Why You Should Never Walk Around a Tree for More Than 45 Minutes at 32 Degrees

Numbers inspire confidence.

Perhaps that is because they argue so little. A 17 just stands there without claiming that it actually meant 18. Sixty-three percent looks like sixty-three percent. And once a decimal point appears, a number acquires that aura of scientific precision against which an ordinary sentence hardly stands a chance.

And yet, by itself, a number tells us surprisingly little.

32 degrees, for example.

Pleasantly warm?

Not necessarily.

At 32 degrees Fahrenheit, water freezes. At 32 degrees Celsius, you might consider finding some shade. And 32 kelvin would be decidedly unsuitable for an extended stay.

The number is the same.

The reality is not.

Or, more precisely: reality can even remain exactly the same while the number changes. Freezing water is at 0 degrees Celsius, 32 degrees Fahrenheit and approximately 273 kelvin.

Anyone who turns those into three different temperatures has not misunderstood the water.

They have misunderstood the scale.

A folding ruler tells much the same story. In German, we still commonly call it a Zollstock – literally an “inch stick” – even though, as far as I know, mine does not display inches at all. Instead, it shows centimetres and millimetres and, if one insists on being technically correct, is called a Gliedermaßstab, a folding measuring rule.

Which of course sounds much nicer.

“Hand me the Gliedermaßstab” possesses precisely the kind of linguistic elegance every home renovation urgently requires.

So Zollstock it remains.

People once measured things in feet, cubits or handspans – in other words, using things that human beings conveniently carried around with them anyway.

Until somebody asked:

“Whose foot?”

Because human beings have the unfortunate habit of not being built according to standard specifications.

Standardisation solved the problem by, as it were, unscrewing the foot from the human being. An English foot now has a fixed length, entirely independent of whether the person doing the measuring wears size 4 or size 12 shoes.

The foot became more precise by ceasing to be a foot.

A certain degree of agreement about units of measurement can also be useful at the butcher’s.

In an old joke, a customer orders:

“I’d like two pounds of meat for schnitzel.”

The butcher corrects him:

“It’s called kilograms now!”

The customer replies:

“Oh? A bit too modern for my taste, but all right – two pounds of kilograms, please!”

All he has done is mix up three things with unusual thoroughness: the number, the unit and the thing being measured.

We do that surprisingly often in less obvious situations.

Things become even more interesting with the Morgen.

A Morgen was once a unit of land area. The idea behind it was roughly the amount of land that could be worked or ploughed in the course of a morning.

Which immediately raises further questions:

Whose morning?

With what plough? What team of animals? On what kind of soil?

Of all places, Gießen has an area known as the “Stolzer Morgen” – the “Proud Morgen.” Whether the name was intended as a comment on the lack of precision in historical units of land measurement may safely be doubted.

It is fitting nonetheless.

Just how dependent such measurements can be on their frame of reference is illustrated by an old joke about two boastful American farmers. One says:

“My farm is so big it takes me an entire day to ride all the way around it.”

The other replies:

“Yeah, I know. I once had a horse that slow, too.”

The same amount of time. Two completely different statements. To one man it describes the size of the farm; to the other, the speed of the horse.

Sometimes the ambiguity is not in the measurement itself, but in a variable we failed to notice we were measuring as well.

It is even possible to calculate something perfectly correctly and still arrive at remarkable nonsense.

In a 30 km/h zone, a police officer pulls over a driver.

“You do realise you were just doing 150 in a 30 zone?”

The driver replies:

“Exactly. There are five of us in the car!”

150 divided by five really is 30.

The arithmetic is flawless.

That does not make it reasonable.

With a ruler, at least, we can eventually reach an agreement.

Time is more interesting.

When it is 12 noon in central Hesse during daylight saving time, the morning is only just beginning in Pennsylvania. There it is 6 a.m.

On Kiritimati in the Pacific, however, that very same moment is already midnight on the following day.

Three people could be talking to one another on the phone.

6 in the morning. 12 noon. Midnight the next day.

None of their clocks is wrong.

Even to the apparently simple question

“What day is it today?”

two of them would give the same answer and the third a different one.

And all of them would be right.

Technology has almost abolished distance.

It has not abolished the frame of reference.

We can now take part in the same conversation at the same moment without even being in the same day.

Perhaps there is already something in this that reaches beyond time zones. Understanding does not require everyone involved to share the same frame of reference. It requires, first of all, that they know other frames of reference exist.

And perhaps we should remember that before mistaking our own position for the zero point of the world.

Not even the number of the year is as self-evident as it looks.

I learned that almost by accident when I was a child.

I was twelve years old in 1983 because, according to the calendar I had grown up with, I was born in 1971.

At the time, I was talking about birthdays with a classmate from Afghanistan who was about the same age as me. His year of birth, if I remember correctly, was 1350.

According to the calendar I had known until then, my classmate should therefore have been several centuries old.

For someone of about twelve, that would have been quite an achievement.

Until then, I had probably never thought of 1971 as the result of a scale at all.

It was simply my year of birth.

Suddenly it was only my year of birth within a particular frame of reference.

Our own numbering of years did not simply fall from the sky either – although, of all systems, this one comes surprisingly close to claiming that it did. It was developed centuries after the period to which its starting point refers.

Jesus and his disciples, when asked what year they were living in, would hardly have replied:

“It is the year 30 AD.”

An archaeological find bearing the inscription

“Manufactured on 34 May, 36 BC”

would therefore be remarkable for several reasons.

And frames of reference can disappear.

When I was a child, everyday life in Europe involved a whole range of currencies that now survive mainly in memories, collections and drawers: the Deutsche Mark, the franc, the lira, the schilling.

What a currency is called makes very little difference to me. For all I care, it could be called Karl-Heinz, as long as I possess enough Karl-Heinz to pay for my daily needs.

I do still own a one-mark coin, though.

Not out of nostalgia.

I use it for shopping trolleys.

Precisely because the coin is no longer of any practical use in ordinary transactions, it is particularly useful for that purpose. I cannot accidentally spend it at an ordinary checkout. But it still fits many shopping-trolley locks.

On the scale of modern-day purchasing power, the coin is largely useless.

On the scale of “How reliably can I obtain a shopping trolley when I go shopping?” it has considerable value.

The loss of its original usefulness has, of all things, created a new usefulness.

Language, too, carries vanished frames of reference around for a while.

A neighbour of mine used to joke that her bank balance was “being displayed in schillings again.” The joke depended on the letter S on a German bank statement: Soll, meaning debit. Conveniently, the S could also be read as Schilling, the former Austrian currency.

The joke worked because both meanings were readily available to everyone.

Today it only works with an explanation.

The schilling has disappeared.

The debit, unfortunately, has not.

Even a joke can lose its frame of reference.

And even where the scale itself is perfectly clearly defined, the result need not be equally clear.

I learned many things at school.

Some of them were even on the curriculum.

One particularly instructive lesson was the way different teachers dealt with grades.

Our religious education teacher was an exceptionally friendly man. He was almost always cheerful, usually had some harmless joke ready and seemed to be one of those people who simply did not want to make life unnecessarily difficult for anyone.

Unfortunately, that very kindness came with a pedagogical drawback.

When grades were being discussed, classmates who were unhappy with the grade he intended to give them would sometimes keep talking until they had persuaded him to give them a better one.

Even the top grade: a 1.

That was probably meant kindly.

For someone who had earned a 2 through their work and then did not negotiate, however, it was rather unfortunate.

Suddenly, a 2 could mean:

The student had mistakenly believed that their performance mattered more to the grade than their negotiating skills.

Our chemistry teacher solved the same problem in precisely the opposite way.

When he first came into our class, he more or less greeted us by informing us that we could forget about anyone ever getting a 1 from him.

I found this lack of confidence in his own abilities deeply troubling.

After all, if someone chooses a profession devoted to passing on knowledge but, before teaching the very first lesson, is already convinced that among a group of roughly twenty mentally healthy children he will fail to convey enough of that knowledge for even one of them to earn the highest grade, that is quite a remarkable assessment of his own abilities.

Perhaps he meant something else.

Perhaps he merely wanted to make it clear that his standards were particularly high.

For the scale, it made little difference.

With one teacher, the top grade was negotiable.

With the other, it was fundamentally unobtainable.

On the school report, it looked exactly the same in both cases.

And so did the 2.

With the latter teacher, perhaps it merely indicated that someone had believed a grade would be awarded on the basis of performance.

Numbers have this pleasant characteristic: afterwards, you cannot see the conditions under which they came into existence.

Years later, I encountered this love of numbers again at university. Questionnaires were supposed to assess the quality of courses, teaching materials or seminar rooms. Students ticked boxes; numbers emerged.

That is perfectly legitimate to begin with. If eighty percent of respondents say a seminar room is too cold, we have learned something: eighty percent of respondents think the seminar room is too cold.

Things become difficult when this quietly turns into something else: the seminar room is unsuitable.

Or when the final result is 4.3 out of 5 – a number with a clear position on a scale, but an unclear relation to what we actually wanted to know.

Did we measure the quality of the teaching?

Or how satisfied the students were?

The two may be related.

They are not the same thing.

On some of those questionnaires, I would therefore have liked an additional option:

□ No idea. Please fund several comparison semesters at other universities first.

Curiously, that option was always missing.

A thermometer can measure with great precision and still fail to answer the question of whether I feel comfortable.

A questionnaire can be evaluated to two decimal places and still measure something that is only partly related to the question we were actually interested in.

Even a perfectly correct number can become the wrong answer if we forget which question it answers.

Perhaps, then, the problem is not measurement itself.

We have to measure.

We have to compare.

Without scales, we could hardly talk sensibly about distances, temperatures, time, speed, money or social developments.

The ruler is not our enemy.

If I need a board one metre long, I am extremely glad that the hardware store and I broadly agree on what a metre is.

“About three of my feet” would present an unnecessary challenge in furniture making.

The problem begins when we forget that we chose a scale.

Or that we assign meaning to a result.

Big and small do not exist without comparison.

Fast and slow need a reference point.

Early and late depend on where and when you are.

Valuable and worthless require the counter-question:

For what?

And sometimes even a perfectly correct calculation can produce nonsense because we have forgotten what the numbers actually refer to.

Which finally brings us to one of the questions this essay has so far left unresolved:

How long are you actually allowed to walk around a tree?

Many years ago, I read in a book with the delightful title Zauberhandbuch für neue Hexen – Magic Handbook for New Witches – that certain spells required you to walk around a tree five times counterclockwise.

Five times.

That is, at least, an impressively precise instruction.

Not four times.

Not six times.

Five times.

On closer inspection, however, this immediately raises further questions.

How large is the tree?

How far from the trunk are you supposed to walk?

What walking speed is prescribed?

And, above all:

What happens if you simply keep going?

Five circuits of a standard commercial tree are completed relatively quickly. Even allowing for a certain tendency to wonder halfway through whether you have already gone around four times or only three, the whole affair should hardly take an hour.

This matters.

Because roundabouts also run counterclockwise.

And, as everyone knows, roundabouts are the poor man’s merry-go-round.

Which inevitably raises the question of why roundabouts do not already have signs saying:

Maximum use: 45 minutes!

There is, of course, a scientific explanation for this as well.

Let us assume that statistical studies have found that the average road user can retain the contents of their stomach for approximately 50 minutes while continuously driving around a roundabout. If we additionally allow for a standard deviation of five minutes, considerations of municipal cost control give us a maximum permitted usage time of 45 minutes.

After all, somebody has to clean up the mess afterwards.

That costs money.

We have an average.

We have a standard deviation.

We even have an economic justification.

What could possibly be unscientific about that?

As the old saying goes:

“Never trust a statistic you haven’t falsified yourself.”

Which is, of course, unfair to statistics.

They do not have to be falsified at all.

It is quite sufficient to measure the right thing and then draw the wrong conclusion from it.

“5-minute standard deviation” sounds considerably more scientific than:

“I’d stop five minutes earlier, just to be on the safe side.”

Five laps around a tree can therefore be, depending on the frame of reference, a brief sporting activity, a peculiar walk or a fully completed act of witchcraft.

For a random passer-by, the second lap may already be sufficient to produce an entirely different result:

“That man has completely lost it.”

Perhaps, then, when confronted with very precise-looking numbers, we should occasionally ask a rather imprecise question:

What exactly was measured?

Because human beings are extraordinarily good at obtaining the correct answer and then forgetting which question it answered.

And sometimes a number even looks completely different in another system.

For example:

101010

On the other hand, there is of course one number that makes all further questions unnecessary.

42

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