All models are wrong

But some are useful (and that’s not the purpose anyway)

If you have ever taken a statistics course that deals with modeling or probabilistic forecasting, you may have heard the aphorism “all models are wrong.” This is generally attributed to George Box, who is noted in a 1976 paper in the Journal of the American Statistical Association, saying:

Since all models are wrong the scientist cannot obtain a “correct” one by excessive elaboration. On the contrary following William of Occam he should seek an economical description of natural phenomena. Just as the ability to devise simple but evocative models is the signature of the great scientist so overelaboration and overparameterization is often the mark of mediocrity. [1]

Box’s aphorism evolved two years later in a paper that was published in the proceedings of a 1978 statistics workshop, to include the contingency “all models are wrong, but some are useful.”

Now it would be very remarkable if any system existing in the real world could be exactly represented by any simple model. However, cunningly chosen parsimonious models often do provide remarkably useful approximations. For example, the law PV = RT relating pressure P, volume V and temperature T of an “ideal” gas via a constant R is not exactly true for any real gas, but it frequently provides a useful approximation and furthermore its structure is informative since it springs from a physical view of the behavior of gas molecules. For such a model there is no need to ask the question “Is the model true?”. If “truth” is to be the “whole truth” the answer must be “No”. The only question of interest is “Is the model illuminating and useful?” [2]

On Exactitude in Science

The idea of exactitude in science is not a new one. Jorge Luis Borges created a fictional reality, and a reductio ad absurdum of the territory-map relation, the science representing real-world objects via maps. In this fictional world, humans developed map-making ability so precise, it could only be printed on the 1:1 scale, with a full-scale map of the empire being the size of the empire itself. Succeeding generations came to judge the map at the scale of the earth itself as cumbersome, and in the western deserts, evidence of the geographic creation appeared in tattered fragments, which are still to be found, sheltering an occasional beast or beggar. [3] Borges’ short paragraph titled On Exactitude in Science:

…In that Empire, the Art of Cartography attained such Perfection that the map of a single Province occupied the entirety of a City, and the map of the Empire, the entirety of a Province. In time, those Unconscionable Maps no longer satisfied, and the Cartographers Guilds struck a Map of the Empire whose size was that of the Empire, and which coincided point for point with it. The following Generations, who were not so fond of the Study of Cartography as their Forebears had been, saw that that vast Map was Useless, and not without some Pitilessness was it, that they delivered it up to the Inclemencies of Sun and Winters. In the Deserts of the West, still today, there are Tattered Ruins of that Map, inhabited by Animals and Beggars; in all the Land there is no other Relic of the Disciplines of Geography. — (purportedly from) Suarez Miranda, Viajes de varones prudentes, Libro IV,Cap. XLV, Lerida, 1658. (published 1947)

The idea of the map-territory relation was not founded by Borges, either, it was an emergent idea which came about under the light of surrealism. Borges borrowed from a 1931 paper by Alfred Korzybski “A Non-Aristotelian System and its Necessity for Rigour in Mathematics and Physics,” which stated:

A map is not the territory it represents, but, if correct, it has a similar structure to the territory, which accounts for its usefulness.

Furthermore, Korzybski attributes inspiration for his paper to the mathematician Eric Temple Bell, who wrote in an epigram “the map is not the thing mapped [5].” This all has grounding in the cultural phenomenon that was surrealism, and famous idea of artist René Magritte, who promoted that “perception always intercedes between reality and ourselves,” and was famous for the “this is not a pipe” painting.

All we have about the world are approximations| What can be said of the similarity between the mathematical modeling of nature and human (fashion) modeling? We use the same word, but the two that take the verb as vocation are not doing the same thing.

Karl Lagerfeld, a German creative director (of Chanel), artist, photographer, and caricaturist said of modeling (not statistical modeling, but subject-modeling):

The secret to modeling is not being perfect. What one needs is a face that people can identify in a second. You have to be given what’s needed by nature, and what’s needed is to bring something new.

Thus, never by modeling the world or our definition of beauty will we achieve perfection — the ideal. Meanwhile, we continue to strive for the ideal using our best language(s) for the job. The best language we have for describing the natural world is mathematics. Naturally, then, mathematicians, who are not known for their creative generation, but their descriptiveness, best describe what they see in their own language. And they do it well. They do it better than the naturalists. Though the naturalists capture a certain complexity better than the mathematicians. A certain complexity borne of irrepeating patterns, jumbled confusion, chaos, intermittencies, and interregna. A complexity which requires at its core the central tenant that the universe is anything but congruous and predictable. Complexity which is not yet captured in traditional mathematics, and which is now the founding idea of the budding Chaos theory.

Modelers on the runway or catwalk are in fact doing something like the mathematical modeler and the naturalist. They are approximating the definition of something which is irreducibly subjective. In this case, the subject in question is approximating an answer to the question “What is (human) beauty?” The best answer to this question is the one which garners the most eyeballs, the most applause, and the most praise.

Are the mathematicians, then, not doing a similar thing, when they gather (ceremonially) at research conferences, and purport their models to be the best model? Are they not engaging in the same ritualistic kind of courtship?

Although a perfect model does not exist, we have not stopped striving for one. Perhaps it is because the idea of the perfect model drives us more than its truth. Thus, the question “Is the ideal achievable?” may not be the right one. Perhaps the purpose of the ideal is simply to orient our behavior and the structure of our society, that they may be stable and predictable across time. If we indeed live in a world without an objectively describable reality, would it not be so that we might develop the ideal instead, to serve in its place?

Although, us meek vertical apes, using our most fallible organs, have managed to describe an ideal world, and this is a heartening fact indeed, we have not yet learned to describe the objective one.