## Paraclimbing Edinburgh

In September I competed for the IFSC Paraclimbing Worldcup in Edinburgh (UK). The two qualification routes went well, but in the finals I fell from the third grip (the grip I’m sitting on in the picture). But I had a great time!

### Belaying

The paraclimbing cup was held at the same time as the ‘able-bodied’ worldcup so there was a large audience and there were a lot of famous climbers walking and climbing around. At some point I heard someone from the organisation, a volunteer who did the belaying during finals, say to a friend “it always makes me nervous when I have to belay one of the big guys.” – with that he probably meant one of the famous climbers – “I’m not going to ask for their autograph or anything, but what if such a big shot gets hurt because I belay him in the wrong way? I would never forgive myself.”

With belaying comes responsibility

Isolation

At the beginning of the climbing finals, the climbers who are going to compete go into isolation. A room in which the climbers remain separated from the rest of the climbing hall. This is to make sure that climbers have no previous knowledge about their route, and so have no advantage over each other (otherwise the first climber is clearly at a disadvantage). The finalists are called from the isolation one by one, and no climber is allowed to talk to other climbers who have completed their climbs.

The isolation in Edinburgh was great! For our warming-up there were hometrainers and rowing apparatuses, and we were even offered sports-massages. When I’m in the isolation I often find it difficult to focus on the match and not be distracted by everything that is going on around me. At such moments I always recite a poem that I like very much. It was written in 18-something by a guy who lost two of his legs to tuberculosis, but never gave up.

Out of the night that covers me,
Black as the pit from pole to pole,
I thank whatever gods may be
For my unconquerable soul.

In the fell clutch of circumstance
I have not winced nor cried aloud.
Under the bludgeonings of chance
My head is bloody, but unbowed.

Beyond this place of wrath and tears
Looms but the Horror of the shade,
And yet the menace of the years
Finds, and shall find me, unafraid.

It matters not how strait the gate,
How charged with punishments the scroll,
I am the master of my fate:
I am the captain of my soul.

[William Ernest Henley (1888)]

## Do Mathematicians Discover or Create?

In my previous blogpost (What Is Mathematics?) we saw that platonists believe that what mathematicians do is discovering things about a world which exist independently of themselves. Intuitionists, on the other hand, believe that mathematicians do not discover but create mathematical theorems*. For many mathematicians Platonism is the obvious choice here. Doesn’t nature just show us how mathematics works? We need only look at triangles drawn in the sand to see that Pythagoras’ theorem is true, don’t we? Isn’t that discovery? Let’s take a look at an example of a mathematical theorem to see whether Platonism is really that obvious.

Suppose that some brilliant mathematician, burning the midnight-oil in her lonely attic, finds some very complex geometrical theorem (say, about the volume of an exotic geometrical shape). It may be that this theorem is not instantiated; it may be that there is nothing in nature which actually has this shape (maybe the shape has a large number of dimensions). It is not so obvious that the geometrical theorem that our lonely mathematician has come up with exists anywhere outside her mind (and the paper she wrote it down on). Why should we believe that the mathematician has discovered anything? That would imply that it was already there before she came along. Might we not say that she just ‘made it up’? Such questions show that it is far from obvious that Platonism is self-evident while Intuitionism is not.

## Why should we care?

Ok, fine.’ I hear you think, ‘discovering and creating are two different things. Platonists and intuitionists have different ideas about what mathematics is. They disagree on whether mathematics necessarily exists outside the mind of the mathematician. But why is that important? Does it matter for how we do mathematics?

Yes, it does. One of the rules that is often used in mathematics is ‘the exclusion of the middle ground’. It is the simple rule that any mathematical proposition is either true or it is false – there is no middle ground. Mathematicians use this rule whenever they prove a proposition by contradiction. That works as follows. Suppose we want to prove a certain proposition. If we can show that the negation of our proposition (the assumption that it is false) leads to a contradiction, then ‘the exclusion of the middle ground’ tells us “assuming the falseness of the proposition leads to a contradiction, so the proposition must be true.”

Intuitionists argue that proof by contradiction isn’t really proof at all. They believe that real proof is constructive. The only thing that is constructed in a case of proof by contradiction is that the negated proposition is not true, not that it is false. Mathematical proof, according to the intuitionist, shouldn’t be based on the metaphysical assumption that the truth is in a sense ‘binary’ (that propositions must be either true or false). In cases where ‘the exclusion of the middle ground’ seems like a useful rule, it must be possible to find real, constructive proofs. Intuitionists, therefore, believe that much of what we call mathematical knowledge is actually based on conjecture (because it is based on ‘the exclusion of the middle ground’-rule).

Can you think of a reason why the intuitionists are skeptical of the ‘exclusion of the middle ground’-rule? (please leave your answer as a comment to this blogpost).

## Intuitionism & P vs NP

Some computer scientists believe that the skepticism of the intuitionists regarding ‘the exclusion of the middle ground’ is connected to the infamous P vs NP problem. In one sentence, the P vs NP problem is finding an answer to the question whether any mathematical problem whose solution can be quickly verified (to be an actual solution), can also be quickly solved. Some say that the difference between constructive and non-constructive proofs as well as the difference between solving problems and verifying solutions can be understood as a difference between bottom-up and top-down mathematical reasoning.

If there is indeed this connection (between the intuitionism debate and the P vs NP problem) then the interpretation of mathematics may be of great importance to, for example, cryptography. Much of the security of financial transactions today relies on certain mathematical problems being very difficult to solve (such as breaking very large numbers down into prime factors). What if whole swathes of mathematical knowledge turn out to be based on conjecture? Of course, these conjectures are not necessarily false, but intuitionist philosophy casts doubt on them. But how will the world’s economy fare, if so much is based on trust in financial systems? Trust which is often based on mathematical proof?

## What is a Mathematician?

Back to the philosophy of mathematics. Can we answer the question from my earlier blogpost ‘What is a Mathematician’? We might say that the platonist believes that all physical things can be a mathematician, as long as they follow (or embody?) the rules of mathematics (whether it be some conscious being, a lifeless calculating machine, or even a falling stone). The intuitionist, on the other hand, believes that a mathematician must be able to apply the rules of mathematics.

I’ll illustrate this difference by comparing mathematics to a game of chess. Let’s say that every possible state of affairs on the chessboard represents a mathematical proposition and that the moves to get to that state of affairs are limited by the rules of mathematics.

The platonist would say that any state of affairs on the chessboard that can be reached by following the rules of mathematics is a mathematical proposition – whether it actually has been reached or not. They just exist. Or they don’t. The intuitionist, on the other hand, would say that mathematical propositions are constructed by the mathematician by following the rules of mathematics in moving about the pieces on the chessboard. Before the mathematician moved the pieces, the proposition simply didn’t exist yet.

Don’t forget to subscribe to the updates on this blog so that you receive the next blogpost automatically! 😉

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More about the philosophy of mathematics:

https://plato.stanford.edu/entries/philosophy-mathematics/#For

*) Wait a minute! What happened to Formalism? Why aren’t we discussing that anymore? To see how Formalism differs from Platonism and Intuitionism it is important to know that formalists believe that mathematics in itself is ’empty’ – consisting in tautologies whose truth is independent of anything material. Understanding this is at the basis of what I wrote about numbers at the end of the post What Is Mathematics?. I’ll discuss the emptiness of mathematics in a later blogpost and here focus on the difference between Platonism and Intuitionism.

Posted in Philosophy of Mathematics | 2 Comments

## Paraclimbing in Imst

Last month* I participated in a paraclimbing competition in Imst, Austria. The wall which we climbed was situated between the beautiful Tiroler mountains.

In the movie below you can see my attempt at climbing the second of the six routes I had to climb. As you can see the number of grips available is large, so that wasn’t a problem. But you’ll also see that there’s quite a bit of overhang, which becomes a problem after a while (when it becomes more difficult to stretch my left arm and unstretch my left leg)

Any expert climbers out there with useful tips?!

I finished fourth in my category. Am I content? Not really: I didn’t win. On the other hand, considering the amount of training I’ve done in the past year and the good coaching I’ve had, I think I couldn’t have done much better (this time). At the end of September there’ll be another paraclimbing competition. This time in Edinburgh. Perhaps the air in the Scottish highlands will take me to a higher level! 😉

*) The reason why I haven’t blogged about this before is that, during the weeks after the event, I’ve been occupied with the defense of my PhD and several surprise-festivities afterwards.

Posted in paraclimbing, Travels | 1 Comment

## What Is Mathematics?

Mathematics is a language. More specifically, mathematics is the language that scientists use to organise and order observations.

For example, physicists may describe falling stones in terms of mathematical concepts like parabolas and perfect spheres and sociologists describe their observations of large numbers of people in terms of normal distributions and differential equations.

However, stones are not perfect spheres and groups of people don’t behave exactly as described by normal distributions. It follows that talking about falling stones is not the same as talking about spheres that follow parabolas. Considerations like these prompted Bertrand Russel to say that

“mathematics may be defined as the subject in which we don’t know what we are talking about, nor whether what we are saying is true.”

So what is the relation between the work of mathematicians and the observations that physicists make? What is the relation between mathematics and physicis?

Mathematicians do not agree on this issue. In this blogpost I want to show you that there are different views on the nature of mathematics. The view that most mathematicians have (and, perhaps without being aware of the discussion, most physicists as well) is called Platonism. Platonism is named after the ancient Greek philosopher. And that’s not a coincidence.

### Platonism

The Plato of old is known for many things, but perhaps most for his ‘theory of forms’. According to the theory of forms the reality that we humans see is made up of imperfect copies of some other ideal reality: the reality of forms. According to Plato, our situation is comparable with that of prisoners that are chained inside a cave. The prisoners don’t see the real world outside the cave, but only the shadows cast by things in this real world on the walls of the cave. It is the task of the philosopher to try to escape the cave and see the real world – the world of perfect forms. The way to do so, according to Plato, is by studying mathematics. Plato’s reality of forms is a reality described by mathematics (in terms of perfect spheres and ideal straight lines).

I know of no present-day scientists who take Plato’s view seriously in the way Plato formulated it. There is, however, a sense in which the theory of forms is still a part of modern philosophy of science. In the modern philosophy of science, Platonism is the name associated with a view on the nature of mathematics. Not all platonists agree on what Platonism exactly is, but for our present discussion we may characterise it as follows:

Platonists believe that what mathematicians do is discovering things about a world that exist independently of themselves. The theorems that mathematicians discover exist independently of the mathematician, and un-proven yet provable theorems exist just as proven theorems do.

### Intuitionism

Although Platonism is probably the most common view on the nature of mathematics, it is not the only possible view. Another way of looking at these matters is that of the intuitionist. The intuitionist regards mathematics as a creative process in which mathematical theorems are created as intuitions in the mathematician’s mind. Mathematical theorems do not exist independently of the mathematician. In other words: mathematical theorems do not exist before they are proved.

### Formalism

Yet another way to look at these matters is that of proponents of Formalism.

Formalists believe that mathematics is merely a formal game in which a fixed set of rules (the rules of logic) are followed to go from axioms to theorems and from those theorems to yet more complex theorems.  The axioms and theorems are about numbers. It may be the case that these numbers refer to things we can observe (tables or chairs or what have you) but numbers themselves are meaningless symbols, and the theorems in higher mathematics are strings of such symbols within the game of mathematics – nothing more.

### Difference

According to the formalist, we saw, mathematical axioms and theorems are about numbers. Neither the intuitionist nor the platonist would disagree. The difference between Formalism, Intuitionism and Platonism lies in the belief about what numbers are. The platonist believes that numbers in some sense exist independently of the mathematician; the intuitionist believes that numbers exist only in the mind (or intuition) of the mathematician; and the formalist believes that numbers are meaningless abstract symbols that don’t exist outside the game of mathematics.

It seems to me that much of the debate introduced above depends on our idea of what a mathematician is. Does it have to be a living creature? …with a PhD perhaps?

The most general answer possible would be that any physical process can be a ‘mathematician’. Read my next blogpost to find out what the consequences are of such a ‘most general answer’. In my next blogpost I will also give you a concrete example in which the different interpretations are illustrated.

Don’t forget to subscribe to the updates on this blog so that you receive the next blogpost automatically! 😉

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More about the philosophy of mathematics:

https://plato.stanford.edu/entries/philosophy-mathematics/#For

Posted in Philosophy of Mathematics | 4 Comments

## PhD Dissertation

I will be defending my PhD dissertation on 13/07/2017 in Utrecht. Below you can find the documents which have kept me busy the past few years:

To find out what this is all about, look at p.41 of my book!

| Tagged , , , | 5 Comments

## The White Rabbit’s Watch

Alice and the Friendly Minotaur have decided to go on a search for the Architect. Alice wants to ask the Architect – who designed the Minotaur’s labyrinth – whether numbers were used in making the labyrinth’s design. She hopes that the Architect can help her to understand what numbers are.

The White Rabbit.

Where should we start looking?’ Alice asked. She had no idea what kind of a person – or creature – this Architect was that the Minotaur talked about, and so she had no idea either where to start looking. The Minotaur thought about this while he rested his chin on one of his hoofs. ‘We need someone who can tell us where the Architect lives.’ He said after a while. ‘We need someone who knows his way around Wonderland.’ He frowned deeply and moved around his eyes as if he tried to look into his own head. And then, suddenly, his eyes brightened and he said with a loud voice ‘we should first go look for the White Rabbit!’ The Minotaur continued in a calmer tone of voice ‘The White rabbit has been a manservant for the Queen for a long time, so he must know many things about Wonderland. Perhaps he can tell us where to find the Architect?’

Alice sighed. ‘But how do we find the White Rabbit?!’ She had the feeling that the Minotaur’s suggestion didn’t help very much, but instead doubled the problem. ‘First we were looking only for the Architect, and now we must find both the Architect and the White Rabbit!’

Everybody in Wonderland knows where to find the White Rabbit,’ the Minotaur mumbled indignantly. ‘As the Queen’s manservant he can always be found at 12 o’clock at the entrance of the Queen’s garden, standing watch and greeting the visitors of the Queen.’

Alice and the Minotaur made their way towards the Queen’s garden. When they were almost at the gate, they ran into the White Rabbit. As the rabbit hurriedly passed them, Alice heard him talking to himself. ‘Oh dear, oh dear, I shall be late!’

Perhaps it is a bit impolite to stop someone who is in a hurry,’ Alice thought, but with the big Friendly Minotaur by her side she was not at all afraid to address the hurrying rabbit. ‘Excuse me, mister Rabbit..?’ The White Rabbit stopped and turned around. It was clear to Alice that he was annoyed. ‘What do you want?’ the Rabbit snapped. The shrill voice of the rabbit didn’t frighten Alice. If anything, it actually made her bolder. Without introducing either herself or the minotaur – without even speaking of the labyrinth – Alice asked ‘can you tell us where the Architect lives?’ The White Rabbit frowned when the little girl addressed him in such a bold manner. But he was too much in a hurry to take offence. He had to be at the entrance of the Queen’s garden at 12 o’clock sharp. The Queen was very intolerant of disobedience.  The rabbit shivered as he thought of what must be the Queen’s favourite command. ‘Off with his head!’

The White Rabbit pointed his walking-cane to the Queen’s garden. ‘the Architect lives several days travelling beyond the garden. When I was as little as you are now, I used to play…’ Then suddenly the rabbit remembered that he was in a hurry. Looking at his pocket-watch he started walking towards the entrance of the Queen’s garden. ‘It is already 12 o’clock. I am late. I must go, or the Queen might be displeased.’

Alice saw the White Rabbit’s watch and noticed that the hands of the watch were not moving. ‘But your watch stands still!’ Alice said while she pointed at the rabbit’s clock. ‘It must be broken.’

‘Broken, pff!’ The White Rabbit snorted. ‘This is a real Gettier-watch! My father gave it to me when I first went to school. It cannot break.’ The rabbit spoke about his watch with such firmness that Alice began to doubt what she had seen. ‘But I’m sure I didn’t see the watch’s hands moving, so how can you be sure that it tells you the right time?’ Alice asked uncertainly. ‘Well, you see, the watch always says its 12 o’clock, and whenever I check that – and I check it often (the White Rabbit wasn’t the Queen’s manservant for nothing) – it is indeed 12 o’clock. That’s why I’m certain!’

The White Rabbit’s words left Alice puzzled. ‘I think that the reason that your watch can’t break is that…it isn’t working.’ Alice said, again with an uncertain voice. The White Rabbit clearly didn’t like it that his watch, which he held so dearly, was so lightly accused of being broken. Alice saw that the White Rabbit became very upset, and when she saw that he even started breathing heavily through his nose, she changed the tone of her voice. ‘If the watch tells you all day that it’s 12 o’clock, then it is right twice a day. I guess you could call that ‘working’,’ Alice said while shrugging her shoulders. ‘Precisely!’ The White Rabbit shouted triumphantly. ‘But it’s not working very well then,’ Alice continued with a soft voice. ‘I heard that!’ the Rabbit responded.

The White Rabbit’s words made Alice feel very uneasy. It all sounded very likely, but if the White Rabbit’s watch is just as good as any other clock, then how can you tell whether any clock is ever working? ‘Or,’ Alice thought, ‘could it be that time in Wonderland is something that is different from what time is at home?’ When Alice wanted to ask  the White Rabbit whether all watches in Wonderland stand still, she found that he had not waited around, but had continued his way to the Queen’s garden. ‘Well,’ she said while prodding the Minotaur, who had fallen asleep during the discussion between Alice and the White Rabbit, ‘at least now we know where to look for the Architect.’

When Alice had helped the Minotaur rub his eyes (due to their sharp-edged hoofs minotaurs can’t rub their own eyes, which is why minotaurs are very slow at waking up in the morning), she told him of her plan. ‘Now that we know that the Architect lives somewhere on the other side of the Queen’s garden we should just walk to the other side’ The Minotaur answered, still a bit sleepy, ‘the Queen is not fond of strangers in her garden. We’d better visit her first, and ask her permission.’

The little girl agreed. As Alice wondered why everybody was so careful not to offend the Queen, she and the Friendly Minotaur followed the path of the White Rabbit. They entered the Queen’s garden…

***

Previous episodes of ‘Numbers in Wonderland’:

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Posted in Numbers in Wonderland | 2 Comments

## Wonderland without Numbers?

$\Leftarrow$ Previously in ‘Numbers in Wonderland’

Alice followed the Friendly Minotaur as he walked deeper into the forest. Any other little girl would have been at least a bit frightened by the shape of the large beast and the shadows cast on him by the looming trees, but Alice was too excited to find out what the Minotaur was going to tell her. Alice really didn’t like numbers, so if the Minotaur would tell her that we don’t really need numbers, that would be great! ‘I will go back to the Hatter and the other tea-party-goers and tell them that I never want anything to do with numbers!’ Alice smiled as she thought of this. ‘no more of those boring math classes for me.’

Suddenly the Minotaur stopped walking. Alice almost bumped in to him, immersed as she was in her thoughts, and she saw that they had reached an open spot in the forest. There were no trees, but only very tall grass. When she looked more closely, she saw that the grass in some places hid a brick wall. Then it struck her. This was the Minotaur’s labyrinth! Ever since Alice had read about the labyrinth in her sister’s books this was how she had imagined the Minotaur’s home to be: as a large, circular and stoney structure. Hidden somewhere far away in the middle of nowhere. The Minotaur turned around and started to speak: “you ask me whether we really need numbers… well I can tell you one thing; I don’t need them. The Architect, who designed my home for me, gave me several lists each of which tells me where to find a particular room. For example, if I want to find the way to my bedroom,” the Minotaur said while pointing one of his hoofs in the direction of the labyrinth, “I take the list with the title ‘bedroom’ and simply follow the directions that are listed. That is how I know where to go. Numbers are difficult things. It’s better to stay away from them.”

The Minotaur’s words puzzled Alice. On the one hand she was delighted to hear someone say that mathematics is not as all-important as her big sister always said, but on the other hand she felt that there was something not quite right about what the Minotaur said. “But,” Alice said, “what kind of directions are there on the Architect’s lists?”

“Well,” the Minotaur answered, “the list tells me to walk a bit and then turn, and then walk a bit more and make another turn… and then hopefully, after a few such bits and turns I am where I want to be.”

Alice didn’t have to think very long about her reply. Being the clever girl that she was, she had come up with the idea that the Minotaur, although he had told Alice to stay away from numbers, made use of numbers himself! Only the numbers that the Minotaur used were cleverly hidden away by the Architect in the lists with directions. Alice mumbled, half to herself, half to the Minotaur “If all that the directions on these lists say is something like ‘walk a bit, and then turn’, then how do you know how much is a bit?” With a stern voice (which Alice had often heard her mother use when Alice and her sister had misbehaved) Alice asked the Minotaur “isn’t there something more on the lists? …distances perhaps? …and aren’t those distances… numbers?”

The Minotaur sighed and lowered his head. After having stared at his feet for a while, he looked again at Alice. The pride Alice had felt upon having discovered the numbers that she believed the Architect had hidden in the Minotaur’s lists disappeared instantly when she saw that the Minotaur had tears in his eyes. “Why are you crying?” She asked, now with a soft voice.

“I never learned how to count,” the Minotaur said while trying to wipe away his tears (which was quite difficult due to the sharp edges of his freshly-trimmed hoofs), “so specially for me the Architect made lists with directions without numbers. Every direction just says: ‘turn right and walk until you can’t go any further’ or ‘turn left and walk until you can’t go any further'”.

“Oh,” said Alice, feeling very sorry for the Minotaur. “But…” she suddenly remembered what the Hatter had told her. The Hatter had said that the Minotaur must know very much about mathematics because the Minotaur’s name, just as the word ‘mathematics’, starts with an ‘M’. “Then why does your name start with an ‘M’?” Alice asked.

“I hope Mr. Hatter has told you” said the Minotaur, who had difficulty with wrapping a handkerchief around one of his hoofs (actually, the Minotaur had never wrapped a handkerchief around a hoof before, because minotaurs almost never cry), “that I am usually called ‘the Friendly Minotaur’. My mother gave me that name because she wanted me always to remember that mathematics can be Free of numbers. And that’s also why she never taught me how to count.”

Alice couldn’t understand how it was possible that someone can’t count. ‘If I couldn’t count’. she thought, ‘I wouldn’t even be able to trim my nails properly.’ Alice always counted her fingers while she was trimming her nails because she was afraid that she might miss one if she didn’t (Alice didn’t realise that since the hoofs of Minotaurs are split only in two they don’t need numbers to trim them without missing any hoof-parts).

‘So we still don’t know whether we need numbers,’ Alice thought while frowning sadly. Again she addressed the Minotaur: “so you don’t need numbers, but that doesn’t mean that nobody needs them. How do you think the Architect made the lists with directions in the first place?”

“Well… I don’t know” the Minotaur said hesitatingly, “maybe we should ask the architect himself? I am told he lives on a steep hill beyond the forest’s rim.”

“let’s go on an adventure then; let’s go to the Architect!” – said Alice.

In the next episode of ‘Numbers in Wonderland’ Alice and the Friendly Minotaur will meet the White Rabbit and they’ll find out that the Rabbit’s watch is a strange thing indeed!

Next episode of ‘Numbers in Wonderland’ $\Rightarrow$ The White Rabbit’s Watch

## Alice and the Friendly Minotaur

‘I wonder what would be left,’ Alice thought to herself, ‘if I take five apples and throw away the apples’. ‘What would it feel like to have five in my hands?’ I guess it would be heavier than three.’

## Mouthguard

I grind my teeth. Even when I’m not thinking about very difficult things – such as what present to give my mother for her birthday – I scrape the teeth of my lower jaw with my upper teeth. The dentist kindly but urgently advised me to stop grinding my teeth, because the teeth on my lower jaw have already been worn down to miniature versions of their upper comrades. But that’s easier said than done – because I’m spastic.