The Infinitesimal Monad - Numberphile 

Her face at the end is just priceless.

"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.

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.

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

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

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.

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

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)

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.

I think she broke Brady. :p

Right in the monads...

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.

Kth!

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

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!)

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!

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.

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

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.

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

THIS IS THE MONAD'S POWAH!

what about k factorial?

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.

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.

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

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

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

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

√2 is WAY too close to 2!

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

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?

I think that mathematicians mean something very different by "bigger" and "closer together" than laymen do in discussions like this. It seems to me that focusing on the ordinal nature of these spaces would be less confusing for the uninitiated.

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

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

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.

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

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

"Because we're mathematicians" 😂

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

Thanks Numberphile for showing us all this beauty!

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.

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.

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

Thank you professor. Great job explaining.

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...

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

The grin at the end just made everything better. XD

This is amazingly interesting and easy to understand..

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

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

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!

So fascinating! Thank you.

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

I love math

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

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

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.

Why do mathematicians always leave out proper quantification?

This is such a great explanation.

This is pretty cool, I've always though about there being a sort of "quantum realm" for numbers inifinitely close and infinitely far. At these numbers, common rules we already know start to break down. They are all part of the same line, but are so tiny or so far away that they pretty mich exist in their own magical dimension. Maths can be pretty awesome like this XD

Are these the surreal numbers?

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

As far as I know a monad is defined by a functor that takes objects in a category C to objects in the same category and two natural transformations usually called mu and epsilon. The triple (T, mu, epsilon) then construct the monad. There are also some laws of associativity and identity that I can’t state by heart. The respective types are: T : C -> C, mu : TT => T, epsilon : 1_T => T. However, how would one go about defining the monad of “infinitesimals” as presented in this video. Is this some other notion of “monad” than the category theoretical one?

6:23 "No"

Square root of 2 is about 1.8? :D

OMG I love listening to this woman! Even if this is largely unimportant to me in my day-to-day life.

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

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.

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

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

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

4:02 sqrt of 2 doesn't go there

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

Love this prof!

yay finally some model theory stuff! Brady! Do one one infinitary logics!

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.

The video fails to mention an use for this.

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

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.

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

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.

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.

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.

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.

Is it accurate to describe the real bounds of positive 0's monad(m) as: 0 < m < 0.0̅ (In case that over-bar doesn't render properly; 0.0 recurring)

Could you do the same with the K-series of numbers... just say there's a number Q which is bigger than any and all K? It seems a bit abstract. What purpose would numbers like these have?

So are there monads around these k values as you infinitely approach them? And monads with in them, ect

She says that if you count backwards from K, you would just keep arriving at numbers that are still greater than any natural number. This would seem to mean that subtraction has to be defined differently for these numbers in order for (K+1)-K to still be greater than any natural number. Any insights on this?

I see why this works and I see that this is somewhat of an interesting topic but does that relate to any other region of mathematics? Do we get any new information or is this just a part of mathematics that stands more or less alone ? For example can you plug those numbers into funtions and get more information about their behaviour or does it help develop integrals and derivations ?

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

Does K have anything to do with 'transfinite numbers' such as Aleph null, Beth null etc?

I love this concept.

I love this sort of theoritical mathamatics. Very Cool stuff

yay im early! Anyways, this channel is amazing, thank you for creating it and keeping it up Brady

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

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!

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

Does this have anything to do with the monads of Pythagoras and Leibniz? Are they're definition of monad different from each other?

I like how pleased she looks at the end

Vids like this are a reminder to me that, although very useful in our perceivable universe, math is still bound to the rules of this universe (incl our limited imagination), and thus can never be a describer of the whole reality...