How little is too little overhead?

A lot of my recent life decisions have been influenced by an incredible observation by Austin Kleon:

Low overhead + “do what you love” = a good life.

“I deserve nice things” + “do what you love” = a time bomb.

This isn’t about [shudder] minimalism. But recognising that unless you’re Beyoncé you can’t do it all. And even then, is your support team doing it all?

There is an incredible amount of satisfaction in working out what it is you want and stripping away the excess. But it can go too far.

I was reminded of this while reading a biography of Paul Erdos, called The Man Who Loved Only Numbers. A man who took it too far:

Erdos structured his life to maximize the amount of time he had for mathematics. He had no wife or children, no job, no hobbies, not even a home, to tie him down. He lived out of a shabby suitcase and a drab orange plastic bag from Centrum Aruhaz (” Central Warehouse” ), a large department store in Budapest. In a never-ending search for good mathematical problems and fresh mathematical talent, Erdos crisscrossed four continents at a frenzied pace, moving from one university or research center to the next. His modus operandi was to show up on the doorstep of a fellow mathematician, declare, ” my brain is open,” work with his host for a day or two, until he was bored or his host was run down, and then move on to another home…

…if it wasn’t mathematics, Erdos wouldn’t be bothered. ” Some French socialist said that private property was theft,” Erdos recalled. ” I say that private property is a nuisance.”

This isn’t stripping away the extraneous in order to do what you love. It’s outsourcing the job of being a human.

Why the west?

Why the ‘West’ has dominated the last couple of centuries is an interesting question with many aspects. It is especially so considering how many important technologies and ideas were either first or concurrently invented elsewhere.

Take zero:

The Western civilizations, for example, failed to come up with it even after thousands of years of mathematical inquiry. Indeed the scale of the conceptual leap achieved by India is illustrated by the fact that the classical world was staring zero in the face and still saw right through it. The abacus contained the concept of zero because it relied on place value. When a Roman wanted to express one hundred and one, he would push a bead in the first column to signify one hundred, move no beads in the second column indicating no tens, and push a bead in the third column to signify a single unit. The second, untouched column was expressing nothing. In calculations, the abacist knew he had to respect untouched columns just as he had to respect ones in which the beads were moved. But he never gave the value expressed by the untouched column a numerical name or symbol.

This is a passage from Here’s Looking and Euclid, which further proposes the failure to see zero was in part due to the West being down its own particular philosophical tangent. Whereas India was a place with a rich history that embraced nothingness:

Indian philosophy embraced the concept of nothingness just as Indian math embraced the concept of zero. The conceptual leap that led to the invention of zero happened in a culture that accepted the void as the essence of the universe.

Author Alex Bellos goes on to assert that it is partly the transmission of these Indian concepts that spurred the scientific revolution.

With the adoption of Arabic numbers, arithmetic joined geometry to become part of Western mathematics in earnest, having previously been more of a tool used by shopkeepers, and the new system helped open the door to the scientific revolution.

Of course the story is much more complicated.

Not only did the use of Arabic numerals and arithmetic require a rich intellectual and practical tradition upon which to glom on to. But the right combination of openness, education, wealth (etc.), and technologies like paper to kick off the revolution.

What is really clear is how important the interplay between civilisations has been for progress. Culture, philosophy, ideas and even biology all crossed borders. Some many times over.

The real question then is, why was the West the main beneficiary?

As always my emphasis.

Why are we not keeping up?

There’s a fascinating line in the opening pages of Here’s Looking at Euclid:

By age 16, school kids have learned almost no math beyond what was already known in the mid-seventeenth century, and likewise by the time they are 18 they have not gone beyond the mid-eighteenth century.

If the point of schooling is to teach the current best understanding of the world, we appear to be failing.

Granted, maths is unlike science in that the old ways are built upon rather than overturned. We still learn Pythagoras theorem but not geocentrism.

This necessitates teaching newer mathematical discoveries in addition to what came before it. But is there really no recent mathematical discovery that would be useful to learn in school? Or at least help us shape more rounded adults, equipped for the modern world? Boolean algebra mayhaps?

I was thinking about this as I came across the following passage in The Lady Tasting Tea:

As measurements became more and more precise, more and more error cropped up. The clockwork universe lay in shambles. Attempts to discover the laws of biology and sociology had failed. In the older sciences like physics and chemistry, the laws that Newton and Laplace had used were proving to be only rough approximations. Gradually, science began to work with a new paradigm, the statistical model of reality. By the end of the twentieth century, almost all of science had shifted to using statistical models… Popular culture has failed to keep up with this scientific revolution. Some vague ideas and expressions (like “correlation,” “odds,” and “risk”) have drifted into the popular vocabulary, and most people are aware of the uncertainties associated with some areas of science like medicine and economics, but few nonscientists have any understanding of the profound shift in philosophical view that has occurred

It’s definitely my experience that people are more comfortable with a deterministic model of the universe than a probabilistic one. But why are so many of us stuck in old paradigms even as our world has become immensely sophisticated and complicated?

If nothing else, a more widespread probabilistic model would lead to more nuanced discussions around new research and challenges like climate change. A more modern understanding of mathematics would help us all get the most of technological advances.

Many of the books I’ve read criticising modern education see it as a relic of the industrial revolution and empire, geared towards producing identical widgets to keep things running smoothly. Can’t help thinking that’s right. And we haven’t changed it much.

As always my emphasis.

Finding joy in mathematics

I have always regretted losing touch with maths during high school. Part of this is undoubtedly my fault. I wasn’t a great, or really even that interested in being, a student until midway through university. But there’s also something a bit broken in how we approach maths. Both in school and life generally.

I’m talking about maths as a purely abstract phenomena. A series of formulas and steps, divorced from how they relate to the real world. Where multiplication is a table to be memorised and trigonometry takes place purely within a textbook.

This kind of mathematics not only strips away a lot of the beauty and joy, but relegates the subject to one only grasped by those who excel in a particular system. It turns maths into something like an ecclesiastical language, almost scary to the unindoctrinated.

This is a shame, really, as Lara Alcock writes in Mathematics Rebooted:

“…mathematical thinking is not magical. It is often thought of that way in our culture, where it is common to have a demanding career or to run a happy and successful household, yet to say, ‘Oh, I am terrible at maths.’ I hear this a lot, and every time it is clear to me that it cannot really be true: this person is obviously a capable thinker. ”

To some extent abstraction is necessary in schools as currently constituted. Students must be judged against something objective, and must be taught at scale. But does this really require so much concentration on the doctrine, to the detriment of the art?

Why is maths largely rote, rather than logic? Abstract rather than practical?

“Your mathematical knowledge might be rusty and full of holes, but people who can function well in our complicated world must be good general thinkers, and mathematics is just general thinking about abstract concepts.”

When you read about the likes of Newton and Galileo, maths jumps out as a tool for problem solving and creativity. In The Triumph of Numbers, I. Bernard Cohen explores how numbers and maths have evolved over time, and describes a plethora of interesting applications.

Including the algebra of morality:

“[Francis Hutcheson] used this algebraic relationship to translate several commonsense notions about morality into mathematical language. The first is that if two people have the same natural ability to do good (A), the one who produces more public good (M) is more benevolent (B). Conversely, if two people produce the same amount of public good, the one with more ability is less benevolent (since it was in that person’s ability to do more). The plus/minus sign in the equation allowed Hutcheson to factor in self-interest.”

“…he concluded from his algebra that “in equal Numbers, the Virtue is as the Quantity of the Happiness, or natural Good.” That is, he taught that “Virtue is in a compound Ratio of the Quantity of Good, and Number of Enjoyers.” This led him to the important conclusion that “that Action is best, which accomplishes the greatest Happiness for the greatest Numbers.” Here is a precursor, by more than 50 years, to Jeremy Bentham’s (1748–1832) utilitarian philosophy of “the greatest happiness for the greatest number.”

This is a maths of reasoning and personal application. He wasn’t trying to calculate the change from a $20 or the tensile strength of a beam.

Similarly, in The Calculus Story, David Acheson produces probably the best explanation of how to calculate the area of a circle, by imagining a polygon with more and more sides (I think I finally understand pi r squared).

As you might be able to tell, I’ve been reading a fair amount of maths books recently, and the thing that strikes me is how differently mathematicians approach the subject than how I was taught in school.

Mathematicians work through subjects largely by reasoning and logic, not necessarily ever more complicated formulae. They also emphasise the problem solving nature of maths, often tasking you to think about a problem and come up with your own generally applicable rule.

Putting this into practice, I have also been studying maths using Brilliant.Org, which has a similar philosophy:

“In school, people are often trained to apply formulas to rote problems. But this traditional approach prevents deeper understanding of concepts, reduces independent critical thinking, and cultivates few useful skills…The capacity to think critically separates the great from the good. We can grow this capacity by trying — and often failing — to solve diverse, concrete problems.”

I have only finished one module on Brilliant and I’m not sure how it will work as a method for the masses. But I will report back in a couple of months.

In the mean time, I’ll leave you with Lara Alcock again, whose book I really recommend:

“School mathematics tends to come in horizontal slices: children learn basic ideas about several topics, then, the next year, they learn slightly more advanced ideas about those topics, and so on. This is entirely sensible. But it means that the vertical links are not very salient, which is important because mathematics can be seen as a highly interconnected network in which more sophisticated ideas build upon more basic ones. So this book’s approach is to focus explicitly on the vertical links. Each chapter starts with an idea that is bang in the middle of school mathematics—primary school mathematics in many cases—then takes a tour upward through related concepts, arriving eventually at ideas that people encounter in more advanced study”