to fill and where the story never really ends whereas as mathematics has to be precise so that all readers get the same message and the story ends with a firm statement, e.g. a proof.

The point was also well made that the mathematics invented (or discovered, perhaps) by the Ancient Greeks is still used today, and always will be, whereas their stories are mostly forgotten.

From a KM perspective, the aim of literature seemed almost to be the antithesis of KM in that the aim is to have uncertainty but in maths it was interesting to hear Marcus du Sautoy describe how he goes from having his original ideas, to sharing them with a small number of people who can understand what he is on about (the arm waving stage) and then writing them down in an academic paper that any mathematician can understand.

Like all good conversations it was inconclusive but it was well worth listening to for an hour and it gave the 200 or so eavesdroppers a few things to think about.

Welcome to the multiverse!

This imagining of the multiverse as snowflake in 196,833 dimensional spaces comes from the excellent Planetary by Warren Ellis and John Cassaday, which was first published as a regular comic in 1999 but my version is the large hardback collection Absolute Planetary.

This gets a worthy mention in my blog now as it was referred to surprisingly (to me!) in one of my regular podcasts this week.

Most of my podcasts are about business, current affairs or science and the most sciency of the science podcasts is Material World from the BBC. Because it is the one that is the most technical it is also the one that I enjoy the most.

One of the topics this week was symmetry, based on the book Finding Moonshine by Marcus du Sautoy (hardly my favourite mathematician but it is good to see mathematics getting some coverage in the media), and this looked at symmetry in various dimension.

In 2 dimensions (flat surfaces) this is rather easy and it's not much harder in three (our world as we see it) but in larger dimensions things get a lot more complex.

The topic was opened by the presenter, Quentin Cooper, quoting from Planetary, i.e. the page above, and then explaining that this is a genuine multiverse theory, thereby combining my loves for science and comics in one act. Superb!