Nothing to See Here, Folks

Image copyright Bodleian Library, University of Oxford

Some pretty cool mathematical history made the news recently. Researchers at Oxford University investigated the Bakhshali manuscript, an ancient Indian text, and using carbon dating they apparently “pin[ned] the moment” of the “discovery of zero”.

Well, no. Dating one particular manuscript to “the 3rd or 4th century [AD]” is not pinpointing anything. And there are other issues.

The story is genuinely interesting, and much of the media reported the conclusions of the (not yet peer-reviewed) research accurately and engagingly. Others, however, muddled the story, particularly in the headlines. In order to clear things up, we can distinguish four related but distinct ideas to which “zero” might refer:

1)(a) The use of some symbol, say , as a placeholder in positional notation. We can then distinguish, for example, 43 and 43 (i.e. four hundred and three).

1)(b) The use of some symbol, say , to represent the number zero, for example in the equation 5 – 5 = .

2)(a) The use of something resembling the symbol 0 as a placeholder (as in 43 versus 403).

2)(b) The use of something resembling the symbol 0 to represent a number (as in 5 – 5 = 0).

All these ideas are of genuine interest, but 1(a) and, particularly, 1(b) much more so. Famously, from about 2000 BC Babylonian mathematicians employed a form of positional notation, using spacing when required to make the positions clear; so, it would be as if we used 43 and 4 3 to indicate forty-three and four hundred and three, respectively. From around 400 BC Babylonian mathematics began to employ a double-wedge symbol as a placeholder. That’s the earliest such occurrence of symbol for “zero”, in any sense, of which we are aware.

It took much longer for zero to be employed as a genuine number. The first known use was in 628 AD, in a text of the Indian mathematician Brahmagupta. He stated algebraic rules of the integers, though in words rather than symbols: a debt [negative] subtracted from zero is a fortune [positive], and so on. The symbolic arithmetic of zero may have followed soon after, though it is not clear (at least to me) even approximately when. By the end of the ninth century, however, the use of the symbol for the number 0 had appeared in both Indian and Arabic arithmetic.

The interest in the Bakhshali Manuscript is its use of (something resembling) the symbol 0: it is the filled-in dot on the bottom line in the photograph above. As for the Babylonians, this dot was employed as a placeholder rather than to represent a number. It had been thought that the Manuscript dated from the ninth century, and more recent than the (placeholder) 0 appearing on the walls of the famous Gwalia Temple, also from the ninth century. The recent carbon dating, however, determined that portions of the Manuscript, including pages that used the dot as a placeholder zero, were much older, dating to around 300 AD. That’s the big news that hit the headlines.

Now, none of that is as mathematically interesting as the still cloudy origins of the number zero. Combined with our knowledge of Brahmagupta, however, this new dating of the Bakhshali Manuscript suggests the possibility that the use of the number 0 in arithmetic occurred centuries earlier than previously suspected. So, not yet the magnificent historical revelation suggested by some newspaper reports, but still very cool.

 

The “Marriage Theorem” Theorem

The Marriage Theorem is a beautiful piece of mathematics, proved in the 1930s by mathematician Philip Hall. Suppose we have a number of men and the same number of women. Each man is happy to marry some (but perhaps not all) of the women, and similarly for each woman. The question is, can we pair up all the men and women so that everyone is happily married?

Obviously this will be impossible if too many people are too fussy. We’ll definitely require, for example, each woman to be happy to marry at least one man. Similarly, if we take any pair of women then there’s no hope if those two women are both just keen on the one and same man. More generally, we can take any collection W the women, and then we can consider the collection M of men who are acceptable to at least one of those women. The marriage condition states that, no matter the collection W, the corresponding collection M is at least as large as W.

If the marriage condition is not satisfied then there’s definitely no hope of happily marrying everyone off. (If the condition fails for some W then there simply aren’t enough acceptable men for all the women in W.) The Marriage Theorem is the surprising result that the marriage condition is all we need to check; if the marriage condition is satisfied then everyone can be happily married.

That’s all well and good. It’s a beautiful theorem, and you can check out a very nice proof at (no pun intended) cut-the-knot. This, however, is a blog about mathematical crap. So, where’s the crap? For that, we head off to Sydney’s University of New South Wales.

It appears that a lecturer at UNSW who has been teaching the Marriage Theorem has requested that students not refer to the theorem by that name, because of the “homophobic implications”; use of the term in student work was apparently marked as “offensive”. How do we know this? Because one of the affected students went on Sky News to tell the story.

And there’s your crap.

But, at least we have a new theorem:

The “Marriage Theorem” Theorem

a) Any mathematician who whines to her students about the title “Marriage Theorem” is a trouble-making clown with way too much time on her hands.

b) Any student who whines about the mathematician in (a) to a poisonously unprincipled pseudonews network is a troublemaking clown with way too much time on his hands.

Proofs: Trivial.

Going off at a Tangent

So Plimpton 322, the inscrutable Babylonian superstar, has suddenly become scrutable. After a century of mathematics historians puzzling over 322’s strange list of Pythagorean triples, two UNSW mathematics have reportedly solved the mystery. Daniel Mansfield and Norman Wildberger have determined that this 3,800-ish year old clay tablet is most definitely a trigonometry table. Not only that, the media have reported that this amazing table is “more accurate than any today“, and “will make studying mathematics easier“.

Yeah, right.

Evelyn Lamb has provided a refreshingly sober view of all this drunken bravado. For a deeper history and consideration, read Eleanor Robson.

Babylonian mathematics is truly astonishing, containing some great insights. It would be no surprise if (but it is by no means guaranteed that) Plimpton 322 contains.great mathematics. What is definitely not great is to have a university media team encourage lazy journalists to overhype what is probably interesting research to the point of meaninglessness.

The Marriage Equality Theorem

Theorem: Let V be the set of valid arguments against marriage equality. Then is empty.

Proof: Let P be a valid argument. Then, by now, someone would have argued P. This has not occurred. (Proof: by exhaustion.) By contradiction, it follows that P does not exist, and thus V is empty. QED.

An alternative, direct proof of the theorem was provided by the California Supreme Court; their proof applied the definition of equality.

Consideration of the many straight-forward corollaries of this theorem are left to the reader.

Three Apples + Two Oranges = Infinite Nonsense

The key findings of Australia’s 2016 National Drug Strategy Household Survey were released earlier this year, and they made for sobering reading. The NDSHS reported that over 15% of Australians had used illicit drugs in the previous year, including such drugs as cannabis, ice and heroin. Shocking, right?

Wrong. Of course.

We’re being silly in a way that the NDSHS reporting was not. Yes, the NDSHS reported that 15% had used illicit drugs at least once (including the possibility of exactly once) in the previous year, but NDSHS also emphasised the composition of that 15%. By far the most commonly used drug was cannabis, at about 10% of the population. Ice use was around 1%, and heroin didn’t register in the summary.

Illicit drug use is a serious problem, and a problem exacerbated by idiotic drug laws. Nothing can be learned, however, and nothing can be solved if one focuses upon a meaningless 15% multicategory. Whatever the specific threats or the reasonableness of concerns over the broad use of cannabis, such concerns pale in comparison to the problems of ice and heroin. The NDSHS makes no such categorical mistake. Unfortunately, there are plenty of clowns who do.

Last week, the Federal Ministers for Social Services and Human Services announced the location of a drug testing trial for job seekers who receive federal benefits. The ironically named Christian Porter and the perfectly named Alan Tudge announced that receipients would be tested “for illicit substances including ice (methamphetamine), ecstasy (MDMA) and marijuana (THC) … People who test positive to drug tests will continue to receive their welfare payment but 80 per cent of their payment will only be accessible through Income Management.” The plan is deliberately nasty and monumentally stupid, and it has been widely reported as such. For all the critical reporting, however, we could find no instance of the media noting the categorical lunacy of effectively equating the use of ice and ecstasy and THC.

Still, one should be fair to Porter and Tudge. They are undeniably dickheads, but Porter and Tudge are hardly exceptional. They are members of a very large group of thuggish, victim-blaming politicians, which includes Malcolm Turnbull, and Peter Dutton, and Adolf Hitler.

It is also notable that this kind of multicategory crap is only practised by social conservatives. It’s not like a nationwide survey on sexual harrassment and sexual assault in universities would ever couch the results in broadly defined categories in such a clouded and deceptive manner. Nope, not a chance.

NAPLAN’s Numerological Numeracy

This year Australia celebrates ten years of NAPLAN testing, and Australians can ponder the results. Numerous media outlets have reported “a 2.55% increase in numeracy” over the ten years. This is accompanied by a 400% increase in the unintended irony of Australian education journalism.

What is the origin of that 2.55% and precisely what does it mean to have “an increase in numeracy” by that amount? Yes, yes, it clearly means “bugger all”, but bugger all of what? It is a safe bet that no one reporting the percentage has a clue, and it is not easy to determine.

The media appear to have taken the percentage from a media release from Simon Birmingham, the Federal Education and Training Minister. (Birmingham, it should be noted, is one of the better ministers in the loathsome Liberal government; he is merely hopeless rather than malevolent.) Attempting to decipher that 2.55%, it seems to refer to the “% average change in NAPLAN mean scale score [from 2008 to 2017], average for domains across year levels”. Whatever that means.

ACARA, the administrators of NAPLAN, issued their own media release on the 2017 NAPLAN results. This release does not quote any percentages but indicates that the “2107 summary information” can be found at the the NAPLAN reports page. Two weeks after ACARA’s media release, no such information is contained on or linked on that page, nor on the page titled NAPLAN 2017 summary results. Both pages link to a glossary, to explain “mean scale score”, which in turn explains nothing. The 2016 NAPLAN National Report contains the expression 207 times, without once even pretending to explain what it means. The 609-page Technical Report from 2015 (the latest available on ACARA’s website) appears to contain the explanation, though the precise expression is never used and nothing remotely resembling a user-friendly summary is included.

To put it very briefly, each student’s submitted test is given a “scaled score”. One purpose of this is to be able to compare tests and test scores from different years. The statistical process is massively complicated and in particular it includes a weighting for the “difficulty” of each test question. There is plenty that could be queried here, particularly given ACARA’s peculiar habit of including test questions that are so difficult they can’t be answered. But, for now, we’ll accept those scaled scores as a thing. Then, for example, the national average for 2008 Year 3 numeracy scaled scores was 396.9. This increased to 402.0 in 2016, amounting to a percentage increase of 1.29%. The average percentage increases from 2008 to 2017 can then be further averaged over the four year levels, and (we think) this results in that magical 2.55%.

It is anybody’s guess whether that “2.55% increase in numeracy” corresponds to anything real, but the reporting of the figure is simply hilarious. Numeracy, to the very little extent it means anything, refers to the ability to apply mathematics effectively in the real world. To then report on numeracy in such a manner, with a who-the hell-cares free-floating percentage is beyond ironic; it’s perfect.

But of course the stenographic reportage is just a side issue. The main point is that there is no evidence that ten years of NAPLAN testing, and ten years of shoving numeracy down teachers’ and students’ throats, has made one iota of difference.

Malcolm the Mathematician

Australia’s Prime Minister tends to be pretty pleased with himself, and plenty of other people seem to think of Malcolm Turnbull as the smartest guy in the room. Perhaps he sometimes he is.* Malcolm didn’t appear so smart, however, when presenting Australia’s proposal to require the tech giants to decrypt their customers’ encrypted messages. When ZDnet reporter Asha McLean suggested that “the laws of mathematics [might] trump the laws of Australia”, Malcolm was unfazed:

The laws of Australia prevail in Australia, I can assure you of that. The laws of mathematics are very commendable but the only law that applies in Australia is the law of Australia.

And yes, the Government’s plan (for want of a better word) is as clueless as Malcolm makes it sound.

We already knew that Malcolm was a scientific clown, an economic illiterate, a coward, a Luddite, an Orwellian thug and a moral midget. So, maybe it shouldn’t be a great surprise when Malcolm also turns out to be an anti-mathematical git.

* If the other people in the room are Peter Dutton and Barnaby Joyce.

The Mysterious Wisdom of the East

According to The Australian newspaper (paywalled), a bunch of “education and policy experts” have headed to China in an attempt to address Australia’s educational woes:

Frustrated by stagnating maths and STEM standards, [they] are travelling to China for lessons on how to boost maths and science in local classrooms. 

Gee, I wonder what they might learn. What secret path to mathematical facility could those inscrutable Chinese possess? A wonderful new app, maybe. Or perhaps Chinese schools flip their classrooms in some really special way. 

But, whatever their secret, it may not help us to learn it. The worth of “importing other countries’ teachings practices” is apparently questionable, “given that education is woven within the cultural fabric of nations.”

There’s plenty woven within (?) the cultural fabric of Australia, but whether one should refer to it as education is open to debate.