Het ideale kerstcadeau!

dit boek is het ideale cadeau voor veel lees- en denkplezier onder de kerstboom!
Uitgeverij Prometheus, 2020

Met voorwoord van Nobelprijswinnaar Gerard ‘t Hooft

Bestellen? klik hier!

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Reign of Error: Chapter Two

Read Chapter One of ‘Reign of Error’

Our cleaning droid, CD-2, is crashing on a blue-greenish planet, but what is happening on the planet below? Read on to find out!

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Reign of Error: Chapter One

Read the prologue to ‘Reign of Error’

“…It all started when the first manned interstellar mission ever recorded was carried out by the society that had produced me. Ever recorded, because scientists had been unable to determine whether the feat had been accomplished by one of the earlier civilizations that had dominated our world. What scientists also had failed to determine seemed to be the last obstacle between us and the stars. No astronaut could be found that was crazy enough to boldly dash into the infinite depths of a universe of which ninety-five percent was missing! They sugarcoated their ignorance by using the euphemism ‘dark matter’ to describe the missing stuff. Finally, after weeks of campaigning, the Space Corporation found, in the Asylum for Depressed Veterans of Cybernetic Warfare, an entire squad of lunatics that was ready to do the job. And they found me. That’s where I learned to play chess.

Space is not all it’s cracked up to be. When the initial enthusiasm about the launch had subsided, interstellar space-travel turned out to be rather boring. Especially for a cleaning droid. In fact, I remember quietly wishing that our ship would run into a dust-storm every now and then. What I didn’t know was that my life was about to become a whole lot more interesting. But at that time there was little more to do than play chess with the crew. The excitement started when our ship collided with an asteroid whose course had been diverted by a sudden solar storm.

At first it looked like we hadn’t taken too much damage. Some bruised crewmen, a few scratches on the hull, that seemed to be all. But it wasn’t all. Two days later our artificial gravity suddenly failed. Guess who had to clean up the mess it made! But that was not the end of it. The following day, just when we passed this blue-greenish, medium-sized planet, our engines began to falter. The captain decided to try to land the ship, so we started a descent into the planet’s atmosphere.

This seemed like a good idea, but with our engines crippled there was no way to counter the planet’s gravitational pull. We rushed through a thick layer of clouds, and ground zero was closing in fast. At last – the lands below were beginning to take shape – someone came up with the idea to throw overboard unnecessary ballast. The thought at first seemed reasonable, until I realized that their favorite chessplaying-mate was to be cast away along with some redundant furniture. Before I could warn them of having to operate the vacuum cleaner themselves, those brutes had already thrown me overboard like a lifeless chunk of metal. Indignation turned into panic when I saw the image of the ship shrink to a miniature as I fell faster and faster towards the surface below…”

Read more about CD-2 in next week’s episode of ‘The reign of Error’. Don’t forget to subscribe to the updates on this blog so that you receive the next episode automatically! 

[fill in your email-address in the box at the upper-right corner of the page (below ‘categories’)]

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The reign of Error: prologue


This summer I’ll continue writing my short story series ‘Alice in Numberland’, but before I do, I’d like to share with you a SF-like story I wrote several years ago. What motivated me to write it was, believe it or not, Edward Gibbon’s ‘Decline and Fall of the Roman Empire’. Gibbon’s work is pretty far from a short story (six volumes; 3500 pages), but it made me wonder: how can it be that one of the greatest empires in the world was ruled by such lunatics? Take Caligula, who made his favourite horse senator; or Nero, who set Rome on fire. Or that time when Commodus was killed by his own bodyguards who then sold the throne to the highest bidder – does it even matter who’s in charge? This blogpost contains the prologue to the story… enjoy!

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Coping with covid II: English translation

I promised that I would write another blogpost (click here for the first) about how I cope with the whole quarantine situation, so here it is. The second part of my strategy is to do a lot of writing. That works very well for me, because it makes me feel less alone, as if I am interacting with those who read my texts.

But this strategy is not as easy as it seems. The moments I need company the most, the moments I am most lonely, are also the moments at which it is most difficult to put myself to work, to actually start writing. So this strategy only works in combination with something else: I need to plan my writing activity. On the evening before a writing day, before I go to bed, I try to come up with an idea about what I want to start writing about the next morning. Only that makes it possible for me to start writing even when I am feeling lonely or sad.  

In the past few days I coped with the quarantine by translating the first chapter of my book about relativity theory into English:

Down the rabbit hole

It is often said that the beginning of the 20th century is an era in which physics has become too complicated for ordinary mortals. We have a clear intuition for the ideas of Newton and his contemporaries (often called classical physics), in which gravity explains why stones fall and the earth moves around the sun. This intuition comes to a sudden end when relativity theory and the theory of quantum mechanics appear.

In Newton’s physics, space and time are abstract but simple concepts, that can be measured with clocks and measuring sticks. Space and time enable us to understand the world around us, because they make it possible to describe any kind of physical change. But in the 20th century we no longer know which measuring sticks are straight and which clocks are synchronous, so it has become a challenge to understand what space and time are, which makes the transition from Newton’s to Einstein’s worldview seem like a radical transformation.  In this book I will show that the transition from classical to modern physics is not as abrupt as it is often presented – Newton and Einstein are more alike than we think.

The classical Newtonian worldview is not as obvious or easy-to-understand as is usually assumed, because much interpretation is needed to get from Newton’s physics (his mathematical equations) to a coherent view of what ‘reality’ is like. When we try to find out what Newton’s formulas tell us about the world outside ourselves, we will see that the concepts that lie at the foundations of classical physics are the same as those on which early 20th century physics is built.

Gerard ‘t Hooft and Alice in Wonderland

A couple of years ago I was at a physics conference with a philosophical bend, a conference about the foundations of spacetime theories. During one of the coffee breaks I had a chat with the keynote speaker of the conference, the Nobel laureate Professor Gerard ‘t Hooft. We agreed on many things – the location of the conference, Varna, Bulgaria, was great, and the weather was perfect. Then he said something that really surprised me: he was not there for the philosophy of space and time. “Then what are you doing here?!” I asked him full of surprise. “I have a new theoretical toy-model for black holes, and I want to discuss that”, he said, shrugging his shoulders.

            Gerard (we have become very good friends since then) is not the only physicist for whom the philosophy of space and time, and philosophy in general, is not the primary reason to visit a conference on physics. I find that difficult to grasp. Why are we interested in physics? Of course, we want technological advance, so we want to know how we can make new discoveries and which experiments are necessary for that, but we also want to find out something about the world – We want to understand the reality that exists independently of us and our experiments.  

            Physics gives us a model of reality outside of us, but that model does not say of itself whether it is a good model. It’s as if you’re trying to check a calculation that you made with a calculator by using the calculator itself. If the calculator made a mistake the first time, for example because of a loose key or because something went wrong in the factory when the calculator was manufactured, then probably the calculator will make the same mistake when checking the calculation. The physicist who wants to check whether their model of reality is a good model, is also checking their own calculations, just as the calculator.

The plight of the physicist is comparable to that of Alice in Wonderland. Alice wants to know whether she has grown after she drank from a small bottle and she tries to find out by holding her hand above her own head. That doesn’t work because she has no external point of reference, like a measuring stick. When physicists try to find out whether their model of reality is a good model, they are doing the same thing as Alice. They do not have an external point of reference, so the best they can do is hold their own hand above their heads.

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What is the philosophy of physics?

Physics and its philosophy

Physics is the search for a mathematical model which describes the phenomena around us. New physics usually starts with observing something that cannot be explained, after which a hypothesis is formulated which explains the observations. The physicist then tries to come up with experiments that show that the hypothesis is true or false, or should be modified.

What is the philosophy of physics?

There are many difficulties with the picture of physics that I just described. What counts as an observation? How can we ever justify a general hypothesis on the basis of a finite number of observations? What is the nature of mathematics? These are questions about the philosophy of physics. On my blog I’m going to address these and similar questions, but here I’d like to discuss just one example of a topic in the philosophy of physics – determinism.

Determinism

The idea of determinism is that if we know all that there is to know about a physical system at one point in time – the position and velocity of all particles in the system – and we know the laws that tell us how the system changes, then we can calculate what is going to happen. As an example, think of a coin toss. We usually say that the probability of heads and tails are ½ because we have no reason to think otherwise.

But a strict believer in determinism might say something else. If we know exactly the situation when the coin was tossed, the position and velocity of all particles at that moment, we can use the laws of physics to calculate the outcome of the coin toss. The probability of an outcome is then either one or zero – it either happens, or it doesn’t. A physical model is deterministic if complete knowledge of the initial situation allows us to predict the outcomes with certainty, while a model that yields uncertainty about the outcomes is called indeterministic.

Sub-quantum theory

It is often stated that, since there is uncertainty in quantum theory, it must be the case that our reality is indeterministic. That conclusion is not justified, since the uncertainty in the quantum theory means only that quantum theory is indeterministic, and not the reality that it describes. It could be that there is some underlying theory that describes a deterministic reality, of which quantum theory is only an approximation. We do not have such a theory yet, but a minority of physicists believe that we will find a deterministic ‘sub-quantum theory’ in the future (e.g. the Cellular Automata version of quantum theory of Gerard ‘t Hooft).   

Why should we care?

Why should we care about the philosophy of physics? If all we want is a better mathematical model, so that we can construct better cars and faster rockets, then why don’t we stick to physics itself? Why do we have to drag in philosophy?

Physics is the attempt to reshape our mathematical model so that it yields the best predictions, but questions about the model (or about different possible models) are philosophical questions. For example, when is a model a good model? When it is as accurate as possible, or when it is as broadly applicable as possible? Therefore, if it has to be decided where to invest money for scientific research, philosophical questions are important.

For me personally, there is a far more important reason: I’d like to understand reality as best as I can. Nothing can be known with absolute certainty, but the second best thing is that the philosophy of physics makes it possible for us to explore the limits of our own knowledge. Questions like “what is the nature of time and space?” and “what happened before the Big Bang?” are clearly connected to physics, but even questions involving our free will and the meaning of life are, in the end, questions in the philosophy of physics. Physics and philosophy are two sides of the same coin – both are applied logic.

[Thanks to Carlo Rovelli for proof-reading the text]

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Coping with Covid

Because the number of Covid infections increased rapidly, the Dutch government has decided that the Netherlands should go into a ‘partial lockdown’. I fully understand why this is necessary, but that doesn’t mean that it doesn’t bother me. I live alone, and although I don’t always show it, I really love company. What is also very important for me, is a well-structured day. What to do when social interaction is actively discouraged and I’m not allowed to go to work every day?

I have come up with several ways to deal with these things, and I have the idea (or hope?) that others might benefit from them as well. That’s why this blogpost will be the first in a series of posts about strategies to cope with the quarantine situation. I know that the situation is far worse in other places, but I hope that perhaps my strategies also work for others.

FIRST STRATEGY: GOING UP IN A FANTASY WORLD

Since my early childhood, I’ve been a bit of a fanboy of Alexander the Great. I had a poster of him in my bedroom and I knew the names of all the battles he had fought and cities he had conquered. My passion for Alexander has not become less over the years: I have visited many places where Alexander has once been, and in my bedroom there is now a map of his expedition.

In these times of quarantining, one of my ways to cope with the loneliness is curling myself up in a warm easychair and start reading a book about the history of Alexander the Great. So much is written about his expedition, that I never have any trouble finding a book about Alexander which I haven’t read yet (when I heard about the new semi-lockdown yesterday, I immediately mail-ordered W.W. Tarn’s 1948 classic ‘Alexander the Great’).

To optimise my Alexander-experience, I set my phone to ‘do not disturb’ for 60 minutes. I choose appropriate background music (The Kaiser Chiefs for Alexander’s youth; Wagner for Alexander’s return-journey), while I surround myself with dictionaries and (historical) atlasses that might assist me when Alexander’s expedition is under way. Sometimes I put a poster up with a detailed map of the territory involved, so that I can really follow the development of the story. All these things together really help me get my mind off things that are happening around me.

I think a similar strategy could work for ‘Lord of the Rings’, ‘Game of Thrones’, stories about ‘Donald Duck’ or ‘Mickey Mouse’ (combine this with Stravinsky!), or perhaps also with romantic literature (Tolstoy with Tchaikovski?). But be sure to think it through! The strategy only has effect if you are doing things in an unusual way. Try to make a real ritual out of it: move your armchair to the centre of the room every afternoon at four o’clock sharp, put on the music, turn off your phone, and dive into your fantasyworld for 60 minutes!

wrap up:

  • choose your fantasyworld
  • reposition armchair
  • set phone to ‘do not disturb’
  • choose suitable background music

Do you have another strategy to get through the corona-crisis? let me know by posting below!

[also: don’t forget to subscribe to this blog (fill in your emailaddress in upper right corner) to find out more about my coping strategies ;)]

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Verloren hoofdstuk: ‘Newton, Astronoom of Astroloog?’

Het is dan eindelijk zover: vanaf deze week ligt mijn boek “Op zoek naar de grenzen van de natuurkunde” in de boekhandel! Door het gedoe rondom corona zal er geen officiële boekpresentatie plaatsvinden, maar dat betekent natuurlijk niet dat de publicatie geruisloos voorbij gaat. Via Twitter, Instagram, Facebook en LinkedIn zullen jullie veel van me horen de komende tijd. Als voorproefje hieronder vast een ongepubliceerd hoofdstuk uit een eerdere versie van het boek.

Het hoofdstuk gaat over de natuurkunde van Isaac Newton, die wordt beschouwd als een van de grondleggers van de moderne wetenschap.

“Was Newton Astronoom of Astroloog?

Toen ik aan de universiteit van Utrecht sterrenkunde studeerde, vertelde onze docent over een foutje dat zijn secretaresse enkele jaren eerder gemaakt had. Op de doctorandesbul van een de studenten van onze docent stonden de woorden “doctorandus in de astrologie”, in plaats van de doctorandus in de astronomie.” Vanaf dat moment ging de grap rond dat er in heel Nederland maar één universiteit is waar je een doctorandus-titel in de astrologie kunt krijgen.

Waarom noemen we astronomie een wetenschap, terwijl we astrologie – de overtuiging dat de stand van de sterren invloed heeft op menselijk handelen –  beschouwen als pseudo-wetenschap? En hoe dacht Newton daarover? Laten we ons eerst richten op de eerste vraag: waarom is astrologie geen echte wetenschap? Het standaardantwoord is dat de uitspraken van astrologen vaak zó vaag en multi-interpretabel zijn dat ze niet met behulp van experimenten kunnen worden weerlegd.

Zwaartekracht in Newtons theorie is een beetje als de portemonnee die wordt weggetrokken in de klassieke ‘practical joke’ – waar is het touwtje en wie trekt eraan?

Uitspraken van astrologen zijn vaak vaag, terwijl de meeste astrologische uitspraken niet weerlegbaar zijn. ‘Vaak’, ‘de meeste’, …zijn er dan ook gevallen waarin astrologie wel wetenschap is? Er is een tweede reden waarom velen astrologie onwetenschappelijk vinden: het ontbreken van een mechanisme dat beschrijft hoe astrologie werkt. Als een astroloog bijvoorbeeld een verband ziet tussen de stand van de sterren tijdens iemands geboorte en het feit dat de geborene graag appels eet, dan is het niet duidelijk hoe de sterren invloed op de persoon in kwestie kunnen hebben gehad. Het is niet duidelijk hoe de correlatie tussen de sterrenstand en de geboorte een oorzakelijk verband met zich meebrengt. Deze kritiek op astrologie hangt samen met het begrip lokaliteit.

Lokaliteit

We zien om ons heen dat objecten invloed op elkaar kunnen uitoefenen als ze elkaar raken (zoals botsende biljartballen). Als we zien dat een object plotseling versnelt of vertraagt, dan gaan we ervan uit dat deze werking wordt veroorzaakt door iets dat het object raakt; we gaan ervan uit dat alle beïnvloeding lokaal (plaatselijk) is. Als er een afstand is tussen het object dat versnelt en het object dat deze versnelling veroorzaakt, dan gaan we ervan uit dat we iets missen van wat er gaande is. Astrologische invloed – de invloed van de stand van de sterren op menselijk handelen – is niet-lokaal, waardoor astrologie heel sterk het gevoel oproept dat we iets missen.

Gedegen wetenschap?

Laten we nu eens kijken naar Newtons werk. Iets dat vaak over Newton gezegd wordt is dat hij astroloog, alchemist en dogmatisch religieus was, maar desondanks gedegen wetenschappelijk werk verrichtte. Is dat wel zo? Zijn wet van de traagheid stelt dat een object waarop geen krachten werken voor altijd op dezelfde manier door zal bewegen (zonder wrijving zou een bal voor altijd verder blijven rollen). Maar hoe is dat te toetsen in een experiment? We moeten dan van een object waarop geen krachten werken nagaan of het steeds even snel blijft gaan, maar dat kan helemaal niet: Op ieder object werken altijd krachten (denk alleen maar eens aan de gravitatiekracht van degene die het object bestudeert)! De wet van de traagheid is niet te weerleggen, omdat de wet gaat over een situatie die we niet in een experiment kunnen nabootsen.

Naast de wet van de traagheid is de zwaartekracht een erg belangrijk onderdeel van Newtons theorieën. Newton zelf zegt dat zwaartekracht een werking-op-afstand veroorzaakt: de aarde en de maan trekken elkaar aan, terwijl er een grote afstand tussen de twee is. Ook dat is verre van wat we tegenwoordig gedegen wetenschap noemen. Net als bij de astrologie is de beïnvloeding niet lokaal, en dus krijgen we weer het gevoel dat we iets missen. Hoe kan Newtons zwaartekracht een niet-lokale werking hebben? Voor alle duidelijkheid: Newton zegt dat er een relatie is tussen de stand van de maan en vallende stenen hier op aarde (beide veroorzaakt door de zwaartekracht), en stelt dat een of andere occulte en mysterieuze kracht die relatie veroorzaakt. Hoe is dat anders dan astrologie?!

Als we Newtons theorieën bekijken in het licht van weerlegbaarheid en lokaliteit, dan komen we tot de verbazende conclusie dat Newton zich altijd met astrologie heeft beziggehouden. Hoe werd er door Newtons tijdgenoten over dit soort zaken gedacht?

Molière

De tijd van Newton (eind 16e en begin 17e eeuw) was de tijd van de Wetenschappelijke Revolutie. Het was een tijd van vernieuwing, niet alleen wat betreft de inhoud van de wetenschap (zoals de wetten van Newton), maar ook wat betreft de wetenschappelijke methode. De ideeën die we hadden over wat telt als echte wetenschappelijke kennis, en hoe we aan deze kennis kunnen komen, veranderden radicaal. Zo werd bijvoorbeeld de waarde die werd gehecht aan (reproduceerbare) experimenten erg groot (denk aan Galileï’s experimenten).

De toneelstukken van Jean-Baptiste Poquelin, beter bekend onder zijn artiestennaam Molière, illustreren deze veranderende tijdgeest. In zijn laatste toneelstuk, ‘Le Malade Imaginaire’ (‘De Ingebeelde Zieke’), steekt Molière de draak met doktoren van de oudere generatie (van vóór de Wetenschappelijke Revolutie), die de slaapverwekkende werking van een medicijn verklaren door te zeggen dat het medicijn een slaapverwekkende kracht, een ‘virtus dormitiva’, heeft. De onderliggende gedachte van Molière is dat doktoren met hun ‘slaapverwekkende kracht’ eigenlijk niets anders doen dan verdoezelen dat ze geen idee hebben waarom het medicijn werkt door een moeilijke Latijnse naam voor ‘slaapverwekkende werking’ te geven: ‘dormire’ is Latijn voor slapen en ‘virtus’ is het Latijnse woord voor kracht – zeggen dat een slaapwekkende werking wordt veroorzaakt door een ‘vitus dormitiva’ is dus allesbehalve een verklaring.

Had Molière aan Newton niet eenzelfde soort verwijt kunnen maken? Newton stelde dat alle massa’s elkaar aantrekken, en dat daarom massa’s ‘zwaar’ zijn. De aantrekkende werking wordt veroorzaakt door een onzichtbare ‘zwaartekracht’ waarover Newton niet verder wilde speculeren. Newton gebruikte het Latijnse woord voor zwaar (‘gravitas’) om zijn kracht te benoemen. Doet hij niet precies hetzelfde als de doktoren van Molière – het kiezen van een dure Latijnse naam om te verdoezelen dat hij geen idee heeft van waarom dingen gebeuren zoals ze gebeuren? …”

Wil je hier meer over lezen, of ben je benieuwd wat Einstein hierover te zeggen heeft?

Bestel en lees dan mijn boek door hier te klikken!

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“Op zoek naar de grenzen van de natuurkunde”

In mei gaat Prometheus mijn boek “Op zoek naar de grenzen van de natuurkunde” publiceren. Een mooie kans om al je zorgen even van je af te zetten en helemaal op te gaan in diepe gedachten over wetenschap en filosofie. Zet Netflix even op pauze, en kijk vast naar het lijstje hieronder, met de termen die in de index (‘zakenregister’) komen. De komende weken ga ik meer bloggen over mijn boek, dus abonneer je op mijn blog door je e-mailadres in te vullen op de startpagina (bij: Follow Blog via Email).

Alice in wonderland

Gerard ‘t Hooft

bewegingswetten

natuurwetten

model

werkelijkheid

rekenmachine

Narcissus

waarneming

subatomair deeltje

normale verdeling

gemiddelde

meetverstoring

cern

meetfout

higgsdeeltje

deeltjesversneller

standaardmodel

Newtons vallende appel

Einstein

Zwaartekracht, universele

Archimedes

Galileï, inquisitie

Stukeley

Relativiteitsbegrip

Relativiteit, Einstein

Relativiteit, Galileï

Grootheid, absoluut/relatief

Postulaat

Bewegingswetten van Newton

Waarnemer

Parabool

Galileï, schip

Constante snelheid

Versnelling

Traagheidskracht, ontstaan

Assyriërs

Atomen

Demokritos

Leukippos

Coördinatenstelsel

Functie

Coördinatenstelsel, oorsprong

substantivalisme

Ruimte, absolute

tijd, absolute

Snelheid, absolute

Ordening, absolute

Ruststelsel

Rust, absolute

Versnelling, absolute

Versnelling, relatieve

Traagheid

Impuls

Massa

Vector

Gewicht

Wetten van Newton

Traagheidswet

Wrijving

Krachtwet

Oneindig

Impulsbehoud, principe van

zwaartekrachtswet

werking-op-afstand

gravitatieconstante

G (zie: gravitatieconstante)

Cavendish

Jan Klaassenspel

Onderbepaaldheid

Celsius

Fahrenheit

sociaal construct

Latour, Bruno

Puntdeeltjes

Atomen, botsen/kruisen

Ontelbaar

Singulariteit

Hilbert, David

Leibniz, Gottfried Wilhelm

Informatie kopieren

Popper, Karl

Theoriegeladen

Lineaire samenhang tussen variabelen

Galileï, experiment Pisa

vrije val

equivalentie van zwaartekracht en traagheidskracht, bij Newton

energie, kinetische

energie, potentiële

hoogtekaart

veld

potentiaalveld

Lagrange, Joseph-Louis

Gps

kleinste werking, Het principe van de

mozaïek

Hawking, Stephen

Tijd als verandering

tijdsymmetrisch

Boltzmann, Ludwig

Clausius, Rudolf

Entropie

Wanorde

Tijdrichting

Evenwichtstoestand

Equilibrium, zie: Evenwichtstoestand

kans, entropie en

ruimte als relatie tussen objecten

standaardmaten

relativiteitstheorie, Einsteins algemene

relativiteitstheorie, Einsteins speciale

lengtecontractie

ruimtetijd

AltaVista

‘Over de elektrodynamica van bewegende lichamen’

Elektrodynamica

Veld, magnetisch

Veld, elektrisch

Geleider

magneet

Maxwell, wetten van

Kracht, elektromagnetische

gedachte-experiment

snelheid, ten opzichte van de absolute ruimte

snelheid, relatieve

lichtsnelheid, voor iedereen hetzelfde

lichtsnelheid, absolute

tijdsdilatatie

lichtklok

gamma (γ)

Lorentz, Hendrik Antoon

Synchronisatieprobleem

Gelijktijdigheid

Skype

Relativiteitspostulaat

Lichtpostulaat

Beschrijving/model

Mechanica, klassiek

Mechanica, van Newton (zie: Mechanica, klassiek)

Equivalentie van zwaartekracht en traagheidskracht, bij Einstein

Equivalentie, Einsteins principe van

Kuipers, André

kromming van ruimte en tijd

ruimtetijd

Eddington, Sir Arthur Stanley

Afbuiging, licht-

Foton

Langeafstandswerking (zie: werking-op-afstand)

Lokaal

Ligo

Zwaartekrachtsgolven

Zwart gat

Gauss, Carl Friedrich

180 graden-regel

Rovelli, Carlo

Scheermes, van Ockham

Cox, Brian

Maudlin, Tim

kwantumgravitatie

Kwantumtheorie

Unificatie

Wheeler-DeWitt-vergelijking

Tijd, als illusie

Snaartheorie

Laplace, Pierre-Simon

Kans, Laplace’s definitie

Kans, het principe van gelijke

Kans als relatieve frequentie van uitkomsten

Aristoteles

Elektronenmicroscoop

‘Metaphysica’

diepte zien

kracht, middelpuntvliedende

Franse revolutie

Dainton, Barry

Looney Tunes

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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!


Pythagoras (c. 570 – c. 495 BC)

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’)

Posted in AUC-Big Questions in Science, Philosophy of Mathematics, Philosophy of Physics | Tagged , , , , , , , , , | 4 Comments