A discontinuity in our sessions!

Unfortunately we need to postpone the next session (on the theme of Continuity and planned for 31st March). Victoria has decided to cancel or postpone current evening events at the Tree House due to the Government programme aimed at delaying the spread of corvid-19. Clearly this is a sensible move in light of the rapid escalation of developments in recent days. It’s a pity – I was just getting into the main story-line for our session – and Leon was practising using Desmos to show us nice examples of the various ways a function can be discontinuous. It’s only a postponement but, in light of current predictions, it is likely we shall not be able to resume until September or so.

Drama at the Tree House

We had some (modest) drama in the last session (Episode 4: Proofs and Fallacies) when some audience members took the part of students in a Lakatos seminar, and Leon and I had a dialogue highlighting the change in perspective on visual proofs in geometry that has occurred over the last 50 years. (You can see this dialogue if you missed it on the webpage for episode 4. )

Discussion following the talk focussed on ways that proofs, and how they are judged, change – both over time and across cultures. The case of the intermediate value theorem was mentioned and so was the issue of computer assisted proofs. I think I should have emphasised more the role of logic in mathematical proofs compared with science and medicine where empirical evidence plays a more dominant role. We showed an example (‘proving’ that all triangles are isosceles) where it was not the logic at fault but the diagram – although only slightly inaccurately drawn – was completely misleading. (The example may be found in E.A. Maxwell, ‘Fallacies in Mathematics’.)

The point of the number line

In episode 3 I forgot to mention that in episode 2 we had made many references to ‘points on the number line’, e.g. that the point marking \sqrt{2} was missing from the (rational) line. The proof of Lemma 1 (episode 3) in the proof of the centroid (as where the three medians meet) made implicit use of the number line.

Very first question

The very first question of our episode 1 was along the lines of, ‘ these patterns you have shown in everyday life, and in mathematics, are they really there?’ . The idea (I think) was that maybe we had imagined /invented them ourselves and then thought we had ‘discovered’ them in the world or in maths. Were they really only our own fictions?

I could not then give a good answer to that. But I think that what we looked at in episode 3 suggests that, at least sometimes, the mathematical patterns really are there. I am thinking of the frequency that we came across surprisingly simple ratios (like 1:2, or 2:1:3) arising in all triangles. Indeed concurrency itself is a kind of pattern (a symmetry) that we could hardly have ‘put there’ ourselves.

Constructions and proofs

Did you have a go at drawing any of those concurrent lines or collinear points? Either with instruments, or with software (e.g. Geogebra). If the constructions always ‘work’ are the proofs necessary? What do they do for us? One difference with the software is being able to move the vertices of the triangle and ‘check’ the concurrency – like doing many experiments of drawing, or is it?

Could there be mistakes in the proofs? (I mean the intended proofs – not just any of ‘my’ mistakes!) Related to the question about ‘accuracy’ of Geogebra, it seems to me the software ‘knows’ more than expected. I could not fool it by drawing a third perpendicular bisector into making them truly ‘concurrent’ – only when it was using its own version of perpendicular bisector would it make the intersection one point (rather than three).

Proofs for ‘Lines and Points’

The last session (Episode 3) was all about triangles. Detailed proofs of most of the results were omitted in the talk because there was not time. I have, at last, posted all the proofs on the webpage for Episode 3 – you will see why we omitted them – they are a bit long! I’ll be more than happy to explain further if you have any (even the simplest!) questions (or corrections/omissions). But they are only 2 pages long (each one).

It’s hard to know quite how surprised we should be about lines being concurrent, and points being collinear, in the study of triangles (and in many other contexts). But when we do get a surprise it’s natural to ask ‘Can we prove it?’, meaning, ‘Can we prove this will always happen in any triangle?’. If it’s a while since you’ve looked at any proofs, I recommend the ‘Centroid’ one – go straight to the Theorem (page 2) and look back at the Lemmas if you need or wish. It involves two examples of reasoning about a pair of similar triangles, so it is a mixture of concepts and logic – good preparation for thinking about proofs in the next episode.

Episode 4 is on Proof

A central feature of mathematics is proof. This is not so much apparent at school – where learning a variety of concepts and techniques, problem solving, and getting answers (right or wrong!) and qualifications are rightly the main emphasis. But in study beyond school, proofs, and methods for construction and evaluation of proofs, are as characteristic of mathematics as experiments are in the natural sciences. We’ll survey a range of examples of proofs (good and bad) – both historical and modern, including proofs generated (or checked) by computer.

This next episode is on Tuesday 25th February.

Correction to slides (Episode 3 Lines …)

There was an error on slide 8 (‘Similar, not congruent’) of the handout some people took after episode 3 yesterday. It’s been corrected in the two versions of the slides on the webpage. Let me know by email if you have any difficulty downloading the slides, or trying the constructions by drawing or by Geogebra (which is definitely fun to learn and to play with!) – you can just google it and download it.

Surprises all round in Episode 3

In most of our episodes I am showing some results which are ‘well known’ to mathematicians but may come as an interesting surprise to others. We shall be looking at four collinear centres of a triangle that divide their common line in a pleasing ratio. The ‘common line’ is the Euler Line marked in red in the nearby diagram. This is a classical result (18/19C). I had a surprise myself while preparing this episode to discover there had been a substantial renewal of interest in these (and very many other) ‘centres’ of a triangle in the 1980s – find out more in Tuesday evening’s episode!