## Irrational Happiness

When I started writing this blogpost a week ago, I wrote this: “When times are as difficult and unpredictable as they are today, I find comfort in mathematics. No matter how things turn out, how many more difficulties come our way, we may be certain that mathematics stays the way it is* – it is our indestructable beacon of rationality. In this blogpost I want to share with you some of the comfort that mathematics gives to me.”

But the longer I think about these things, the more I realise that it is not this comfort that I want to share. I realise that when I am at home alone, sitting in my ‘internest’, writing this blog, I forget about all the trouble around me. I stop worrying about the coronavirus or about my recent divorce. Mathematics makes me happy, that’s what I want to share!

Just like me, the followers of Pythagoras in the sixth century BC really loved their numbers. When one of them discovered that there are numbers, like √2, which you can’t write out in all their decimals, because they have infinitely many of them (√2=1.41421356237…), the other Pythagoreans threw the poor fellow into the sea. But his death hasn’t helped. Numbers with infinitely many decimals exist, and √2 is one of them.

To prove that √2 has infinitely many decimals, we must show that it can’t be written as the division of two whole numbers – that √2 is an irrational number (so-called because it is not the ratio between two whole numbers).

If mathematical proof isn’t your favourite pass-time (not even in quarantine) then perhaps you should skip to the paragraph ‘Hamlet’s nutshell’.

## Talk contradictory to me baby!

The proof that √2 is an irrational number is a proof by contradiction. We assume that the opposite of what we want to prove, the statement “√2 is the ratio between two whole numbers”, is true, and from that we derive a contradiction, so that we know that the assumption that we started with is false.

So we start with the assumption that √2 is the ratio between two whole numbers. Let’s begin by writing the ratio in terms of its smallest divisors, the lowest values of m and n for which the ratio stays the same (so that 3/6 becomes 1/2, 5/15 becomes 1/3 and 2/1200 becomes 1/600). Let’s call these smallest divisors m and n. To derive the contradiction that we need for our proof, we will show that the m and the n that go into √2 are even numbers, which means that $\frac{m}{n}$ is not the ratio between two smallest divisors (because both terms in the ratio can be divided by two). But the assumption that we started with is that m and n are the smallest divisors, so if our reasoning is correct, our starting assumption must be wrong.

Let’s begin with this:

$\sqrt{2}=\frac{m}{n}$

square both sides to get

$2 = \left( \frac{m}{n} \right)\textsuperscript{2} = \frac{m^2}{n^2}$

then multiply both sides by $n^2$ to get

$2n^2=m^2$

If m and n are arbitrary whole numbers, then so are $m^2$ and $n^2$. So if $n^2$ is equal to some arbitrary whole number multiplied by two, $m^2$ must be an even number (because any number multiplied by 2 is an even number). Ok, so we know that $m^2$ is even, but what about m? Is m also even, if $m^2$ is even? m appears in our equation in its squared form, but whatever the value of $m^2$, there are only two possibilities: m is either even or odd (mathematicians like stating the obvious). But how does that help us? How can we explore these two possibilities?

We’re going to use a little trick here: we know that a number is an even number if it can be divided by 2, so if m is an even number, we may write $m=2k$, where k is again some arbitrary number. Think about it. If some arbitrary number is one of these “1, 2, 3, 4, 5…” then two times that number is in the list “2, 4, 6, 8…”. This gives us a list of even numbers, but how do we get to the odd numbers? The second part of our trick is to add 1 to every number in the list of even numbers, so that we get a list with odd numbers (“3, 5, 7, 9…”), so we know that we can write any odd number as $2k+1$. Try it yourself, fill in “2, 4, 6, 8…” for k in $2k+1$ and you get “3, 5, 7, 9…”.

Back to our two possibilities. We know that $m^2$ is even, but what about m? Is m even or odd? Where do these possibilities lead us?

## Possibilities

If 1: m is even, we know that $m=2k$ for an arbitrary k, so that $m^2= \left( 2k \right)^2=4k^2$, which is again an even number because it is divisible by two. We now know that $\sqrt{2}=\frac{m}{n}$ is consistent with $m^2$ and m being both even.

If 2: m is odd, then we may write $m=2k+1$ for some arbitrary number k, so that $m^2= \left( 2k + 1 \right) \left( 2k+1 \right) = 4k^2 + 4k+1$. This expression has the form “even number + 1”, because $4k^2 + 4k$ is even, so $4k^2 + 4k+1$ must be odd. We see that the second possibility leads to saying “if m is odd, then $m^2$ is also odd.”

Before we started talking about the two possibilities, we asked “what can we say about m if we know that $m^2$ is even?” We see that assuming that m is odd leads to an $m^2$ which is odd, so possibility 2 is not a possibility at all! The only remaining option is possibility 1: if $m^2$ is even, then m is also even.

Are we there yet? We started this blogpost by saying that it is impossible that $\sqrt{2}=\frac{m}{n}$ if m and n are smallest divisors because $\sqrt{2}=\frac{m}{n}$ implies that m and n are even, but we haven’t shown that yet. We know that from $m^2 = 2n^2$ it follows that both m and $m^2$ are even, but what about n? Let’s use our ‘little trick’ again. Since we know that m is an even number, we may write $m = 2k$. We also know that $m^2=2n^2$, which gives us $4k^2=2n^2$. Divide both sides by 2, and we see that $2k^2=n^2$, which tells us that $n^2$ is an even number. We know from our earlier reasoning that if the square of an arbitrary number is even, then so is the number itself (if $m^2$ is even then m is even). We just showed that $n^2$ is even, so n itself must also be even.

Phiew! We have finally reached the contradiction. Both m and n are even, so they are not the smallest divisors in $\sqrt{2}=\frac{m}{n}$. This contradicts our starting assumption, so that assumption must be false, so √2 is not a rational number.

Q.E.D.**

Something keeps nagging, though. We have shown that both m and n are even, which tells us that m and n are not the smallest divisors in $\sqrt{2}=\frac{m}{n}$, but what does that tell us? We assumed that they were smallest divisors, and that’s why there was a contradiction. What if we don’t make this assumption? What if we assume that there are numbers m and n such that $\sqrt{2}=\frac{m}{n}$, for some m and n that are not the smallest divisors in $\frac{m}{n}$?

What we should realise is that both m and n could be any number, so if we say $\sqrt{2}=\frac{m}{n}$, then this equation must hold for all choices of m and n, including the choice where m and n are smallest divisors. m and n being the smallest divisors is not an extra assumption – it is part of the assumption that m and n are arbitrary numbers.

Can’t we make a claim about √2 that is a bit weaker? What if we say that m and n are numbers for which $\sqrt{2}=\frac{m}{n}$ does not have smallest divisors? Will that allow us to say that √2 is a ratio between whole numbers?

Nice try, but it won’t work. Or rather, it works, but then the m and n that go into √2 are not numbers as we know them. Take any pair of positive, whole numbers, and make a rational number out of them $\left(\frac{m}{n}\right)$. Depending on the choice you make, it is either possible to divide both n and m by some other number (so that, for example, $\frac{3}{6}$ becomes $\frac{1}{2}$, and $\frac{4}{16}$ becomes $\frac{1}{4}$) or you can’t do that, in which case you have the smallest divisors ($\frac{1}{2}$, $\frac{5}{8}$, $\frac{6}{7}$). The possibility left for m and n that neither have nor are smallest divisors of √2, is to choose for m and n some trans-finite numbers, such as $\aleph_0$, which represents, among other things, the cardinality of the set of all integers. But the moment we start talking about trans-finite numbers, we leave the realm of the real and the rational numbers. The ratio which trans-finite numbers can give us is not a rational number.

## Hamlet’s nutshell

It makes me a bit dizzy when I start thinking about trans-finite numbers (if you’ve skipped the proof, just believe me when I say that trans-finite numbers are stranger than fiction). Math-induced dizziness is a pleasant dizziness, much as that due to a glass of whisky. It reminds me of something that Shakespeare’s Hamlet says:

“I could be bounded in a nutshell, and count myself a king of infinite space”

No matter what happens, no matter what comes our way or how isolated I become, in my mind I can instantly travel to the farthest reaches of human understanding. Simply by following the rules of logic.

*) which is not to say that it’s impossible that mathematicians discover new theorems or new mathematical relations. What I mean is that mathematics, as logical reasoning about numbers, will continue, no matter what.

**) Nerd-speak for ‘I told you so’. Also sometimes “quod erat demonstrandum” (Latin for ‘…which needed to be proved’)

## Back in Utrecht

My wife has decided that she wants to split up with me, so I have moved back to my old home in Utrecht (the Netherlands). I am very sad, because I really saw a future for us together in Sweden, and writing this actually brings tears to my eyes. When I am as sad as I am now, I often try to think of Monty Python’s song ‘Always look on the bright side of life’. But this time that thought provides little comfort, because ‘Always look on the bright side of life’ is the song that was played during our marriage ceremony, so the song reminds me of her smiling and singing.

Having said that, I think there is a lot of brightness awaiting me. I have started teaching a course on electromagnetism at a university in Amsterdam, I’m writing a popular book on relativity theory which will appear in May (with a foreword written by Nobel prizewinner Gerard ‘t Hooft), and the Dutch climbing & mountaineering federation has made me Dutch paraclimbing ambassador!

And what about Sweden? I’m not letting go of my mission to spread paraclimbing in Scandinavia. In October I have given a talk about paraclimbing in Norway, and at the end of April there will be a sports camp in Sweden, where a wide variety of parasports (adaptive sports) are represented. I have been invited by the Swedish climbing federation to come and represent paraclimbing. You haven’t heard the last of me yet!

Dutch open nationals, Sittard; November 2019

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## Changes!

I have great news! No, I’m not pregnant, I’m moving to Sweden. My wife accepted a job as a teacher at an international school in Älmhult (near Malmö), so from August 1 I’ll be driving my tricycle in Viking-country.

One remark before I tell you more about my plans for Sweden: this will be my last blogpost on paraklimmer.wordpress.com; from now on I’ll only post on my English paraclimbing website so follow this link, and click ‘Follow Blog via Email’ to keep getting updates!

What am I going to do in Sweden? I’m going to continue my job as an editor for a journal in physics (Foundations of Physics) and I’ll continue to do research in the philosophy of physics at the Linnaeus University in Växjö.

And what about the climbing? Will that come to an end?

Of course not! I’m going to start a new chapter of my mission – to spread the word about paraclimbing. When I started paraclimbing in the Netherlands three years ago, there was only one other paraclimber, now there’s eight of us (see image below). At the moment there are only two paraclimbers in Sweden, that will change!

But before I move to Sweden, there are two things that I have to take care of:

1. I’m organising an international conference in honour of the Nobel laureate Gerard ‘t Hooft, to be held at the university of Utrecht July 11-13 (hey, that’s Thursday!)
2. The morning after the conference dinner on Saturday I’m traveling to Briançon (Fra) to compete in the Worldchampionship paraclimbing!

## WK 2019: Briançon, 16/17 juli

De organisatie van het WK paraklimmen 2019 heeft bekend gemaakt dat het WK dit jaar toch niet in Tokio zal zijn. Het wordt gehouden in Briançon (Fr), en niet in augustus, maar op 16/17 juli. Ik vind dat jammer en vervelend: jammer omdat ze in Tokio waarschijnlijk maar weinig blonde Friezen te zien krijgen; en vervelend omdat mijn trainingsplanning nu in de soep loopt.

Maar wat dit alles volgens mij nog het duidelijkst laat zien is dat paraklimmen nog niet ‘één van de grote jongens’ is: ik kan me moeilijk voorstellen dat het WK voetbal zo kort van tevoren een maandje en 1000 km wordt verplaatst. Het is dus wel duidelijk dat mijn missie – ervoor zorgen dat paraklimmen serieuzer, groter en bekender wordt – nog in de kinderschoenen staat.

Op dat front is trouwens ook succes geboekt. Toen ik in 2015/16 begon met paraklimmen, bestond het Nederlandse paraklimteam uit twee personen. Inmiddels zijn we met veel meer (we hebben zelfs alweer een nieuw teamlid dat niet op de foto staat; binnenkort meer daarover):

Heb je zelf een handicap en zou je graag een keertje mee willen trainen, of ken je iemand die daar misschien in geïnteresseerd zou zijn? Schrijf je hier in voor de open training op 8 mei bij Mountain Network Nieuwegein.

Voor wie niet iedere dag RTV Utrecht kijkt: ze hebben een reportage over me gemaakt (klik hier)!

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## A Quantum Leap of Faith

Earlier today, one of my students asked me this: “So in quantum mechanics an electron can be in different orbits around a positively charged nucleus, and each of these orbits require different levels of kinetic energy for the electron. Electrons can leap from one orbit to another, either emitting or absorbing a photon as they go, but they cannot exist in between orbits. That’s odd. Any theory about such electron-orbit-changes must describe what happens during such a quantum leap. How can it be that at a certain moment an electron leaps from one orbit to a higher orbit without traversing the orbital space in between? Because that’s what quantum theory tells us, right?”

Bohr’s Copenhagen interpretation is famously silent about this. We should only speak, according to Bohr, about what we see, what we observe. In class I was very critical of Bohr, but his ideas are not silly: We never know for certain that an electron is in a certain orbit. We see electrons in certain locations which we might extrapolate with a certain probability to certain orbits. The same goes for leaps from one orbit to a higher orbit: we don’t see the actual leap, but only a sequence of locations which we can extrapolate with

a certain probability to a certain leap. What happens in between observations? Physics is silent about that.

How do the proponents of other interpretations of quantum theory answer the question what happens during a quantum leap? Some would say that an unobserved electron is in a superposition of orbits until it is observed; others would say that the electron just moves from one orbit to another – traversing the in-between-orbits space – we just can’t see that.

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## Tokyo augustus 2019?

Het is nog steeds niet helemaal zeker of het WK paraklimmen in Tokio in augustus doorgaat, maar ik ga er in 2019 in ieder geval hard tegenaan!

## The Order of Rovelli’s Time – does time exist?

What is time? Does it even exist? In this paper I will discuss the three main phases of Carlo Rovelli’s philosophy of time, and I will conclude with an analysis of some of the criticism levelled against his view.

1. The End of Time
2. Time doesn’t exist?!
3. Time is subjective
4. We Need More Time

## 1.      The End of Time

The first part of Rovelli’s narrative is a stepwise demolition of the traditional view of time. This traditional view is due to Newton, who viewed time as an objective (ie. the same for everyone) quantity that is needed to describe change, but which is itself in no way influenced by this change. Newton’s time has a specific direction, which means that there is an objective difference between past and future, a difference that is the same for all observers.

Rovelli shows how little of this traditional view is left in modern physics. Relativity theory tells us that clocks tick at different rates in different places; thermodynamics shows us that the direction of time depends on our perspective (so it is not objective) and that there is no good reason to believe that time is anything more than merely change (I’ll explain this in the following paragraphs).

Finally, Rovelli acquaints us with his relational interpretation of quantum mechanics. For his central claim, however, his interpretation of quantum mechanics is not essential, so I will not treat it in this paper.

## 1.      Time doesn’t exist?!

Rovelli is often portrayed as making the grand claim that time does not exist. But what does that even mean? We can illustrate Rovelli’s claim in a simple example: say we use a clock to keep track of a runner in a race. We are used to thinking about such a situation as follows: To measure the change in position of the runner we use the parameter ‘time’, and changes in the time-parameter are described in terms of the changing positions of the hands of the clock. In such a way the changes in the clock can be used to measure the change in position of the runner with the help of the parameter ‘time’. The physical description we end up with has three ingredients: the runner, the clock, and the parameter ‘time’.

This sounds pretty obvious, but there is a different way of looking at the situation. Instead of expressing both the change in position of the runner and the changing positions of the hands of the clock in terms of the variable ‘time’, we could choose to describe the changes in the runner in terms of changes in the clock, and the other way around, the changes in the clock in terms of changes in the runner. In that way we don’t need the time-variable anymore! Our alternative description has only two ingredients: the runner and the clock.

That’s the idea behind Rovelli’s claim. But the alert reader will have noticed that the conclusion we just reached (we don’t need time) does not necessarily imply that Rovelli’s claim is justified (there is no time). Although the phrase “time doesn’t exist” can be heard in many interviews Rovelli gives, the claim he actually makes in his book “The Order of Time” is more subtle. He writes: time does not exist as a fundamental entity. There is change; we see change all around us. But that’s all there is to it. There isn’t some fundamental entity or parameter underlying or describing this change.

We might rest here, and either agree or disagree with Rovelli that we have no need of a variable time and that therefore time doesn’t exist as a fundamental entity. Something keeps nagging though, for what does ‘fundamental’ mean? Within the community of philosophers of science there is no consensus about this. Scientific articles about ‘fundamentality’ appear regularly, and in May 2018 there was even a conference in Geneva devoted solely to the topic. Sadly, no consensus was reached in the course of this conference (Prof. Vallia Allori; personal correspondence).

From what Rovelli writes I get the idea that what he means is that an entity is fundamental in the context of physics if it is a necessary element of physics. Since we don’t need ‘time’ for the description of physical situations (as the example of the runner and the clock shows), it is not a fundamental element of physics. In his book Rovelli describes the Wheeler-DeWitt equation, an equation central to any attempt at unifying quantum theory with relativity. This equation is independent of time – a fact that serves as the main motivation for Rovelli to explain how it is possible that change happens in a universe devoid of time.

## 2.      Time is subjective

According to Rovelli, we live in a world in which time doesn’t exist as a fundamental entity. In the third phase of his argumentation, Rovelli shows us what the role is of change in a world without time, and how our experience of time emerges from that change.

The laws that govern the motion of particles – the laws of Newton – are time-symmetric. Were we to make a movie of two colliding particles (or billiard balls), then it doesn’t matter whether we play the movie from beginning to end or the other way around (starting at the end and going to the beginning). Both movies represent the same collision, and neither of the two movies seem odd to us. What is considered ‘past’ and what ‘future’ depends on how we watch the movie: It seems as if nature itself has no preferred direction. The laws of motion are the same whether we play the movie from beginning to end or in reverse.

Then where does our feeling come from that time has a specific direction? We all know that ice melts and hot tea cools down; to see those processes in reverse would be odd indeed! In these processes there is clearly a direction – a direction that everybody will agree on: it is the direction in which the arrow of time points. Most phenomena in nature clearly follow the arrow of time: the smell of a rose spreading through the entire house (it won’t just stay in one room), milk diffusing when poured into a cup of tea (it won’t stay on or near the spoon), etc, etc. What do all these phenomena have in common? They show that nature is inclined to flow to increasingly chaotic (and therefore more probable) states.

This statement needs some unpacking. What do we mean by chaotic? And why are states which are more chaotic also more probable? To answer these two questions, we must introduce the distinction between two types of physical states: microstates and macrostates. Microstates are not (as the term might suggest) states of very small systems, but they are states described in terms of microscopic constituents. Think, for example, of a certain volume of gas. A microstate of the gas would be a description in terms of the positions and velocities of all the particles of which the gas consists. Such a description lies beyond the reach of any human endeavour (as the number of particles in gasses are typically in the order of 1023), so physicists resort to describing the macrostate of the gas: a description in terms of macroscopic quantities, such as temperature, volume and pressure.

Any one macrostate will be associated with many microstates. To see why, consider again the box with gas: often it doesn’t matter for the temperature or the pressure (the macroscopic quantities) if some particle A is on the right and another particle B on the left or vice versa. A-right&B-left and B-right&A-left are different microstates, both of which manifest themselves as the same macrostate. Chaos measures this: a macrostate is more chaotic if it is associated with more microstates. We can now understand why more chaotic states are more probable: due to the random motion of particles (assuming nonzero temperature) any physical system continually flows from one microstate to another. Macrostates that have more microstates associated with them will be ‘visited’ more often. Random thermal motion assures that more chaotic states are more probable.

We now have our arrow of time, and it has a clear direction: that of increasing chaos (or entropy, as physicists call it). But Rovelli argues that in gaining a clear direction (by looking at macrostates), we have lost objectivity. Rovelli argues as follows. A description in terms of macroscopic variables represents a choice. We mentioned temperature, volume and pressure as possible macroscopic variables, but there are many more choices. In fact, there is an infinity of choices (mass, colour, location, you name it). The characterisation of a macrostate presupposes a particular coarse-graining – a bit like representing the physical system at a certain resolution (you necessarily lose information). The direction in which chaos (entropy) increases depends on our definition of chaos, which in turn depends on our choice of macroscopic variables. The direction of the arrow of time, we conclude, depends on our perspective!

### 3.      We Need More Time

So what do we make of this? Some philosophers of science that are very critical of Rovelli are convinced that there must be an objective arrow of time that can be agreed upon by everyone. Arguing that the arrow of time is dependent on our perspective isn’t necessarily wrong, they say, but it defeats the whole purpose of doing science! It is the job of scientists to find out as much as they can about the objective world – the world that exists independently of us. If Rovelli argues that objective time doesn’t exist and that the time we perceive is subjective, then what Rovelli’s argument shows, according to his critics, is that science isn’t doing its job properly. If the concept of time as it is used in modern physics is indeed subjective, then we haven’t found the real thing yet: we need more time (try to sell that in an interview!).

It seems to me that the reasoning above, “’real’ time is objective; the time we perceive is subjective; so the time we perceive isn’t ‘real’ time”, is perfectly reasonable. It also has little to do with Rovelli’s stance that time is subjective. It is about the starting point of Rovelli’s analysis, not about the analysis itself. If you are dogmatic in not wanting to let go of the concept of objective time, then Rovelli’s argument can’t force you.

Is there nothing that can be said against the content of Rovelli’s analysis? Is there really no hope of retaining the idea of objective time without resorting to dogmatism? I think there is. Let us return to the context of the definition of chaos (entropy). Rovelli argues that characterising macrostates is necessarily a subjective affair (this is important for Rovelli, because the subjectivity of time depends on it). But he cannot prove that there is not a way to objectively characterise macrostates. He claims that the subjectivity of the characterisation lies in the choice of macroscopic variables, but that presupposes that there isn’t some set of preferred variables. If there are such preferred variables then the choice of variables wouldn’t be different for different observers anymore, because all of them would choose the same variables – what was a subjective choice turns out to be objective. It is the task of the scientist to find out which these preferred variables are, because they describe real, objective time. Science is saved. Or so the critics might argue.

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## Galilean Relativity

At the basis of Newtonian mechanics lies an idea we call galilean relativity, which is the idea that different observers will come up with the same laws of motion as long as their reference frames are not accelerating relative to each other. The reason that acceleration is ‘special’, is because Newton defined force in terms of acceleration: If force equals mass times acceleration, then an accelerating observer will see other forces than a non-accelerating observer.

When two people see something happening, and one of them is moving while the other isn’t, they will see different things. Imagine people playing a game of tennis inside a moving ship, and compare them with someone on the shore looking at the game.

Differently moving observers see different things, yet these observers agree that the laws of physics are the same, and that momentum is conserved – How can that be?

## WK Innsbruck 2018 – Resultaat

Ik ben helaas tijdens het klimmen van de tweede route gediskwalificeerd omdat ik op een metalen bout ging staan (zie foto; kijk goed onder de rechter voet).

Door mijn slechte zicht en kleurenblindheid had ik de bout aangezien voor de rode greep die er bij in de buurt zit.

Dat was jammer, want tot daar aan toe ging de route best goed.

Ik ben daardoor geëindigd op positie nummer tien van de veertien.

Zoals we in Friesland zeggen: koe minder!

(kon slechter)

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## WK Paraklimmen, Innsbruck 2018

Het NL Paraklimteam (Renske Nugter, Mees Vooijs, onze coach Berber Brouns en ikzelf) is zondag in alle vroegte afgereisd naar het Oostenrijkse Innsbruck, in het bergachtige Tirol.

In Tirol doen we mee aan het Wereldkampioenschap paraklimmen.

Op de foto hiernaast is de wand te zien waarop de routes zijn uitgezet die we morgen (dinsdag) moeten klimmen.

Renske en ik moeten morgen de rode route klimmen, en Mees op woensdag. De wand is behoorlijk overhangend (zo’n 35 graden), dus het zal een behoorlijke uitdaging worden!

Als deze twee kwalificatieroutes goed genoeg gaan, hebben we een plaats in de finale, die op donderdag is.

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