part0016

The other dogma associated with modern science is its singular obsession with presenting rigorous mathematical treatments and having them take precedence over rigorous scientific argument; thereby greatly reducing the likely applicability and accessibility of the science itself. Lord Kelvin once said: “Until you can measure something and express it in numbers, you have only the beginning of understanding.” There is little doubt he was right; mathematics is a very powerful tool that is the key to the quantitative understanding that is generally applied in physics to an initial qualitative appreciation, and which is usually more useful. (I must point out, however, that Kelvin was not always correct – he apparently said “X-rays will prove to be a hoax.” which was, I fancy, an error. But I would agree with him regarding the benefits of the metric system of measurement and his recommendation for its widespread adoption.) It is also not surprising that the great man who invented physics also invented the powerful mathematical tool that can be used to quantify and analyse so many phenomena – the calculus (and I don’t mean the calculus dentists talk about). It does, however, have to be admitted that Newton was somewhat eccentric – or prone at times to neuroses that would today be classified as mental illness. It seems he was subject at times to some degree of paranoia and that this tended to make him wish not to publish his findings or to do so in a relatively inaccessible way. This may explain why, although he invented calculus he did not publish it for twelve years (but did in that time discuss his invention with others, conversations that other mathematicians may have heard of – thereby giving them an opportunity to claim the invention for themselves). Regarding the slight inaccessibility of his papers, this may have been increased by his tendency to write them in Latin (although to be fair this was usual for scientists/natural philosophers of his time), but it also appears that in presenting his mathematics he avoided the use of algebra (which we employ today) in favour of the more traditional ratios and proportions – which would have made his mathematical proofs less accessible. It is, however, worth making a point in relation to this which illustrates the astonishing apparent prescience of Newton. In mechanics we think of his famous second law as F = ma, but Newton preferred to denote the force in terms of a change in momentum divided by the time taken: F = (m2 v2 – m1 v1 )/t. If we are considering very high velocities near to c, then according to accepted modern physics the mass of the body will increase – this effect is accommodated in F = (m2 v2 – m1 v1 )/t, but not in F = ma; so that in the 1600s Newton seems to have anticipated relativistic time dilation! I will in fact have much more to say on relativity and time dilation, later on.

Looking back over the long period since Newton, it is difficult to fully appreciate the influence he has had on physics (and science in general) and on all the technologies (e.g. engineering) that have been facilitated by it over the years. I do not think it is unreasonable to say that the whole of the scientific method, and its subsequent publication, have been and still are, heavily influenced by his approach. Consequently, the eccentricities mentioned above may be a capital cause of the criticisms that could be levelled at modern science (particularly physics) practice and subsequent reporting. The mechanistic approach to science that Newton pioneered, and which has proved so useful in our development of machines (e.g. trains and cars) over recent centuries, may not in fact be appropriate to revealing fundamental knowledge on the universe. More specifically, the tendency in physics to try to analyse particles/fields on an ever-smaller scale in the hope that this will reveal some ‘general truths’ of physics seems to be failing to generate useful outcomes, i.e. outcomes with high impact. Reasons for this may be, firstly, that there is no guarantee that looking at an increasingly small scale is producing more fundamental knowledge; instead, in reality all that seems to be happening is that more complex structures and behaviours are becoming apparent. If progress towards fundamental understanding were being made, instead of this we would expect to discover simplification that would indicate unification of our understanding of physics – rather like Einstein was seeking in his unsuccessful quest for a unified field theory. The second reason for the deficiency of the current ‘bottom up’ approach to physics is that observing behaviour on the scale of quantum dynamics does not imply that the behaviour can be directly translated to our macroscopic man-size scale. For example, quantum mechanics shows that superposition can occur at the quantum scale - an elementary particle such as a photon can be in two different positions at the same time - but this is certainly not something we observe on our human scale. Again, more on this later.

But if a mechanistic ‘bottom up’ approach is not working, what other methodologies should we apply in modern physics? We already have an answer to this in the form of a rigorous analysis of the physics involved in example physics scenarios – or, in other words, using our knowledge of physics to solve various ‘thought experiments’. Yes, Albert Einstein made good use of the visualized thought experiment (which he called Gedankenexperiment ) as a means for understanding physical issues and for explaining his concepts to others. And that is quite a recommendation when you consider that he developed the theory of general relativity that describes how matter and energy interact with time and space, which has been checked experimentally. With Einstein, the physics ideas and logical reasoning came first and then the mathematical framework followed. But if you were to write a paper based on your carefully reasoned thought experiments, in order to shed light on an important question of physics, it is very likely publishers would refuse to print it if it did not contain a rigorous mathematical treatment. In fact, it is quite possible that they would be more likely to publish it if it contained the rigorous maths, even if it were not to make really significant points about physics. Freeman Dyson made some interesting comments somewhat relevant to this, when he described first meeting Richard Feynman. He said that prior to meeting Feynman he had been publishing papers that essentially comprised presentation of some ‘pretty maths’, but when they met he realised that Feynman was actually trying to understand the physics of what was going on. Dyson has also said: “Dick Feynman, who was my mentor as a physicist, had very little math. He never really thought in terms of mathematics; he had a very concrete imagination. He drew pictures instead of making calculations, and somehow got the right answers.”

There is a tendency for authors and publishers to prefer publication of technical papers that display a good deal of relatively abstruse, and so quite inaccessible, mathematics. One reason for this is that it helps to build a mystique around the subject which authors and publishers often believe boosts the impressiveness and gravitas of the work in question (this is, I believe, what Freeman Dyson was referring to when he mentioned pretty maths and is similar to the aversion that some clinicians have to simple solutions to medical problems, which is discussed later). Another reason for preferring the inclusion of relatively esoteric maths is simply a tendency to follow in the tradition established by Newton, which was discussed earlier – this is perhaps somewhat understandable given the giant of modern science that he was.

Therefore, to summarize this aspect of the dogma associated with modern science, we can say that maths is an extremely powerful tool in physics, but it is a facilitating tool – not an end in itself. The maths we employ in Newtonian mechanics, which is based on Euclidian geometry, does not have an inherent preternatural quality, but instead has great value because it is extremely useful for solving mechanics problems. If we developed another type of maths, based on non-Euclidian geometry, we might find this to be of no use in solving practical problems or describing the universe, and hence it would have no inherent value for us. But maths as we know it is useful; if we use an analogy of building a road, if we did not have maths available in physics it would be like trying to build the road using a teaspoon. What maths is to physics is what a JCB is to roadbuilding; but detailed analysis/endless variations on the JCB will provide very limited information on the nature of the road.

Another way of expressing the idea of the maths not having intrinsic general applicability to our world, but instead only being suitable to abstract calculations was put forward by Poe in his short story ‘The Purloined Letter’: "I dispute the availability, and thus the value, of that reason which is cultivated in any especial form other than the abstractly logical. I dispute, in particular, the reason educed by mathematical study. The mathematics are the science of form and quantity; mathematical reasoning is merely logic applied to observation upon form and quantity. The great error lies in supposing that even the truths of what is called pure algebra are abstract or general truths. And this error is so egregious that I am confounded at the universality with which it has been received. Mathematical axioms are not axioms of general truth…” Of course, Poe is best known for his gothic horror stories, such as ‘The Cask of Amontillado’ and ‘The Fall of the House of Usher’ (the latter being somewhat metaphysical and comprising what may be his most important story); and who can forget the pieces of doggerel that he penned, such as ‘The Raven’: “Once upon a midnight dreary, While I pondered, weak and weary…” (I seem to remember from Star Trek , that at one stage even the unemotional Mr Spock found this famous rhyme upon his lips – I believe it was the episode ‘Charlie X’.

But Charlie was forcing him to say the words – as well as the first two lines from Blake’s ‘The Tyger’. I was entertained.) By the way, one of Poe’s perhaps less well known works is his essay Eureka , where he starts off by saying that: "I design to speak of the Physical, Metaphysical and Mathematical – of the Material and Spiritual Universe: of its Essence, its Origin, its Creation, its Present Condition and its Destiny ". A pretty tall order you might think; but then, he was a genius. He is said to have considered it his greatest work and to claim it was more important than the discovery of gravity. That might be over-stating things a little I fancy, but I do find Eureka to be rather astonishing.

Sometimes of course, a rigorous mathematical treatment is appropriate and needed, but it is often found that this occurs when trying to solve problems in relatively simple situations. When the situation becomes more complex (as is usually the case in the real world), the closed form solution becomes too complicated or limiting and we need to look for other solutions. An example of this is provided by the calculation of strain distributions in engineering components that are exposed to mechanical stresses (forces). In the past there were attempts to employ classical stress strain calculations, based on closed form geometrical analyses, to calculate the strain distributions within components. However, it turns out that, other than for very simple geometries, such analyses lead to highly complex equations. For parts with geometries similar to those needed to be useful in practice, it is unlikely that closed-form solutions would be available. The ‘other solution’ in this case is simply to employ Finite Element Analysis (FEA), which is a commonly used iterative computer technique that has been shown to enable useful stress/strain analyses of a wide range of complex component geometries. It is necessary to mention that some maths is employed in FEA, but it is not complex, abstruse, or impenetrable. Each element in the FEA mesh is linked to its neighbour by means of an equation (for simple elastic simulations it would be the linear equation of a spring – for simulations of plastic deformations a more complex equation would be needed). However, the important point is that the ‘cleverness’ and so power of the technique does not lie with any particular complex equation, or set of equations; but instead is associated with the patterns formed when all of the elements combine and are iteratively calculated by the FEA programme.

A similar situation arises with the new computer modelling that is demonstrating an ability to automatically facilitate the undertaking of complex tasks in the real world, specifically deep learning and in particular convolutional neural networks . This is a large subject, but to summarise, it can provide a very powerful method for classification or solution of real-world problems (e.g. face recognition), by employing relatively crude simulations of the functions of the human eye (convolution) and brain (neural network). Deep learning comprises a genuinely impressive development in artificial intelligence that has demonstrated pattern recognition capabilities well above any competing methods; and is thereby opening up possibilities for automating a wide range of tasks which, up to now, have been considered too difficult to automate. The resulting systems may well prove to be lucrative for the inventors and their companies, but perhaps more importantly, they offer potential for liberating human workers from the labour and dangers associated with a large number of difficult, dangerous, and/or dirty jobs.

It is worth noting that this facility is being provided to us not by developments in classical mathematics that have a preternatural inherent knowledge of the world somehow associated with them, but rather by employing relatively simple mathematical algorithms, but utilising them in large numbers in highly inter-connected networks – where the model is in the network. The only drawback to such an approach is that it does require access to prodigious amounts of computing power. But now, fortuitously, we find such power to be commonly available to us all and at relatively low cost – through breakthroughs in the performance of graphics processor units (GPU); and the affordability of modern GPU technology which has been driven by none other than the demands of the computer gaming market. Wasn’t it Feynman who said that to get the best out of people at work you must also let them play? (I seem to remember that one of his ways of playing at work was to spin plates on long sticks – which is not something I have tried as of yet and am not sure that such an activity would be appreciated by Hospitality Services.)