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The Joy of Calculation

I recently had the pleasure of finding a copy of the manual for my favorite calculator. I know it is incredibly nerdy to have a favorite calculator (and even more nerdy to read the manual), but it really got me thinking.

The manual subtly sold an incredible point of view: the engineer’s view. The manual appears trivial at the surface but is in fact a very good rhetoric pushing a fascinating point of view: you can infer things quickly. This led me to think about a number of technical viewpoints (engineers point of view, scientists point of view and lastly mathematicians point of view). They are all lumped together as “quantitative” but they are radically different.

Listen to this (from the beginning of the HP15C calculator manual):


Notice the emphasis on the physical activity of calculation. The emphasis is not on equations, mathematics or physics. The calculation is deliberately described as key strokes. No attempt is made to justify any of the steps or numbers used. The point being made is: if you are agile and ready (have the correct fore-knowledge) you can calculate. If you can calculate you can know things. Robert Heinlein made this point about slide-rules in his science fiction story: “Have Spacesuit- Will Travel.” And likely a similar joy can be felt while accounting on an abacus.

This is the engineer’s view: the world continuously gives up many small and simple clues as to what is going on around you. These are like “tells” in poker. You can reason from them and build incredible things using them. The smallness and simplicity of the techniques are pure comfort.

In Michael Lewis’s “Liars Poker” the author mentions a moment when he knows that everything he is being told about the market is a lie. He knows this because he attempts to convert one statement about the market into another using his calculator. When he attempts the conversion (figuring out something he was not supposed to know from clues coming from something he was told) it does not add up. Importantly he describes working this out on his calculator- not using a sophisticated computer model or a spreadsheet. He is comfortable in his heterodox position because he calculated it by hand in small and simple steps.

This joy in comparing one conclusion to another (using a calculator) differs from the idealized scientist’s view in that there is no derivation or application of deeper laws. The engineer’s view is: if you can remember it or guess at it then you don’t need to derive it.

Some of the great scientists (Enrico Fermi) and mathematicians (Stanislaw Ulam) became masters of the engineering view and could dazzle with it.

One of Fermi’s famous stunts was measuring the yield of a nuclear bomb test by observing how far scraps of paper were moved. Fermi may have worked from first principles, but he could also have used a simple pre-prepared trick. If he had observed how far scraps of paper had moved in an earlier conventional bomb test (which he now knew the yield of) and then applied a simple engineering trick called “dimensional analysis” that let him reason the amount of work observed (how far the slips of paper were moved) depended linearly on the bomb yield and decreased as the cube of how far away he was from the explosion. So all he did was compute the ratio of of how far the slips moved in each test and then divide this three times in succession by the ratio of how far way he was from the center of each test. Merely being able to divide told Fermi something (the new bomb yield) before he was officially allowed to know it. Notice how he did not need to use any facts about the bombs being tested, the speed of sound, atmospheric pressure, density or temperature.

Such reasoning may seem crude- but it is far more informative and far more exciting than the published work of many lesser scientists. The bulk of most merely poor scientific work (as opposed to outright wrong work) is of the form: “here are some pointless measurements I got by applying an expensive new instrument in exactly the situations the manufacturer designed it for.” Or “here are some manipulations that seem original since I don’t feel I have to cite any non-physicists.”

I side with the mathematicians (not the engineers or even scientists) and I think it is safe to say that mathematicians (who have their own particular view) are more sympathetic to the engineer’s view than to the scientist’s view.

One joke that has been told about me is that I am not happy at a presentation unless there is an equation on the board. This is typical of mathematicians. The excitement comes from the opportunity to “kick the times.” Once you remove enough details an equation is a simple statement of the form “A=B” (to borrow the title of a wonderful book by Marko Petkovsek, Herbert Wilf and Doron Zeilberger). An equation is a welcome moment of concreteness in contrast to the many painful abstractions that are necessary for much of mathematics. The dirty secret is that mathematicians perk up when an equation is on the board not because they like equations- but they are hoping to plug in values for “A” and “B” such that the equation is shown to be false. My branch of mathematics (theoretical computer science) is more a competitive than a cooperative field. One measure of audience interest in my field was if somebody to grab the magic marker out of your hand to try and write down a counter-example to what you were trying to demonstrate. Gian-Carlo Rota tells a similar tale where someone in a mathematical audience grabs the chalk and tries to complete the presentation.

One reason I side with the mathematicians and not the engineers is: if pressed too far the engineer’s view goes wrong. The way it goes wrong is found in the thick classic comprehensive engineering handbooks. These books attempt to store and systematize all of a given field’s engineering knowledge. Once you attempt to become comprehensive and are devoting all of your intellect to memorizing and applying the standard approximations and estimates you are lost.

I also do not side with the scientists because mathematicians have no sympathy for trying to “buy your way out of solving a hard problem” by running an expensive experiment. Mathematicians do work with data (even messy data) but we call this “application” not “proof.”

To me the best view is: if you can derive anything then you do not need to remember anything.

Categories: Mathematics Opinion

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Data Scientist and trainer at Win Vector LLC. One of the authors of Practical Data Science with R.

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