WitCH 48: Thesis Not Good

This combo WitCH comes courtesy of mystery correspondent, tjrb. They flagged three multiple choice questions from the 2018 Algorithmics exam (here, and examination report here), and we’ve added a fourth. tjrb also remarks, “There are probably a lot more errors in this paper (and the other algorithmics papers), but these were the most strikingly incorrect”.

For Q2, the examination report indicates that 41% of students gave the intended answer of A. By way of explanation, the report then remarks,

“Cobham theorised that problems that are feasibly computable (also known as easy problems) are those that are decidable in polynomial time.”

For Q6, the report indicate that both A (51%) and C (33%) were “accepted”, but is otherwise silent.

The report is silent on Q12 and Q16, except to indicate the intended answers: C (94%) and A (66%), respectively.

WitCH 46: Paddling in the Gene Pool

The question below is from the first Methods exam (not online), held a few days ago, and which we’ll write upon more generally very soon. The question was brought to our attention by frequent commenter Red Five, and we’ve been pondering it for a couple days; we’re not sure whether it’s sufficient for a WitCH, or is a PoSWW, or is just a little silly. But, whatever it is, it’s pretty annoying, so what the hell.

WitCH 45: Learning from Our Betters

OK, we’ll get back into this slowly, and let others do the work. (Yes, at some point soon we’ll write about the seventy million knuckle-draggers who voted for Trump.)

A couple days ago The Conversation published a math ed article by some familiar maths ed folk:

Fewer Australians are taking advanced maths in Year 12. We can learn from countries doing it better.

To be fair, and making our way past the pithy title, we’re not sure the article is crap: we’re just not sure what it is. See how you go.

WitCH 44: Estimated Worth

This WitCH is from Cambridge’s 2020 textbook, Mathematical Methods, Unit 1 & 2. It is the closing summary of Chapter 21A, Estimating the area under a graph. (It is followed by 21B, Finding the exact area: the definite integral.)

We’re somewhat reluctant about this one, since it’s not as bad as some other WitCHes. Indeed, it is a conscious attempt to do good; it just doesn’t succeed. It came up in a tutorial, and it was sufficiently irritating there that we felt we had no choice.

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.