Building a Bridge to the Twentieth Century

Predictably, last week’s talk ran short of time, and we were forced to skip some slides. The most regrettable omission was a slide titled “How to Teach …”, the motivation for which was to talk about the man in the photograph above, and about the photograph.

Our approach to teaching is, shall we say, eccentric. We won’t comment on the effectiveness of our teaching but, if “method” is too strong a word, there is an underlying idea. This idea is best captured by Ralph Waldo Emerson, writing upon writing: “The way to write is to throw your body at the mark when your arrows are spent”. Even if it indicates one way to teach, however, Emerson’s quote is of course not a dictum on teaching. Teaching is communication, and every teacher has to determine for themselves how they can best communicate ideas to their students.

Which brings us, almost, to the man in the fuzzy photograph. For the twenty years we were involved in the popularisation of mathematics, including the giving of and arranging of presentations, we were privileged to witness a number of great teachers. The brilliant John Conway was a stand-out, of course, as was Art Benjamin. But there were also two Australian mathematicians that were truly and particularly memorable.

The first mathematician was Mike Deakin. We mentioned Mike in last week’s talk, as one of our go-to guys when we started LunchMaths at Monash, and he gave a number of beautiful talks. Before that, Mike was, for decades, an editor, proofreader, janitor and mega-contributor for Monash’s mathematics magazine, Function.

The other mathematician was, finally, the man in the fuzzy photograph above: that is E. R. Love, who was professor of mathematics at the University of Melbourne for about three hundred years. In 1992, when Professor Love was 80, Terry Mills encouraged us to invite Professor Love to give a talk to the mathematics department at LaTrobe, Bendigo. We did so and Professor Love accepted. Declining multiple offers to be driven, Professor Love took the train to Bendigo and gave an absolutely beautiful talk on Legendre functions. Afterwards, over lunch, Professor Love entertained all with stories of Cambridge in the 30s.

Why write about Mike Deakin and, especially, Professor Love? Well, why not, of course; Deakin and Love were great contributors to Australian mathematics and deserve to be remembered and honoured. There was a specific reason, however, why we thought they were relevant to our talk, and why we particularly regret not having included acknowledgment of Professor Love: they were great teachers in a manner ceasing to exist. They were great lecturers.

Mike Deakin, who was an undergraduate at the University of Melbourne and then a Masters student under Professor Love, reminisces here on Professor Love’s teaching:

Love, in particular, was a superb lecturer. It was said of him that he was a menace because he made his subject seem so straightforward and logical that one missed seeing its difficulties.

The point is not that Mike Deakin and Professor Love were popular lecturers; the point is that they lectured in a careful, scholarly manner that is being lost. Their lectures had no gimmicks, had none of the crazy showmanship of the Mathologer, or of the writer of this blog. They simply lectured, conveying carefully crafted ideas to an audience willing and keen to listen. And, the point is that almost no one now recognises this, or cares, or can even properly understand. Almost no one under the age of fifty can realise that what is being lost is an art form, and an extremely beautiful and valuable one.

The title of this blog post is a play on Neil Postman‘s book titled Building a Bridge to the Eighteenth Century, which was in turn a play on a Clintonism. Postman’s excellent, and final, book was written in 1999. It was concerned with society’s inability to understand and to cope with technology, and the consequent loss of tradition and authority, of wisdom and plain meaning. Subtitled How the Past can Improve our Future, Postman’s book argued that we should look back to the 18th century, to the Enlightenment, for guidance into the future.

And now, twenty years later? The idea of building a bridge to the eighteenth century seems utterly fantastic, and perhaps always was. Twenty years on, and there is scarcely a memory of the twentieth century. The photo above was the best, the only photo we could find of Professor Love.

Mike Deakin and E. R. Love are dead, and they are being forgotten. The scholarly tradition they represented, the gift they gave, is being lost. And no one cares.

UPDATE

Gareth Ainsworth has contacted us, noting that Scotch College had an obituary for E. R. Love, which included a short biography and a photograph.

Video: Mathematics in Hell

Below is the video of our recent LunchMaths talk. You can comment/correct below and/or at the YouTube link.

A big thanks to Lawrence and Emma-Jane for arranging the talk, and for making the zooming as painless as possible. A couple of aspects that I intended to talk about, and some probably valuable clarification, were only covered in the Q and A. I’ll leave it be except in reply to comments, except for one aspect that I really regret not getting to and which I’ll cover in a separate post ASAP.

Zooming into Friday: Mathematics in Hell

Because we’re so in love with technology, and because we’re so short of things to do and, mainly, because we’re so, so stupid, we’ve agreed to give a LunchMaths/MUMS talk via Zoom this Friday.

The details are below, and this link is supposed to work. Attempt to enter at your own risk.

UPDATE (24/08) 

The video of the talk has been uploaded and can be viewed on YouTube and/or on this post.

A Simple Message to Primary Schools About Multiplication Tables

Dear Primary Schools,

If your students are not learning their multiplication tables, up to 12, by heart, then you are fucking up.

If you think giving your students a grab-bag of tricks replaces multiplication tables, then you are fucking up.

If you think orchestrating play-based, student-centred theatricalities replaces multiplication tables, then you are fucking up.

If you think quoting the Australian Curriculum gives you license to not teach multiplication tables, then you are fucking up.

If you think quoting some education twat gives you license to not teach multiplication tables, then you are fucking up.

Thank you for your attention.

WitCH 41: Zero Understanding

This is way unimportant in comparison to the current idiocy of the neoliberal nutjobs. But, as they say in the theatre, the shitshow must go on.*

We had thought of taking this further whack at Bambi a while back, but had decided against it. Over the week-end, however we were discussing related mathematics with Simon the Likeable, and that has made us reconsider:

Get to work.

*) Mostly Andrew Lloyd Webber productions.

UPDATE (9/8)

We were working on an update to polish off this one, when Simon the Likeable pointed out to us the video below. It could easily be its own WitCH, but it fits in naturally here (and also with this WitCH).

We’ll give people a chance to digest (?) this second video, and then we’ll round things off.

UPDATE (12/08/20)

OK, time to round this one off, although our rounding off may inspire objection and further comment. We will comment on four aspects of the videos, the third of which we regard the most important, and the fourth of which is mostly likely to cause objection.

The first thing to say about these videos is that, as examples of teaching, they are appalling; they are slow and boring and confusing, simultaneously vague and muddyingly detailed. In particular, the “repeated addition” nonsense is excruciating, and entirely unnecessary. You want us to think of division as “how many”, then fine, but don’t deliver a kindergarten-level speech on it.

Eddie Woo’s video has the added charm that at times no one seems to give a damn what anyone else is saying; particularly notable is the 6:00 mark, where the girl suggests “Therefore it’s [i.e. 1/0 is] undefined?”, the very point Eddie wants to make, and Eddie pointedly ignores her so he can get on with his self-aggrandizing I’m-So-Wonderful performance. Dick.

The second thing to say is that the Numberphile video is littered with errors and non sequiturs, the highlights being their dismissing infinity as an “idea” (as if 3 isn’t), and their insane graph of \boldsymbol{x^x}. We’ll go through this in detail when we update this WitCH (scheduled for sometime in 2023).

The third thing to say is that the videos’ discussion of the impossibility of defining 1/0 gives a fundamentally flawed view of mathematical thought. The entire history of mathematics is of mathematicians breaking the rules, of doing the impossible. (John Stillwell has written a beautiful book, in fact two beautiful books, on the history of mathematics from this perspective.) As such, one should be very careful in declaring mathematical ideas to be impossible. So, 1/0 may generally not be defined (at school), but is it, as Eddie declares, “undefinable”?

Of course taken literally, Eddie’s claim is silly; as we suggested in the comments, we can define 1/0 to be 37. The real question is, can one define 1/0 in a meaningful manner? There are reasonable arguments that the answer is “no”, but these arguments should be laid out with significantly more care than was done in the videos.

The first argument for the (practical) undefinability of 1/0 is that we’ll end up with 1/0 = 2/0, leading to 1 = 2. What is really being claimed here? Why is 1/0 = 2/0, and why should it lead to 1 = 2?

The heart of this approach is asking whether 0 can have a multiplicative inverse. That is, is there a number, let’s call it V, with 0 x V = 1? Of course V couldn’t be an everyday real number (not that real numbers are remotely everyday), but that’s neither here nor there. It took a hugely long time, for example, for mathematicians to leave the safety of the world of everyday (?) integers and to discover/create an inverse for 3.

Well, what goes wrong? If we have such a number V then 1/0 stands for 1 x V. Similarly 2/0 stands for 2 x V. So, does it follow that 1 x V = 2 x V? No, it does not. V only has the properties we declare it to have, and all we have declared so far is that V x 0 = 1.

Of course this is cheating a little. After all, we want V to be an infinityish thing, so let’s concede that 1 x V and 2 x V will be equal. Then, if we assume that the normal (field) rules of algebra apply to V, it is not hard to prove that 1 = 2. That assumption is not necessarily unreasonable but it is, nonetheless, an assumption, the consequences of that assumption require proof, and all of this should be clearly spelled out. The videos do bugger all.

The second argument for the undefinability of 1/0, at least as an infinity thing, is the limit argument, that since tiny numbers may be either positive or negative, we end up with 1/0 being both \boldsymbol{+\infty} and \boldsymbol{-\infty}, which seems a strange and undesirable thing for infinity to do. But, can we avoid this problem and/or is there some value, in a school setting, of considering the two infinities and having them equal? The videos do not even consider the possibilities.

The fourth and final thing to note is that, as we will now argue, we can indeed make sense of 1/0 as an infinity thing. Moreover, we believe this sense is relevant and valuable in the school context. Now, to be clear, even if teachers can introduce infinity and 1/0, that doesn’t imply they necessarily should. Perhaps they should, but it would require further argument; just because something is relevant and useful does not imply it’s wise to give kids access to it. If you’re collecting wood, for example, chain saws are very handy, however …

First, let’s leave 1/0 alone and head straight to infinity. As most readers will know, and as has been raised in the comments, mathematicians make sense of infinity in various ways: there is the notion of cardinality (and ordinality), of countable and uncountable sets; there is the Riemann sphere, adding a point at infinity to the complex plane; there is the real projective line, effectively the set of slopes of lines. Cardinality is not relevant here, but the Riemann sphere and projective line definitely are; they are both capturing 1/0 as an infinity thing, in contexts very close to standard school mathematics. And, in both cases there is a single infinity, without plusses or minuses or whatever. Is this sufficient to argue for introducing these infinities into the classroom? Perhaps not, but not obviously not; infinite slopes for vertical lines, for example, and with no need for a plus or minus, is very natural.

What about the two-pronged infinity, the version that kids naturally try to imagine, with a monster thing at the plus end and another monster thing at the minus end? Can we make sense of that?

Yes, we can. This world is called the Extended Real Line. You can watch a significantly younger, and significantly hairier, Marty discussing the notions here.

The Extended Real Line may be less well known but it is very natural. What is \boldsymbol{\infty + \infty} in this world? Take a guess. Or, \boldsymbol{\infty +3}? It all works just how one wishes.

But what about when it doesn’t work? You want to throw \boldsymbol{\infty  - \infty} or \boldsymbol{0\times  \infty} or \boldsymbol{\frac{\infty}{\infty}} at us? No problem: we simply don’t take the bait, and any such “indeterminate form” we leave undefined. In particular, we make no attempt to have \boldsymbol{\infty} be the multiplicative inverse of 0. And, then, modulo these no-go zones, the algebra of the Extended Real Line works exactly as one would wish.

Can these ideas be introduced in school, and for some purpose? No question. Again, whether one should is a trickier question. But as soon as the teacher, perhaps in hushed and secretive tones, is suggesting \boldsymbol{\infty + \infty =\infty} or \boldsymbol{\frac1{\infty} =0}, then maybe they should also think about this in a less Commandments From God manner, and let \boldsymbol{\infty} come properly out of the closet.

Finally, what about 1/0 in the Extended Real Line? Well, the positive or negative thing is definitely an issue. Unless it isn’t.

There are many contexts where we naturally restrict our attention to the nonnegative real numbers. And, in any such context 1/0 is not at all conflicted or ambiguous, and we can happily declare \boldsymbol{\frac10 =\infty}. The exact trig values from 0 to 90 is just such a context: in this context we think it is correct and distinctly helpful to write \boldsymbol{\tan(90) = \infty}, rather than resorting to a what-the-hell-does-that-mean “undefined”.

That’s it. That’s a glimpse of the huge world of possibilities for thinking about infinity that Numberphile and Woo dismiss with an arrogant, too-clever-by-half hand. Their videos are not just bad, they are poisonously misleading for their millions of adoring, gullible fans.