1 2 3… Infinity!

Zoals eerder aangekondigd heb ik op 17 februari meegedaan aan de “YouReCa Challenge 2016”, een Science Slam georganiseerd door de KU Leuven in het Depot. Er waren vijf presentaties in het Engels over wetenschappelijke onderwerpen, maar op zo’n manier gebracht dat het ook voor niet-wetenschappers te volgen was. Het was tegelijk ook een wedstrijd, met een vier-koppige jury en een publieksprijs. Beide hoofdprijzen werden gewonnen door Pieter Thyssen, met zijn presentatie over tijdreizen.

Op deze website vind je foto’s van de avond en onderstaande video geeft een sfeerverslag met fragmenten van alle presentaties.

Opnames van alle presentaties staan nu ook online:

Mijn bijdrage was een presentatie van 8 minuten over de vraag of we oneindig kunnen tellen. Ik heb de videoregistratie van mijn presentatie aangevuld met de slides. Aangezien mijn Engelse dictie te wensen overlaat, heb ik er op YouTube nu ook Engelstalige ondertitels aan toegevoegd. ;-) (Die moet je wel nog zelf aanzetten door onderaan rechts in de video op het rechthoekige symbool voor Subtitles/Ondertitels te klikken.)

Onder de vouw vind je de Engelstalige transcriptie met aanvullende informatie in voetnoten.

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“1 2 3… Infinity!”(1)

One, two, many, many-one, many-two, many-many.(2)

When children learn to count, at some point, they know one and two, but anything more than that is just ‘many’. We may smile at this, but as adults we have a similar relationship with infinity.(3) It is perplexing.

But don’t worry, I can explain it to you in just two words. Infinite means… Not Finite. That’s all it means! (Yeah.) So, we have this mindboggling concept of bigger than anything we can imagine. We could have called it Ultra-Mega-thingamajig,(4) but the scientists were like “Nah, we’re just gonna call it: not finite.”

That’s so frustrating: they don’t even tell us what it is, only what it’s not. But, it’s also brilliant, because it leaves room for our imagination to fill in the details. It’s like UFOs, unidentified flying objects are way more interesting than identified ones, or, you know: birds, planes, Mega Mindy.(5) Likewise with infinity: it has inspired generations of mathematicians, philosophers, artists.(6)

So, now, do we know anything more about infinity than that it is not finite? My question tonight is: Can we count infinity? Yeah, some of you laugh, but I know some of you are thinking: “NO, ‘counting infinity’, is she barking mad?”

If you have this intuition, you are in good company. Galileo Galilei thought like this. He considered the natural numbers: one, two, three, and so on. There are infinitely many finite numbers. That’s already borderline paradoxical! But Galilei went on; he thought also about the even numbers: two, four, six, and so on.(7) There are infinitely many of those too. So can we compare them?(8)

On the one hand, Galilei said, clearly there are equally many even numbers and natural numbers. There is a one-to-one correspondence between them: one corresponds to two, two corresponds to four, three corresponds to six, and so on.(9) It’s like stepping on a bus where exactly all seats are taken. You don’t need to count the passengers, to know that there are equally many passengers as there are seats.

Okay, but on the other hand, Galilei said, clearly there are fewer even numbers than there are natural numbers. The even numbers are just a part of the natural numbers.(10) It’s like stepping on a bus full of teenagers, with their big, odd bags on the seat next to them. You don’t need to count the passengers, to know that there are fewer passengers than seats.

But now, two sizes cannot be at the same time the same and different. That is a paradox, from which Galilei concluded that we cannot measure or compare infinite sets. They are all ‘many’.

Maybe some of you think that there really are equally many even numbers as natural numbers. Or maybe this is what you have been taught in mathematics class. If you have this intuition, then you are in good company. The German mathematician Cantor developed a theory for this. If you want to be on team Cantor, I have to tell you the secret knock to get into the club house. It goes like this: one, two, … No, wait, this is going to take forever! I’d better tell you the password. It is ‘cardinality’. It is the word Cantor used for size – the notion of size that takes this one-to-one correspondence to be decisive. So the even numbers have the same cardinality as the natural numbers. But Cantor also showed that there do exist larger sets! The real numbers, for instance, have a larger cardinality than the natural numbers.(11) Since then, mathematicians have accepted that some infinities are bigger than others. Cardinality says there is many and there’s many-many.

But maybe some of you still think that there really are fewer even numbers than there are natural numbers. If you have this intuition, then you are in good company. I have this intuition too, but more importantly: Vieri Benci, a mathematician who works in Pisa, just like Galilei did, developed a theory for doing this.(12) If you want to be on team Benci, I’ll tell you the password: it is ‘numerosity’. Numerosity is Benci’s word for size, which takes the fact that the even numbers are just a part of the natural numbers to be decisive. Numerosity agrees with cardinality that there is many and there is many-many, but in between there is many-one, many-two, and so on.

Now, the paradox is avoided because we use two words for size to pick out two different aspects of size. Cardinality is more coarse-grained, whereas numerosity is more fine-grained.(13)

Oh, all this thinking about infinity… It reminds me of shadow boxing. Shadow boxers do not fear that they will be attacked by an invisible opponent. They just want to be trained for when they meet a real-world attacker. Likewise, thinking about infinity is a great exercise: in thinking straight about confusing concepts, for dodging the paradoxes and aiming at clarity.

This is useful in science. Just think about the visible Universe: it contains many many galaxies, each of which contains many, many stars. These are huge, but finite numbers. Yet, some physical models tell us that outside the visible part the Universe is truely infinite. And we know that it is expanding. So, it’s already infinite and getting even bigger?! Maybe it will continue to do so for eternity.

If we are part of this infinity getting infinitely bigger, we’d better start shadow boxing.

One, two, many, many-one, many-two, many-many.(14)

~

Footnotes

  1. The title “One Two Three… Infinity: Facts and Speculations of Science” is also the title of a 1947 popular science book by George Gamow. (I’ve never read it.)
  2. Inspired by the way the trolls count in Discworld books by Terry Pratchett – simplified a bit for the current purpose and resonating with the myth of indiginous tribes counting “one, two, many” (see also xkcd).
  3. There is a lot of literature on the relation between ‘indefinitely large’ and ‘infinite’. I wrote a short article related to this topic in the context of probability: “Ultralarge and infinite loteries” (part of my thesis and published in a proceedings volume).
  4. Thingamajig is the same as whatchamecallit. ;-)
  5. Mega Mindy” is the title of a Flemish TV series for kids, referring to the main character who is a superheroine. When I showed this Science Slam video to my son (who does not speak English), he only understood this word, and kept asking during the rest of it when I would be saying “Mega Mindy” again. ;-D
  6. Yes, this is an image by M.C. Escher: “Circle limit I“.
  7. The original version of Galileo’s paradox uses the perfect squares (1, 4, 9, 16, and so on), but for ease of presentation I used the even numbers instead.
  8. The theories that I discuss in this presentation are about assigning sizes to sets of numbers. In this presentation I use loose talk, so I say things like “counting the natural numbers”, rather than “assigning a measure to the set of natural numbers”.
  9. One-to-one correspondence or bijection requires there to be exactly one even number for each natural number and vice versa.
  10. Our coach advised me not to use the term “proper subset”, so I used “just a part” instead.
  11. Obviously, there was no time to illustrate Cantor’s diagonal argument. Still, maybe I could have added that there are infinitely many bigger infinities. ;-)
  12. Vieri Benci is a contemporary mathematician that I really admire. I applied his numerosity theory (article together with M. Di Nasso) to probability theory (in “Fair infinite lotteries“, together with L. Horsten) and later we co-authored on this topic (“Non-Archimedean probability“, arXiv-version here; there is another article accepted but not online yet in BJPS).
  13. Yes, it is more complicated than this. Although I depict cardinality theory and numerosity theory as compatible, some people do not like numerosity theory at all. For a really great article that discusses some of the history (including Galileo’s paradox) in an accessible way, check out Paolo Mancosu’s 2009 paper “Measuring the size of infinite collections of natural numbers: was Cantor’s definition of infinite number inevitable?
  14. Ending where we started. Our coach taught us that in Dutch this is called a “tangconstructie”. Literally this translates to “pincer construction”, but I do not know whether there is an English term for it at all.

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2 Reacties

  1. simongelten

    Mooie presentatie, zie die nu pas voor de eerste keer. Kwam hier terecht via je tekst over Vieri Benci. Je dictie valt best mee hoor. Ik ga eens wat van deze Benci lezen. Zijn theorieën interesseren me wel (en het is waarschijnlijk goed voor mijn Italiaans. Als middelbare scholier werd ik al gekweld door de vraag of de verzameling hele getallen groter was dan de deelverzameling even getallen, en hoe dat te combineren was met het idee van oneindigheid.

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  2. Sylvia Wenmackers (Auteur bericht)

    Over dat laatste aspect raad ik echt het artikel van Mancosu aan (zie eindnoot 13). Maar bronnen lezen is natuurlijk nog leuker. Veel plezier!

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