Nomological machine

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/ˈnɒməˈlɒʤɪkᵊl (adj.)
Philosophy: Relating to or denoting principles that resemble laws, especially ones describing brute facts of the universe: things that are not explainable by theory, but are just “so”.

Philosopher Nancy Cartwright coined the term “nomological machine” to describe the limited conditions that must prevail for scientific laws to work. A nomological machine:

“... is a fixed (enough) arrangement of components, or factors, with stable (enough) capacities that in the right sort of stable (enough) environment will, with repeated operation, give rise to the kind of regular behavior that we represent in our scientific laws” [1]

As a piece of marketing, this is a terrible, obscurant — if technically accurate — label.[2] A better name would be “regularity machine” or even just a model: a device designed to generate regularities predicted by the theory by filtering out the inconvenient chattering, debris and crosstalk we get in real life, to extract the pure, untrammelled outcomes your theory would predict if the planet’s cussedly intractable impurities didn’t get in the way.

A “nomological machine” is a hermetically-sealed simple system, carefully designed to generate the specific outcome a scientific theory predicts. It is not a means of proving the theory so much as articulating it.

So, for example, take Newton’s second law of motion, F=ma. The force (F) acting on an object is equal to its mass (m) times its acceleration (a).

The nomological machine might be something like this: a perfectly elastic one kilogramme ball, in a frictionless vacuum, to which we apply a force of one Newton, which therefore accelerates at 1 metre per second squared.

The conditions in which this machine operates — zero friction, perfect elasticity, Euclidean spacetime, a non-inertial frame of reference, no other interfering causal effects — never prevail “in the wild”. In practice, there is always friction, interference, unknowns and inexactitude. We can never be sure of our measurements — was it exactly a Newton? Was the force was applied perfectly flush? Was the speedo was correctly calibrated? Being pragmatic folk, we we expect the prediction to be “near enough” but don’t expect accuracy to the micrometre. It is too hard to calculate, and we don’t have the data in any case.

Newton’s neat formula, with all these unrealistic conditions, is a nomological machine. If the observed universe does not quite come up to brief, we put its shortcomings down to observational errors and the lack of conditions required to satisfy the model. The nomological machine is not properly represented.

Most of the time, this does not matter. It is said that, when calculating Apollo module trajectories, NASA scientists used Newton rather than Einstein, because the relativistic maths was too hard, the effects would have been swamped by the margin for error in data observations, and it was safer and easier to just make mid-course corrections in any case.[3]

A rolling ball with no force acting upon it will eventually stop. This, so the theory goes, is only because of the corruptions of reality. So too, a crisp packet blowing this way and that across St. Mark’s square. Once you have discounted all the contaminating effects of the real world; the friction, convection, dust, drafts and so on — all of which are subject to their own equally scientific, equally certain laws, just in this case uncalculated — it still does, we assume obey scientific canon — but good luck proving it. For every lunar module, crisp packet, or every rolling ball, for every mass that ever accelerates in our imperfect human world, we give our models the benefit of a large and practically untestable doubt.

We presume that observed divergence is purely a function of lack of data and calculating wherewithal.

Are we justified in extrapolating laws that hold for nomological machines to the real world? Do these imaginary regularity generators really tell us how wind-blown crisp packets, or any of the other myriad quotidian physical effects we see and take for granted every day, behave, or are we just taking this on trust? Is this a conjuring trick? To find out, read Cartwright’s book. By way of hint, it is called, How the Laws of Physics Lie.

There are limits: if at some point a nomological machine doesn’t, even roughly, equate to observation, we just say it is wrong. The nomological machine F=25ma is wrong. Objects don’t accelerate anything like that fast. We would reject that nomological machine. We would say it is falsified.[4]

Analogical machines

Now: there are theory-based models of lifenomological machines — and life-based models of theory — for a laugh, let’s call these “analogical machines” — in which we force real-word artefacts to generate theoretical results. The former are things like F=ma; the latter are things like flipping coins and rolling dice, which we use as randomisers or to introduce a specific statistical risk into a game or a calculation. It is important not to confuse them.

Between these two classes the “map” and “territory” are transposed. In science, the map is the nomological machine: it is an abstract simplification of an intractable real-world territory. Lots of extraneous detail is missing, so we must remember to account for it when we use it to navigate.

With an analogical machine it is the other way round: the “real-world” dice are the map, and the territory is a theoretical probability. But it is a 1:1 scale map: as far as engineering permits it is identical to the territory. Machined dice falling on a flat, hard, constrained surface are not meant to represent “the real world”. They represent the idealised Platonic utopia of theory, free of friction and caprice, where abstract objects yield obediently to expected statistical outcomes.

Tumbling dice

There are two kinds of dice. Hypothetical dice, which are used to illustrate probabilities, ergodicity and the like — “imagine you rolled a dice ten million times” kind of thing — and actual dice, which we use to force probabilistic outcomes we need for other purposes. These are nomological machines. They are designed to explore and articulate the implications of a mathematical or scientific theory.

When we roll actual dice and flip actual coins — when we play monopoly or need to agree who kicks off — we are using analogical machines. We use them to practically obtain the probability we want: in this way they emulate a nomological machine. This is life imitating art imitating life, in a way. As long as our actual dice have six equal, evenly weighted sides and a flat constrained surface, they will be close enough to do the trick. Even though when we roll them, their trajectories are chaotic and fully impossible to predict, we still know the probabilities of the outcome. Such is the nomological machine we are emulating: all the excellent, unpredictable, randomising, chaos of the throw will be eventually be wiped out and replaced by a probability. On a flat, hard surface, one side must come to rest face-up. There are six equal sides. Each therefore has a ⅙ probability.

We calculate that probability in the abstract, using nomological machines. As long as our actual dice are well machined, it will be, basically, true of every single die. We do not need to experiment with lots of different dice and calculate an average to arrive at this conclusion. Every individual die must, within minimal tolerance, yield a ⅙ probability. If, over time, our dice don’t do that, we have not falsified probability theory: we have found some defective dice.

All dice, to count as dice, are functionally identical. Hold this thought: statistics is designed to work on populations that are functionally identical.

Map and territory as an immutable dualism: crossing and re-crossing the threshold

But hold map and territory — model and reality — as an immutable dualism. Map, territory. Model, reality. Online, offline. Formal, informal. Narnia, the real world.

We live in the territory: to abstract from territory to map, is to cross a threshold from the ordinary world to a model realm. This is a mythical, metaphorical journey. It is the same as the hero’s journey into a magical world, as Joseph Campbell outlined it The Hero with a Thousand Faces.[5] As we cross it we abstract from an intractable, analog actuality to a simplified digital essence, in the process giving up a colossal weight of “extraneous” information. What counts as extraneous is determined by the model. But unlike the fictional archetype, the magical model world cannot change the real world. Things that are true in the model kingdom are not necessarily true in the mundane world. Crossing back over the threshold, the lossed information is not restored. We can extrapolate, interpolate, approximate to emulate that information, and substitute something like it — in each case using the mathematical tools and amulets and swords we discovered in the magical model realm — but should we cross back to the mundane, the magic would drain away.

Maddeningly the magic often seems to work, sort of, in ordinary use, but it is hard to tell whether this is magic, or if it is just behaving like a normal, non-magical sword. (I once had an electric bike and it took me a week to realise the motor wasn't working. That kind of thing). It is only on those rare occasions when a normal sword won’t do — when you could really use a special sword, that you find it isn’t magic. This can be a “Wylie Coyote hanging in mid air off the cliff” situation.

There is a great temptation to steal back into the magical kingdom, where the magic works. And, the longer we stay there, in Narnia, the more we fall under its spell: the more we build out our book of magical incantations; the more we extrapolate from its own terms and logical imperatives the more impressive the model world seems to be. But the magical world is our own creation: the spells work because we define the rules and customs and principles therein. We are building our own memory palace.

We flesh out these theoretical implications without grounding them back to the territory they were originally supposed to be representing, we risk amplifying the variance between this simple magic all world and the complex, ornery world outside. As we push back the “limitations” of the model we cannot see the buried differences between this increasingly fantastical map and the boring old territory.

The map of theoretical physics has long since departed from the theoretical possibility of such a practical re-grounding. There is no possible real-world evidence for string theories, the multiverse, dark matter or the cosmological constant — the cosmological constant exists only to account for a gap in the evidence. For some of these things the very act of seeking evidence would destroy it. This is quite the skepticism-defeat device, by the way. as powerful as anything found in religion. These are all pure functions of extrapolation from the model. If the model is wrong, all this fantastical superstructure, also, is wrong. Yet the whole superstructure the investment in it, the careers, the billion-dollar particle accelerators, the industrial academic complex behind it — these exist in the real world. These are, seemingly, reason enough to believe, notwithstanding the apparently, unfalsifiably bonkers things these things, with a straight face, tell us must be true.

This is not to say any of this higher-order theoretical physics is not true or correct. We laypeople have no reason to doubt the maths. But mathematics is the business of working out an internal logical consistency.[6] Mathematics is a closed logical system; a linguistic game. It is the language in which we articulate our scientific and financial models. But it is the language of science, not itself a science. It has nothing to say about the territory.

First, be sure you know which domain is which. Are you trying to fit the world to a model — as you do when flipping a coin or rolling dice — or a model to the world?

Volatility calculations, Black-Scholes formulae, You can abstract fit real world to the model a normal distribution is a

For real-world events to confirm to normal distributions, standard deviations, and confident probabilities they must meet the criteria of the nomological machine. All potential events must known and independent of each other and our observation of them.

If a motivated agent intervenes it can upset the observed behaviour of the system. If you have all that all risks can be calculated and probabilities assigned.

See also


  1. Nancy Cartwright. The Dappled World – A Study of the Boundaries of Science. (Cambridge University Press, 1999)
  2. Like academics, lawyers learn to use the arcane vocabulary of the power structure while on the bottom rungs of the profession as a means of climbing up it: it is a credentialing strategy and part of the tribal identification ritual. By the time they get high enough to influence how the upcoming generations write, they have often forgotten how to write clearly and simply themselves. Cartwright is a brilliant thinker, but her writing is dense and academic.
  3. This would please Gerd Gigerenzer. There is a real lesson here for business managers techno-futurists.
  4. This is a very, very skin deep reading of the philosophy of science, I know, but bear with me.
  5. meta-irony: Campbell’s theory is of course a model, a carefully filtered monomythical model of the countless fables, legends and morality tales — all doubtless, per the model, similar but, in the analog particular, different, that he found in the oral and cultural traditions he surveyed.
  6. Bertrand Russell explicitly set out to formulate the complete set of mathematical axioms at the beginning of the 20th century with Principia Mathematica, and was fairly disappointed by Kurt Gödel proving it to be logically impossible.