“Daddy,” my eldest daughter asked me, some years ago now (at a time when Europe was but a short train ride away and a welcome escape from the grey winters of Surrey), “What’s the Spanish word for thank you?”
“Gracias.” I replied, pleased by her inquisitiveness “And denada means you’re welcome.”
Warming to the conversation, she went on “Oh. And what’s thank you in French?”
“And what do they say for you’re welcome?”
I paused for a moment (but only, it must be said, a moment) reflecting on the fact that I had no idea how to express that concept – my ability with the French language extending little further than ordering coffee and croissants for breakfast – before telling her, with all my fatherly sincerity “The French have no phrase for that.”
Now yes, I admit it was a cheap knee-bend to Francophone stereotypes, and a ‘Dad joke’ to cover my linguistic ignorance…and it was probably inappropriate for me to let an impressionable child go on believing this for as long as I subsequently did.
But it does introduce an interesting and important concept – our ability to describe something has no bearing on its reality. Even if my statement were true and the French had, through some curious artifact of linguistic heritage, failed to develop a phrase capable of expressing gratitude, it would not change the fact that such feelings could – and do – exist. Language describes reality. It does not – outside of the most extreme hardline views of social constructivism – define it.
Mathematics too is essentially a language – a language, moreover, that we can use to describe the physical reality of the universe. Most of the time. As with the example of spoken language above though, the critical caveat is that however well mathematics describes physical behaviour, again, it does not define it.
Sir Phillip Bin, the fictional hero of Mark Evans’ radio comedy ‘Bleak Expectations’, muses wistfully on the days before Sir Isaac Newton ‘invented’ gravity, when people falling from great height would ‘simply drift gently and harmlessly to the ground’.
Such satirical diversions aside, Newtonian mechanics works pretty well in describing the interactions of macroscopic objects under the conditions of our everyday experience. But gravitational attraction between two bodies doesn’t fall off in proportion to the square of the distance between them because that’s the way the equation is written – rather, the equation seeks to empirically describe the behaviour that occurs.
As Einstein recognised in his theories of general and special relativity, under certain circumstances – far removed from the world of everyday experience – objects behave in ways that are incompatible with Newtonian physics. In formulating expressions to account for this relativistic behaviour, Einstein did not change the nature of the universe – he simply gave us a new form of language by which to describe the poetry of our existence.
Similarly, the remarkable duality of electrons – whereby they can be shown through physical experiment to possess the characteristics of both a continuous wave function and a discrete physical particle – is only a paradox in the context of the ways in which we have come to describe these sub-atomic features. Fundamentally, the electron is what it is, and if theories are unable to fully account for its behaviour, it is a reflection of the inadequacy of our mathematical approximations for reality, not proof of some cosmic trick set up to titillate a Vegas audience on the quantum scale.
Perhaps the most interesting example of this concept in action, however, is the search for an ultimate physical ‘theory of everything’. The properties of electromagnetism, strong nuclear and weak nuclear attraction, and gravity – the fundamental forces that define and control interactions of matter and energy throughout the universe – converge at high energy, and it is theorized that all four derive from a common underlying property. But just what this is remains a point of hard debate, as none of the individual equations that are so successful in describing the behaviour of each of these forces on the macroscopic level of the everyday can adequately cope with the conditions of this theoretical point of convergence.
This does not mean that there are somehow four separate overlapping layers making up the Universe that don’t quite fit together perfectly where they join, like some kind of badly put together set of existential DIY shelves. Rather, the theory runs that there is one reality, where all aspects of the physical behaviour that we observe in the universe must somehow derive from the fundamental character of matter and energy. The failure lies in the mathematical language in our possession – it’s not just that it’s tricky to calculate the results, standard mathematics is literally unable to describe reality under those conditions.
The ‘theory of everything’ that can account for the emergence and existence of these separate forces is one of the great challenges at the business end of modern physics where the big kids of theory get serious. Tackling this problem however requires not just a dab hand with a slide rule, but the creation – literally – of entirely new forms of mathematics, incorporating additional physical fields and interactions, and even extra dimensions of space.
For the record, I should confess that I’m not one of those big kids – a real physicist would have stolen my mathematical lunch money and sent me crying for home long before we even got to string theory – which I understand is regarded as one of the more accessible (and promising) of these approaches. As secret shames go, I can appreciate that this is not exactly stupendous, but I’ve been happily married for 16 years and don’t get out to as many wild parties as I used to.
The point is, I’m fine with that. I don’t need to understand the higher order branches of mathematics – the high linguistics of the Physicist’s hymnal – to appreciate the reality and significance of what they are trying to achieve in understanding the nature of reality. I wish them well, and look forward to the day that Google produces a Mathematics-English translator so I can appreciate the beauty of their work.
I’m sure even the French would be grateful for that.