Paradox: The incoherence of common sense

My musings on mathematicians and engineers were provoked by my Love’s reaction to something I saw in a quotes file:

There’s no way to develop an ambitious, broad-ranging, self-consistent metaphysical system without doing serious violence to common sense somewhere.
Eric Schwitzgebel

When I saw that, I laughed. It sums up what I’ve always thought about metaphysics. It sums up what almost everyone thinks about analytic philosophy.

I quoted it to my Love, who was trained as a pure mathematician. (For those of you who have never spent time with a pure mathematician: They make Mr Spock seem illogical.) She smiled and said,

Of course, sweetheart. Everything in mathematics, everything in science, did serious violence to the common sense of its time. That’s why we remember Galileo and Newton and Euler and Einstein. They defied common sense. Common sense is always wrong, unless it’s based on science that did violence to the common sense of its time.


The perils of quotes files: They lack context.

After that conversation with my Love, I read the whole interview with Professor Schwitzgebel. He said essentially the same thing as my Love said. He’s not criticizing metaphysics. He’s criticizing common sense. I still think metaphysics (other than Kant) is mostly silly, but he’s devastatingly right about common sense.

In context, Professor Schwitzgebel says,

Common sense is incoherent in matters of metaphysics. There’s no way to develop an ambitious, broad-ranging, self-consistent metaphysical system without doing serious violence to common sense somewhere. It’s just impossible. Since common sense is an inconsistent system, you can’t respect it all. Every metaphysician will have to violate it somewhere.

Common sense is an acceptable guide to everyday practical interactions with the world. But there’s no reason to think it would be a good guide to the fundamental structure of the universe. Think about all the weirdness of quantum mechanics, all the weirdness of relativity theory. The more we learn about such things, the more it seems we’re forced to leave common sense behind. The same is probably true about metaphysics.

You don’t even need to get into the weirdness of quantum mechanics. The Sun orbits the Earth? Common sense. A heavier stone falls faster than a lighter stone? Common sense. Species were as God created them in the Garden of Eden? Common sense. Newtonian mechanics? Crazy. Invisible animals cause disease? Insane! Send pictures through the air? Get this guy a straitjacket.


Even in the most abstract pursuits, there’s a place for common sense. Professor Schwitzgebel again:

But here’s the catch: Without common sense as a guide, metaphysics is hobbled as an enterprise. You can’t do an empirical study, for example, to determine whether there really is a material world out there or whether everything is instead just ideas in our minds coordinated by god. You can’t do an empirical study to determine whether there really exist an infinite number of universes with different laws of physics, entirely out of causal contact with our own. We’re stuck with common sense, plausibility arguments, and theoretical elegance – and none of these should rightly be regarded as decisive on such matters, whenever there are several very different and yet attractive contender positions, as there always are.

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Mathematician and engineer

My fiancée is trained as a pure mathematician. If you’ve ever spent time with a mathematician, you understand why Pythagoras and his gang were considered a strange, unworldly religious cult. Mathematicians aren’t like you and me.

When my fiancée was eight or nine, her mother bought her a boxed set of books called, The World of Mathematics. The description in the catalog made it sound like a book of puzzles and games. Instead, it’s a collection of essays and papers, some historical, some philosophical, some theoretical, some practical. Some are out-and-out funny – Bishop Berkeley’s The Analyst: A DISCOURSE Addressed to an Infidel MATHEMATICIAN. WHEREIN It is examined whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith.

All are far over the head of an eight- or nine-year-old; most are aimed at a mathematically literate college graduate. Many assume an understanding of calculus. And yet, she read it over and over until she understood it.

Her favorite essay? G H Hardy’s A Mathematician’s Apology. (You can read it here.) It’s what inspired her to become a mathematician. Hardy says that the joy of mathematics isn’t that it’s useful. It’s that it’s beautiful. The beauty isn’t in the usefulness of the thing proved, it’s in the elegance of proof itself.

Hardy was a number theorist. To Hardy, much of the charm of Number Theory was that it had no immediate use. It was pure elegance. Mathematics purely for the joy of Mathematics.

Hardy was a pacifist; he wrote the Apology in 1940, as the Second World War was raging and the Great War still fresh in his mind. He was pleased that Number Theory couldn’t be used to make bombs or poison gas. The joke (if there is one) was on Hardy: Number Theory is the basis for modern cryptography and code-breaking. Polish mathematicians had already used it to crack the Nazi Enigma code machine. Alan Turing and his gang would build on that to crack more sophisticated code machines. General Eisenhower said the Enigma intelligence was “decisive” in defeating the Nazis.

Hardy not only inspired my Love to be a mathematician, he inspired her to become a number theorist. This summer, she gave me a copy of a textbook Hardy wrote with E M Wright, Introduction to the Theory of Numbers. It’s an incredibly elegant book, accessible to anyone who passed ninth grade algebra. Even an engineer can see why it enthralled her.

You might think that Number Theory means proving things about numbers. If so, you’d be wrong. She proved things about constructs that have some attributes of numbers, but have strange and interesting pathologies. As she describes it, it’s taking something familiar (the integers) and then removing the elements that make it familiar.

I don’t pretend to understand any of it. One afternoon, I picked up one of her books, entitled A Course in Arithmetic. “Aha,” I thought, “Arithmetic. I can understand that!”

I was wrong. There’s nothing in there that you or I would recognize as arithmetic. The only numbers are page numbers. Should you so desire, you can read the original French or an English translation on the internet.


Euclid alone has looked on Beauty bare.
Let all who prate of Beauty hold their peace,
And lay them prone upon the earth and cease
To ponder on themselves, the while they stare
At nothing, intricately drawn nowhere
In shapes of shifting lineage; let geese
Gabble and hiss, but heroes seek release
From dusty bondage into luminous air.

O blinding hour, O holy, terrible day,
When first the shaft into his vision shone
Of light anatomized! Euclid alone
Has looked on Beauty bare. Fortunate they
Who, though once only and then but far away,
Have heard her massive sandal set on stone.
— Edna St Vincent Millay, Euclid alone has looked on Beauty bare


She insists that she’s not a mathematician now: Mathematicians prove things; she hasn’t proved anything since she was in graduate school. Her partners tell me that’s false: She has proved dozens of theorems fundamental to her business. They insist that she could write as many as 20 ground-breaking, publishable papers over a weekend.


Engineering is as ruthlessly pragmatic as Mathematics is ruthlessly unworldly. Mathematics is logical. Engineering is empirical.

Engineers delight in teasing mathematicians.

Who cares if you can prove it? The only thing that matters is, does it work?

Thanks for the rules of thumb!

Mathematicians are horrified at what engineers do with Mathematics. They are particularly horrified at engineers’ use of dot notation for derivative. (You are not expected to understand this.)

Still, the joy of engineering is also in creating something elegant – and tangible and useful.

I started my career as a design engineer. There’s a purity to design engineering, which one doesn’t really understand until one visits a jobsite where one’s design is being executed. (Or, as any design engineer will tell you, being butchered.) Then you realize that,

The map is not the territory.
— Alfred Korzybski

I worked while I pursued my graduate degrees. My interesting papers are on integration of complex subsystems. They are abstract, theoretic and analytic.

My work career quickly took a different direction. I left the desk for the field. I loved taking on difficult practical problems. The more bizarre and difficult the problem, the better I liked it. I wasn’t married, I had no social life, I didn’t have any ties or distractions. I could throw myself into problems, working on them every waking hour. Even while sleeping: I got some of my best ideas while asleep.

The problems that seemed most intractable – and interesting – involved integration of complex subsystems. My academic career was heavy on abstraction, but the abstractions helped me think clearly about solving concrete problems.


My fiancée and I work in overwhelmingly male-dominated fields.

Mathematics is not so male-dominated as it once was. But in the uses to which my fiancée puts Mathematics, the decisions are made by men: CEOs, CFOs and heads of corporate strategy for major multinationals; managers of high-risk international ventures; senior bankers and investment bankers. She’s usually the only girl in the room, and she has to prove that she’s smarter than all the boys.

Engineering and construction are overwhelmingly masculine and testosterone-laden. A girl engineer is a rare thing, especially a girl engineer in charge. I’m usually the only girl in the room, and I have to prove that I’m smarter than all the boys.

At that level respect is critical. Man or woman,

Respect isn’t given. It must be earned.

More than that, it has to be earned anew on every job.