Universality

After dinner on the 15ᵗʰ of April 1726, William Stukeley was drinking tea with Isaac Newton under the shade of some apple trees at Newton’s lodgings. Stukeley’s handwritten memoirs recount the 83-year old Newton’s reminiscences that evening, of his epiphany some 60 years previously on universal gravitation:

Amidst other discourse, he told me he was just in the same situation, as when formerly the notion of gravitation came into his mind. “Why should that apple always descend perpendicularly to the ground,” thought he to himself; occasioned by the fall of an apple, as he sat in contemplative mood. “Why should it not go sideways, or upwards? but constantly to the Earth’s center? Assuredly, the reason is that the Earth draws it. There must be a drawing power in matter. And the sum of the drawing power in the matter of the Earth must be in the Earth’s center, not in any side of the Earth. Therefore does this apple fall perpendicularly or toward the center. If matter thus draws matter, it must be in proportion of its quantity. Therefore the apple draws the Earth, as well as the Earth draws the apple.”

Newton’s 1687 Law of Gravitation explained its universal action, applicable to all matter throughout the cosmos, but famously could not deduce its causal properties1. That derivation was the profound achievement of Albert Einstein’s (1915) General Theory of Relativity. Einstein understood gravity not as a force acting across space but as mass defining the curvature of spacetime and spacetime defining the momentum of mass.

Many universalities continue to defy derivation. Feigenbaum’s constant ≈ 4.669 describes the rate of convergence to chaotic dynamics for any of a large class of functions that model populations with self-limiting growth2. Although its origin has been well understood for some 50 years, its deeper analytical structure remains mysterious.

Other universalities are unobservable, although probably real. All extant lifeforms are generally agreed to have a single Last Universal Common Ancestor (LUCA), to which they could in principle be traced back through evolutionary time. In other words, all organisms share a common ancestry, exemplified by Darwin’s 1837 notebook sketch of a ‘tree of life’.

Whereas universalities get discovered or inferred, a unification of systems requires construction as a coherent set of principles. What the unification may lack in subjective beauty it more-than makes up for in universal utility. The Standard Model of particle physics achieves an extraordinary feat in combining quantum mechanics with special relativity, to explain how elementary particles determine the composition of all perceived matter and all its governing forces except gravitation. It has been powering a worldwide revolution in technology since the late-20ᵗʰ century, even while physicists continue to disagree wildly on what quantum mechanics tells us about reality.

Unifications, being products of human willpower, often promise more than they deliver. The as-yet unfulfilled Grand Unified Theory conjectures a merging of electromagnetic, weak nuclear and strong nuclear forces into a single electronuclear force at high energies. We may never have instruments powerful enough to discover it. And nothing has come of a half-century of efforts to unify the Standard Model of these three fundamental forces with the General Theory of Relativity for gravitational force.

The Langlands Programme – the grand unified theory of mathematics – was born as a conjecture in a letter penned by Robert Langlands to André Weil in 1967, and it remains largely unproven to date. A few astounding proofs under its banner only accentuate the magnitude of the challenge. So far so daunting … And yet, these advances are now finding traction with some previously absurd thought experiments. It makes no sense to contemplate a unification of physics, which concerns the realities of the known Universe, with pure mathematics, which operates within universes of its own creation, right? Even that bridge now appears feasible, if not yet sensible, with surprising correspondences appearing between abstract geometry and quantum mechanics.

Why do we have this compulsion to seek commonalities? Very often less is more for our rational minds, in quotidian practicalities as much as esoteric or baroque ambitions. If you saw your neighbour using a drill and a screwdriver, you might want to show off the virtues of your own drill with screwdriver attachment. Universality often has an intrinsic aesthetic appeal, but not always. Perhaps your neighbour would be less impressed, for example, to see you eating with a spork.

If you met your doppelgänger, you might want to know whether you have relatives in common. You probably do have an unrelated doppelgänger somewhere amongst the 8+ billion other people in the world. You certainly have a common ancestor at some point in history, regardless of the resemblance and probably at least as recently as 2,000 years ago. And at some future time within the next 6,000 years, you – yes you, gentle reader! – will become either an ancestor of everyone or an ancestor of no one. Go on then, nest-makers – have that baby, and get yourself a ticket to the greatest lottery on Earth.

• Isaac Newton’s 1713 second edition of the Principia concludes in the penultimate paragraph with his frustration at knowing only how gravity acts, and not what causes it (translated from the Latin here).
• Mitchell Feigenbaum first saw his number 4.669201… (it has indefinite accuracy) in ‘logistic’ maps produced by a simple equation for the growth of a self-limiting population. He was using this equation to explore how the behaviour of mature populations varies across a spectrum of severity in self-limitation. For individuals with inherently slow self-replication rate, r, their population grows monotonically (smoothly) to its density limit, the carrying capacity K (e.g., an inoculate of phytoplankton in weak sunlight). For larger r (e.g., an inoculate benefitting from stronger sunlight), the population grows more sharply until it alternately overshoots and undershoots K in damped cycles that eventually settle to K. A still larger r will take the population to a 2-period ‘limit cycle’ of constant over- and under-shoots that never settles to K. Increase r a small amount more, and the cycling around K bifurcates from 2 periods (+ / −) to 4 periods (+ / − / + + / − −); with a further and even smaller increment in r it bifurcates again to 8 periods, and so on in a cascade of period-doubling bifurcations leading into chaotic behaviour. Feigenbaum found that each successive bifurcation arrived with somewhere between 4 and 5 times less increment in r than the last, converging on the ratio 4.669201… at infinite periodicity. This is the critical point at which population sizes transition from ordered periodicity into turbulent chaos. The logistic function powering these dynamics describes a parabolic rise then fall in the rate of population growth with density, and Feigenbaum had cranked up its steepness parameter, r, all the way to inducing chaotic behaviour in the mature population. Trigonometric functions such as a sinewave also describe humped population growth, and he found that they too converged to precisely the same ratio at the transition into chaos. Extraordinarily, the ratio applies to any non-linear function that expresses a rising then falling rate of population growth. Feigenbaum offered a theoretical explanation in his seminal 1978 paper, but not an analytical derivation, and none has been forthcoming since.

C.P. Doncaster, Timeline of the Human Condition, star index