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.



Posted in Epistemology, Philosophy of Physics, Probability, Ruimte-tijd filosofie, spacetime, time | Tagged , , | 4 Comments

What Does ‘=’ Mean?

It is often said that in mathematical equations, such as ‘2+2=4’, the symbol ‘=’ represents the fact that on both sides of the equation there is the same thing. That’s wrong: on the one side there are three symbols, while on the other there is just one.


Even in the equation ‘1=1’ the same-thing-on-both-sides-reading is wrong, because there there are two separate things (the ‘1’ on the left and the ‘1’ on the right).

So how should we understand the ‘=’-symbol? It might seem reasonable to say that in the equation ‘2+2=4’ we should read ‘=’ as meaning that the things that the numbers refer to are the same (in number). If the numbers in the equation refer to apples, for example, then the equation merely states that if you add two apples to two apples you end up with four apples. The ‘=’-symbol tells you that two apples added to two apples is the same as four apples.

But this view is problematic. Mathematics is usually understood as a language or formalism that is independent of reality. Mathematics is, to put it disrespectfully, merely a bunch of tautologies, whose truth depends on the definitions that we ourselves choose. We can of course apply mathematical language/formalism to things we see, but then we are no longer doing mathematics – we’ve turned into physicists.

We can’t just say that the number of apples referred to by the symbols are the same, because mathematical terms have no meaning outside of mathematics.

so what is “=”?


Posted in Epistemology, Philosophy of Mathematics | 2 Comments

Strange Encounters

The man from Dutch Railways (NS) had actually grabbed and stopped my tricycle and walked around to stand in front of me. My front wheel was between his feet, so I couldn’t turn away. He looked at me and said: “I could take you from your tricycle easily if I wanted to.” I was perplexed. Was this guy actually threatening me?! “yes, that’d be easy,” I said, “because my left arm and leg are paralysed.” My response did not have the effect I had hoped for. He smiled at me and said: “can you prove that?”

Because of my disability I use a tricycle to get around. Inside buildings, and even inside shops, when I have to walk a long distance, I cycle my way. That of course raises many eyebrows – you don’t see a bearded man on a tricycle in the supermarket every day. I understand that, and I don’t mind telling people that I’m on a tricycle because of a brain tumour, not even if I have to do that every day. But what happened to me on the particular morning I’m telling you about, was simply astonishing.

The story starts several months ago. I had been teaching the whole day, and I was trying to find the train to Utrecht, to get back home. So I tricycled my way through the station, when an employee of the Dutch Railways (NS) told me to step off from my tricycle and proceed on foot. Normally I would have told him about my brain tumour, but since I was tired, I told him that I don’t ride a tricycle for fun. “Then why do you ride a tricycle?” he asked. “Because I can’t walk” I told him. He wasn’t convinced. He put his hands in his sides and took on what they call the ‘power-pose’. “I’d like to see that” he said with a smile. I couldn’t believe my ears. I became very angry, but I couldn’t think of the proper way to answer him. I just shook my head in disbelief and decided to ignore him. I cycled past him, and found my way to the train to Utrecht – and never looked back. I did my best to forget about what had happened, and I didn’t think about the events very much afterwards.

Until today.

Again I was on my way to the train travelling to Utrecht. I was tricycling across the central hall of the train station when my eyes met those of the NS employee that I had met before. I was about to cycle past him, but apparently he didn’t remember our previous encounter, because again he told me to get off my tricycle. Without turning my head I cycled on, towards where the platforms are. I had hoped that this would be the end of our second encounter, but I was sorely mistaken. The man ran after me and grabbed the frame of my tricycle.

“How can I prove that I have a brain tumour!? I can’t just cut myself open, can I?!” I felt that I was losing my temper, because I really didn’t know what to do or say. A colleague of the man was standing several meters away from us, but didn’t interfere. It felt as if they were waiting for me to become aggressive. I knew that, so I knew that that’s what they’d expect. At a certain point the man looked aside, towards the entrance of the railway station. when he turned his head, he also moved his foot a little bit. And that’s when I escaped. I got on the train, and went home.

As I write this, I feel both proud and ashamed. Proud, because I hadn’t started shouting or become aggressive, but also ashamed, for escaping…

Posted in Personal, Travels | 5 Comments

Einstein & Kant: Synthetic Relativity?

curved space ball

Einstein’s approach to physics may be compared to Kant’s approach to philosophy. Where Kant derived things about the world we perceive from the possibility of human knowledge, Einstein derived things about the world we perceive from the possibility of human physics. At the basis of Kant’s philosophy lies the thought that the world must be such that our knowledge of that world is possible. Kant said that if knowledge is possible, then the world we perceive must have certain features*.  In a way that appears similar, Einstein said that if physics is possible then the world we perceive must have certain features. At the basis of Einstein’s physics lies the thought that the world must be such that physical knowledge of that world is possible.

Kant’s Synthetic A Priori

Let’s start with Kant. To understand Kant’s view it is important to understand that he made a distinction between propositions that are analytic and propositions that are synthetic. A proposition is analytic if its its truth-status can be judged by analysing the definitions of the terms used in the proposition (a standard example of an analytic proposition is “a square has four sides”). A proposition is synthetic if more than mere terminological analysis is required: we must make an observation. A standard example of a synthetic proposition is “the apple is red”.

Kant called analytic propositions a priori, which means that he believed that the truth-status of such propositions can be judged prior to any observation. It won’t surprise anyone that Kant believed that most synthetic statements are a posteriori. Wait a minute… most? why not all? Kant believed that besides the analytic a priori and synthetic a posteriori there is yet a third category of propositions: the synthetic a priori. Synthetic a priori propositions tell us something about the world around us, yet can be known to be true independent of observation.

Einstein’s Relativity

How is Einstein’s approach to physics comparable to this? Einstein analysed physics and came up with the idea of relativity: to be able to do physics, it must be possible for different observers to agree on what the physical laws are. In other words: physical laws must be the same in all reference frames. The thought of relativity was not new. Galileo already observed that if you do an experiment in a uniformly moving lab (say, the cargo hold of a steadily moving ship) or in a lab at rest (on the shore) you will find the same physical laws. However, for Galileo that was a result of how we do physics. The innovative thing about Einstein’s approach is that the thought that physical laws must be the same in all reference frames is no longer a result, but lies at the basis of physics – it has become a postulate.

Is Einstein’s Relativity Postulate Synthetic A Priori?

A debated question about the relation between the approaches of Einstein and Kant is whether Einstein’s postulates are synthetic a priori. Einstein regarded relativity as a postulate – doesn’t that mean that he believed that relativity is a priori?


‘Relativity’ by M.C. Escher (1953)

Being the empiricist that he was, Einstein did not think of his postulates as synthetic without observations telling us so. Only experiment can tell us whether the postulates we choose as the basis of our theories “latch on to the blueprint of reality”. Period. Case closed.

But suppose, for the sake of the argument, that someone with sound common sense but a scientifically untrained mind (a tabula rasa, if you will) were given the following task: try to find a system of laws or rules that can be used by a group of people to make accurate predictions about the things that surround them. What will she find?

Thought Experiment

One might argue that our scientifically untrained friend comes up with a principle of relativity, as laws and rules are most useful if they hold for everyone in the group**. But after that our untrained friend would have to say for who (which reference frames) relativity holds. It might seem as if she should assume relativity for all reference frames, as Einstein did. But Einstein never did that. Einstein only assumed relativity for all reference frames that move either uniformly or acceleratedly relative to each other. He assumed nothing about reference frames that differ from each other in other respects. Here one could think of different movements (eg. irregular, or even discontinuous) or other parameters (such as size, colour or mass).

In other words, our friend doesn’t have any idea what relativity is (what it means that all laws are the same), because for that it is necessary to say what parameters are relevant in describing physical laws. For Einstein it was Newton’s definition of force in terms of acceleration that singled out different states of motion as relevant for reference frames. Only with a definition of force in hand our friend would know which parameters should be the invariants.




*) Where Kant seems to have gone wrong is in thinking that the knowledge of his time (featuring absolute simultaneity and euclidean geometry) was the only possible knowledge.

**) Or she could come up with some kind of subjective relativity: Physical laws are not necessarily the same in all reference frames, but physical laws which can be associated with more reference frames are considered to be better.

Read about Hans Reichenbach’s take on Kant & Modern Physics here.

Posted in Epistemology, Philosophy of Physics | Tagged , , , , | 1 Comment

Wikimedia2017 lezing – Feiten bestaan!?

Vragen aan het Publiek

Steek uw hand omhoog als u denkt dat de uitdrukking klopt.

  • ‘feiten bestaan niet’
  • ‘feiten kunnen waar of onwaar zijn’
  • ‘feiten kunnen waargebeurd zijn’
  • ‘een verhaal kan waargebeurd zijn’

Volgens de Nederlandse Van Dale is een feit “iets dat werkelijk is of heeft plaatsgehad.” Met andere woorden: iets is een feit als het is gebeurd en het is geen feit als het niet is gebeurd. Dat klinkt helder.

Gemaakte feiten (I)

De zaken worden ingewikkelder als we naar de herkomst van het woord ‘feit’ kijken (nu hoor ik u denken: “doe dat dan ook niet”, maar dat is al te laat). Ons woord ‘feit’ stamt af van het Latijnse ‘factum’, dat zoveel als ‘gemaakt’ betekent. Dat is belangrijk, omdat het een andere kijk op feiten geeft. Feiten zijn niet gewoon ‘wat er gebeurt’, maar eerder onze interpretatie van wat er gebeurd is.  Maar wat betekent dat dan, dat feiten gemaakt worden?


Laten we, om te zien hoe feiten geïnterpreteerd kunnen worden, kijken naar autisme in de VS (Trump staat hiernaast afgebeeld omdat we kijken naar autisme in de VS; Trump heeft natuurlijk niets met autisme te maken).

In de jaren ’70 schatten wetenschappers dat een op de 2000 kinderen autistisch is. Nu, anno 2017, is dat één op de 70. Dat is bijna 30 keer zoveel! Er is veel gediscussieerd over deze vermeende ‘autisme-epidemie’. Het kan natuurlijk zo zijn dat dat de jeugd van tegenwoordig de modernisering van de maatschappij niet meer kan bijhouden, dat in 40 jaar de wereld zo is veranderd dat het logisch is dat autisme zo vaak voorkomt.

Maar er zijn ook andere verklaringen mogelijk. Er zijn altijd mensen met autisme geweest, maar die werden niet altijd als iemand met autisme beschouwd. Autisme als naam voor een specifieke aandoening, is ooit ‘uitgevonden’. Wetenschappers denken dat de wonderbaarlijke toename van het aantal mensen met autisme ten dele wordt veroorzaakt door een groeiend bewustzijn dat er zoiets bestaat als autisme. Dat klinkt misschien vreemd. Of iemand autistisch is of niet, dat is toch een feit? Hoe kan ons bewustzijn van autisme daar nu iets mee te maken hebben?

Denk eens aan een huisarts die nog nooit heeft gehoord van autisme. Zij zal de diagnose ‘autisme’ niet stellen. Als we kijken naar de jaren waarin het aantal gevallen het meeste stijgt, dan blijkt dat de stijging in veel gevallen kan worden verklaard zonder te verwijzen naar de daadwerkelijke toestand van personen. Omgevingsfactoren lijken dan verantwoordelijk voor de stijgingen. Zo werd er bijvoorbeeld in 1991, een jaar waarin het aantal gevallen erg steeg, een nieuwe ziektewet ingevoerd die families met autistische kinderen recht gaf op subsidie. Ook is over de jaren meerdere keren de definitie van autisme (de symptomen die iemand moet hebben om als ‘autist’ te worden beschouwd) veranderd.

Ook al zouden we op deze manier iedere stijging in het aantal mensen met autisme kunnen verklaren zonder het over de personen zelf te hebben, dan nog zijn medici het erover eens dat autisme een bestaand fenomeen is met een biologische achtergrond. Deze biologische achtergrond is een ingewikkeld verhaal over hoe atomen en moleculen zich gedragen in onze hersenen. ‘Oké, zul je nu misschien denken, ‘hoeveel autisten er precies zijn is dus niet een feit. Maar dat verhaal over atomen en moleculen, dat zijn toch feiten?’

Gemaakte feiten (II)

Nou… nee. Ons tweede voorbeeld van de interpretatie van feiten laat zien dat ook de uitspraak ‘atomen bestaan’ geen feit is, maar een interpretatie van feiten. Laten we eens kijken naar de waarneming van atomen. Natuurkundigen gebruiken daarvoor een elektronenmicroscoop. Een elektronenmicroscoop meet de dikte van een oppervlak door met een naald over dat oppervlak te gaan en te meten hoe dichtbij het oppervlak is. Als het oppervlak bestaat uit atomen dan meet de elektronenmicroscoop dat het oppervlak regelmatig dichtbij komt (zie afbeelding 2).


afbeelding 2: de naald van een elektronenmicroscoop ‘voelt’ atomen

Maar stel nu dat atomen niet bestaan; dat er in het hele universum een soort veld bestaat met ‘bobbels’ erin (zie afbeelding 3). Dan zou een elektronenmicroscoop hetzelfde zien als wanneer er atomen bestaan.

We weten dus helemaal niet zeker of atomen wel bestaan! Het idee dat materie uit atomen bestaat is slechts een (heel erg nuttige) aanname.





afbeelding 3: een universeel materie-veld

Waarom doen we dit soort aannames? Als we het niet zeker weten, waarom nemen we dan aan dat atomen bestaan? Het antwoord op deze vragen is eigenlijk heel simpel: we moeten bepaalde aannames wel doen, anders is wetenschap onmogelijk.

Wetenschap: Model van de Werkelijkheid

Een gangbare opvatting is dat wetenschap een zoektocht is naar het wiskundige model dat het beste past bij de werkelijkheid die we waarnemen. Zo denken we bijvoorbeeld dat het model dat het beste past bij een steen die wordt afgeschoten met een katapult een wiskundige parabool is (zie afbeelding 5).


Parabola afbeelding 5 & 6: een katapult & een parabool

In deze opvatting van wetenschap als modellering van de werkelijkheid kunnen we duidelijk aangeven wat een feit nu eigenlijk is. Ook aan het begrip ‘waarheid’ kunnen we een heldere betekenis geven. Een feit, zoals we in de eerste zin van dit essay al zagen, is “iets dat werkelijk is of heeft plaatsgehad.” In onze opvatting over wetenschap horen feiten dus thuis in de werkelijkheid.

Wat is ‘waarheid’ in onze opvatting? Kunnen we ‘waarheid’ eenduidig definiëren? In onze opvatting is het van het grootste belang dat het (wiskundige) model dat we kiezen zo veel mogelijk lijkt op wat er in het echt gebeurt. ‘Waarheid’ is dan niets anders dan een juiste afbeelding van iets uit de werkelijkheid (bijvoorbeeld een vliegende kanonskogel) op iets uit het gekozen model (bijvoorbeeld de wiskundige parabool).

Antwoorden aan het Publiek

We kunnen met behulp van deze opvatting over wetenschap de vragen beantwoorden die ik eerder aan het publiek voorlegde.

  • ‘Feiten bestaan niet.’

[Er gebeuren dingen; er bestaat een werkelijkheid, dus feiten bestaan. MAAR: feiten zoals wij ze waarnemen zijn noodzakelijkerwijs ‘gekleurd’ door interpretatie.]


  • ‘Feiten kunnen waar of onwaar zijn.’

[Dit is een ‘categorische fout’; feiten zijn dingen die gebeuren in de werkelijkheid terwijl de waarheid gaat over het verband tussen de werkelijkheid en onze modellen daarvan.]


  • ‘Feiten kunnen waargebeurd zijn.’

[Dit is een tautologie. Iets is een feit of het is geen feit. Als iets een feit is, dan is het per definitie waargebeurd.]


  • ‘Een verhaal kan waargebeurd zijn.’

[Nee, een verhaal kan worden verteld (of geschreven of geschilderd of…), een verhaal kan niet gebeuren.]



Maar wat klopt er dan wel?! zult u zich misschien afvragen.

‘een op feiten gebaseerd verhaal’



  • feiten kunnen niet gemaakt worden
  • feiten kunnen op verschillende manieren worden gemodelleerd en dus geinterpreteerd.
  • We kunnen er nooit zeker van zijn of iets een feit is.
  • Waarover we zeker (denken te kunnen) zijn, zijn geen feiten.

Appendix – vragen uit het publiek

  • bestaat de waarheid in jouw opvatting over wetenschap?

Ze bestaat wel, maar de waarheid is onbereikbaar (en misschien ook wel niet te vatten in een model).


  • wat zou Kant gezegd hebben?

Kant zou de splitsing tussen de werkelijkheid en onze modellen daarvan nog een stap verder doorgevoerd hebben. Kant geloofde namelijk dat er achter de werkelijkheid zoals wij die waarnemen nog een diepere werkelijkheid ligt die voor ons onkenbaar is (hij noemde dat “das Ding an sich”) omdat wij nu eenmaal alleen maar kennis hebben van de werkelijkheid zoals wij die waarnemen.


  • wat zou Wittgenstein gezegd hebben?

Wittgenstein (in zijn vroege werk; de tractatus) zou zich kunnen vinden in ons idee van wat feiten zijn. Ook zou hij zeggen dat de verdere stap van Kant te ver gaat: je moet het niet hebben over onkenbare dingen. “Waarover men niet spreken kan, daarover moet men zwijgen” schreef hij.


Posted in Epistemology, Philosophy of Mathematics, Philosophy of Physics | Tagged , , , , | 4 Comments

Paraclimbing Edinburgh

edinb sit

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!

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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.

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Posted in Philosophy of Mathematics | 2 Comments

Paraclimbing in Imst

Imst site

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?!

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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 Russell 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.”

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PhD Dissertation

I have defended my PhD dissertation on 13/07/2017 in Utrecht. Below you can find the documents which have kept me busy for several years:


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



Posted in Philosophy of Mathematics, Philosophy of Physics, Probability | Tagged , , , | 5 Comments