BAD MATHEMATICS

A Madness for all Seasons

November 20, 2017

5:58 pm

Australia, exams, Further Mathematics, statistics, VCAA, VCE

Our fourth post on the 2017 VCE exam madness will be similar to our previous post: a quick whack of a straight-out error. This error was flagged by a teacher friend, David. (No, not that David.)

The 11th multiple choice question on the first Further Mathematics Exam reads as follows:

Which one of the following statistics can never be negative?

A. the maximum value in a data set

B. the value of a Pearson correlation coefficient

C. the value of a moving mean in a smoothed time series

D. the value of a seasonal index

E. the value of a slope of a least squares line fitted to a scatterplot

Before we get started, a quick word on the question’s repeated use of the redundant “the value of”.

Bleah!

Now, on with answering the question.

It is pretty obvious that the statistics in A, B, C and E can all be negative, so presumably the intended answer is D. However, D is also wrong: a seasonal index can also be negative. Unfortunately the explanation of “seasonal index” in the standard textbook is lost in a jungle of non-explanation, so to illustrate we’ll work through a very simple example.

Suppose a company’s profits and losses over the four quarters of a year are as follows:

$\begin{tabular} {| c | c | c | c |}\hline {\bf\phantom{S}Summer \phantom{I}} &{\bf\phantom{S}Autumn \phantom{I}} &{\bf\phantom{S}Winter \phantom{I}} &{\bf\phantom{S}Spring \phantom{I}} \\ \hline {\bf \$6000} & {\bf -\$1000} & {\bf -\$2000} & {\bf \$5000}\\ \hline \end{tabular}$

So, the total profit over the year is $8,000, and then the average quarterly profit is $2000. The seasonal index (SI) for each quarter is then that quarter’s profit (or loss) divided by the average quarterly profit:

$\begin{tabular} {| c | c | c | c |}\hline {\bf Summer SI} &{\bf Autumn SI} &{\bf Winter SI} &{\bf Spring SI} \\ \hline {\bf 3} & {\bf -0.5} & {\bf -1.0} & {\bf 2.5}\\ \hline \end{tabular}$

Clearly this example is general, in the sense that in any scenario where the seasonal data are both positive and negative, some of the seasonal indices will be negative. So, the exam question is not merely technically wrong, with a contrived example raising issues: the question is wrong wrong.

Now, to be fair, this time the VCAA has a defense. It appears to be more common to apply seasonal indices in contexts where all the data are one sign, or to use absolute values to then consider magnitudes of deviations. It also appears that most or all examples Further students would have studied included only positive data.

So, yes, the VCAA (and the Australian Curriculum) don’t bother to clarify the definition or permitted contexts for seasonal indices. And yes, the definition in the standard textbook implicitly permits negative seasonal indices. And yes, by this definition the exam question is plain wrong. But, hopefully most students weren’t paying sufficient attention to realise that the VCAA weren’t paying sufficient attention, and so all is ok.

Well, the defense is something like that. The VCAA can work on the wording.

Further Madness

by marty

November 19, 2017

12:41 pm

UPDATE (30/10/22)

The examination report has finally been amended to acknowledge the obvious error, albeit in a snarky “no harm, no foul” manner. The fundamental nonsense of the question remains unacknowledged. As commenter DB has noted, the examination report also makes the hilarious claim,

The overwhelming majority [of students] answered the question in the manner intended without a problem.

We’re not sure the “minority” of 72% of students who scored 0/1 on the question would agree.

Some Special Madness

by marty

November 17, 2017

2:46 pm

4 Comments on Some Special Madness

VCE

Australia, integration, limits, school mathematics, Specialist Mathematics, VCAA, VCE

Our second post on the 2017 VCE exam madness concerns a question on the first Specialist Mathematics exam. Typically Specialist exams, particularly the first exams, don’t go too far off the rails; it’s usually more “meh” than madness. (Not that “meh” is an overwhelming endorsement of what is nominally a special mathematics subject.) This year, however, the Specialist exams have some notably Methodsy bits. The following nonsense was pointed out to us by John, a friend and colleague.

The final question, Question 10, on the first Specialist exam concerns the function $\boldsymbol{f(x) = \sqrt{\arccos(x/2)}}$ , on its maximal domain [-2,2]. In part (c), students are asked to determine the volume of the solid of revolution formed when the region under the graph of f is rotated around the x-axis. This leads to the integral

$V \ = \ \pi \int\limits_{-2}^{2} \arccos(x/2)\, {\rm d}x\,.$

Students don’t have their stupifying CAS machines in this first exam, so how to do the integral? It is natural to consider integration by parts, but unfortunately this standard and powerful technique is no longer part of the VCE curriculum. (Why not? You’ll have to ask the clowns at ACARA and the VCAA.)

No matter. The VCAA examiners love to have the students to go through a faux-parts computation. So, in part (a) of the question, students are asked to check the derivative of $\boldsymbol{x\arccos(x/a)}$ . Setting a = 2 in the resulting equation, this gives

$\frac{{\rm d}\phantom{x}}{{\rm d}{x}}\left(x\arccos(x/2)\right)= \arccos(x/2) - \dfrac{x}{\sqrt{4-x^2}}\,.$

We can now integrate and rearrange, giving

$\aligned V \ &= \ \pi\left\Big[\!x\arccos(x/2)\!\!\right\Big]\limits_{-2}^{2} \quad +\quad \pi \int\limits_{-2}^{2} \dfrac{x}{\sqrt{4-x^2}}\, {\rm d}x\\[2\jot] \ &= \ 2\pi^2\quad +\quad \pi \int\limits_{-2}^{2} \dfrac{x}{\sqrt{4-x^2}}\, {\rm d}x\,.\endaligned$

So, all that remains is to do that last integral, and … uh oh.

It is easy to integrate $\boldsymbol{x/\sqrt{4-x^2}}$ indefinitely by substitution, but the problem is that our definite(ish) integral is improper at both endpoints. And, unfortunately, improper integrals are not part of the VCE curriculum. (Why not? You’ll have to ask the clowns at ACARA and the VCAA.) Moreover, even if improper integrals were available, the double improperness is fiddly: we are not permitted to simply integrate from some –b to b and then let b tend to 2.

So, what is a Specialist student to do? One can hope to argue that the integral is zero by odd symmetry, but the improperness is again an issue. As an example indicating the difficulty, the integral $\boldsymbol{\int\limits_{-2}^2 x/(4-x^2)\,{\rm d}x}$ is not equal to 0. (The TI Inspire falsely computes the integral to be 0, which is less than inspiring.) Any argument which arrives at the answer 0 for integrating $\boldsymbol{x/(4-x^2)}$ is invalid, and is thus prima facie invalid for integrating $\boldsymbol{x/\sqrt{4-x^2}}$ as well.

Now, in fact $\boldsymbol{\int\limits_{-2}^2 x/\sqrt{4-x^2}\,{\rm d}x}$ is equal to zero, and so $\boldsymbol{V = 2\pi^2}$ . In particular, it is possible to argue that the fatal problem with $\boldsymbol{x/(4-x^2)}$ does not occur for our integral, and so both the substitution and symmetry approaches can be made to work. The argument, however, is subtle, well beyond what is expected in a Specialist course.

Note also that this improperness could have been avoided, with no harm to the question, simply by taking the original domain to be, for example, [-1,1]. Which was exactly the approach taken on Question 5 of the 2017 Northern Hemisphere Specialist Exam 1. God knows why it wasn’t done here, but it wasn’t and the consequently the examiners have trouble ahead.

The blunt fact is, Specialist students cannot validly compute $\boldsymbol{\int\limits_{-2}^2 x/\sqrt{4-x^2}\,{\rm d}x}$ with any technique they would have seen in a standard Specialist class. They must either argue incompletely by symmetry or ride roughshod over the improperness. The Examiners’ Report will be a while coming out, though presumably the examiners will accept either argument. But here is a safe prediction: the Report will either contain mealy-mouthed nonsense or blatant mathematical falsehoods. The only alternative is for the examiners to make a clear admission that they stuffed up. Which won’t happen.

Finally, the irony. Look again at the original integral for V. Though this integral arose in the calculation of a volume, it can still be interpreted as the area under the graph of the function y = arccos(x/2):

But now we can consider the corresponding area under the inverse function y = 2cos(x):

It follows that

$V \ = \ \pi \int\limits_{-2}^{2} \arccos(x/2)\, {\rm d}x \ = \ \pi \int\limits_{0}^{\pi} \left[ 2\cos(x) - (-2)\right] \, {\rm d}x \ = \ 2\pi^2\,.$

Done.

This inverse function trick is standard for Specialist (and Methods) students, and so the students can readily calculate the volume V in this manner. True, reinterpreting the integral for V as an area is a sharp conceptual shift, but with appropriate wording it could have made for a very good Specialist question.

In summary, the Specialist Examiners guided the students to calculate V with a jerry-built technique, leading to an integral that the students cannot validly compute, all the while avoiding a simpler approach well within the students’ grasp. Well played, Examiners, well played.

There’s Madness in the Methods

by marty

November 15, 2017

2:11 pm

3 Comments on There’s Madness in the Methods

VCE

analysis, Australia, differentiation, exams, limits, Mathematical Methods, school mathematics, VCAA, VCE

Yes, we’ve used that title before, but it’s a damn good title. And there is so much madness in Mathematical Methods to cover. And not only Methods. Victoria’s VCE exams are coming to an end, the maths exams are done, and there is all manner of new and astonishing nonsense to consider. This year, the Victorian Curriculum and Assessment Authority have outdone themselves.

Over the next week we’ll put up a series of posts on significant errors in the 2017 Methods, Specialist Maths and Further Maths exams, including in the mid-year Northern Hemisphere exams. By “significant error” we mean more than just a pointless exercise in button-pushing, or tone-deaf wording, or idiotic pseudomodelling, or aimless pedantry, all of which is endemic in VCE maths exams. A “significant error” in an exam question refers to a fundamental mathematical flaw with the phrasing, or with the intended answer, or with the (presumed or stated) method that students were supposed to use. Not all the errors that we shall discuss are large, but they are all definite errors, they are errors that would have (or at least should have) misled some students, and none of these errors should have occurred. (It is courtesy of diligent (and very annoyed) maths teachers that I learned of most of these questions.) Once we’ve documented the errors, we’ll post on the reasons that the errors are so prevalent, on the pedagogical and administrative climate that permits and encourages them.

Our first post concerns Exam 1 of Mathematical Methods. In the final question, Question 9, students consider the function $\boldsymbol{ f(x) =\sqrt{x}(1-x)}$ on the closed interval [0,1], pictured below. In part (b), students are required to show that, on the open interval (0,1), “the gradient of the tangent to the graph of f” is $(1-3x)/(2\sqrt{x})$ . A clumsy combination of calculation and interpretation, but ok. The problem comes when students then have to consider tangents to the graph.

In part (c), students take the angle θ in the picture to be 45 degrees. The pictured tangents then have slopes 1 and -1, and the students are required to find the equations of these two tangents. And therein lies the problem: it turns out that the “derivative” of f is equal to -1 at the endpoint x = 1. However, though the natural domain of the function $\sqrt{x}(1-x)}$ is [0,∞), the students are explicitly told that the domain of f is [0,1].

This is obvious and unmitigated madness.

Before we hammer the madness, however, let’s clarify the underlying mathematics.

Does the derivative/tangent of a suitably nice function exist at an endpoint? It depends upon who you ask. If the “derivative” is to exist then the standard “first principles” definition must be modified to be a one-sided limit. So, for our function f above, we would define

$f'(1) = \lim_{h\to0^-}\frac{f(1+h) - f(1)}{h}\,.$

This is clearly not too difficult to do, and with this definition we find that f'(1) = -1, as implied by the Exam question. (Note that since f naturally extends to the right of x =1, the actual limit computation can be circumvented.) However, and this is the fundamental point, not everyone does this.

At the university level it is common, though far from universal, to permit differentiability at the endpoints. (The corresponding definition of continuity on a closed interval is essentially universal, at least after first year.) At the school level, however, the waters are much muddier. The VCE curriculum and the most popular and most respected Methods textbook appear to be completely silent on the issue. (This textbook also totally garbles the related issue of derivatives of piecewise defined (“hybrid”) functions.) We suspect that the vast majority of Methods teachers are similarly silent, and that the minority of teachers who do raise the issue would not in general permit differentiability at an endpoint.

In summary, it is perfectly acceptable to permit derivatives/tangents to graphs at their endpoints, and it is perfectly acceptable to proscribe them. It is also perfectly acceptable, at least at the school level, to avoid the issue entirely, as is done in the VCE curriculum, by most teachers and, in particular, in part (b) of the Exam question above.

What is blatantly unacceptable is for the VCAA examiners to spring a completely gratuitous endpoint derivative on students when the issue has never been raised. And what is pure and unadulterated madness is to spring an endpoint derivative after carefully and explicitly avoiding it on the immediately previous part of the question.

The Victorian Curriculum and Assessment Authority has a long tradition of scoring own goals. The question above, however, is spectacular. Here, the VCAA is like a goalkeeper grasping the ball firmly in both hands, taking careful aim, and flinging the ball into his own net.

UPDATE (20/09/20)

Above, we hammered Q9(c) on the 2017 Mathematical Methods, Exam 1. We regret not having hammered also the idiotically misleading diagram, but another issue has arisen, pointed out to us by frequent commenter SRK.

In Q9(b), students were asked to show that the derivative of $\boldsymbol{ f(x) =\sqrt{x}(1-x)}$ is $\boldsymbol{ (1-3x)/(2\sqrt{x})}$ . as we noted, the question was pointlessly verbose in classic VCAA style, but no big deal; an easy 1-mark question. What could go wrong?

Well, what went wrong is that 2/3 of students scored 0/1 on this very easy question. How? The Examination Report explains:

When answering ‘show that’ questions, students should include all steps to demonstrate exactly what was done, but many students often left steps out. A common pattern was to go straight from the first line of differentiation immediately to the final line, with no indication of obtaining a common denominator.

For fuck’s sake.

The stark incompetence of VCAA is often stunning. And, the nasty, meaningless pedantry of the VCAA is often stunning. But, on a question like this, when you see the two in seamless combination, that’s when you realise that you’re in the presence of true greatness.

Nothing to See Here, Folks

by marty

November 5, 2017

7:20 pm

Leave a comment on Nothing to See Here, Folks

history of mathematics

Arabic mathematics, history of mathematics, Indian mathematics, media, numerals, zero

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 4♣3 (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 Treachery of Images

by marty

October 25, 2017

12:36 pm

4 Comments on The Treachery of Images

exams

calculus, CAS, exams, Mathematical Methods, VCAA

Harry scowled at a picture of a French girl in a bikini. Fred nudged Harry, man-to-man. “Like that, Harry?” he asked.

“Like what?”

“The girl there.”

“That’s not a girl. That’s a piece of paper.”

“Looks like a girl to me.” Fred Rosewater leered.

“Then you’re easily fooled,” said Harry. It’s done with ink on a piece of paper. That girl isn’t lying there on the counter. She’s thousands of miles away, doesn’t even know we’re alive. If this was a real girl, all I’d have to do for a living would be to stay at home and cut out pictures of big fish.”

Kurt Vonnegut, God Bless you, Mr. Rosewater

It is fundamental to be able to distinguish appearance from reality. That it is very easy to confuse the two is famously illustrated by Magritte’s The Treachery of Images (La Trahison des Images):

The danger of such confusion is all the greater in mathematics. Mathematical images, graphs and the like, have intuitive appeal, but these images are mere illustrations of deep and easily muddied ideas. The danger of focussing upon the image, with the ideas relegated to the shadows, is a fundamental reason why the current emphasis on calculators and graphical software is so misguided and so insidious.

Which brings us, once again, to Mathematical Methods. Question 5 on Section Two of the second 2015 Methods exam is concerned with the function $V:[0,5]\rightarrow\Bbb R$ , where

$\phantom{\quad} V(t) = de^{\frac{t}3} + (10-d)e^{\frac{-2t}3}\,.$

Here, $d \in (0,10)$ is a constant, with $d=2$ initially; students are asked to find the minimum (which occurs at $t = \log_e8$ ), and to graph $V$ . All this is par for the course: a reasonable calculus problem thoroughly trivialised by CAS calculators. Predictably, things get worse.

In part (c)(i) of the problem students are asked to find “the set of possible values of $d$ ” for which the minimum of $V$ occurs at $t=0$ . (Part (c)(ii) similarly, and thus boringly and pointlessly, asks for which $d$ the minimum occurs at $t=5$ ). Arguably, the set of possible values of $d$ is $(0,10)$ , which of course is not what was intended; the qualification “possible” is just annoying verbiage, in which the examiners excel.

So, on to considering what the students were expected to have done for (c)(ii), a 2-mark question, equating to three minutes. The Examiners’ Report pointedly remarks that “[a]dequate working must be shown for questions worth more than one mark.” What, then, constituted “adequate working” for 5(c)(i)? The Examiners’ solution consists of first setting $V'(0)=0$ and solving to give $d=20/3$ , and then … well, nothing. Without further comment, the examiners magically conclude that the answer to (c)(i) is $20/3 \leqslant d< 10$ .

Only in the Carrollian world of Methods could the examiners’ doodles be regarded as a summary of or a signpost to any adequate solution. In truth, the examiners have offered no more than a mathematical invocation, barely relevant to the question at hand: why should $V$ having a stationary point at $t=0$ for $d=20/3$ have any any bearing on $V$ for other values of $d$ ? The reader is invited to attempt a proper and substantially complete solution, and to measure how long it takes. Best of luck completing it within three minutes, and feel free to indicate how you went in the comments.

It is evident that the vast majority of students couldn’t make heads or tails of the question, which says more for them than the examiners. Apparently about half the students solved $V'(0)=0$ and included $d = 20/3$ in some form in their answer, earning them one mark. Very few students got further; 4% of students received full marks on the question (and similarly on (c)(ii)).

What did the examiners actually hope for? It is pretty clear that what students were expected to do, and the most that students could conceivably do in the allotted time, was: solve $V'(0)=0$ (i.e. press SOLVE on the machine); then, look at the graphs (on the machine) for two or three values of $d$ ; then, simply presume that the graphs of $V$ for all $d$ are sufficiently predictable to “conclude” that $20/3$ is the largest value of $d$ for which the (unique) turning point of $V$ lies in $[0,5]$ . If it is not immediately obvious that any such approach is mathematical nonsense, the reader is invited to answer (c)(i) for the function $W:[0,5]\rightarrow\Bbb R$ where $W(t) = (6-d)t^2 + (d-2)t$ .

Once upon a time, Victorian Year 12 students were taught mathematics, were taught to prove things. Now, they’re taught to push buttons and to gaze admiringly at pictures of big fish.

The “Marriage Theorem” Theorem

by marty

October 9, 2017

11:38 am

2 Comments on The “Marriage Theorem” Theorem

media

graph theory, media, theorems

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

by marty

September 15, 2017

12:18 pm

2 Comments on Going off at a Tangent

history of mathematics

Babylonian mathematics, media, Pythagorean triples, trigonometry

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

by marty

September 15, 2017

1:15 am

2 Comments on The Marriage Equality Theorem

media

equality, media, politics, theorem

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

by marty

August 28, 2017

12:17 pm

3 Comments on Three Apples + Two Oranges = Infinite Nonsense

media

drugs, media, statistics

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.