2 + 2 = 4 and mathematical models

One of the strangest aspects of 2020 was a number of people arguing that 2 + 2 = 4 wasn’t necessarily true, and that it might be the case that 2 + 2 = 5. In fact, 2 + 2 = 4 is not only true throughout the universe, it is true in every possible alternate universe as well. A great many silly arguments were made by people trying to defend 2 + 2 = 5. But there is a deeper response here that relates to how applied mathematics works, and a few people have been trying to express that.

In general, applied mathematics questions have the form A → B, where A is some real-world situation, and B our desired but unknown answer. We abstract our real-world situation A to the mathematical object X, so that X → Y is the mathematical analogue of our real-world A → B. Mathematical questions have clearly-defined answers (although finding them is not always easy). We can then take our mathematical answer Y and reverse-translate it to the real world, giving something that we claim is a good approximation to B.

In the mathematical world of X and Y, everything is crisp and clear (and 2 + 2 = 4). In the real world, things are messy and ill-defined. Furthermore, in going from A to X we introduce simplifications and approximations which may mean that the mathematical answer Y is not “fit for purpose.”

For example, our real-world question might be “what is the distance from Melbourne to Sydney?” There are three ambiguities here: What is “Melbourne”? What is “Sydney”? And what is “distance”?

The conventional location marker for the city of Melbourne, Australia is the Old Melbourne General Post Office (now a shopping centre, at 37°48′49″S, 144°57′48″E). Likewise, the conventional location marker for the city of Sydney is the Sydney General Post Office (at 33°52′4″S, 151°12′27″E). Let’s use those coordinates. But what is “distance”? If “distance” means “as the crow flies,” then a simple answer might be to find the great-circle distance on a sphere approximating the Earth (let’s use the equatorial radius of 6,378.137 km). This distance can be calculated fairly easily as 714.2 km, which might be a close enough answer for many purposes.

A better mathematical model might be distance on the WGS reference ellipsoid model of the Earth. This gives the slightly lower value 713.8 km (according to Google Earth or raster::pointDistance).

Alternatively, “distance” might mean “by road,” in which case we need a computer representation of Australia’s road network. For this question, Google Maps reports a distance of 878 km via the M31. The expected travel time by car (9 hours and 7 minutes when I looked) might be even more useful.

It may therefore be the case that out of various mathematical answers Y, many are not “what you wanted.” But that failure to abstract correctly does not invalidate the mathematical truths involved in X → Y. In particular, it does not invalidate 2 + 2 = 4. It just means that you picked up the beautiful crystal knife of mathematics and cut yourself with it.

As Korzybski liked to say, “the map is not the territory.”

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Thermodynamic visualisation

This plaster model was made by the great James Clerk Maxwell in 1874 (the photograph was by taken by James Pickands II, 1942). This historic artefact is one of three copies, held in museums around the world, including the Cavendish and the Sloane Physics Laboratory at Yale.

The model shows the relationship between volume, energy, and entropy for a fictitious water-like substance, based on theoretical work by Josiah Willard Gibbs. The lines connect points of equal pressure and of equal temperature. Maxwell found the model a useful aid in his research. The model prefigured modern visualisation techniques – today we would use computer software to visualise such surfaces, like this:

The Double Helix, 60 years later

This photograph (by “Alkivar”) shows a reconstruction of the double helix model of DNA, constructed by James Watson and Francis Crick in 1953. The metal plates are molecular models of bases (some of these plates are original). The model is located in the Science Museum, London (see also their photograph).

This model not only led to one of the greatest-ever breakthroughs in biology (see the original 1953 paper, as PDF), but also demonstrated that “playing with models” was an effective way of doing chemistry. The discovery built on X-ray crystallography work by Maurice Wilkins, Rosalind Franklin, and Raymond Gosling (see their papers and the famous X-ray photograph produced by Gosling). Chemical investigations by Erwin Chargaff and others also produced essential information.

The breakthrough by Watson and Crick is commemorated by, among many other things, the Cambridge stained glass window shown below (located in the dining hall of Gonville and Caius College, photo by “Schutz”).