F=ma

On Christmas Day 1642, a baby boy was born to a well-off Lincolnshire family in Woolsthorpe Manor. His childhood was somewhat chaotic; his father had died before he was born, and his mother remarried (to a stepfather he came to acutely dislike) when he was three. He was later to run away from school, discovered he hated the farming alternative and returned to become the school’s top pupil. He was also to later attend Trinity College Cambridge; oh, and became arguably the greatest scientist and mathematician of all time. His name was Isaac Newton.

Newton started off in a small way, developing binomial theorem; a technique used to expand powers of polynomials, which is a kind of fundamental technique used pretty much everywhere in modern science and mathematics; the advanced mathematical equivalent of knowing that 2 x 4 = 8. Oh, and did I mention that he was still a student at this point? Taking a break from his Cambridge career for a couple of years due to the minor inconvenience of the Great Plague, he whiled away the hours inventing calculus, which he finalised upon his return to Cambridge. Calculus is the collective name for differentiating and integrating, which allows one to find out the rate at which something is occurring, the gradient of a graph and the area under it algebraically; plus enabling us to reverse all of the above processes. This makes it sound like rather a neat and useful gimmick, but belies the fact that it allows us to mathematically describe everything from water flowing through a pipe to how aeroplanes fly (the Euler equations mentioned in my aerodynamics posts come from advanced calculus), and the discovery of it alone would have been enough to warrant Newton’s place in the history books. OK, and Leibniz who discovered pretty much the same thing at roughly the same time, but he got there later than Newton. So there.

However, discovering the most important mathematical tool to modern scientists and engineers was clearly not enough to occupy Newton’s prodigious mind during his downtime, so he also turned his attention to optics, aka the behaviour of light. He began by discovering that white light was comprised of all colours, revolutionising all contemporary scientific understanding of light itself by suggesting that coloured objects did not create their own colour, but reflected only certain portions of already coloured light. He combined this with discovering diffraction; that light shone through glass or another transparent material at an angle will bend. This then lead him to explain how telescopes worked, why the existing designs (based around refracting light through a lens) were flawed, and to design an entirely new type of telescope (the reflecting telescope) that is used in all modern astronomical equipment, allowing us to study, look at and map the universe like never before. Oh, and he also took the time to theorise the existence of photons (he called them corpuscles), which wouldn’t be discovered for another 250 years.

When that got boring, Newton turned his attention to a subject that he had first fiddled around with during his calculus time: gravity. Nowadays gravity is a concept taught to every schoolchild, but in Newton’s day the idea that objects fall to earth was barely even considered. Aristotle’s theories dictated that every object ‘wanted’ to be in a state of stillness on the ground unless disturbed, and Newton was the first person to make a serious challenge to that theory in nearly two millennia (whether an apple tree was involved in his discovery is heavily disputed). Not only did he and colleague Robert Hooke define the force of gravity, but they also discovered the inverse-square law for its behaviour (aka if you multiply the distance you are away from a planet by 2, then you will decrease the gravitational force on you by 2 squared, or 4) and turned it into an equation (F=-GMm/r^2). This single equation would explain Kepler’s work on celestial mechanics, accurately predict the orbit of the ****ing planets (predictions based, just to remind you, on the thoughts of one bloke on earth with little technology more advanced than a pen and paper) and form the basis of his subsequent book: “Philosophiæ Naturalis Principia Mathematica”.

Principia, as it is commonly known, is probably the single most important piece of scientific writing ever written. Not only does it set down all Newton’s gravitational theories and explore their consequences (in minute detail; the book in its original Latin is bigger than a pair of good-sized bricks), but he later defines the concepts of mass, momentum and force properly for the first time; indeed, his definitions survive to this day and have yet to be improved upon.  He also set down his three laws of motion: velocity is constant unless a force acts upon an object, the acceleration of an object is proportional to the force acting on it and the object’s mass (summarised in the title of this post) and action and reaction are equal and opposite. These three laws not only tore two thousand years of scientific theory to shreds, but nowadays underlie everything we understand about object mechanics; indeed, no flaw was found in Newton’s equations until relativity was discovered 250 years later, which only really applies to objects travelling at around 100,000 kilometres per second or greater; not something Newton was ever likely to come across.

Isaac Newton’s life outside science was no less successful; he was something of an amateur alchemist and when he was appointed Master of the Royal Mint (a post he held for 30 years until his death; there is speculation his alchemical meddling may have resulted in mercury poisoning) he used those skills to great affect in assessing coinage, in an effort to fight Britain’s massive forgery problem. He was successful in this endeavour and later became the first man to put Britain onto the gold, rather than silver, standard, reflecting his knowledge of the superior chemical qualities of the latter metal (see another previous post). He is still considered by many to be the greatest genius who ever lived, and I can see where those people are coming from.

However, the reason I find Newton especially interesting concerns his private life. Newton was a notoriously hard man to get along with; he never married, almost certainly died a virgin and is reported to have only laughed once in his life (when somebody asked him what was the point in studying Euclid. The joke is somewhat highbrow, I’ll admit). His was a lonely existence, largely friendless, and he lived, basically for his work (he has been posthumously diagnosed with everything from bipolar disorder to Asperger’s syndrome). In an age when we are used to such charismatic scientists as Richard Feynman and Stephen Hawking, Newton’s cut-off, isolated existence with only his prodigious intellect for company seems especially alien. That the approach was effective is most certainly not in doubt; every one of his scientific discoveries would alone be enough to place him in science’s hall of fame, and to have done all of them puts him head and shoulders above all of his compatriots. In many ways, Newton’s story is one of the price of success. Was Isaac Newton a successful man? Undoubtedly, in almost every field he turned his hand to. Was he a happy man? We don’t know, but it would appear not. Given the choice between success and happiness, where would you fall?

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There is an art, or rather, a knack, to flying…

The aerofoil is one of the greatest inventions mankind has come up with in the last 150 years; in the late 19th century, aristocratic Yorkshireman (as well as inventor, philanthropist, engineer and generally quite cool dude) George Cayley identified the way bird wings generated lift merely by moving through the air (rather than just by flapping), and set about trying to replicate this lift force. To this end, he built a ‘whirling arm’ to test wings and measure the upwards lift force they generated, and found that a cambered wing shape (as in modern aerofoils) similar to that of birds was more efficient at generating lift than one with flat surfaces. This was enough for him to engineer the first manned, sustained flight, sending his coachman across Brompton Dale in 1863 in a homemade glider (the coachman reportedly handed in his notice upon landing with the immortal line “I was hired to drive, not fly”), but he still didn’t really have a proper understanding of how his wing worked.

Nowadays, lift is understood better by both science and the general population; but many people who think they know how a wing works don’t quite understand the full principle. There are two incomplete/incorrect theories that people commonly believe in; the ‘skipping stone’ theory and the ‘equal transit time’ theory.

The ‘equal transit time’ theory is popular because it sounds very sciency and realistic; because a wing is a cambered shape, the tip-tail distance following the wing shape is longer over the top of the wing than it is when following the bottom surface. Therefore, air travelling over the top of the wing has to travel further than the air going underneath. Now, since the aircraft is travelling at a constant speed, all the air must surely be travelling past the aircraft at the same rate; so, regardless of what path the air takes, it must take the same time to travel the same lateral distance. Since speed=distance/time, and air going over the top of the wing has to cover a greater distance, it will be travelling faster than the air going underneath the wing. Bernoulli’s principle tells us that if air travels faster, the air pressure is lower; this means the air on top of the wing is at a lower pressure than the air underneath it, and this difference in pressure generates an upwards force. This force is lift.

The key flaw in this theory is the completely wrong assumption that the air over the top and bottom of the wing must take the same time to travel across it. If we analyse the airspeed at various points over a wing we find that air going over the top does, in fact, travel faster than air going underneath it (the reason for this comes from Euler’s fluid dynamics equations, which can be used to derive the Navier-Stokes equations for aerofoil behaviour. Please don’t ask me to explain them). However, this doesn’t mean that the two airflows necessarily coincide at the same point when we reach the trailing edge of the wing, so the theory doesn’t correctly calculate the amount of lift generated by the wing. This is compounded by the theory not explaining any of the lift generated from the bottom face of the wing, or why the angle wing  is set at (the angle of attack) affects the lift it generates, or how one is able to generate some lift from just a flat sheet set at an angle (or any other symmetrical wing profile), or how aircraft fly upside-down.

Then we have the (somewhat simpler) ‘skipping stone’ theory, which attempts to explain the lift generated from the bottom surface of the wing. Its basic postulate concerns the angle of attack; with an angled wing, the bottom face of the wing strikes some of the incoming air, causing air molecules to bounce off it. This is like the bottom of the wing being continually struck by lots of tiny ball bearings, sort of the same thing that happens when a skimming stone bounces off the surface of the water, and it generates a net force; lift. Not only that, but this theory claims to explain the lower pressure found on top of the wing; since air is blocked by the tilted wing, not so much gets to the area immediately above/behind it. This means there are less air molecules in a given space, giving rise to a lower pressure; another way of explaining the lift generated.

There isn’t much fundamentally wrong with this theory, but once again the mathematics don’t check out; it also does not accurately predict the amount of lift generated by a wing. It also fails to explain why a cambered wing set at a zero angle of attack is still able to generate lift; but actually it provides a surprisingly good model when we consider supersonic flight.

Lift can be explained as a combination of these two effects, but to do so is complex and unnecessary  we can find a far better explanation just by considering the shape the airflow makes when travelling over the wing. Air when passing over an aerofoil tends to follow the shape of its surface (Euler again), meaning it deviates from its initially straight path to follow a curved trajectory. This curve-shaped motion means the direction of the airflow must be changing; and since velocity is a vector quantity, any change in the direction of the air’s movement represents a change in its overall velocity, regardless of any change in airspeed (which contributes separately). Any change in velocity corresponds to the air being accelerated, and since Force = mass x acceleration this acceleration generates a net force; this force is what corresponds to lift. This ‘turning’ theory not only describes lift generation on both the top and bottom wing surfaces, since air is turned upon meeting both, but also why changing the angle off attack affects lift; a steeper angle means the air has to turn more when following the wing’s shape, meaning more lift is generated. Go too steep however, and the airflow breaks away from the wing and undergoes a process called flow separation… but I’m getting ahead of myself.

This explanation works fine so long as our aircraft is travelling at less than the speed of sound. However, as we approach Mach 1, strange things start to happen, as we shall find out next time…

“Lies, damn lies, and statistics”

Ours is the age of statistics; of number-crunching, of quantifying, of defining everything by what it means in terms of percentages and comparisons. Statistics crop up in every walk of life, to some extent or other, in fields as widespread as advertising and sport. Many people’s livelihoods now depend on their ability to crunch the numbers, to come up with data and patterns, and much of our society’s increasing ability to do awesome things can be traced back to someone making the numbers dance.

In fact, most of what we think of as ‘statistics’ are not really statistics at all, but merely numbers; to a pedantic mathematician, a statistic is defined as a mathematical function of a sample of data, not the whole ‘population’ we are considering. We use statistics when it would be impractical to measure the whole population, usually because it’s too large, and when we instead are trying to mathematically model the whole population based on a small sample of it. Thus, next to no sporting ‘statistics’ are in fact true statistics as they tend to cover the whole game; if I heard during a rugby match that “Leicester had 59% of the possession”, that is nothing more than a number; or, to use the mathematical term, a parameter. A statistic would be to say “From our sample [of one game] we can conclude that Leicester control an average of 59% of the possession when they play rugby”, but this is quite evidently not true since we couldn’t extrapolate Leicester’s normal behaviour from a single match. It is for this reason that complex mathematical formulae are used to determine the uncertainty of a conclusion drawn from a statistical test, and these are based on the size of the sample we are testing compared to the overall size of the population we are trying to model. These uncertainty levels are often brushed under the carpet when pseudoscientists try to make dramatic, sweeping claims about something, but they are possibly the most important feature of modern statistics.

Another weapon for the poor statistician can be the mis-application of the idea of correlation. Correlation is basically what it means when you take two variables, plot them against one another on a graph, and find you get a nice neat line joining them, suggesting that the two are in some way related. Correlation tends to get scientists very excited, since if two things are linked then it suggests that you can make one thing happen by doing another, an often advantageous concept, and this is known as a causal relationship. However, whilst correlation and causation are rarely not intertwined, the first lesson every statistician learns is this; correlation DOES NOT imply causation.

Imagine, for instance, you have a cold. You feel like crap, your head is spinning, you’re dehydrated and you can’t breath through your nose. If we were, during the period before, during and after your cold, to plot a graph of one’s relative ability to breath through the nose against the severity of your headache (yeah, not very scientific I know), these two facts would both correlate, since they happen at the same time due to the cold. However, if I were to decide that this correlation implies causation, then I would draw the conclusion that all I need to do to give you a terrible headache is to plug your nose with tissue paper so you can’t breath through it. In this case, I have ignored the possibility (and, as it transpires, the eventuality) of there being a third variable (the cold virus) that causes both of the other two variables, and this is very hard to investigate without poking our head out of the numbers and looking at the real world. There are statistical techniques that enable us to do this, but they are for another time.

Whilst this example was more childish than anything, mis-extrapolation of a correlation can have deadly consequences. One example, explored in Ben Goldacre’s Bad Science, concerns beta-carotene, an antioxidant found in carrots, and in 1981 an epidemiologist called Richard Peto published a meta-analysis (post for another time) of a series of scientific studies that suggested people with high beta-carotene levels showed a reduced risk of cancer. At the time, antioxidants were considered the wonder-substance of the nutrition, and everyone got on board with the idea that beta-carotene was awesome stuff. However, all of the studies examined were observational ones; taking a lot of different people, seeing what their beta-carotene levels were and then examining whether or not they had cancer or developed it in later life. None of the studies actually gave their subjects beta-carotene and then saw if that affected their cancer risk, and this prompted the editor of Nature magazine (the scientific journal in which Peto’s paper was published) to include a footnote reading:

Unwary readers (if such there are) should not take the accompanying article as a sign that the consumption of large quantities of carrots (or other dietary sources of beta-carotene) is necessarily protective against cancer.

The editor’s footnote quickly proved a well-judged one; a study conducted in Finland some time afterwards actually gave participants at high risk of lung cancer beta-carotene and found their risk of both getting the cancer and of death were higher than for the ‘placebo’ control group. A later study, named CARET (Carotene And Retinol Efficiency Trial), also tested groups at a high risk of lung cancer, giving half of them a mixture of beta-carotene and vitamin A and the other half placebos. The idea was to run the trial for six years and see how many illnesses/deaths each group ended up with; but after preliminary data found that those having the antioxidant tablets were 46% more likely to die from lung cancer, they decided it would be unethical to continue the trial and it was terminated early. Had the Nature article been allowed to get out of hand before this research was done, then it could have put thousands of people who hadn’t read the article properly at risk; and all because of the dangers of assuming correlation=causation.

This wasn’t really the gentle ramble through statistics I originally intended it to be, but there you go; stats. Next time, something a little less random. Maybe

Art vs. Science

All intellectual human activity can be divided into one of three categories; the arts, humanities, and sciences (although these terms are not exactly fully inclusive). Art here covers everything from the painted medium to music, everything that we humans do that is intended to be creative and make our world as a whole a more beautiful place to live in. The precise definition of ‘art’ is a major bone of contention among creative types and it’s not exactly clear where the boundary lies in some cases, but here we can categorise everything intended to be artistic as an art form. Science here covers every one of the STEM disciplines; science (physics, biology, chemistry and all the rest in its vast multitude of forms and subgenres), technology, engineering (strictly speaking those two come under the same branch, but technology is too satisfying a word to leave out of any self-respecting acronym) and mathematics. Certain portions of these fields too could be argued to be entirely self-fulfilling, and others are considered by some beautiful, but since the two rarely overlap the title of art is never truly appropriate. The humanities are an altogether trickier bunch to consider; on one hand they are, collectively, a set of sciences, since they purport to study how the world we live in behaves and functions. However, this particular set of sciences are deemed separate because they deal less with fundamental principles of nature but of human systems, and human interactions with the world around them; hence the title ‘humanities’. Fields as diverse as economics and geography are all blanketed under this title, and are in some ways the most interesting of sciences as they are the most subjective and accessible; the principles of the humanities can be and usually are encountered on a daily basis, so anyone with a keen mind and an eye for noticing the right things can usually form an opinion on them. And a good thing too, otherwise I would be frequently short of blogging ideas.

Each field has its own proponents, supporters and detractors, and all are quite prepared to defend their chosen field to the hilt. The scientists point to the huge advancements in our understanding of the universe and world around us that have been made in the last century, and link these to the immense breakthroughs in healthcare, infrastructure, technology, manufacturing and general innovation and awesomeness that have so increased our quality of life (and life expectancy) in recent years. And it’s not hard to see why; such advances have permanently changed the face of our earth (both for better and worse), and there is a truly vast body of evidence supporting the idea that these innovations have provided the greatest force for making our world a better place in recent times. The artists provide the counterpoint to this by saying that living longer, healthier lives with more stuff in it is all well and good, but without art and creativity there is no advantage to this better life, for there is no way for us to enjoy it. They can point to the developments in film, television, music and design, all the ideas of scientists and engineers tuned to perfection by artists of each field, and even the development in more classical artistic mediums such as poetry or dance, as key features of the 20th century that enabled us to enjoy our lives more than ever before. The humanities have advanced too during recent history, but their effects are far more subtle; innovative strategies in economics, new historical discoveries and perspectives and new analyses of the way we interact with our world have all come, and many have made news, but their effects tend to only be felt in the spheres of influence they directly concern- nobody remembers how a new use of critical path analysis made J. Bloggs Ltd. use materials 29% more efficiently (yes, I know CPA is technically mathematics; deal with it). As such, proponents of humanities tend to be less vocal than those in other fields, although this may have something to do with the fact that the people who go into humanities have a tendency to be more… normal than the kind of introverted nerd/suicidally artistic/stereotypical-in-some-other-way characters who would go into the other two fields.

This bickering between arts & sciences as to the worthiness/beauty/parentage of the other field has lead to something of a divide between them; some commentators have spoken of the ‘two cultures’ of arts and sciences, leaving us with a sect of sciences who find it impossible to appreciate the value of art and beauty, thinking it almost irrelevant compared what their field aims to achieve (to their loss, in my opinion). I’m not entirely sure that this picture is entirely true; what may be more so, however, is the other end of the stick, those artistic figures who dominate our media who simply cannot understand science beyond GCSE level, if that. It is true that quite a lot of modern science is very, very complex in the details, but Albert Einstein was famous for saying that if a scientific principle cannot be explained to a ten-year old then it is almost certainly wrong, and I tend to agree with him. Even the theory behind the existence of the Higgs Boson, right at the cutting edge of modern physics, can be explained by an analogy of a room full of fans and celebrities. Oh look it up, I don’t want to wander off topic here.

The truth is, of course, that no field can sustain a world without the other; a world devoid of STEM would die out in a matter of months, a world devoid of humanities would be hideously inefficient and appear monumentally stupid, and a world devoid of art would be the most incomprehensibly dull place imaginable. Not only that, but all three working in harmony will invariably produce the best results, as master engineer, inventor, craftsman and creator of some of the most famous paintings of all time Leonardo da Vinci so ably demonstrated. As such, any argument between fields as to which is ‘the best’ or ‘the most worthy’ will simply never be won, and will just end up a futile task. The world is an amazing place, but the real source of that awesomeness is the diversity it contains, both in terms of nature and in terms of people. The arts and sciences are not at war, nor should they ever be; for in tandem they can achieve so much more.

The Myth of Popularity

WARNING: Everything I say forthwith is purely speculative based on a rough approximation of a presented view of how a part of our world works, plus some vaguely related stuff I happen to know. It is very likely to differ from your own personal view of things, so please don’t get angry with me if it does.

Bad TV and cinema is a great source of inspiration; not because there’s much in it that’s interesting, but because there’s just so much of it that even without watching any it is possible to pick up enough information to diagnose trends, which are generally interesting to analyse. In this case, I refer to the picture of American schools that is so often portrayed by iteration after iteration of generic teenage romance/romcom/’drama’, and more specifically the people in it.

One of the classic plot lines of these types of things involves the ‘hopelessly lonely/unpopular nerd who has crush on Miss Popular de Cheerleader and must prove himself by [insert totally retarded idea]’. Needless to say these plot lines are more unintentionally hilarious and excruciating than anything else, but they work because they play on the one trope that so many of us are familiar with; that of the overbearing, idiotic, horrible people from the ‘popular’ social circle. Even if we were not raised within a sitcom, it’s a situation repeated in thousands of schools across the world- the popular kids are the arseholes at the top with inexplicable access to all the gadgets and girls, and the more normal, nice people lower down the social circle.

The image exists in our conciousness long after leaving school for a whole host of reasons; partly because major personal events during our formative years tend to have a greater impact on our psyche than those occurring later on in life, but also because it is often our first major interaction with the harsh unfairness life is capable of throwing at us. The whole situation seems totally unfair and unjust; why should all these horrible people be the popular ones, and get all the social benefits associated with that? Why not me, a basically nice, humble person without a Ralph Lauren jacket or an iPad 3, but with a genuine personality? Why should they have all the luck?

However, upon analysing the issue then this object of hate begins to break down; not because the ‘popular kids’ are any less hateful, but because they are not genuinely popular. If we define popular as a scale representative of how many and how much people like you (because what the hell else is it?), then it becomes a lot easier to approach it from a numerical, mathematical perspective. Those at the perceived top end of the social spectrum generally form themselves into a clique of superiority, where they all like one another (presumably- I’ve never been privy to being in that kind of group in order to find out) but their arrogance means that they receive a certain amount of dislike, and even some downright resentment, from the rest of the immediate social world. By contrast, members of other social groups (nerds, academics [often not the same people], those sportsmen not in the ‘popular’ sphere, and the myriad of groups of undefineable ‘normies’ who just splinter off into their own little cliques) tend to be liked by members of their selected group and treated with either neutrality or minor positive or negative feeling from everyone else, leaving them with an overall ‘popularity score’, from an approximated mathematical point of view, roughly equal to or even greater than the ‘popular’ kids. Thus, the image of popularity is really something of a myth, as these people are not technically speaking any more popular than anyone else.

So, then, how has this image come to present itself as one of popularity, of being the top of the social spectrum? Why are these guys on top, seemingly above group after group of normal, friendly people with a roughly level playing field when it comes to social standing?

If you were to ask George Orwell this question, he would present you with a very compelling argument concerning the nature of a social structure to form a ‘high’ class of people (shortly after asking you how you managed to communicate with him beyond the grave). He and other social commentators have frequently pointed out that the existence of a social system where all are genuinely treated equally is unstable without some ‘higher class’ of people to look up to- even if it is only in hatred. It is humanity’s natural tendency to try and better itself, try to fight its way to the top of the pile, so if the ‘high’ group disappear temporarily they will be quickly replaced; hence why there is such a disparity between rich and poor even in a country such as the USA founded on the principle that ‘all men are created free and equal’. This principle applies to social situations too; if the ‘popular’ kids were to fall from grace, then some other group would likely rise to fill the power vacuum at the top of the social spectrum. And, as we all know, power and influence are powerful corrupting forces, so this position would be likely to transform this new ‘popular’ group into arrogant b*stards too, removing the niceness they had when they were just normal guys. This effect is also in evidence that many of the previously hateful people at the top of the spectrum become very normal and friendly when spoken to one-on-one, outside of their social group (from my experience anyway; this does not apply to all people in such groups)

However, another explanation is perhaps more believable; that arrogance is a cause rather than a symptom. By acting like they are better than the rest of the world, the rest of the world subconsciously get it into their heads that, much though they are hated, they are the top of the social ladder purely because they said so. And perhaps this idea is more comforting, because it takes us back to the idea we started with; that nobody is more actually popular than anyone else, and that it doesn’t really matter in the grand scheme of things. Regardless of where your group ranks on the social scale, if it’s yours and you get along with the people in it, then it doesn’t really matter about everyone else or what they think, so long as you can get on, be happy, and enjoy yourself.

Footnote: I get most of these ideas from what is painted by the media as being the norm in American schools and from what friends have told me, since I’ve been lucky enough that the social hierarchies I encountered from my school experience basically left one another along. Judging by the horror stories other people tell me, I presume it was just my school. Plus, even if it’s total horseshit, it’s enough of a trope that I can write a post about it.

Determinism

In the early years of the 19th century, science was on a roll. The dark days of alchemy were beginning to give way to the modern science of chemistry as we know it today, the world of physics and the study of electromagnetism were starting to get going, and the world was on the brink of an industrial revolution that would be powered by scientists and engineers. Slowly, we were beginning to piece together exactly how our world works, and some dared to dream of a day where we might understand all of it. Yes, it would be a long way off, yes there would be stumbling blocks, but maybe, just maybe, so long as we don’t discover anything inconvenient like advanced cosmology, we might one day begin to see the light at the end of the long tunnel of science.

Most of this stuff was the preserve of hopeless dreamers, but in the year 1814 a brilliant mathematician and philosopher, responsible for underpinning vast quantities of modern mathematics and cosmology, called Pierre-Simon Laplace published a bold new article that took this concept to extremes. Laplace lived in the age of ‘the clockwork universe’, a theory that held Newton’s laws of motion to be sacrosanct truths and claimed that these laws of physics caused the universe to just keep on ticking over, just like the mechanical innards of a clock- and just like a clock, the universe was predictable. Just as one hour after five o clock will always be six, presuming a perfect clock, so every result in the world can be predicted from the results. Laplace’s arguments took such theory to its logical conclusion; if some vast intellect were able to know the precise positions of every particle in the universe, and all the forces and motions of them, at a single point in time, then using the laws of physics such an intellect would be able to know everything, see into the past, and predict the future.

Those who believed in this theory were generally disapproved of by the Church for devaluing the role of God and the unaccountable divine, whilst others thought it implied a lack of free will (although these issues are still considered somewhat up for debate to this day). However, among the scientific community Laplace’s ideas conjured up a flurry of debate; some entirely believed in the concept of a predictable universe, in the theory of scientific determinism (as it became known), whilst others pointed out the sheer difficulty in getting any ‘vast intellect’ to fully comprehend so much as a heap of sand as making Laplace’s arguments completely pointless. Other, far later, observers, would call into question some of the axiom’s upon which the model of the clockwork universe was based, such as Newton’s laws of motion (which collapse when one does not take into account relativity at very high velocities); but the majority of the scientific community was rather taken with the idea that they could know everything about something should they choose to. Perhaps the universe was a bit much, but being able to predict everything, to an infinitely precise degree, about a few atoms perhaps, seemed like a very tempting idea, offering a delightful sense of certainty. More than anything, to these scientists there work now had one overarching goal; to complete the laws necessary to provide a deterministic picture of the universe.

However, by the late 19th century scientific determinism was beginning to stand on rather shaky ground; although  the attack against it came from the rather unexpected direction of science being used to support the religious viewpoint. By this time the laws of thermodynamics, detailing the behaviour of molecules in relation to the heat energy they have, had been formulated, and fundamental to the second law of thermodynamics (which is, to this day, one of the fundamental principles of physics) was the concept of entropy.  Entropy (denoted in physics by the symbol S, for no obvious reason) is a measure of the degree of uncertainty or ‘randomness’ inherent in the universe; or, for want of a clearer explanation, consider a sandy beach. All of the grains of sand in the beach can be arranged in a vast number of different ways to form the shape of a disorganised heap, but if we make a giant, detailed sandcastle instead there are far fewer arrangements of the molecules of sand that will result in the same structure. Therefore, if we just consider the two situations separately, it is far, far more likely that we will end up with a disorganised ‘beach’ structure rather than a castle forming of its own accord (which is why sandcastles don’t spring fully formed from the sea), and we say that the beach has a higher degree of entropy than the castle. This increased likelihood of higher entropy situations, on an atomic scale, means that the universe tends to increase the overall level of entropy in it; if we attempt to impose order upon it (by making a sandcastle, rather than waiting for one to be formed purely by chance), we must input energy, which increases the entropy of the surrounding air and thus resulting in a net entropy increase. This is the second law of thermodynamics; entropy always increases, and this principle underlies vast quantities of modern physics and chemistry.

If we extrapolate this situation backwards, we realise that the universe must have had a definite beginning at some point; a starting point of order from which things get steadily more chaotic, for order cannot increase infinitely as we look backwards in time. This suggests some point at which our current universe sprang into being, including all the laws of physics that make it up; but this cannot have occurred under ‘our’ laws of physics that we experience in the everyday universe, as they could not kickstart their own existence. There must, therefore, have been some other, higher power to get the clockwork universe in motion, destroying the image of it as some eternal, unquestionable predictive cycle. At the time, this was seen as vindicating the idea of the existence of God to start everything off; it would be some years before Edwin Hubble would venture the Big Bang Theory, but even now we understand next to nothing about the moment of our creation.

However, this argument wasn’t exactly a death knell for determinism; after all, the laws of physics could still describe our existing universe as a ticking clock, surely? True; the killer blow for that idea would come from Werner Heisenburg in 1927.

Heisenburg was a particle physicist, often described as the person who invented quantum mechanics (a paper which won him a Nobel prize). The key feature of his work here was the concept of uncertainty on a subatomic level; that certain properties, such as the position and momentum of a particle, are impossible to know exactly at any one time. There is an incredibly complicated explanation for this concerning wave functions and matrix algebra, but a simpler way to explain part of the concept concerns how we examine something’s position (apologies in advance to all physics students I end up annoying). If we want to know where something is, then the tried and tested method is to look at the thing; this requires photons of light to bounce off the object and enter our eyes, or hypersensitive measuring equipment if we want to get really advanced. However, at a subatomic level a photon of light represents a sizeable chunk of energy, so when it bounces off an atom or subatomic particle, allowing us to know where it is, it so messes around with the atom’s energy that it changes its velocity and momentum, although we cannot predict how. Thus, the more precisely we try to measure the position of something, the less accurately we are able to know its velocity (and vice versa; I recognise this explanation is incomplete, but can we just take it as red that finer minds than mine agree on this point). Therefore, we cannot ever measure every property of every particle in a given space, never mind the engineering challenge; it’s simply not possible.

This idea did not enter the scientific consciousness comfortably; many scientists were incensed by the idea that they couldn’t know everything, that their goal of an entirely predictable, deterministic universe would forever remain unfulfilled. Einstein was a particularly vocal critic, dedicating the rest of his life’s work to attempting to disprove quantum mechanics and back up his famous statement that ‘God does not play dice with the universe’. But eventually the scientific world came to accept the truth; that determinism was dead. The universe would never seem so sure and predictable again.

NUMBERS

One of the most endlessly charming parts of the human experience is our capacity to see something we can’t describe and just make something up in order to do so, never mind whether it makes any sense in the long run or not. Countless examples have been demonstrated over the years, but the mother lode of such situations has to be humanity’s invention of counting.

Numbers do not, in and of themselves, exist- they are simply a construct designed by our brains to help us get around the awe-inspiring concept of the relative amounts of things. However, this hasn’t prevented this ‘neat little tool’ spiralling out of control to form the vast field that is mathematics. Once merely a diverting pastime designed to help us get more use out of our counting tools, maths (I’m British, live with the spelling) first tentatively applied itself to shapes and geometry before experimenting with trigonometry, storming onwards to algebra, turning calculus into a total mess about four nanoseconds after its discovery of something useful, before just throwing it all together into a melting point of cross-genre mayhem that eventually ended up as a field that it as close as STEM (science, technology, engineering and mathematics) gets to art, in that it has no discernible purpose other than for the sake of its own existence.

This is not to say that mathematics is not a useful field, far from it. The study of different ways of counting lead to the discovery of binary arithmetic and enabled the birth of modern computing, huge chunks of astronomy and classical scientific experiments were and are reliant on the application of geometric and trigonometric principles, mathematical modelling has allowed us to predict behaviour ranging from economics & statistics to the weather (albeit with varying degrees of accuracy) and just about every aspect of modern science and engineering is grounded in the brute logic that is core mathematics. But… well, perhaps the best way to explain where the modern science of maths has lead over the last century is to study the story of i.

One of the most basic functions we are able to perform to a number is to multiply it by something- a special case, when we multiply it by itself, is ‘squaring’ it (since a number ‘squared’ is equal to the area of a square with side lengths of that number). Naturally, there is a way of reversing this function, known as finding the square root of a number (ie square rooting the square of a number will yield the original number). However, convention dictates that a negative number squared makes a positive one, and hence there is no number squared that makes a negative and there is no such thing as the square root of a negative number, such as -1. So far, all I have done is use a very basic application of logic, something a five-year old could understand, to explain a fact about ‘real’ numbers, but maths decided that it didn’t want to not be able to square root a negative number, so had to find a way round that problem. The solution? Invent an entirely new type of number, based on the quantity i (which equals the square root of -1), with its own totally arbitrary and made up way of fitting  on a number line, and which can in no way exist in real life.

Admittedly, i has turned out to be useful. When considering electromagnetic forces, quantum physicists generally assign the electrical and magnetic components real and imaginary quantities in order to identify said different components, but its main purpose was only ever to satisfy the OCD nature of mathematicians by filling a hole in their theorems. Since then, it has just become another toy in the mathematician’s arsenal, something for them to play with, slip into inappropriate situations to try and solve abstract and largely irrelevant problems, and with which they can push the field of maths in ever more ridiculous directions.

A good example of the way mathematics has started to lose any semblance of its grip on reality concerns the most famous problem in the whole of the mathematical world- Fermat’s last theorem. Pythagoras famously used the fact that, in certain cases, a squared plus b squared equals c squared as a way of solving some basic problems of geometry, but it was never known as to whether a cubed plus b cubed could ever equal c cubed if a, b and c were whole numbers. This was also true for all other powers of a, b and c greater than 2, but in 1637 the brilliant French mathematician Pierre de Fermat claimed, in a scrawled note inside his copy of Diohantus’ Arithmetica, to have a proof for this fact ‘that is too large for this margin to contain’. This statement ensured the immortality of the puzzle, but its eventual solution (not found until 1995, leading most independent observers to conclude that Fermat must have made a mistake somewhere in his ‘marvellous proof’) took one man, Andrew Wiles, around a decade to complete. His proof involved showing that the terms involved in the theorem could be expressed in the form of an incredibly weird equation that doesn’t exist in the real world, and that all equations of this type had a counterpart equation of an equally irrelevant type. However, since the ‘Fermat equation’ was too weird to exist in the other format, it could not logically be true.

To a mathematician, this was the holy grail; not only did it finally lay to rest an ages-old riddle, but it linked two hitherto unrelated branches of algebraic mathematics by way of proving what is (now it’s been solved) known as the Taniyama-Shimura theorem. To anyone interested in the real world, this exercise made no contribution to it whatsoever- apart from satisfying a few nerds, nobody’s life was made easier by the solution, it didn’t solve any real-world problem, and it did not make the world a tangibly better place. In this respect then, it was a total waste of time.

However, despite everything I’ve just said, I’m not going to decide that all modern day mathematics is a waste of time; very few human activities ever are. Mathematics is many things; among them ridiculous, confusing, full of contradictions and potential slip-ups and, in a field whose age of winning a major prize is younger than in any other STEM field, apparently full of those likely to belittle you out of future success should you enter the world of serious academia. But, for some people, maths is just what makes the world makes sense, and at its heart that was all it was ever created to do. And if some people want their life to be all about the little symbols that make the world make sense, then well done to the world for making a place for them.

Oh, and there’s a theory doing the rounds of cosmology nowadays that reality is nothing more than a mathematical construct. Who knows in what obscure branch of reverse logarithmic integrals we’ll find answers about that one…