Mathematics: A Beautiful Thing Made Ugly
Has Social Media taken the Misunderstanding and Mis-application of Mathematics to a new low?
A Beautiful Thing
My father, who had a fairly basic school education, was largely responsible for giving me an appreciation of mathematics. Somewhat by accident, World War Two took him deeper into numbers. In pre-war England he was a bookkeeper for a cosmetic company. With Hitler threatening invasion he decided to join the army. His flat feet made him unsuitable for combat duties so the recruiter, noting that he was a bookkeeper, said, “You know about numbers, you can join the engineers”. Thus he joined the Corps of Royal Electrical and Mechanical Engineers and was made to study algebra, calculus, trigonometry and complex numbers, all of which fascinated him. He applied what he had learned to help keep Britain’s war-time radar in working order. After the conflict he moved to Australia where he used his knowledge, including the mathematics, to design and supply technology for the bush which was just being opened up for agriculture by other returning servicemen. Television eventually became the staple of his business, but before its advent he made two-way radios for firetrucks and 32V-240V DC-AC inverters because the rural electricity grid was still only in its infancy.
To my father, myself and many others, mathematics was and is, not only useful, but an intriguing and beautiful thing. Beyond the simple arithmetic of the every-day, there lie patterns, new ways of thinking and many wonderful and sometimes absurd outcomes to problems. The absurd often resolves itself in a novel idea having yet unheard of applications. Some of us were blessed in having our eyes opened to some of these:
- the Möbius strip, a piece of paper with only one side,
- imaginary numbers, numbers that have no real substance to them but have a myriad of uses in the real world,
- the thousands of lines, points and circles associated with triangles,
- chaos theory and its beautifully wild and colourful representations,
- the properties of some quirky and unusual numbers like 1089, amicable numbers, sociable numbers, hailstone numbers, perfect numbers, transcendental numbers and really weird ones like the Euler–Mascheroni constant.
Sadly, it appears from what I have seen over the years, and now observe in the media, and in social media particularly, is that what people have learned about mathematics in formal education largely melts away upon graduation. Many may have not seen the wonders of mathematics. They knew enough to pass examinations, but even that knowledge proved ephemeral. Furthermore, as graduates, they have continued to pick up incorrect views and run with them. The symptoms range from a complete lack of understanding about the foundations and language of numbers through to bizarre ideas about probability and statistics. To lose any sense of the curiosity about numbers, to not retain mathematical knowledge and understanding, or worse, to take up a phony substitute, does, I believe, have a detrimental effect on individuals and the whole of society. In social media, not only do I observe this degradation, but I see a reticulated conduit for mathematical ignorance flowing into every part of society.
The beauty is forgotten. Good practice is forsaken. The mind is no longer stirred. We are in danger.
Before considering the topic questions let us consider a slightly different question: what happens to mathematics itself after formal education? Does the gaining of a degree or diploma suddenly mean the ground rules of mathematics are changed? Does its language morph into something less precise? (I fear this is happening within the education system itself but that is the subject of another article.) Can graduates, with impunity, suddenly dismiss or misapply the BODMAS (aka PEDMAS) rule and ignore parentheses and instruction order? To many, even the equals sign, “=”, seems to have become a symbol of vague implication rather than exact sameness. They sometimes want to do silly things like divide a real number by zero and call the answer ‘infinity’¹, and then treat it like a number. Some get confused with concepts like ⅓ equaling 0.333… (Remember recurring decimals? The ‘…’ here mean that the 3s go on forever) and ⅔ = 0.666… and when you tell them to add ⅓ to ⅔ and they get 1 but adding 0.333… to 0.666… they tell you it’s 0.999… and there is no way that is equal to 1. (Okay give me a number that exists between 0.999… and 1. Ummm… Now there is a formal proof for this equality, but not for this time or forum.) Don’t even let me get started on pi (π) being equal to ²²/₇ and the “Law of Averages”.
The Basis of Mathematics
Mathematics is an extension of logic and is based on axioms. Axioms are agreed, often self-evident, truths which cannot themselves be proven. We apply the rules of logic (which are themselves axiomatic) to the basic, agreed truths of mathematics to derive further truths. The validity of axioms is a subject of philosophy, not of logic. Unless we want to enter into philosophical debate, the arena for correct and effective mathematics has been set for us. Axioms and the rules used to manipulate them are unchangeable (although they may be added to as new fields arise) if mathematics is to remain useful and, dare I say, beautiful.
The Ugliness.
The Internet, and social media in particular, have greatly added to the problem. While there are sites dedicated to promoting good and interesting mathematics, the nature of the medium engenders a "that’s good enough” attitude. The “lowest common denominator”, to borrow a phrase from the topic subject, becomes the standard. You only have to read the comments in good mathematics site forums and it is quickly evident that many have entrenched views that fly in face of what the site is sharing. Once such a respondent has put an idea on a forum there are others who follow with apparently no desire to learn. They merely want to add fuel to the fire of ignorance.
Like the workers of Metropolis there are many who toil in darkness (ignorance) unaware that there is more for the human mind to feed upon in the daylight (knowledge).
Anarchic Methods
Generally, most people have no reasoned justification to hang onto formal algebra, calculus and trigonometry after graduation. My contention is that, if that is the case, they should not mess with it. But now those with mathematical amnesia have a forum - the Internet. It abounds with mathematical and logical puzzles which are poorly formulated and only exacerbate the problem: the sequence for which you are allowed only one correct answer, fallacious antecedents, the invalid use of the equals sign, the slightly complex arithmetic problem without parentheses. And so, since they have forgotten the correct rules and since someone else made up their own rules to set the problem, they say “I’ll make up my own to solve it.” Sure, you can invent your own axioms and rules, something mathematicians do all the time, but your rules and axioms are not likely to be consistent² and will not necessarily correctly solve a problem in what we might call the normal, everyday world. This is solution by anarchy but there is another equally bad idea.
Solution by Democracy.
Social media usually presents these problems to the world as amusements. This is instructive as the word “amuse” comes from “a” meaning “without” and “muse” meaning “thought”. These mathematical tidbits invite a section of their audience to take a trip down memory lane to a time when, perhaps, they understood such things. But, alas, that destination is overgrown with weeds, its paint is peeling off and it is shrouded in darkness. It offers little light for the problem before them. As the correct answer eludes many of them, they again invent their own rules, solve the problem according to those rules and post a solution of their own liking. As many individuals post such solutions a consensus unfolds revealing the “correct” answer. My reading of such lists is that many, using incorrect bases, find a solution like someone else’s and add it to the list as proof they got it right. In turn others, reading the solutions, look simply at the frequency of various solutions and figure that the majority answer must be correct.
Part of the problem is that there is no umpire or arbiter for mathematics in social media. Except for a group of online mathematical aficionados, who, I consider, would eschew such past-times in favour of the real thing, and a number of mathematically oriented websites like Wolfram Alfa, which are somewhat passive, there is nowhere to turn in order to verify any ordinary persons’ claim to mathematical fame.
Mathematical problem solving by anarchy or democracy has real consequences.
Consequence 1. The lie of a high IQ and other personal snares.
Most puzzles in social media are traps for showing poor understanding. They do not excite the intelligence. They may be fun but so can real and well put together problems.
Mathematical ability is not necessarily a measure of intelligence. Ability to think logically may be. Creativity certainly is. Social intelligence, the ability to steer through and manage the mosaic of social relationships and environments life throws at us, definitely is. Social media tells lies that pander to that part of us that wants to be recognised as bright, witty, someone to be looked up to. It promotes trivial and incorrect problem solving exercises to make its audience feel accepted or even better than others. It does little to educate, enhance or correct mathematical ability or to increase intelligence. Basically, it promotes ignorance ³.
Can mathematics improve intelligence? Mathematics, per se, will do little for IQ but the processes it uses, the creativity it engenders and the challenges it provokes will. As Andrea Kuszewski points out:
Intelligence isn’t just about how many levels of math courses you’ve taken, how fast you can solve an algorithm, or how many vocabulary words you know that are over 6 characters. It’s about being able to approach a new problem, recognize its important components, and solve it — then take that knowledge gained and put it towards solving the next, more complex problem. It’s about innovation and imagination, and about being able to put that to use to make the world a better place. This is the kind of intelligence that is valuable, and this is the type of intelligence we should be striving for and encouraging. (Andrea Kuszewski, March 7, 2011https://blogs.scientificamerican.com/guest-blog/you-can-increase-your-intelligence-5-ways-to-maximize-your-cognitive-potential/)
Having a poor understanding of mathematical principles can also lead to being taken in. Here is where mathematics and social intelligence coincide. It might the bargain that is no bargain. It might be a get-rich-quick scheme, for example, a pyramid (Ponzi) scheme. It may be simply considering that their are predictable ways of winning a lottery⁴ .
Furthermore, if you get it wrong on the Internet you have tainted your identity (possibly your most prized social possession) which can be extremely hard to undo. Consider this problem which, when it appeared in one of its many incarnations on LinkedIn, evinced the following responses.
- From a business consultant : “For an equation without brackets, proceeding left to right without any contradicting instructions, then 6 -1 = 5 x 0 = 0 + 2 = 2 / 2 = 1” . I am not sure I would consult with him about anything that even vaguely involved money.
- A computer systems architect, although he got the correct answer was, in a public forum, way off in his understanding of how mathematics is taught: “This is another one with different answers depending on where you were taught mathematics and order of operations (and whether or not you can do maths without a calculator). In my case it is BODMAS where the multiplication and division are resolved before addition and subtraction, hence the answer is 7. It is _not_ done as a sequence of operations left to right in isolation.” I may be wrong, but I believe mathematics is taught very much the same way world-wide. There is only one way of solving this.
Consequence 2. Scary stuff for us all.
We play with the exactness of mathematics at our peril. If we assume we know something in this area without real and tested skills we create dangers for ourselves, others, and possibly the world. The portrayal in certain quarters, especially in social media, that mathematics is somehow an innate skill, a measure of intelligence, leads some people — people who have control of our welfare — to think they can make judgements from or about the mathematics that regulates that welfare.
Threats to Internet safety.
We have recently seen a slew of law enforcement officials, politicians and even celebrities, weighing in on the argument about having back-doors into the encryption algorithms that protect our data on the Internet and elsewhere.
It is evident that most have forgotten enough mathematics to ensure they can no longer understand the underlying algorithms that encrypt data. Such encryption is based on, and tested by, sound mathematics. The idea of weakening encryption to help catch criminals and terrorists may be admirable but at what cost? Toying with the precise mathematics of cryptography diminishes its usefulness. Weakening the algorithms for law enforcement means weakening them for anybody and everybody. Loss of usefulness in encryption means we all lose out to the malicious characters and causes in our world. (There are also good, non-mathematical reasons not to demolish this means of secrecy provided in modern-day communications).
Losing a basic understanding of mathematics and replacing it with some tepid substitute, backed up by the standards of social media, disqualifies anyone from formulating valid opinions about such fields as cryptography. The minds that gave rise to today’s encryption methods over a period of more than two millennia were steeped in mathematics. If the names Martin Hellman, Whitfield Diffie, Neal Koblitz, Victor Miller, Ron Rivest, Adi Shamir, Leonard Adleman, Euclid, Sun Tzu, Leonhard Euler, Évariste Galois and Alan Turing mean nothing to a public figure then I (respectfully) suggest they study their contributions to mathematics before before making pronouncements about encryption.
Miscarriages of Justice.
Mathematics, particularly statistics, plays an important evidentiary role in modern court proceedings. Lawyers, judges and juries need to know their limits when it comes to assessing guilt or innocence based on mathematical knowledge. Many great and fair legal minds would already be on the back foot when interpreting evidence in the light of mathematics, after all, most chose law over mathematics in senior education. Add to that the milieu of mathematical lies and trivialization that abounds in social media and we can only see a downward trend — the anarchy and democracy applied to ‘fun’ mathematics will turn otherwise factual evidence into lies and trivia.
I cite the following three cases, not because they had great social media coverage, but because they show what can happen in a system where mathematics is treated without rigour and sometimes with disdain. Social media is such a system.
Lucia de Berk, a Dutch nurse was arrested in 2001 after the apparent poisoning of a 6-month-old baby in her care at a hospital in The Hague. After de Berk’s initial arrest, what was thought to be a pattern of suspicious deaths and near-deaths of patients under her care, was uncovered. Statistics played a vital part in the investigation and subsequent trials. A court appointed statistician, somewhat an amateur, had calculated that it would be a one in 342 million chance that it was pure coincidence she had been on duty when all the incidents had occurred. After two trials she was convicted of nine murders and 3 attempted murders and sentenced to life with no parole. What the judges in these cases could not see, by dint of lack of mathematical knowledge, was that the statistical processes used were flawed. Jailed for life in 2004 she was eventually released to allow an appeal in 2008 and in 2010 she was exonerated and compensated be the state. Is nine years of someone’s life that easily taken by poor mathematics? Yes, it seems.
On October 3, 2011, Amanda Knox and Raffaele Sollecito, who had been convicted of the murder of British exchange student Meredith Kercher, were acquitted in an Italian court. An earlier appeal judge had ruled out re-testing a minuscule DNA sample found on a knife at the scene, stating that, “The sum of the two results, both unreliable… cannot give a reliable result.” He was wrong. Whilst intuitive, it is not mathematically accurate. Repeating relatively unreliable tests can make them more statistically reliable. Retesting did occur when the acquittal went to retrial and found to be in accused’s favour, but it was not until some time later the the Supreme Court of Cassation (the highest court in Italy) ruled that Knox and Sollecito were innocent of involvement in the murder. Had the original appeal Judge had a little statistical knowledge, multiple tests may been called for earlier and the accused’s pain would have been spared.
Another case is that of Sally Clark. She was convicted, in Britain, of the murder of her two sons in 1999. Both of Clark’s children had died from cot-death (SIDS) and not long after she was arrested for their murder. The case against her was based on a seemingly indisputable statistic. An eminent pediatrician testified that the chance of the death by cot-death of two children from the same family was one in 73 million. She was convicted and languished in prison for four years until an appeal was allowed. The prime witness had, among other things, failed to take into account that the two deaths were not independent. He had simply multiplied the independent chances of two such deaths occurring. Had he retained enough mathematics, or had an unbiased statistician been called to refute his testimony, it is doubtful Sally Clark would have spent those years in prison.
What these cases, and perhaps many others that will never be questioned, show is that a correct knowledge of mathematics is essential to be able to properly use it in determining guilt or innocence. The social media approach to mathematics leads, not to instruction, not to understanding, but to ignorance and trivialization. This engenders a poor use of mathematics where it is critical. Judges should know their own limits, and possibly the limits of a jury’s knowledge, and, I opine, appoint mathematical experts in the pursuit of justice.
Ignorance = Cleverness Undone
Or its inverse(?) Cleverness = Ignorance done up! Pseudo-mathematics is often used to make someone look clever. Sometimes it fails miserably.
Look at the graphic opposite. Again it was from LinkedIn. It was a poor choice of analogy. I think what was intended was r ∝ nᵐ. The author did not define the function f. The point he was trying to make could have done without it. It all depends on the nature of the function f. If you consider that we could define f(x)=0 then no matter what n and m are r will always be zero — no resistance at all! I would suggest not using mathematical analogies without proper understanding as it may put the audience off.
Traps seen in social media include enticing would-be mathematicians to solve problems that are poorly formulated. Some, like the one opposite, seem to be put there to incite arguments. I would suggest that posing such a problem only adds to the milieu of bad ideas about mathematics. It does nothing for the thinking mind.
Problem solving ability can, to a large degree, be taught.
How many people would recognise that the first line of this puzzle is superfluous data. The answer is the difference of the second two. This kind of thinking is taught in most high schools around the world. It gets lost after graduation.
In Conclusion
Math is not for an elite. But it is for trained minds. Its universal truths, if correctly understood, can entertain, educate and enhance our intelligence if we build on a proper foundation. Everyone benefits.
If we continue, via social media, to feed mathematical swill to the general population under guise that it will feed their intellect and educate them to understand real world problems we are in peril. Peril of, not only diminishing man’s effective use of mathematics to solve those problems, but of never seeing its beauty.
Footnotes:
- There are mathematical systems, for example, the surreal numbers, where infinity can be treated as a number but not the real numbers.
- Mathematical systems often have paradoxes. These may lead us to think there is an inconsistency in the system. However, could they not be be merely a part of the system or even something that will, in time, be resolved in some higher truth. Famous among these is Russell’s Paradox and Hilbert-Bernays_paradox
- There are many well put together mathematics websites as well as videos in Youtube, Vimeo etc. but reading the comments related to this material often leaves me with the impression that people leave these sites as ignorant as when they came.
- There is, of course, one way to ‘win’ the lottery — simply buy all the tickets. It is, however, financially unwise.
