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!



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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.’

clock-12-00-clipart-etc-sfclyj-clipartAlice 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.

Luckily the Rabbit – who had by now become very angry with Alice – was somewhat afraid of the Minotaur, so instead of just walking away, he tried once more to convince Alice that his watch was working properly. ‘Tell me something about which you are absolutely certain.’ The White Rabbit demanded. Alice thought about this for a while, and suddenly remembered something that she had told her sister the day before, when they were going to have some tea. ‘where is that blue teapot?’ she had asked her sister, ‘I’m certain I’ve put it somewhere.’ Alice smiled as she thought of this, and she told the White Rabbit what she had told her sister. ‘I am absolutely certain that I’ve put our blue teapot somewhere yesterday.’ Now it was Alice’s turn to feel triumphant. ‘Well,’ said the White Rabbit, ‘I don’t see why what my watch tells me is any different. You are certain because no matter where you’ve left your teapot, you’ll always have left it somewhere. No matter where you’ve left it, that is always true. Just as what the watch says is always true.’

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’:

  1. Numbers in Wonderland
  2. Alice and the Friendly Minotaur
  3. Wonderland without Numbers?

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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


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Alice and the Friendly Minotaur

\Leftarrow Previously in ‘Numbers in Wonderland

‘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.’

Continue reading

Posted in Numbers in Wonderland, Philosophy of Mathematics | 3 Comments



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.

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Numbers in Wonderland

madteapartyAfter following the White Rabbit down the rabbit-hole and meeting all kinds of strange and wondrous animals, Alice finds herself at the mad tea party. Her three companions – the Hatter, the Dormouse and the March Hare – keep asking Alice difficult questions, which make her feel very annoyed.

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Paraclimbing plans for 2017

hilversum_npl2017 is going to be a great year for me. I’m getting married to the woman I love; in a month or two I hope to finish my PhD and, on top of that, there’s going to be a lot of climbing. Last September’s Worldchampionship in Paris was the first international climbing-competition that I participated in. I did better then I had expected two months before, so the competition really whetted my appetite. The next Worldchampionship will be in 2018 in Innsbruck (AUT). But I’m not going to wait that long!


In 2017 the IFSC (International Federation for Sportsclimbing) organises a paraclimbing Worldcup: a series of climbing events at different locations. At the time of writing the IFSC  has yet to announce the locations and dates of the worldcup-events, but there are rumours about three of the locations. There will probably be paraclimbing events in Edinburgh (UK) and Sheffield (UK) sometime around September 2017 and another will be organised in Imst (AUT), but the precise date of the event in Imst hasn’t yet been made public. I plan to take part in all three events. Last year – before my joining the Dutch paraclimbing team – there was a paraclimbing event organised in Campitello di Fassa (IT). I’d really like to compete in Campitello di Fassa, so I hope that they’ll organise another event this year!


Another trip which I’m considering would take me to Boston (MA). On June 23rd the Brooklyn Boulders climbing community in Somerville will host the USA Climbing National championships. I would very much like to visit the city of the MIT and the Boston Tea Party. It would also be a great opportunity for me to get a feeling for different styles (formats) of competition: at the Worldchampionship in Paris I had to climb ‘on-sight’; in Boston they’ll be climbing ‘red-point’ (I’ll explain the difference in a later blogpost).

Because I don’t know the precise dates at which I’ll have to get the best out of myself (or even what counts as ‘the best’; because that depends on the competition-format) it’s difficult to make plans about training. It’s difficult to set short-term goals if you don’t yet know your long-term goals. That is why I’ll write about my goals and training-methods in a later blogpost. Don’t forget to subscribe to the updates on this blog! 😉

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Bestaat Lege Ruimte?

Wat zou er gebeuren als uit de ruimte plotseling alle objecten zouden verdwijnen? Blijft er dan lege ruimte over, of is er helemaal niets meer? Met andere woorden: is de lege ruimte zelf ook een soort ‘object’? Continue reading

Posted in Philosophy of Mathematics, Philosophy of Physics, Ruimte-tijd filosofie, space, spacetime | 2 Comments

Paraclimbing World championship – Paris, 2016


UPDATE: I ended fourth in my category. That’s less than I had hoped for, but a lot more than I had expected (I was the only Dutch finalist). The next worldchampionship (Innsbruck, 2018) I’ll do better! Continue reading

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Does Science Describe Reality?

In the philosophy of science there is a debate about whether scientific theories tell us what the world is really like, or whether scientific theories are nothing more than ‘tools’ or ‘instruments’ – useful for making predictions, but not for telling us what the world is really like. This debate is called the realism-debate. Most working scientists are realists: they believe that scientific theories tell us what the world is really like. They argue that realism is the only philosophy of science that can explain the success (in terms of the accuracy of predictions) of science. For example, they argue that “the theory of atoms allows us to predict that a gas expands when it is heated; wouldn’t that be a mystery if atoms did’t exist?” Continue reading

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