Making Sense of Infinity (with Asaf Karagila) - Numberphile Podcast

From high school drop-out to set theorist, Asaf Karagila shares his journey towards infinity.
Asaf is a UKRI Future Leaders Fellow at the University of East Anglia.
Asaf's blog - karagila.org
Asaf's Twitter - / asafkaragila
Some Infinity stuff on Numberphile - • Infinity on Numberphile
You can support Numberphile on Patreon - / numberphile ) like these people - www.numberphil...
With thanks to MSRI - www.msri.org
And with thanks to UKRI Future Leaders Fellowship - MR/T021705/1
PODCAST TRANSCRIPTS: www.numberphil...

[@diribigal]

It's the Math Stackexchange guy!

[@mksybr]

Thats how im here too lol

[@littleM9779]

I took two (and a half) classes with Asaf on Set Theory when he was in Jerusalem . Hands down the best teacher I've ever had, now I'll never know if Set Theory is actually the most fascinating area of mathematics, or if it was just the way Asaf taught it, making it feel like we were talking in the most concrete possible way about the most abstract possible subjects. I continued to do more "applied maths" in Computer Science and Biology, but to this day every few months I'll pull out his lecture notes, read a few sections and just savor the feeling of holding a model of Set Theory in my head for a few days.

[@mihailmilev9909]

Beautiful

[@mihailmilev9909]

@moi2833 yeah same

[@xyz.ijk.]

IMO, this was your best published podcast. So well done!

[@mihailmilev9909]

And it was also my first!

[@ty6339]

43:00 He may not give lectures, but exchanges stacks a lot. Even better.

[@fedesartorio]

Fantastic podcast! Brady’s questions were always on point and the whole thing was a very interesting journey.

[@LeDabe]

1 hour of mario cart video (with a podcast in the background)

[@davis.yoshida]

I'll give it a listen even though he once closed my stackexchange question as a duplicate :(

[@pepepepe5802]

Damn! If someone does that to my questions, I cry and shii bro that must've hurt your feelings. I hope you're doing well :'(

[@icew0lf98]

thanks for this broadcast, as it encourages me into my somewhat rebelious path to pure math phd

[@hansolo6831]

It inspired me to drop out of high-school

[@krumpy8259]

It would be cool if he makes Videos on KZbin to teach us set theory

[@wobblysauce]

Yep, you don't need to have qualifications, you be good at a thing, and others can have all the qualifications and still not get it. Most schools are set on getting fixed money per student rather than teaching the students.

[@kingpin1199]

i know this guy from stack exchange

[@jelllogrl]

As a high school drop out in computer science/theory/academia it's nice to hear from others with a similar situation!

[@hfelippejr]

I remember attending a seminar on Flow theory back in December 2019 -- its first-ever seminar, if I'm not mistaken. Nice to hear of the theory's developments.

[@antman7673]

Cool dude. I like his takes.

[@TheoEvian]

I am a recent japanese literature Phd graduate. I am now dealing with fellowships and postdoc positions and I feel very stressed (it feels like that just putting my project together is super hard) and so I am glad that other people had similar experiences :D

[@rmsgrey]

While I agree with Asaf's point that (from a purely pedagogical standpoint) you can get better value by writing the exam after you know what you've actually taught the students, letting you pick examples that engage the specific students, there is also a purely pedagogical argument for writing the exam first, so that you can check you're covering all the points you want to ensure students pick up from the course, and then tailoring the course to the exam. There's also the third option of writing an exam before the course starts and then revising it later to reflect what happens during the course - in which case the first draft can be very similar to the previous year's exam (assuming it's not an entirely new course). In general, different teachers are going to find different approaches - exam first or exam last - more useful and productive for them. Of course, the potential for industrial action and/or bus accidents is a purely pragmatic argument in favour of having an exam on file, even if it's not necessarily the final form of the exam.

[@NightmareAndNinja]

⁹9⁹⁹⁹⁹⁹

[@mihailmilev9909]

bus accidents?

[@rmsgrey]

@@mihailmilev9909 As in being run over by one. There's a concept coming out of software development, which Wikipedia calls the "bus factor", which refers to the minimum number of people that need to be lost in the same bus crash (or otherwise become unavailable to continue contributing to the project) to make it impossible to continue development. A non-software example is the recipe for coca cola, which mythically is known by only two people (the current official position of the company is that it's only known to "a few") - and legend has it that those two are not allowed to travel on the same plane to avoid accidents... In the case of a single teacher both teaching and examining a course, if anything happens to them with no exam (and mark-scheme) on file, it's then much harder (if it's even possible) for someone to take over the course.

[@AnimusInvidious]

The implicit assumption that dropouts can't be academic is so facepalmy.

[@nicholaskarras2759]

Being a dropout at some level is indicative of someone who has directly gone against the schooling system in pursuit of other things, or that the school system has failed them and they've been prevented from going down that path further, it makes logical sense that most people who drop out wouldn't want to, or maybe even consider academia, and then there is the fact beinf a dropout is often times prohibitive to actually getting into further education depending on where you live.

[@AnimusInvidious]

🤦

[@heyho4488]

i think i know that advisor he talked about. Martin did the same thing to me ^^

[@frankharr9466]

That's a twisty road. And yes, it's very dispiriting when you're wrong all the time. It's worse when you later find out that you were very close to being right.

[@hauntedmasc]

oof that background is motion-sickness land

[@rafaelazo75]

Pure maths is just applied set theory, however set theory is amazing.

[@ez_is_bloo]

This was so damn helpful

[@MagicGonads]

how could you have watched it if you posted that 1 minute after upload?

[@ez_is_bloo]

@@MagicGonads I haven't watched in full it but I needed a video explaining set theory to me. I havr a playlist for these and there's finally another numberphile video about it. My most trusted source I can never get enough because the Wikipedia articles are so hard to grasp.

[@ez_is_bloo]

It's also more casual and a peek at a set theorists life

[@gigagerard]

Story starts at 47 minutes.

[@michaelempeigne3519]

I would like to see video instead of lines going up and down on screen

[@wynautvideos4263]

i dont think you get what a podcast is lol

[@michaelempeigne3519]

@@wynautvideos4263 i asked that question before. @gidean_karp

[@michaelempeigne3519]

@@wynautvideos4263 i asked that question before and got no sufficient answer.

[@RobinDSaunders]

@@michaelempeigne3519 Podcasts are designed to be listened to, not watched. The decision to put it on KZbin is just for convenience.

[@whitb62]

I wish the interviewer actually asked him more questions about Set Theory and Infinity…

[@OriAriel94]

בני גורן המנייק

[@EladLerner]

לגמרי! הבנתי בדיוק למה הוא התכוון.

[@ozargaman6148]

אחי לפני כמה ימים גיליתי שהמורה שלי לפיזיקה כותב שאלות לבני גורן

[@ozargaman6148]

אחי מה ישראל זה ממש לא מדבר אלא אם כן אתה גר באילת

[@vturiserra]

Are telling the story of your whole life and making sense of infinity the same? The title of this video is deceptive.

[@zaonpt]

meth or math?

[@guest_informant]

For me the problem comes with claims such as, (A) 1+1/2+1/4+ ... = 2. I'd say that that equals sign is different in an important way from the equals sign in (B) 1 + 1 = 2. You can define that equality in that way, but let's distinguish between (A) and (B). Maybe people get worked up about infinity because we are told that, "1+1/2+1/4+ ... *equals* 2" rather than "We are going to re-define or extend the definition of the equals sign such that 1+1/2+1/4+ ... *is equal by definition to* 2." In some sense that's a much weaker claim. It's not an insight into the nature of reality which is how it's often presented (though, granted, it might turn out to be) - it's just a definition. Related: I think Aristotle distinguished between potential and actual infinities. 1,2,3,... is a potential infinity. When you start asserting that you are going to work with the cardinality of that infinite set then you are working with an "actual" infinity.

[@billy5030]

isn't it more just 1 + 1/2 + 1/4 + ... is another way of writing the limit of an infinite sum, rather than re-defining the equals sign? I don't think the equals sign is re-defined or extended in definition at all.

[@guest_informant]

​@@billy5030 I'd say that one apple plus one apple equals two apples is a reasonable and regular use of the equals sign. That's how we grow up understanding and using it*. As soon as you apply it to infinite sums I'd say you're re-defining it. It might be some sort of logical extension of the idea of equality, but I think you can make strong case that it's not the same. It's not at all clear to me that it's even possible to carry out an infinite sum, you can define the outcome, but it is a definition. It seems to me there's plenty to think about here, and simply asserting that 1+1/2+1/4+ ... *equals* 2 hides a lot of what is important and interesting. eg (from Wikipedia) The objections to Cantor's work were occasionally fierce: Leopold Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth". Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory", which he dismissed as "utter nonsense" that is "laughable" and "wrong". On the other hand: Hilbert, "No one shall expel us from the paradise that Cantor has created." Take your pick :-) I've spent a while trying to get to the bottom of why people react so strongly _emotionally_ to statements about infinity. My best guess is that it's Opinion or Definition treated as Knowledge, Certainty, and Truth. (Put like that it sounds quasi-religious so no wonder people get worked up.) I'm definitely anti-infinity. Apparently there's a school, the Constructivists, I'm trying to figure out how I get in. *Though even that is not totally clear eg (A) 6+4=10 and (B) 6+4 = 5x2 In (A) the = sign means "carry out this calculation and write down the answer", as I understand it this is how most students first encounter and understand "=" ie as an instruction (B) is perhaps what true equality means, often really confuses students when they first encounter it

[@MartinPoulter]

@@guest_informant You're drawing a distinction without a difference. The sense in which the infinite sum equals 2 is the same sense in which 6+2 = 5x2. You're right that it should be read as a declarative statement of equality rather than an imperative - that's what "equation" means. Note that you can do the infinite sum without invoking an infinite quantity. Yes, people reacted emotionally to Cantor's proof and they still react emotionally today in KZbin comments: that tells you something about human psychology, not about maths. If one side has a deductive proof and the other has insults, and you're thinking the side with insults must be right, something's gone wrong.

[@RaymondBaker]

@@guest_informant the constructivist still do things with with infinite sequences and series; it is still possible to build up real analysis in constructivism. See "Real Analysis: A Constructive Approach Through Interval Arithmetic".

[@guest_informant]

@@RaymondBaker I'm disappointed. Is there a hardcore fundamentalist wing of the Constructivists?

[@bazsnell3178]

1+2+3+4....+ many numbers as big as you want to to go with infinite time available OBVIOUSLY = infinity. 1+half+quarter+eighth+ sixteenth, and so on, does NOT equal 2 and never will do. Agreed, it seems to approach the sum as 2 in the long run, but it doesn't get there.There's always that infinitesimal little bit that is lacking from the goal of 2. So 'equals' means 'is EXACTLY the same as', and any other definitions of 'equals' are merely false.

[@Lexivor]

If you do it an infinite number of times it is exactly 2 though.

[@rmsgrey]

The magic trick is hidden in the ellipsis - the "..." at the end of "1+1/2+1/4+...". If you accept that as meaning "take the value after an infinite number of steps" or "take the limit that the partial sums converge to", then "1+1/2+1/4+...=2" is an exact equality. It's only if you insist both that the "..." means "stop after some finite number of steps" and that equality has to mean precise equality that you get a problem - insisting equality must be exact doesn't cause any issues on its own (and is the standard interpretation)

[@MrLove8736]

Great point that a lot of people miss about that second sequence. If we are dealing with actual infinity and not just potential infinity, to get to 2, if you do 1+ 1/2+1/4+1/8.. etc to 1/∞, that will get you to 2- (1/∞).. you have to do 1+ 1/2+1/4+1/8.. to +4/∞ + 2/∞ + 1/∞+ 1/∞ (with two infinitesimals at the end). Similarly, the inverse sequence 1+2+4+8+16.. etc, for the same amount of terms as the one that is just short of 2, will not get to exactly infinity, instead it will be ∞-1. You can see this an indication of this in the summation "S0?1-r" where S+ 1/1-2=1/-1 = -1. From this, we can deduce that infinity is even, a power of 2, and that infinity -1 is odd and a multiple of 5.

[@MrLove8736]

@@rmsgrey I think Baz's point is that the sequence can never converge to exactly 2, by definition there is always a little piece missing no matter how many times you do it. The sequence will always add up to 1+ (n-1/n). If we accept actual infinity as a definite limit that is not just some ambiguous amount that changes depending on context, ∞-1/∞ cannot be exactly 1. It is infinitesimally smaller than 1. For those who don't see this, do it with smaller fractions and add them up each time: 1+1/2 +1/2 =2. 1 +1/2 +1/4 +1/4 = 2. 1 + 1/2 + 1/4 +1/8 +1/8 = 2. Etc. The last fraction always appears twice for it to equal exactly 2.

[@rmsgrey]

@@MrLove8736 Your arguments only work if there is a last term of the sequence, and it's something other than 0. But if the last term is anything other than 0, then you can halve it and get the next term. Instead, we generally define the result of an infinite process as being equal to the value it converges on (its limit) if there is such a value, and that value is well-defined. So the result of repeatedly halving 1 is the value 0 (you keep getting closer to 0 every time you halve, you can get as close as you like to 0 just by halving enough times, and 0 is the only value you can halve without changing) and the result of adding up all those numbers is 2, and the amount by which it's less than 2 is 0.

[@Arthur0000100]

downvoted before listening. enough with "infinities". Please just provide at least semi-original content. EDIT: listened through it. sorry this is pretty much about nothing. even worse. will consider unsubscribing if this continues

[@RobinDSaunders]

It's about the actual work that mathematicians do, which mathematics as a subject cannot be separated from. Many, perhaps most, people are more strongly drawn to a subject by learning about what it looks like "from the inside": the human aspects. As for infinities: whatever your philosophical stance, the role that concepts of infinity have played in mathematical discovery and the development of our understanding cannot be overstated. This ties in with what I said above: mathematical discovery and understanding are processes carried out by human beings, and what separates it from arbitrary manipulation of symbols is the meaning we are able to assign to those symbols, the conceptual understandings we develop alongside them. It may be tempting to imagine some Platonic ideal notion of mathematics existing in isolation from the subject's human aspect - and certainly reality, with its heavily mathematical character, exists whether or not we are around to study it. But the subject of mathematics itself, the structure we assign to it, our evolving understanding of how and why certain concepts are more important and meaningful, and others less so - all of that is our creation.

[@gigagerard]

We have an interviewer who laughs because he doesn't understand the simplest maths. If we go that road it is all lost.

[@pmcate2]

What did you expect? These podcasts are about the mathematicians more so than their work. If this continues? I'm curious as to what content you are specifically referring to, because this channel has always been solid.

[@sumansharma8990]

Infinity is wonderful and when you say “enough with ‘infinities’” it makes me think you already know everything about infinity. Reality check: you don’t

[@adizmal]

How butthurt is this guy.