The Infinitesimal Monad - Numberphile

[@EebstertheGreat]

"Monad" is one of those confusing words that has substantially different meanings in philosophy, programming, algebra, analysis, and category theory. Most of these definitions are barely related to each other.

[@igguks]

+EebstertheGreat Actually the ones in programming and category theory are the same thing. Or at least very closely related.

[@EebstertheGreat]

skuggi Those two are closely related, yes. They are also related to the monad of linear algebra, which can be shown to be a special case. They are however almost totally unrelated to the monads of non-standard analysis and philosophy.

[@2Cerealbox]

+EebstertheGreat Thanks for that. I was wondering what possible connection infinitesmals had to do with Haskell (the programming language, not the mathematician).

[@jasondoe2596]

Well, considering that "monad" is the Greek word for "unit", it somewhat makes sense that it would be meaningful in so varied contexts/domains :)

[@lamudri]

+Ryan N Interestingly, since Haskell is non-strict by default, the type Nat defined by 〈data Nat = Z | S Nat〉 is actually the type of co-natural numbers, and includes infinite element inf defined with 〈inf = S inf〉. One can do things like 〈take 6 $ take inf $ [0..]〉, which will work okay. The real natural numbers are defined by 〈data Nat = Z | S !Nat〉, with the ‘!’ for strictness. A thorough treatment of this is given in Agda, where the distinction between (finite) data and (possibly infinite) codata is important for proving that functions are total.

[@danielellis2874]

I like watching numberphile, most of the concepts i can't even begin to understand, but it feels like it resets my brain and that feels nice

[@yuridecastro9496]

Her face at the end is just priceless.

[@Yomama5923]

+Yuri de Castro I love the self-satisfied look of a mathematician because they know they just blew our minds.

[@ValDominator]

**rekt**

[@rtyuik7]

+Jonathan Gutsymon its got a lil bit of "yeah...chew on THAT"

[@DarkAngelEU]

+Jonathan Gutsymon I love the self-satisfied look of a scientist because they know they just blew our minds. ftfy ;) She reminds me of my math teacher, she always had that face and I believe she did because there's always something you're gonna discover and just shatter the belief you put in this system. Science is about proving you're wrong, infinitesimally

[@Yomama5923]

rtyuik7 LOL, oh my gosh, yes!

[@tomalator]

whenever i asked my math teachers about things like this, they just say "you cant do that." I feel like i have been lied to.

[@nicholasfazzolari3647]

+Thomas Becker That is precisely the problem with mathematics educators. It would be better to say "Go ahead, but you better be able to prove it.", or "Do you think that someone already proved that? Try to find out!". You know, create an environment where experimentation, learning and competition are welcome. After arithmetic and elementary algebra math starts to break the linear learning pattern. People educating pre-college students should assert that to their students.

[@JedaiFou]

+Thomas Becker I tell my students (high school) that ask those questions (not many of them unfortunately) that "you can extend the reals to have something like that but that creates lots of problems" and that we won't treat the subject in class so if they want to know more to look up the hyperreals on Internet. We often just don't have time to answer those questions in details, plus we risk creating confusion in most students which is damn hard to correct... It is generally better (in a pedagogy for the masses way) to present as simple and straightforward an explanation as possible even at the cost of some inexactitudes. (Which is why Maths teachers will tell you that "nothing" has a negative square before you're introduced to the complex, I try to use "no real numbers" but I'm sure this doesn't make much difference and that I'm guilty of the occasional overreach).

[@jameswhistler968]

+JedaiFou says, "I tell my students (high school) that ask those questions that 'you can extend the reals to have something like that but that creates lots of problems'..." Which problems are you thinking of?

[@JedaiFou]

James Whistler Of course, I'm not really thinking of mathematical problems, but rather that "just introducing" infinitesimals isn't an alternative to standard analysis, you have to introduce all hyperreals (infinitesimals, infinites and other hyperreals) and way to do calculus on them and with them... We may be able to replace standard analysis with non-standard but it's not like non-standard analysis is *that much* simpler than standard. Thus we would have to do a complete replacement and there's no way to introduce properly the hyperreals to high-schoolers in a few minutes.

[@jameswhistler968]

JediFou says, "it's not like non-standard analysis is that much simpler than standard." I disagree. You tell me how you use epsilons and deltas to prove that lim_{x --> 1} 1/x = 1, and I'll show you how to do it with infinitesimals. "and there's no way to introduce properly the hyperreals to high-schoolers in a few minutes." Why in the world should it only take a few minutes?

[@CrimsonWingBlackbird]

This is the best explanation of hyperreal numbers that I have seen. Perhaps a video series could be done on calculus, with both the limit definition and the hyperreal definition? That would be pretty interesting.

[@whatno5090]

For those curious - N* can be constructed as follows: Start with a special kind of set U of sets of natural numbers (i.e. members of U are themselves "properties" that natural numbers can have). U is called a free ultrafilter. Constructing it is a pretty big pain and is quite technical. Then, a "number" K in N* can be described as a function f : N -> N, where two such functions f and g are said to describe the same "number" if the set of all n such that f(n) = g(n) is a member of U. Similarly, f represents some number which is smaller than g if the set of n such that f(n) < g(n) is a member of U.

[@hockeynewfoundland]

At 0:46 the animation says "For any ∈ n N ..." when it should read For any n ∈ N ...".

[@numberphile]

+hockeynewfoundland mea culpa

[@rootofminusone]

noticed that

[@simorote]

+hockeynewfoundland actually they are the same logically speaking, although i will concede that it is an unusual notation.

[@borisjo13]

+simorote For any element of n N? That is just wrong.

[@Clashsoft0]

+borisjo13 x ∈ S is technically just infix notation for ∈(x, S) or ∈ x S in prefix notation.

[@matthewsnell3789]

I really like this lady and the way she presents things. Her pedagogical skills seem much better than some others that appear on this channel.

[@Yulenka-]

I love the attitude "I'm a model theorist, I can do whatever I want" ❤

[@villanelo1987]

I think she broke Brady. :p

[@juliaswallow8637]

i think i heared few times cracks from his brains... :)

[@arpyzero]

Oh, axiom of choice, you and your well-ordering!

[@Tassdo]

But this is not a well-ordering, is it? If I take the set {1/1, 1/2, 1/3,...} this does not have a smallest element, just like in the reals.

[@ПавлоСурженко]

@@Tassdo PR0 probably meant that too construct N* or R* mentioned in the video you need ultrafilter lemma. And its proof requires Zorn's lemma, which is indeed equivalent to axiom of choice and well-ordering theorem.

[@Orenotter]

An easy way to think of these... A line segment has measurable length. A 2-D shape is equivalent to infinity lines, but it has a measurable area. A 3-D form is equal to infinite 2-D shapes. It's area is infinity, but that infinity area equates to measurable volume. A tesseract, likewise, has infinite volume which equates to measurable hypervolume. This isn't just theoretical. It's part of everyday life.

[@RedInferno112]

+Orenotter Cancelling out dem infinities.

[@sciencoking]

Right in the monads...

[@OlivierFRscooter]

+Dennis W that's pretty clever

[@frtard]

+Dennis W I was just typing 'Kicking math in the monads' when I had this strange urge to scroll down...

[@PowerElemental]

There's a talk on KZbin called 'monads and gonads'. It's about a different kind of monad though.

[@zarikegouws5004]

thank you for the idee

[@mal2ksc]

Weeeeeeeeeee! Monads and strife.

[@slpk]

I feel like half of the people doing these numberphile videos should be locked away in a asylum or something. I'd be tempted to join them, though.

[@JarrettWilliams99]

+MichaelKingsfordGray underrated comment

[@thedoublehelix5661]

mathematicians am I right?

[@Triantalex]

??

[@FishKungfu]

Kth!

[@NiacinWaterTaffy]

+Fish Kungfu Kth + 1!

[@carn109]

N*th!

[@Smittel]

(L*sqrt(K)+1)th...

[@john_hunter_]

Knd

[@numberphile]

+Fish Kungfu well played

[@NikolajLepka]

Here I was hoping this would actually talk about monads... in the category theory sense of the word

[@garethdean6382]

Finally! It's near impossible to find a layman's discussion of monads out there. Also, 4th. (I missed out on that bronze medal by *this* much!)

[@VoltzLiveYT]

+Gareth Dean You're probably thinking of a completely different class of monads.

[@garethdean6382]

VoltzLiveYT No, and that's the problem. I was introduced to these through a convoluted path starting with a Sci-Fi book, but when younger me wanted more information all I got was programming and philosophy.

[@VoltzLiveYT]

Gareth Dean That is exactly the misconception I had assumed you had. Monads in FP are different from Category theory are different from number theory.

[@garethdean6382]

VoltzLiveYT It was headache-inducing when I didn't know there were mathematical and non-mathematical uses of the word. Here was this mindblowing concept that I wasn't quite sure about, but it was math of some kind. I rush off to Wikipedia, the only source of information back then... to find an article about creation theories. Had I got the name wrong or just seriously misunderstood that book? It was years before I encountered the concept again under another name. It's nice to have this video to point to if ever the subject comes up again.

[@MarioWenzel]

+VoltzLiveYT Actually, FP monads are the monads from Category theory

[@BlinkLed]

This has got to be my favorite area of mathematics. I love being able to measure infinitesimals and infinities, saying with certainty how much bigger one is than another.

[@BuFFoTheArtClown]

Wait... I thought the patriarchy was supposed to keep women OUT of STEM? How did this very intelligent woman slip through our fingers? We need to have a meeting about this immediately. Men, you know where to meet.... See you there! (This is a sarcastic statement)

[@littlebigphil]

+BuFFoTheArtClown To the patriarchy-mobile!

[@aimcfarl]

Yes let's pretend systematic patriarchy doesn't exist using anecdotal evidence and ignoring evidence from large scientific studies

[@jb76489]

aimcfarl apparently we can do that so long as said studies are disproving the wage gap

[@aimcfarl]

Which studies disprove the wage gap, it's generally found to be around $0.07 in like work and that increases as they move to higher positions plus the wage gap is one part of the problem, glass ceilings still exist as does a large amount of sexism and lack of support for parental care (which is a problem for both sexes but women get the brunt of it)

[@littlebigphil]

I'd like to point out that no one needs to prove that the wage gap isn't due to sexism. The burden of proof lies on those who claim it is due to sexism.

[@juliusreiner5733]

I was trying to think, how is there a numberphile video I haven’t seen. I started watching in 2013 and went back and saw them all and have watched them all since. Turns out this one was dropped on my first day of college, when I was enrolling as a pure math major in large part due to this channel!

[@djyotta]

Now I understand why it's so important to scale your infinitesimal correctly when integrating.

[@stumbling]

I love this stuff! I feel compelled to testify on behalf of mathematics and computer programming. It is programming that really got me interested in mathematics, and I would encourage everyone to learn programming as it provides an outlet for so many interesting and useful things. I was always fairly good at maths but it always felt like a chore at school and I had no inclination to study it in my free time, but then I started learning to code and quickly found my limitations. I couldn't reach into the monitor and mould the worlds I wished to create, I had to do so from afar, I had to learn mathematics to reach into this realm that alone I could not penetrate. This is the same limitation that the first astronomers must have felt, they could not reach the stars but for the aid of mathematics. Programming has shown me the power of mathematics and its true nature as a tool to achieve what otherwise would be impossible or incredibly laborious. I have not ventured very far into the world of mathematics but already I am amazed at what I have found, and with programming I am able to witness the effects with my own eyes.

[@metleon]

Did anyone else immediately go "No, that's where the square root of 3 goes."

[@Rukalin]

THIS IS THE MONAD'S POWAH!

[@UMosNyu]

+Linkaru Holy ... I am feeling it!

[@AlcomIsst]

+Linkaru *MONAD BOY!!*

[@AllHailZeppelin]

√2 is WAY too close to 2!

[@oz_jones]

*not to scale

[@Euquila]

2! = 2

[@Bratjuuc]

Who cares? It was not about precision in the first place

[@anshulagrawal633]

people who thinks while they simultaneously writes would have bad handwriting so co-operate.

[@shruggzdastr8-facedclown]

Yeah, it's closer to 2-1/K

[@MCPhssthpok]

I'd really like to see you do a video on Conway's surreal numbers.

[@Not_Biohazard]

what about k factorial?

[@jameswhistler968]

+Biohazard : K! is pretty cool because it has all the natural numbers (and more) as factors.

[@JorgetePanete]

*SPECIAL K*

[@johnvonhorn2942]

+Jorge C. M. Brilliant!

[@LittlePeng9]

I love prof's look at 6:44 while Brady seems to be processing what he just heard.

[@Argimak]

"It's getting tighter and tighter into that zero". If that doesn't sound dirty then I don't know what does.

[@Marceau.]

+Fel public toilets

[@ThomasGiles]

+Fel Sounds "naughty," in fact. ;P

[@mancheaseskrelpher8419]

+Thomas Giles **jaw drops**

[@photondance]

+Fel I have to admit, I divided by zero when I heard that; just a little.

[@jimi02468]

It doesn't sound dirty if you don't have a perverted mind.

[@tobiaszb]

RIP Conway with surreals. You can add them and multiply, so there is a monad near each real number. K + 1/K consider. Do we need aditional assumtions to have a monad arround 1/K?

[@lakshaymd]

"Because we're mathematicians" 😂

[@Blazagg]

Great video, I'm always interested when it comes to math becoming somewhat philosophical

[@NamelessHobo]

You should do some more videos with her. She's fun to watch.

[@QuantumConundrum]

5:48 "and some physicists like to have" oooooooooo, we put some many things under the rug, you have no idea.

[@Hack3r91]

We punch mathematics in the guts and we make it fit in our theories by kicking it in the kidneys, don't we?

[@hughdepaor]

the sound of the marker on the paper ... eeeeuuuuggghh my brain

[@UnashamedlyHentai]

+Hugh Sleeman Power I am not aloooonnnee!

[@getsomebud]

ASMR.

[@xfgjhxfj]

+Hugh Sleeman Power I can't watch it's too cringy

[@GuyWithAnAmazingHat]

+Alex Ehler (THOUGHTSEIZE) Actually it's the complete opposite effect, misophonia, a negative reaction to sounds in a similar way some of us has positive reactions in the form of ASMR.

+Hugh Sleeman Power To me, the sound is totally okay. Except the extra squeeky sound the marker makes sometimes like at 5:22.

[@bcisbwbdkvivjebeorogidb]

Brady just seems so done with math at the end. And she is just so smug like "Just another day blowing minds."

[@heoTheo]

Seems like the thing we did at the school yard. One that says the largest number wins. and one would say "a billion" and I would say " billion gazilliond" and he would say "infinite" and I would say "infinite +1" Just realising that I was a model theorist and already using compactness theorem.

[@douggwyn9656]

+heoTheo Infinity plus 1 is not larger than infinity.

[@janinja1000]

+heoTheo infinity plus one is still infinity

[@ChaoticNeutralMatt]

@@douggwyn9656 I'm sorry for the irrelevant comment at this point, wow this is ancient.. But I would say it depends on the infinity. Some infinities can be larger, but both are infinite. Just another type of 'set'.

[@FatLingon]

This was facinating. I also watched the video over at the Numberphile2 channel. Please do more on this topic, more in depth. Feels like we only scratched the surface.

[@S1nwar]

so 1/k is closer to 0 than any 1/N could be. that means you can add the entire 1/K number line to any real number and get a new set which consists of all real numbers + all 1/K numbers in between...

[@noimnotnice]

+S1nwar I wanted to brush this off at first but this is essentially the only reason I could think of to introduce this concept in the first place.

[@TheOlliemath]

+S1nwar yeah, in this model the real numbers are more like scattered solitary stars separated by a black sky of infinitesimals than the continuum we like to think of

[@dave-id2eo]

+S1nwar Basically right. If you fix a real number r, the whole r+1/K line is infinitesimally close to r.

[@XouZ88]

+S1nwar Isn't the sum of N -1/12 though?

[@noimnotnice]

XouZ This is probably a joke, but no - it isn't.

[@lucainvernizzi9715]

The grin at the end just made everything better. XD

[@slowfreq]

what about this set K makes it bigger than the real number set? is it just because we said it's bigger?

[@namenotincluded23]

+slowfreq That is exactly why. It is by definition bigger. The same idea is used with i, as well. It is simply defined by the equation i^2 = -1. You can be surprised by how much application can arise by abstract thinking like that.

[@XetXetable]

+slowfreq K is not bigger than the set of reals, it is bigger (has a higher measure) than anything IN the set of reals. Beyond that, what namenotincluded23 said.

[@littlebigphil]

+namenotincluded23 That seems rather weird. How do you determine differences in the size of sets that are larger than countably infinite sets.

[@XetXetable]

+littlebigphil One can define the difference in terms of functions. If set X is smaller than set Y, then there will exist an injective function from X to Y. This holds even for infinite sets. Using this and related concepts, sizes can be established in general. This also means a mathematician has to be involved in the process to construct our function, there's no simple check one can do in general.

[@RFC3514]

+slowfreq - K is the number. The set is N*, and it's being compared to N (the set of naturals).

[@jagaray]

Wow....I've never seen these concepts before, blown my mind. Great video.

[@nishkaarora6343]

"because, we're mathematicians." like puh-leeeeeeeeeezzzz xD

[@Brotcrunsher]

This was amazing! The Monad thing was blowing my mind.

[@coloneldookie7222]

So it's like Klein bottles...you can explain it well enough, but there is no currently tangible way to truly represent it with the way we know the world works so far.

[@wood-eye]

+Colonel Dookie Interesting...

[@TwistedLemniscate]

+Colonel Dookie Perhaps there's no way of representing infinitesimals, but infinitesimals ARE a tool (and a very useful tool!) used by humans to be able to represent the world itself. I'd call it a "beautiful fiction", but isn't all mathematics a fiction anyway?

[@T.Ty7]

+TwistedLemniscate can't really represent anything in the observable world with abstract things like infinity, only in theory

[@codediporpal]

+Colonel Dookie Besides the natural numbers, it's all fantasy.

[@coloneldookie7222]

***** And so can infinitesimals. My point still stands.

[@streetleveltech]

I remember my first brush with these concepts when I was in junior high school and reading Gamow's "One Two Three...Infinity." A mind-blowing experience for a kid who was just being introduced to basic algebra.

[@NoLongerBreathedIn]

Aw, I was hoping for monoids in the category of endofunctors.

[@larrykuenning5754]

1:57 "It's still a natural number." Not if the natural numbers are defined by the Peano postulates. If you include "natural numbers" that you can't get to from 0 by repeatedly adding 1, this breaks the principle of mathematical induction. I guess you can make up a new concept of "natural numbers" but only if you're careful never to let it get confused with the old concept -- and this would work better if you just gave it a new name.

[@RSLT]

But why add one to reach the largest size? Consider the set of all natural numbers. By creating subsets and using induction, you can show that the set of all subsets of natural numbers (the power set) is larger than the set of natural numbers itself. This is akin to comparing the number of molecules in a small sea to those in a vast ocean. No matter how many molecules are in the sea, the ocean contains even more. Similarly, there are more subsets of a set than there are elements in the set, demonstrating that some infinities are indeed larger than others.

[@larrykuenning5754]

@@RSLT: Did this reply get attached to my comment by accident? Because it doesn't seem to reply to anything I said. I certainly didn't say anything like "add one to reach the largest size" -- there is no largest size. Similarly power sets are not relevant to anything I said. My point was that "natural numbers" is already a defined concept, and the classic definition is the Peano postulates -- one of which says that the numbers you can reach by starting at 1 (or 0) and repeatedly adding 1 are all of the natural numbers there are. K is not reachable this way, so no theorem that depends on mathematical induction can apply to K. When you want more numbers than the naturals, you define new sets, such as the negative numbers, the rational numbers, the real numbers, the complex numbers, or the transfinite numbers. All this is standard math. But if you pick an arbitrary number from one of these sets, one that's outside the definition of natural numbers, you mustn't say "and this one is a natural number too" unless you're just deliberately trying to cause confusion.

[@sean3533]

Ahh mathematics, where we can define the universe, or make contrasting definitions of the same thing to make it sound like we're not understanding.

[@StevenShields29]

I like this lady--a LOT! And I love that she is not just young, but she has been doing this for YEARS!

[@StevenShields29]

+StevenShields29 And yes, she goes WAY over my head, lol!

[@noamtashma2859]

am I the only one that thought that this would somehow be about Haskell? Haskell ftw

[@Evan490BC]

That's a different kind of Monad...

[@josematias2010]

I like those too

[@ffhashimi]

This is amazingly interesting and easy to understand..

[@PassionPopsicle]

I love math

[@xoblyxanier]

Great video! I like the amount of your questions you left in. Neither too many nor too few.

[@MarioWenzel]

Why do mathematicians always leave out proper quantification?

[@XetXetable]

+Mario Wenzel It's not mathematicians in general, just the one's that play fast and loose with foundations.

[@MarioWenzel]

MichaelKingsfordGray Well, especially in that example, properly using the universal quantifier and the existential quantifier would have gone a long way for people with some understanding of the issue. But math-people are usually bad at using them properly. They often introduce some variables that are actually existentially quantified but they use syntax that implies that they are free.

[@MarioWenzel]

+MichaelKingsfordGray "proper quantification": Proper use of the existential and universal quantifier to correctly bind non-free variables in order to produce a correct definition but since you haven't shown a problem my argument or shown where I am wrong, I guess I know which troll not to feed.

[@JedaiFou]

+MichaelKingsfordGray His explanation was in no way a world salad (see en.wikipedia.org/wiki/Free_variables_and_bound_variables : those are words familiar to logicians and CS theorists) and he is perfectly right that mathematicians tends to use literals with no considerations given to their quantifications and their status. This is generally left implicit but it means that this is one more obstacle for a mathematician to get into a new field he don't yet know the convention thereof.

[@idlikemoreprivacy9716]

Thanks Numberphile for showing us all this beauty!

[@BvBCrafty]

okay, im done with math now, bye! :D

[@Yomama5923]

+BvBCrafty Haha. Right? It's a fun concept, but I don't think this has any practical applications... at least, nothing I could understand. Sometimes I wonder if mathematicians just enjoy making up and being in their own world.

[@Yomama5923]

Alex Ceceña I take it that you're a mathematician? I disagree that it's an insult. My only point is that I (personally) would enjoy knowing that whatever new concept I'm toiling over would actually help people or solve something with known application. But, I say, if people enjoy just thinking about math or extrapolating on it even though it has no known application, that's fine. You aren't hurting anyone and you should be free to pursue your own happiness.

[@noamtashma2859]

+Jonathan Gutsymon of course we enjoy making stuff up and being in our world. how would anybody do math otherwise?

[@genevaconventionsviolator3994]

+Jonathan Gutsymon Well, I'm a deeply Platonic mathematitian so i personally do :) To me math is far more fundamental and meaningful than the "real world" and seek to explore the realms of abstract mathematics WITHOUT having to think about our contrained 3 dimensional world. But that's just me I guess

[@noamtashma2859]

Alex Ceceña indeed.

[@karlisbikis5013]

There is a typo at 0:46. It's ∈nℕ but it should be n∈ℕ

[@Hythloday71]

This so called separate 'size' scale seems meaninglessly related to 'size'. Seems no more motivated than saying I'm going to have a number scale with sausages on their shoulders, and each has a bigger sausage on their shoulder than does the regular numbers. I don't mean to disparage, this was great and fine, just I note the ambiguity in this video in/ this system as to what size, magnitude, quantity actually is.

[@XetXetable]

+Hythloday71 This is no different than the notion of size between the integers and the rationals. You don't use rational numbers to count sheep, and you don't use integers to measure lengths. You wouldn't use this system for either. Mostly, this method is used to circumvent some complicated proofs using ordinary systems. Basically, it ends up being easier some times to go from a to infinity (k), and then to b, rather than going directly from a to b.

[@Hythloday71]

You use a subset of the rationals to count sheep and you could use the integers to measure length. Both have property of cardinality and a natural relation to each other.

[@XetXetable]

+Hythloday71 They do not have cardinality. You clearly don't know what that means. You cannot have a set with 5/3 elements. You cannot characterize something of length 1/2 using the integers. These are the wrong choices of structure. Also, this new system has the same "natural relation", being the existence a monotone injection.

[@joeybf]

+Hythloday71 There is a natural order (reflexive, antisymmetric and transitive relation) on N by defining "a less than or equal to b" as meaning "there is a natural number c such that a + c = b". Then K is defined to be a number that is bigger (in this order) than any standard natural number, and the order extends uniquely to this extended set of natural numbers. This is a consequence of Peano's axioms. Granted, "size" in this context is not a completely intuitive notion, but it can be formalized. Also, you might be interested to know that there are other useful orders on the standard set of natural numbers. For example, in the divisibility order, 1 is the smallest natural number, while 0 is the biggest.

[@jtoonzkun6480]

+XetXetable Minor nitpick. While you can't use whole numbers to measure lengths like 1/2 in the usual procedures we collectively use, you COULD give equivalent information using ratios of whole numbers. So to express that a length is one half the length of your unit, you just say that are in a 1 to 2 ratio. You COULD interpret that as just being a different spin on rational numbers, and maybe it sort of is, but the idea is that we can fit the short length two times onto the unit, which is a pretty straight forward comparison using whole numbers. This whole tangent is mostly irrelevant though. Feel free to ignore.

[@beefjerkykoolaid]

Love this prof!

[@OKay5067]

Square root of 2 is about 1.8? :D

[@krowwweee2918]

I just watched it until the middle and had to stop for a second and read comments. I think I just had my mind blown by this video :D

[@beeble2003]

1:58 "It's still a natural number" That's an extremely misleading answer. It's not a natural number at all. Rather, it's an object that you might accidentally allow into the natural numbers if you didn't define them very carefully. In more formal terms, it's an element of a non-standard model of the first-order Peano axioms. But the second-order Peano axioms have no non-standard models and the natural numbers are defined to be the unique model of the second-order axioms.

[@pumpkinpie6930]

I like the squeaky noise the marker makes against the dry paper.

[@giorgigogashvili2549]

4:02 sqrt of 2 doesn't go there

[@TheMakersRage]

So the number line stretches off to infinity, but there are in-between parts that don't stretch as much towards infinity... thanks Cantor.

[@mancheaseskrelpher8419]

6:23 "No"

[@rikschaaf]

Yay, finally a good explanation to "infinity" and 1/"infinity"

[@willistrong185]

well i think the square root of 2 doesn t go there:)

[@MisterMajister]

Really good video (the mic setup/audio was a bit off though) of understanding big numbers. I love the idea that the real integers never reach K-1, K, K+1, no matter how many times you add 1. Still, they are real as well? This is where numbers and infinity are so mind blowing!

[@moritzkockritz5710]

I cannot watch this video because of the sound the pen makes...

[@AbsolutGB96]

You should do a Numberphile 2 episode all about the maths James did in his PhD and the research he did after completing it for a while; in the meantime, this was a great video!

[@jameswhistler968]

+AbsolutGB : Are you talking about me?

[@DjVortex-w]

The video fails to mention an use for this.

[@givememore4free]

a

[@user-ri3up7ru4g]

+WarpRulez It's here because it's interesting.

[@douggwyn9656]

+PacMonster0 The Planck length is not infinitesimally small. The universe is not expanding into anything other than itself, and in some models it has finite size. Don't confuse those mathematical fantasies with actual existing conditions.

[@DjVortex-w]

***** This has absolutely nothing to do with planck lengths. It's a well-defined value.

[@RDWize]

+Doug Gwyn I think what he meant was, in most physical models, the Planck length is so small that its usually approximated to be infinitesimal which allows you to take integrals.

[@xanthirudha]

The part that bends my mind 3:20 ..."you do it forever"

[@Kibadda123]

Brady, I think there was a slight error in your video, when you typeset 'n in N'; in your video it showed 'in n N'. Just a slight error, but it's ok. No hate. I like your videos, keep it up!

[@numberphile]

+Kibadda123 I've added an annotation and something in the description - but I am also resigned to a lifetime of comments about my mistake! :)

[@jdgrahamo]

+Numberphile Once upon a time I used to work in a bar cellar. I noticed that some of the kegs had M/T written on them in chalk. When I asked my boss what M/T meant, he looked at me as if I was stupid and said, 'There's nothing in". Perhaps I could be forgiven for thinking ' n ∈ ' meant 'any'.

[@TheRexTera]

+Numberphile Welcome to the internet my friend.:-p

[@numberphile]

+TheRexTera luckily I make very few mistakes.... ha ha!

[@Pelmenji]

"Why not? Because we're mathematicians!" is the best excuse for everything, ever.

[@psionic0]

It's something like a translation of origin to infinite (if we want consider infinite as a defined number). Then you can perform any operation you perform in Natural, Integer or Real numbers, but you can't reach 0 (or any other finite number), like you can't reach infinite stating from finite numbers.

[@nosuchthing8]

Numberphile, best you tube station ever, even better than n star.

[@unvergebeneid]

I love her face in the last shot: "FU, Brady, you chew on that!"

[@HeraldoS2]

Just a suggestion. Could you explain the hypothesis of the continium, Martin's axiom and why they contradict? I mean in someway those things are what are behind these ideas.

[@tamptus3479]

I would like to see more with Carol Wood. May be a bit Modeltheory or remarks on forcing. Remarks on countabe Models of ZFC or remarks on V = L or some about the Life of A. Robinson

[@ChannelJeffrey]

Another great video. You have introduced me to some really brilliant well spoken folks. Thanks. (The sound quality was a little bit off though)

[@OfficialKruz]

Considering how close this borders on breaching the subject, you guys should make a video on the surreal numbers! Preferably with John Conway involved!

[@alexandertownsend3291]

This comment is awkward now. He is no longer with us.

[@OfficialKruz]

@@alexandertownsend3291 yeah well, 6 years does that. They've done plenty to honor him and his legacy.

[@frankkeith7286]

Thank you professor. Great job explaining.

[@RBLXbranefreez]

This is the kind of math that satisfies my pondering mind.

[@fabibi_ha]

In the video they talked about counting backwards from K and never going to end up at an integer. Is there a possibility to end up at L (half K)? It is not clear to me because i never learnt subtracting of such weird numbers

[@archivedtransience]

+Fabian Hägele No, because L is also infinitely big, so you can never reach it counting backwards from K, just like how you can never count up from zero and reach infinity ~ K is infinitely big, so K/2=L is also infinitely big, and the distance between them (also L) is also infinitely big.

[@VYScuti]

Wait so are they not ordinal numbers?

[@MuffinsAPlenty]

No, they are unrelated to ordinal numbers.

[@binbots]

It is hypothesized that there is infinities of different sizes. By making infinity an actual number on the number line (like we did with 0) maybe we can start to make sense of this concept. Examining the number line from left to right the largest number would be infinity. But this infinity would need to contain every number on the number line including negative numbers. So true infinity is negative infinity + positive infinity. This number equals 0 but is also the complete opposite of the 0 we know. The 0 we know is neither + or - and flips the number line from - to +. This new 0 contains all numbers both + and - therefore is (+-0) and flips the number line from + to -. If such a number could exist then the next number on the number line would be (+-0) + 1 (which would be the largest negative number). Then (+-0) + 2 etc etc….all the way back to zero. Then the number line repeats over and over again forever. It continues to cycle between 0 and (+-0) creating a larger infinity that contains an infinite amount of infinities. -1 = the smallest negative number. 0 = nothing. 1 = the smallest positive number. ((+-0)-1) = the largest positive number. (+-0) = infinity. ((+-0)+1) = the largest negative number. …-1,0,1…((+-0)-1),(+-0),((+-0)+1)…-1,0,1… By making infinity a actual number on number line we can eliminate some of its unusual behaviour. For example: instead of infinity - 5 = infinity, now it can equal ((+-0)-5) or the fifth largest positive number. Instead of infinity + 5 = infinity, now it can equal ((+-0)+5) or the fifth largest negative number.

[@Macieks300]

are there any real numbers between 0 and 1/k?

[@ProxyMohawk]

+Macieks300 These aren't real numbers we're talking about. They're hyperreals. But yes. There are infinitely many real numbers between 0 and 1/k. Including 1/(k+1) and 1/(k-1) and such.

[@Macieks300]

+Kogasa Tatara so 1/(k+1) is a real number?

[@douggwyn9656]

+Macieks300 No, the reciprocal of a nonzero real number is a real number, so 1/k etc. aren't real numbers.

[@ProxyMohawk]

+Macieks300 I meant hyperreal rather than real. 1/(k+1) is not a real number.

[@jameswhistler968]

+Kogasa Tatara says, "There are infinitely many real numbers between 0 and 1/k. Including 1/(k+1) and 1/(k-1) and such." No need to hurry so much with math, just to say incorrect things. The answer to Maciek's question, as stated, is no. There are other hyperreals between 0 and 1/K (as you point out), but 1/(K-1) is not one of them.

[@huntermatthews3407]

This is such a great explanation.

[@CacchiusMan]

So this should be the ideas of not-stardand analysis created by Abraham Robinson many years ago, right? You negate Archimedes principle about natural numbers so that N is up limited and then you can take back on the real number line (and create a new set of numbers called hyperreals) the infinite and the infinitesimals as numbers and treat analysis and calculus using relationship between infinitesimals as the first mathematicians like Leibniz used to do (not a coincidence that Leibniz's phylosophy uses a lot the concept of monad). Where can I read more about the topic?

[@CacchiusMan]

Thank you very much! :)

[@01FNG]

I don't see what's new here. It is a fact that every number is that number + 0 K is just another 0, we just put a point on a line and start expanding on both directions from that point. 0-2 0-1 0 0+1 0+2 is same as K-2 K-1 K K+1 K+2 We just renamed our reference point to K and nothing (should) mathematically change. And if we take another reference point which is infinatly away from K then 1/k Approaches that point (which we could name 0 or even K2). So all we've been talking about here is 0.

[@douggwyn9656]

+GOT TOSA No, K is definitely not any kind of a zero and is not equal to zero. K+0=K, but that's not K's fault.

[@Schindlabua]

Mind you that we are not talking about a mathematical constant K here, it stands for any (so-called hyperreal) number, so the reference point analogy doesn't really hold as there is none. But yeah, the maths works out.

[@stefanozurich]

I haven't ever heard a use for this.

[@VicvicW]

So 1/K is not 0? By the same logic then, K-1/K is infinitely close to one, yes? But it is not one. What does this mean for 0.9 recurring? Does it mean it is not 1? Is it a different case? I assume because these K-n/K numbers are so close to 1 yet can be described in the fraction form shown, they are therefore not in the same position.

[@notoriouswhitemoth]

The number of infinities is uncountably infinitely more than the number of finite numbers. There's even a hypothetical infinity that accounts for all possible numbers, both finite and infinite, even those whose existence is mutually exclusive, represented by "Ω".

[@zahbaz]

A few questions come to mind. If K= K, we have K/K = 1. Meaning, if N* is a group, then the operation can't be multiplication. Clearly 1 isn't the identity and 1/K isn't the inverse in the group. So, is N* a group? What is it?

[@douggwyn9656]

+zahbaz There is more than one operation! The group operator is addition, with identity 0. The category of hyperreals is an extension field of the field of real numbers.

[@MuffinsAPlenty]

+zahbaz Neither N nor N* are groups. Under the binary operation addition, both are _monoids_, which is a special kind of _semigroup_. A semigroup is any nonempty set along with an associative binary operation on the set. (i.e., in a semigroup, we require closure and associativity, but there need not be an identity element or inverse elements). In a monoid, we also require the existence of an identity element. In a monoid, you can have elements which don't have inverses. This is the case for both N and N*. For example, 0 is the only element of N which has an additive inverse (there is nothing in N that you can add to 2 to get 0). Similarly, 0 is the only element of N* which has an additive inverse. You can even get fancier and say that both N and N* are semirings. Much like a ring is an Abelian group with an extra operation following certain properties, a semiring is a commutative monoid with an extra operation satisfying those same certain properties. And Doug Gwyn is correct that R* is a field.

[@violetemmott3353]

Me: *excited to learn how computers deal with limits* Numberphile: "around zero we have... a monad"

[@maverator]

The missing ingredient in this video is how any of it means anything in our reality. Maybe I'm just dumb, but I don't see how this entire discussion doesn't equate to "imagine infinity. now imagine something infinitely bigger". Obviously I'm not a mathematician. Sometimes people ask if math is invented or discovered. Part of me intuitively feels that it's discovered. Then this video changes my mind.

[@zakerysimpson5363]

I like how pleased she looks at the end

[@rogerwitte]

This was a fun presentation of (1) Nonstandard models of peano arithmetic and (2) the surreal numbers [at least, so far as I can tell - it is very careful not to mention either of those systems]. I tuned in to the video hoping to improve my understanding of 'monads' (from category theory) but 'monad' was just a name drop here. I enjoyed watching, even though the topic discussed was different from the topic that I expected/hoped for.

[@MuffinsAPlenty]

Yeah, monad is one of those words with multiple meanings. The meaning in nonstandard analysis (used in this video) is different from the meaning in category theory.