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

Thats how im here too lol

Beautiful

@moi2833 yeah same

And it was also my first!

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 :'(

It inspired me to drop out of high-school

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bus accidents?

@@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.

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.

🤦

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

@@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.

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

i dont think you get what a podcast is lol

@@wynautvideos4263 i asked that question before. @gidean_karp

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

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

לגמרי! הבנתי בדיוק למה הוא התכוון.

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

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.

​@@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

@@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 RU-vid 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.

@@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".

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

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

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)

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.

@@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.

@@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.

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.

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

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.

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

How butthurt is this guy.