I'd love to see a full course on nonstandard analysis including more formal definitions and deriving results from "standard" analysis using this approach.
Take my course, Math 5154/CompSci 5203/CompE 5803/Philos 4354, at Missouri S&T. One of the required textbooks is “Nonstandard Analysis for the Working Mathematician”. Once you’ve succeeded in that course, you will be ready to study some nonstandard analysis.
@@danielthonk7481 At least it would include a correct treatment of the construction discussed in this video. You prefer to learn it wrong, I guess. Have fun with that.
Well, of course we want more hyperreal number videos from you, Michael/Stephanie, as these are so fascinating, and novel to me. Oh, and as I said before, you both are THE best youtube math channel in existence, okay?
In fact, John Conway's *surreal numbers* (you can search for them on the web) are the most natural extension of real numbers, as they include hyperreal numbers, ordinal numbers, and many others, while being very simple to describe and manipulate. May be one day you could find a way to explain them in your wonderful chalkboard approach of maths concepts by solving problems ? Amazing video anyway. Thank you Michael !
Please, add more content about nonstandard analysis! It really feals like a whole new world to explore - and so much more intuitive and fun than epsilontic calculation. I am enjoying this so much. Sadly it was only mentioned as a side-note during my time at the university.
yes michael, i'd love to see integrals being defined using hyperreal numbers. this is great video, the production is very professional. the only thing i miss from your older stuff are those backflip transitions lol
This is how we were taught mathematical analysis in college (computer engineering degree in Italy). Fun thing is that it makes the informal notation for limits that one would write on a napkin into something that's actually mathematically correct and well founded.
I ran across nonstandard calculus during my second semester, taking my first course in the ordinary limit-based one and for a while was a bit obsessed about it, so this video is a nice reminder of the "old times" 🙂 That being said, some people (used to?) tout it as a simpler way to teach calculus, involving fewer limits, less quantifiers and so on. I don't agree with this at all, since to do anything at all with them, you need a so-called transfer principle, which basically states that any first-order formula in the language of the real numbers which is true in the actual real numbers is also true in the hyperreal numbers when translated appropriately. This is what enables one to prove things like that addition is commutative, that every number has an integer part and so on. Unfortunately, it is very easy to screw up with the transfer principle. Example 1: Every real number is less than some natural number, so by the transfer principle, every hyperreal number is less than some natural number, right? But that is obviously false -- well, I forgot to translate! The correct statement is "every hyperreal number is less than some hypernatural number", and what even are those (the fixed points of the hyperreal extension of the integer part function, obviously 😀)? Example 2: every bounded set of reals has a least upper bound, so every bounded set of hyperreals has a least upper bound. Also obviously false, since the set of infinitesimals is bounded (by 1, e.g.), yet definitely doesn't have a least upper bound. This time, I accidentally used a second-order formula, referring to sets of reals! So in my view, you're just exchanging the difficult, yet manageable concept of limits for a logical concept of the transfer principle which is at least as hard to use correctly. Case in point, your proof that d/dx e^x = e^x is incorrect! There is no way to justify the claim you implicitly used that st(t)^ε = st(t^ε) -- definitely not using the transfer principle, since st is not part of the language of real numbers. Incidentally, that claim is false in a similar way the claim that lim (x -> 0) f(x)^g(x) equals lim (y -> 0) (lim (x -> 0) f(x))^g(y) is false.
in my(layman) opinion, st(t)^eps is valid but st(t)^omega is not valid because eps is closed (by real numbers as faces) but omega is open. It means that eps connected to real numbers but omega disconnected. Or,for natural numbers all vice verse(zero is open but infiniy is closed). In other words. Line is not shape (has no closed interior and open exterior). epsilon and omega solve this problem. Number line became common shape, where numbers could exist.
I think of the hyperreals as a discrete infinite dimensional vector space over the reals, is this appropriate? The ordering relation is just the lexicographic ordering from higher dimensional index to lower dimensional index, addition is just vector addition, multiplication is distribution over a linear combination of dimensional index shift operators. The only thing I can't wrap my head around is non-integer exponentiation on this space. for real-exponent you can 'complete' the space by adding a continuity of dimensions, so eps^(1/2) is a dimension half way between the eps dimension and the 1 dimension, but what does eps^eps mean? Maybe you have to 'recursively' (a second order theory?) hyper-complete the space for each layer of exponent, for example on this layer eps^eps requires that we also do the same hyper-real construction on top of our completion of the dimensions. Although, let's say this idea does work without the 'transfer principle', any proofs you do now require analogies to limits over real valued vector spaces, and limits over the dimensional index as well, which is quite complicated.
@@MagicGonads I don’t think the space you described is necessarily complete, and the hyperreals are complete. As an example of the space you described, one could take epsilon from the hyperreals and adjoin it to R. This would be an infinite dimensional field extension, and would even be ordered since it inherits the order from hyperreals
I would like to see more of these hyperreal number videos. Not just defining integration, but also simple stuff like proving that exponent rules still work with infinitesimal values.
My calculus professor gave us a brief introduction to non-standard analysis, he seemed enthusiastic about it, also because he saw it as a fruitful attempt to reintroduce Leibniz's notation. And in fact it would be interesting to see other videos about it, above all for the clarity with which you, professor, are able to illustrate even rather complicated concepts
I feel it the other way around : one way of constructing real number from the rational numbers, is to use a "good" set of bounded rational sequences and do the quotient of this, actually, ring with a maximal ideal. I don't know a lot of hyper real number, but the "st" function looks a lot like the rest of a quotient operation. And because of that, it gives me the impression that all the properties of limits (and so derivatives and primitives) are more a consequence of some underlying existence of an algebraic structure than topology.
The limit is a convoluted definition of this. Plus every great mathematician and physicist thinks in terms of infinitesimals. That is how Euler got all his results, with great intuition of infinitesimals Modern calculus teaches that good intuition is somehow a bad thing and forces limits. You could teach yourself all of calculus with just the intuition of infinitesimals, without taking a course or having a teacher guide you, because it is straight forward Then the 'rigor' people got involved and said infinitesimals weren't legit, but they always were; they just had bad intuition
@@pyropulseIXXI what would sin(1/epsilon) be? What kind of operation one can do with infinitesimals? I don't know much about the hyperreals but rigor is important to know whether what you write can make sense or not.
@@qwertyqwerty-jp8pr Any operation that can be done on the reals can be done on the hyperreals; this is the key property. Sine is defined for real numbers only, so sin(1/epsilon) is clearly undefined. Thus, if a (b+c) = ab + ac, then this works if the numbers are all hyperreals. You can just axiomatize this, if you to teach in pure intuition, such as 'Transfer Principle' and 'Closure Principle.' I taught myself calculus using the intuition of infinitesimals; it is very power and very obvious. Calculus was built on infinitesimals, and they abandoned that intuition in favor of limits because of 'rigor.' (or rather, because of a failure to rigor-tize infinitesimals, which was done in 1960 by Abraham Robinson). Now, intuitive and super obvious things tend to have insanely difficult proofs. Just because a system exists in which the rigor is easier does not mean we abandon the intuition that leads to such amazingness. Just look at the real numbers; hardly anyone actually proofs the extension under Dedekind cuts, so where is your supposed rigor? It is just assumed to exist as an axiomatic statement. People just assume the reals exist in an intro calc course, so why abandon good intuition by not assuming the hyperreals exist in an intro calc course? If you do, you can do rigorous calculus proofs that would take a course in *_real analysis_* to be able to do Likewise, one can do this with the hyperreal numbers, for the purposes of teaching intro calc, and posit that the operations that work on the reals also work on the hyperreals. We rely on our intuitive understanding of the reals in order to work with them, and develop extreme proficiency, within calculus. Again, one can do this with infinitesimals and obtain an unrivalled mastery of calculus with such an intuition. Two axioms are: *(1)* - Every fact of traditional mathematics remains true. *(2)* - If there is an object with some property, then there is also a standard object with this property This is called the *Closure Principle.* Bam; if you need absolute rigor, just read _Non-Standard Analysis,_ by Abraham Robinson. This is beyond the rigor level oft found in a course of _standard analysis._ And with the *_Closure Principle_* above, teaching via infinitesimals is above the rigor of an introductory calculus course, and it is far more intuitive. This is why Calculus should be taught in the form of infinitesimals. This is how Euler thought, and his results speak for themselves. I suggest the book _Analysis with Ultrasmall Numbers;_ it allows rigorous proofs at a college introductory to calculus level. I also suggest _Infinitesimal Calculus,_ by Henle & Kleinberg I suggest reading Euler's two books on calculus; his insights are amazing and can be replicated, in an inferior manner, if you, too, start thinking in terms of infinitesimals
@@qwertyqwerty-jp8pr If sine is defined only on the reals, then sin(1/ε) is undefined; 1/ε is a hyperreal number, equal to ω. sin(ω) is not defined. sin(ε) is defined, since st(ε) = 0, and thus, sin(ε) You'd need to take the standard part of 1/ε, which is the closest real number to t, but this is undefined, as there is no closest real number. this is because for any real number you give me as the 'closest' real number, I can find another real number that is closer. Say you give me _a;_ I can give you _a_ + 1. Now, if it was 1/ε + 1, then the closest real number would be 1
I'd love more videos on the hyperreals-alternative number systems are always cool, and it would be interesting to see what derivatives/integrals are harder or easier using the hyperreals.
I love this video. I think nonstandard analysis is much more interesting, useful, and clear for learners and users of calculus. The epsilon-delta definition of a limit always seems to be the hardest part of tutoring calculus. 3B1B did a lot to help in this area, but many people still don't get it. I've heard many physicists, engineers, and UA-camrs talking about a derivative as how a "tiny" change to x results in a "tiny" change to f(x). And while talking about integrals, they talk about how the infintely many "tiny" changes in x result in the infinitely many changes to f(x) and add up to the standard integral. Basically, what I'm saying is that nonstandard analysis matches our intuition much better than standard calculus.
I know I've been critical in the past but I was really impressed that you gave a brief introduction to the hyper reals. I was confused by the simplification math of the last e^x example though but aside from that it was pretty clear. I'd like see more examples and information on derivatives, integrals and other information on hyper reals or nonstandard analysis in general and eventually learn how to use it. I hope you make more related videos in the future. Thanks for the video.
This is the first time I've seen you using camera transitions instead of clearing the board on a static camera and I just wanted to say it's very good for my ADHD and I didn't think that I'd want that but it's really helping me follow you for much longer.
I would like to see your summary of defining an integral using hyperreals! This nonstandard approach is a modern one that (slightly) simplifies proofs!
A followup on calculus in the hypereals would be fun. 👍 Maybe prove the fundamental theorem of calculus using infinitesimals for example. Also another fun related tangent might be to delve into the Surreal Numbers, an invention of John Conway similar to the hyperreals which had some applications in game theory.
I got finished with my AP Calculus BC test a couple days ago, now I'm watching this. Very interesting, and I am that I am, partly hair, partly ham, but always cool.
This is super interesting! Limits are difficult to explain rigorously to many of the people who need to use calculus - epsilon-deltas are much more conceptually tricky than derivatives and integrals, IMO. So schools often just handwave their way through the foundations of calculus leaving students feeling like "so what was the point of THAT?". Whereas it looks like the hyper-real derivations of a lot of basic calc identities are really easy to follow with just a basic foundation in high school algebra. This would great for teaching! I feel like part of the problem is that by the time someone has studied enough math to actually *care* about the hyper-reals, they've probably spent so much time with epsilon-deltamanship that it's no longer counterintuitive at all. Even after watching this, epsilons still feel like the 'natural' way to think about 'very small numbers'. So I think a lot of this seems superfluous to people who actually study analysis, so it gets discarded. Anyway I'm increasingly convinced that nonstandard analysis is unfairly maligned. It'd probably be easier to teach calc this way at least at the high school level - even if I *still* have no intuition for what the hell an infinitesimal is or why we'd need to work with numbers provably too small to matter.
In the nilpotent formulation, it's conceptually simple to just use nilsquares, but when doing higher and/or multivariate derivatives it becomes cumbersome. Just saying "standard part" takes care of all the H.O.T.s in one go. It's similar to saying "limit" but it's more obvious what's happening
The formal definitions are kind of trippy and depend on the Axiom of Choice. I read a short-ish book on Non-Standard Analysis that covers math major Real 1 (or "advanced calc") from the NSA setup. It feels a lot like the usual limits and epsilon-delta, but through a decoder ring. Overall, I think standard won out for good reason, but it's kind of interesting that it works and occasionally you get strange new proofs like the Tychonoff product theorem for compact topological spaces.
I love the fact that hyperreal numbers sort of validates how Newton (or was it Leibniz?) did calculus which was at some point regarded as lacking rigor.
His choice of the “smallest” infinite ordinal is unfortunate,as it is provable that there cannot be such a smallest non-standard “integer”. This is a consequence of “overspill” - if there were such an element k, then {n
Can we push the process by introducing numbers smaller than any hyperreal number and bigger than zero ? Or maybe the system of hyperreal numbers (or maybe surreal numbers) is already closed by this process and so cannot be expanded again ?
So glad to see more attention on nonstandard analysis. For the uninitiated, differentiability and nonstandard differentiability are equivalent although NA uses stronger axioms in its foundations. The main benefit is in some of the simplifications it makes to standard proofs and also how it carries forward the classical intuition of the early infinitesimal calculus of Leibniz, Euler, Gauss, Fourier and Newton into modern mathematics (although Newton had a slightly different approach). You can also do probability with nonstandard analysis (see Nelson) and measure theory (see Loeb measures), and it's attracted the attention of analysts like Terence Tao. Since you have a background in algebra, it might also be a cool idea to cover smooth infinitesimal analysis which achieves similar results with actually fewer axioms than standard ZFC mathematics, using a foundation of category theory and a weaker form of logic.
I would love to see the video about integrals! Could you also make a video about Conway's surreal numbers and surreal analysis? It is similar topic but the approach is slightly different. I am also interested what can hyperreal or surreal analysis offer us that would be different or at least more elegant then standard real analysis because this approach to derivatives looks like it is just a matter of convention and definition but conceptually it is really close to standard limit based definition
I'd love to see the more rigorous side of hyperreals! Maybe show some analysis style theorems involving them, especially where they might be differences!
A few years back I got to read non-standard analysis note from Goldbring. People say that it might not produce a brand new theorem, but this approach always fascinates me. It's really fun and challenging when it comes to more complicated objects. If I remember, there were some cool results in number theory which were first verified using nonstanard analysis, which I also think is really awesome.
@Michael Penn Around 8:55 you tell us that an "infinitesimal times any real number is also infinitesimal". Could you work through why that is in more detail? Intuitively, if multiplication by 0 is always zero, then I guess it makes some kind of sense that multiplication by some epsilon would always render an infinitesimal--but it's not obvious enough that it goes without saying to me. What makes this work in detail, and are the sizes of the infinitesimals then only decided by their real coefficients? Sorry for the (probably) very basic questions (not a mathematician)... Thanks in advance.
(I'll use "e" except writing epsilon) The thing of beeing infenitesimal can be written by writing (separetely, otherwise it wouldn't be a first order sentence) sentence for any positive real number r: |e|0 |e|0: |ex|
Great video. I think a lot of physics informally use infinitesimals than limits when they derive stuffs so this video helps. Yes to integral calculus with non-limits.
Why is binomial expansion used for proving the power rule instead of logarithmic differentiation? Binomial expansion only proves the case when x is a positive integer, whereas logarithmic differentiation can prove the case for when x is any real number
What is f(x + epsilon), as you put it? In other words, is there a natural way to extend (co-extend) real functions to hyperreal space HYP? Can we put a topology on HYP so that R is dense and open? What can be said about the usual branches of Real Analysis in this new space? E.g. is there a Fourier Analysis on it? What’s the best we can say about the hyperreal numbers as an algebraic structure? Is it algebraically closed? Can we define hyperreal manifolds in the obvious way (charts take value in HYP^n instead of R^n)? If so, are there classification theorems for one, two or three-dimensional hyperreal manifolds?
There is extension of every function f:A→R where A={x in R: phi(r1,...,rn)} (r1,...,rn some real numbers. phi is sone formula or sentence) to function f*:A*→R* where A={x in R*: phi(r1,...,rn)} R* cannot be algrebraicly closed. R fulfill ¬ ∃x x²+1=0 hence R* fulfill the same Hence it's not algebraicly closed
Is hyperreal maths equivalent to limit-based? I.e. is there a mapping/translation between both approaches? If not, where do both deviate? I.e. what can be done with one but not the other?
I would really enjoy an integral video and maybe also a note on where we could learn more about this topic like math stack exchange posts or even books. Thanks!
This all stems from the work Abraham Robinson did at Princeton in the 1960s - essentially making rigorous what Newton did and implicitly responding to Biship Berkley's pisstake of Newton's infinitesimals as " souls departed". Robinson approached the issue from logic and his book "Non-standard analysis" really can't be read without a grounding in a mathenatical logic and set theory. Kurt Godel is actually quoted in the preface of the book. There have been a few courses in the US and one or two in Europe which retrace Robinson's work to show how sometimes the non-standard proof is easier (sometimes it isn't). I haven't looked at this stuff for over 30 years so I can't remember the details.
Very nice presentation. I like that you focus on presenting the main idea instead of overwhelming the listener with details of definition. Hyper real numbers are truly weird and can you prove that they actually exist? Is this definition sound? They are very attractive, because they simplify calculus a lot. Physicists often use this idea in explanations without spelling it out. Therefore, I am not surprised but I remain puzzled as I have been for a long time. Maybe, some day someone will makes sense of these numbers like Gauss and Euler made sense of the imaginary numbers by placing them on the y axis in the 2D vector space.
@@kilianklaiber6367 You provably cannot construct an "explicit" model of the hyperreal numbers. To be technical, to do so, you need a free ultrafilter of the set of natural numbers and to construct that, you need the axiom of choice. So simplifying a bit, the situation is "hyperreal numbers exists only if we make some strong assumptions about the foundations of mathematics and even then, nobody knows (and cannot ever know) what they really look like" 🙂
Frankly, I do not understand your comment and what this all has to do with the axiom of choice. The axiom violated by the hyperreal numbers is the axiom of Archimedes, according to my understanding...
@@kilianklaiber6367 That is correct, the hyperreal numbers are what is called non-Archimedean, but that is not in itself a problem. There are plenty of number systems which are non-Archimedean, with infinities and infinitesimals, and which we can explicitly construct, one example being the Levi-Civita field (has an article on Wikipedia). But those systems don't have properties which are essential if want to use them for calculus. The hyperreal field has those properties, we can consistently assume that hyperreal numbers exists, but if we want to *prove* they exists and that all our results we proved using them are valid, we need the axiom of choice. And because of this, we'll only ever know they exists, without ever having a description of their complete structure.
Indeed the def with limit in the real number is cleaner but using the hyperreal number is a great way to simplify the notations as soon as we know the link to the the real numbers.
I don’t think the \omega used here necessarily is the ordinal \omega ? I know he says it is, but at least if arriving via nonstandard models of the natural numbers, the infinite values are *not* ordinals.
*YES, PINKY! THE HYPERREALS!* I've been living for this moment ever since I've read an obscure intro on the subject in some murky corner of the internet 😛
I loved this video! I tried to find it again to share with a friend, but it was hard to find, as hyperreal/infinitesimal numbers weren't mentioned in the title or description. 😅
How does 0 minus an infinitesimal work? For example: (0 minus epsilon) times 3 Is the standard part still just 0? Or is there a concept of negative 0 in this space? Similarly, what about division by 0 +/- epsilon? Are we essentially able to calculate a result as if we were taking a limit from one side in this space?
0-epsilon is some infenitesimal non zero real number* infenitesimal is infenitesinal so standard part of this is 0. Not there is no concept of negative 0. Here same first order logic sentences works so sentences like (for any x) x≠0=> 0/x=0, x*0=0, x≠0=>x^0=1, x+0 =x, etc. Holds for all hyperreal numbers
I've read about this about 17 years ago while starting the math career (as a curiosity), the Spanish Carlos Ivorra Castillo has a really nice book on this stuff (I'm from Argentina). Didn't get in contact with this nonstandard analysis anymore for some years but it's nice to revisit it!
I don't think there is much about any Isomorphism about anything in this case (Isomorphism is between sets, not functions. And R and R* aren't isomorphic). But in general (via transfer principle) the things that work in standard Analysis will work as same in non-standard analysis
Sir can we prove that there does not exist any real number between an infinitesimal and zero in the non Archimedean field and hence infinitesimals exists?.
I'd love to see a full course on nonstandard analysis including more formal definitions and deriving results from "standard" analysis using this approach.
Yes
Take my course, Math 5154/CompSci 5203/CompE 5803/Philos 4354, at Missouri S&T. One of the required textbooks is “Nonstandard Analysis for the Working Mathematician”. Once you’ve succeeded in that course, you will be ready to study some nonstandard analysis.
@@writerightmathnation9481 nah bro just promoted his math course 💀
@@danielthonk7481
At least it would include a correct treatment of the construction discussed in this video. You prefer to learn it wrong, I guess. Have fun with that.
It's pretty difficult to develop Non Standard Analysis rigorously, for starters it requires the use of Axiom of Choice.
That’s quite the limitation
A video about integrals, in this fashion, sounds lovely.🙂
we must see how to integ8 with infinitesimal approach
Well, of course we want more hyperreal number videos from you, Michael/Stephanie, as these are so fascinating, and novel to me.
Oh, and as I said before, you both are THE best youtube math channel in existence, okay?
I must have missed something... Who is Stephanie?
@@writerightmathnation9481 Stephanie is the editor for Michael's videos.
@@titan1235813
Thanks. I missed that somehow. :)
In fact, John Conway's *surreal numbers* (you can search for them on the web) are the most natural extension of real numbers, as they include hyperreal numbers, ordinal numbers, and many others, while being very simple to describe and manipulate. May be one day you could find a way to explain them in your wonderful chalkboard approach of maths concepts by solving problems ? Amazing video anyway. Thank you Michael !
RIP Conway
I just wrote a reply and it disappeared. I don’t understand why???
@@MMarcuzzo Fuckin' COVID
@@writerightmathnation9481 If you sort the comments by "Newest first" then you will see it again.
@@writerightmathnation9481 Spam filter. It's not any specific word, it's some complicated algorithm.
I didn’t watch this video because I was in honors calculus and this is how I was taught. But I love that you created this.
Nonstandard analysis provides an excellent approach to introduce Calculus and their applications to high school students.
Please, add more content about nonstandard analysis! It really feals like a whole new world to explore - and so much more intuitive and fun than epsilontic calculation. I am enjoying this so much. Sadly it was only mentioned as a side-note during my time at the university.
yes michael, i'd love to see integrals being defined using hyperreal numbers. this is great video, the production is very professional. the only thing i miss from your older stuff are those backflip transitions lol
Finally! someone making sense with infinity!
Please more Hyperreal videos! 🙏
A course or a series would be epic!
This is how we were taught mathematical analysis in college (computer engineering degree in Italy). Fun thing is that it makes the informal notation for limits that one would write on a napkin into something that's actually mathematically correct and well founded.
I ran across nonstandard calculus during my second semester, taking my first course in the ordinary limit-based one and for a while was a bit obsessed about it, so this video is a nice reminder of the "old times" 🙂
That being said, some people (used to?) tout it as a simpler way to teach calculus, involving fewer limits, less quantifiers and so on. I don't agree with this at all, since to do anything at all with them, you need a so-called transfer principle, which basically states that any first-order formula in the language of the real numbers which is true in the actual real numbers is also true in the hyperreal numbers when translated appropriately. This is what enables one to prove things like that addition is commutative, that every number has an integer part and so on. Unfortunately, it is very easy to screw up with the transfer principle.
Example 1: Every real number is less than some natural number, so by the transfer principle, every hyperreal number is less than some natural number, right? But that is obviously false -- well, I forgot to translate! The correct statement is "every hyperreal number is less than some hypernatural number", and what even are those (the fixed points of the hyperreal extension of the integer part function, obviously 😀)?
Example 2: every bounded set of reals has a least upper bound, so every bounded set of hyperreals has a least upper bound. Also obviously false, since the set of infinitesimals is bounded (by 1, e.g.), yet definitely doesn't have a least upper bound. This time, I accidentally used a second-order formula, referring to sets of reals! So in my view, you're just exchanging the difficult, yet manageable concept of limits for a logical concept of the transfer principle which is at least as hard to use correctly.
Case in point, your proof that d/dx e^x = e^x is incorrect! There is no way to justify the claim you implicitly used that st(t)^ε = st(t^ε) -- definitely not using the transfer principle, since st is not part of the language of real numbers. Incidentally, that claim is false in a similar way the claim that lim (x -> 0) f(x)^g(x) equals lim (y -> 0) (lim (x -> 0) f(x))^g(y) is false.
in my(layman) opinion, st(t)^eps is valid but st(t)^omega is not valid because eps is closed (by real numbers as faces) but omega is open.
It means that eps connected to real numbers but omega disconnected. Or,for natural numbers all vice verse(zero is open but infiniy is closed).
In other words.
Line is not shape (has no closed interior and open exterior). epsilon and omega solve this problem.
Number line became common shape, where numbers could exist.
Can’t you prove the transfer principle once you have a construction like the ultra filter construction?
@@JM-us3fr yeah the transfer principle is Los' theorem from model theory. You prove it by induction on complexity of formulas
I think of the hyperreals as a discrete infinite dimensional vector space over the reals, is this appropriate?
The ordering relation is just the lexicographic ordering from higher dimensional index to lower dimensional index,
addition is just vector addition, multiplication is distribution over a linear combination of dimensional index shift operators.
The only thing I can't wrap my head around is non-integer exponentiation on this space.
for real-exponent you can 'complete' the space by adding a continuity of dimensions, so eps^(1/2) is a dimension half way between the eps dimension and the 1 dimension,
but what does eps^eps mean? Maybe you have to 'recursively' (a second order theory?) hyper-complete the space for each layer of exponent, for example on this layer eps^eps requires that we also do the same hyper-real construction on top of our completion of the dimensions.
Although, let's say this idea does work without the 'transfer principle', any proofs you do now require analogies to limits over real valued vector spaces, and limits over the dimensional index as well, which is quite complicated.
@@MagicGonads I don’t think the space you described is necessarily complete, and the hyperreals are complete. As an example of the space you described, one could take epsilon from the hyperreals and adjoin it to R. This would be an infinite dimensional field extension, and would even be ordered since it inherits the order from hyperreals
I would like to see more of these hyperreal number videos. Not just defining integration, but also simple stuff like proving that exponent rules still work with infinitesimal values.
A proof for that is by transfer: for all a,b in R m: e^(a+b) = e^a*e^b. Extend R to R* and you’re done
came here to divide by dx.
GIVE ME MY INTEGRAAAAAALS!
My calculus professor gave us a brief introduction to non-standard analysis, he seemed enthusiastic about it, also because he saw it as a fruitful attempt to reintroduce Leibniz's notation. And in fact it would be interesting to see other videos about it, above all for the clarity with which you, professor, are able to illustrate even rather complicated concepts
Can't believe I could actually keep up with this one! Nice
Nice room - I like the decor and the multiple chalkboards
Though this was hyper-interesting, I somehow feel this is just a convoluted definition of a limit.
Than you, professor.
I feel it the other way around : one way of constructing real number from the rational numbers, is to use a "good" set of bounded rational sequences and do the quotient of this, actually, ring with a maximal ideal. I don't know a lot of hyper real number, but the "st" function looks a lot like the rest of a quotient operation. And because of that, it gives me the impression that all the properties of limits (and so derivatives and primitives) are more a consequence of some underlying existence of an algebraic structure than topology.
The limit is a convoluted definition of this. Plus every great mathematician and physicist thinks in terms of infinitesimals. That is how Euler got all his results, with great intuition of infinitesimals
Modern calculus teaches that good intuition is somehow a bad thing and forces limits. You could teach yourself all of calculus with just the intuition of infinitesimals, without taking a course or having a teacher guide you, because it is straight forward
Then the 'rigor' people got involved and said infinitesimals weren't legit, but they always were; they just had bad intuition
@@pyropulseIXXI what would sin(1/epsilon) be? What kind of operation one can do with infinitesimals? I don't know much about the hyperreals but rigor is important to know whether what you write can make sense or not.
@@qwertyqwerty-jp8pr Any operation that can be done on the reals can be done on the hyperreals; this is the key property. Sine is defined for real numbers only, so sin(1/epsilon) is clearly undefined.
Thus, if a (b+c) = ab + ac, then this works if the numbers are all hyperreals. You can just axiomatize this, if you to teach in pure intuition, such as 'Transfer Principle' and 'Closure Principle.'
I taught myself calculus using the intuition of infinitesimals; it is very power and very obvious.
Calculus was built on infinitesimals, and they abandoned that intuition in favor of limits because of 'rigor.' (or rather, because of a failure to rigor-tize infinitesimals, which was done in 1960 by Abraham Robinson).
Now, intuitive and super obvious things tend to have insanely difficult proofs. Just because a system exists in which the rigor is easier does not mean we abandon the intuition that leads to such amazingness.
Just look at the real numbers; hardly anyone actually proofs the extension under Dedekind cuts, so where is your supposed rigor? It is just assumed to exist as an axiomatic statement. People just assume the reals exist in an intro calc course, so why abandon good intuition by not assuming the hyperreals exist in an intro calc course? If you do, you can do rigorous calculus proofs that would take a course in *_real analysis_* to be able to do
Likewise, one can do this with the hyperreal numbers, for the purposes of teaching intro calc, and posit that the operations that work on the reals also work on the hyperreals. We rely on our intuitive understanding of the reals in order to work with them, and develop extreme proficiency, within calculus. Again, one can do this with infinitesimals and obtain an unrivalled mastery of calculus with such an intuition.
Two axioms are:
*(1)* - Every fact of traditional mathematics remains true.
*(2)* - If there is an object with some property, then there is also a standard object with this property
This is called the *Closure Principle.*
Bam; if you need absolute rigor, just read _Non-Standard Analysis,_ by Abraham Robinson. This is beyond the rigor level oft found in a course of _standard analysis._
And with the *_Closure Principle_* above, teaching via infinitesimals is above the rigor of an introductory calculus course, and it is far more intuitive.
This is why Calculus should be taught in the form of infinitesimals. This is how Euler thought, and his results speak for themselves.
I suggest the book _Analysis with Ultrasmall Numbers;_ it allows rigorous proofs at a college introductory to calculus level.
I also suggest _Infinitesimal Calculus,_ by Henle & Kleinberg
I suggest reading Euler's two books on calculus; his insights are amazing and can be replicated, in an inferior manner, if you, too, start thinking in terms of infinitesimals
@@qwertyqwerty-jp8pr If sine is defined only on the reals, then sin(1/ε) is undefined; 1/ε is a hyperreal number, equal to ω.
sin(ω) is not defined.
sin(ε) is defined, since st(ε) = 0, and thus, sin(ε)
You'd need to take the standard part of 1/ε, which is the closest real number to t, but this is undefined, as there is no closest real number.
this is because for any real number you give me as the 'closest' real number, I can find another real number that is closer. Say you give me _a;_ I can give you _a_ + 1.
Now, if it was 1/ε + 1, then the closest real number would be 1
Funny, I'm currently attending a seminar on nonstandard analysis where I'll be presenting dual numbers followed up by hyperreal numbers.
OH perfect topic!!!! Please consider doing the rigorous definitions and the ultrafilter stuff!!!
Oh no, I want integrals with that so badly
Excellent, nonstandard analysis is a topic I've been studying. Would love for there to be more videos on the subject.
Yes, please; it would be interesting to see integrals treated this way.
BROOO this is amazing, keep doing this
I'd love more videos on the hyperreals-alternative number systems are always cool, and it would be interesting to see what derivatives/integrals are harder or easier using the hyperreals.
I love this video. I think nonstandard analysis is much more interesting, useful, and clear for learners and users of calculus.
The epsilon-delta definition of a limit always seems to be the hardest part of tutoring calculus. 3B1B did a lot to help in this area, but many people still don't get it.
I've heard many physicists, engineers, and UA-camrs talking about a derivative as how a "tiny" change to x results in a "tiny" change to f(x). And while talking about integrals, they talk about how the infintely many "tiny" changes in x result in the infinitely many changes to f(x) and add up to the standard integral.
Basically, what I'm saying is that nonstandard analysis matches our intuition much better than standard calculus.
I know I've been critical in the past but I was really impressed that you gave a brief introduction to the hyper reals. I was confused by the simplification math of the last e^x example though but aside from that it was pretty clear. I'd like see more examples and information on derivatives, integrals and other information on hyper reals or nonstandard analysis in general and eventually learn how to use it. I hope you make more related videos in the future. Thanks for the video.
This is the first time I've seen you using camera transitions instead of clearing the board on a static camera and I just wanted to say it's very good for my ADHD and I didn't think that I'd want that but it's really helping me follow you for much longer.
I have a non-standard analysis book written by Abraham Robinson from 1960s that is also signed by him
I would like to see your summary of defining an integral using hyperreals!
This nonstandard approach is a modern one that (slightly) simplifies proofs!
Yes please on the hyperreal definition of the integral!
At 15:55, why are we allowed to pass the epsilon exponent through the standard part function?
Hi Dr. Penn!
Very cool!
I've always been fascinated by this but need get actually had the guts to learn it on my own. So please keep this going
Continue the great work , big love from IRAQ 💖🇮🇶
A followup on calculus in the hypereals would be fun. 👍 Maybe prove the fundamental theorem of calculus using infinitesimals for example.
Also another fun related tangent might be to delve into the Surreal Numbers, an invention of John Conway similar to the hyperreals which had some applications in game theory.
Yes, integrals and solving differential equations. Perhaps, you could solve Riemann's hypothesis if there is space left in the blackboard.
At 11:45 you also have to replace the function from R to R with a function from R* to R*
"Are you a limit?"
"Well yes, but no"
Yes please! More videos on the Hyperreals 🙏 This is such an interesting topic that ks very much missing from the standard curriculums ❤
honestly this is the kinda stuff that i think about in my head when learning calculus but this put that into words that could be written
I got finished with my AP Calculus BC test a couple days ago, now I'm watching this. Very interesting, and I am that I am, partly hair, partly ham, but always cool.
This is super interesting! Limits are difficult to explain rigorously to many of the people who need to use calculus - epsilon-deltas are much more conceptually tricky than derivatives and integrals, IMO. So schools often just handwave their way through the foundations of calculus leaving students feeling like "so what was the point of THAT?". Whereas it looks like the hyper-real derivations of a lot of basic calc identities are really easy to follow with just a basic foundation in high school algebra. This would great for teaching!
I feel like part of the problem is that by the time someone has studied enough math to actually *care* about the hyper-reals, they've probably spent so much time with epsilon-deltamanship that it's no longer counterintuitive at all. Even after watching this, epsilons still feel like the 'natural' way to think about 'very small numbers'. So I think a lot of this seems superfluous to people who actually study analysis, so it gets discarded.
Anyway I'm increasingly convinced that nonstandard analysis is unfairly maligned. It'd probably be easier to teach calc this way at least at the high school level - even if I *still* have no intuition for what the hell an infinitesimal is or why we'd need to work with numbers provably too small to matter.
In the nilpotent formulation, it's conceptually simple to just use nilsquares, but when doing higher and/or multivariate derivatives it becomes cumbersome. Just saying "standard part" takes care of all the H.O.T.s in one go. It's similar to saying "limit" but it's more obvious what's happening
I love the new style. Keep up the good work ♡
So more beautiful
The formal definitions are kind of trippy and depend on the Axiom of Choice. I read a short-ish book on Non-Standard Analysis that covers math major Real 1 (or "advanced calc") from the NSA setup. It feels a lot like the usual limits and epsilon-delta, but through a decoder ring. Overall, I think standard won out for good reason, but it's kind of interesting that it works and occasionally you get strange new proofs like the Tychonoff product theorem for compact topological spaces.
I love the fact that hyperreal numbers sort of validates how Newton (or was it Leibniz?) did calculus which was at some point regarded as lacking rigor.
Well now I want to see those definitions without the quotation marks. Real hardcore textbook definitions.
Be careful what you wish for...
Is there any place where this makes things easier (besides saving on writing) than just using limits like normal?
His choice of the “smallest” infinite ordinal is unfortunate,as it is provable that there cannot be such a smallest non-standard “integer”. This is a consequence of “overspill” - if there were such an element k, then {n
Can we push the process by introducing numbers smaller than any hyperreal number and bigger than zero ? Or maybe the system of hyperreal numbers (or maybe surreal numbers) is already closed by this process and so cannot be expanded again ?
I’m really curious how one might simplify differential geometry using hyperreals.
So glad to see more attention on nonstandard analysis. For the uninitiated, differentiability and nonstandard differentiability are equivalent although NA uses stronger axioms in its foundations. The main benefit is in some of the simplifications it makes to standard proofs and also how it carries forward the classical intuition of the early infinitesimal calculus of Leibniz, Euler, Gauss, Fourier and Newton into modern mathematics (although Newton had a slightly different approach). You can also do probability with nonstandard analysis (see Nelson) and measure theory (see Loeb measures), and it's attracted the attention of analysts like Terence Tao.
Since you have a background in algebra, it might also be a cool idea to cover smooth infinitesimal analysis which achieves similar results with actually fewer axioms than standard ZFC mathematics, using a foundation of category theory and a weaker form of logic.
I'd love a series of videos about nonstandar calculus. 🤓👍
I would love to see the video about integrals! Could you also make a video about Conway's surreal numbers and surreal analysis? It is similar topic but the approach is slightly different. I am also interested what can hyperreal or surreal analysis offer us that would be different or at least more elegant then standard real analysis because this approach to derivatives looks like it is just a matter of convention and definition but conceptually it is really close to standard limit based definition
I'd love to see the more rigorous side of hyperreals! Maybe show some analysis style theorems involving them, especially where they might be differences!
Oooh, finally getting around to these is VERY interesting to me! Definitely very interested to see some integral stuff!
A few years back I got to read non-standard analysis note from Goldbring. People say that it might not produce a brand new theorem, but this approach always fascinates me. It's really fun and challenging when it comes to more complicated objects. If I remember, there were some cool results in number theory which were first verified using nonstanard analysis, which I also think is really awesome.
@Michael Penn Around 8:55 you tell us that an "infinitesimal times any real number is also infinitesimal". Could you work through why that is in more detail? Intuitively, if multiplication by 0 is always zero, then I guess it makes some kind of sense that multiplication by some epsilon would always render an infinitesimal--but it's not obvious enough that it goes without saying to me. What makes this work in detail, and are the sizes of the infinitesimals then only decided by their real coefficients? Sorry for the (probably) very basic questions (not a mathematician)... Thanks in advance.
(I'll use "e" except writing epsilon)
The thing of beeing infenitesimal can be written by writing (separetely, otherwise it wouldn't be a first order sentence) sentence for any positive real number r:
|e|0
|e|0:
|ex|
@@antares2804 Thanks for the clarification. Makes perfect sense.
Cool classroom! I like the paint job, it reminds me a bit of Oliver Byrne's Euclid.
Definitely I want to see how to calculate the integrals with the hyper real numbers
Finally! 🎉🎉
Great video. I think a lot of physics informally use infinitesimals than limits when they derive stuffs so this video helps. Yes to integral calculus with non-limits.
GIVE ME THOSE INTEGRALS!!!!!
Why is binomial expansion used for proving the power rule instead of logarithmic differentiation? Binomial expansion only proves the case when x is a positive integer, whereas logarithmic differentiation can prove the case for when x is any real number
You didnt prove many details but still the video and the exposition was great❤❤
What is f(x + epsilon), as you put it? In other words, is there a natural way to extend (co-extend) real functions to hyperreal space HYP? Can we put a topology on HYP so that R is dense and open? What can be said about the usual branches of Real Analysis in this new space? E.g. is there a Fourier Analysis on it? What’s the best we can say about the hyperreal numbers as an algebraic structure? Is it algebraically closed? Can we define hyperreal manifolds in the obvious way (charts take value in HYP^n instead of R^n)? If so, are there classification theorems for one, two or three-dimensional hyperreal manifolds?
There is extension of every function f:A→R where A={x in R: phi(r1,...,rn)} (r1,...,rn some real numbers. phi is sone formula or sentence) to function f*:A*→R* where A={x in R*: phi(r1,...,rn)}
R* cannot be algrebraicly closed.
R fulfill ¬ ∃x x²+1=0 hence R* fulfill the same Hence it's not algebraicly closed
Very interesting video! I sure would like to see the part about integrals. Thanks for all your work!
I kind of like this more than the limit right now. Unfortunately 0
It is correct, just here you haven't formal construction or any other justification of existance of such numberd
@@antares2804 By “doesn’t sound correct” I meant it’s not intuitive.
Is hyperreal maths equivalent to limit-based? I.e. is there a mapping/translation between both approaches?
If not, where do both deviate? I.e. what can be done with one but not the other?
Yes they are equivalent approaches
Thanks
I would really enjoy an integral video and maybe also a note on where we could learn more about this topic like math stack exchange posts or even books. Thanks!
This all stems from the work Abraham Robinson did at Princeton in the 1960s - essentially making rigorous what Newton did and implicitly responding to Biship Berkley's pisstake of Newton's infinitesimals as " souls departed". Robinson approached the issue from logic and his book "Non-standard analysis" really can't be read without a grounding in a mathenatical logic and set theory. Kurt Godel is actually quoted in the preface of the book. There have been a few courses in the US and one or two in Europe which retrace Robinson's work to show how sometimes the non-standard proof is easier (sometimes it isn't). I haven't looked at this stuff for over 30 years so I can't remember the details.
Wow. That was really cool.
It feels like magic but cooler. So excited
The topic sounds interesting. Are we going to learn about infinitesimal numbers, i. e. numbers greater than zero but smaller than any real number!?
Very nice presentation. I like that you focus on presenting the main idea instead of overwhelming the listener with details of definition.
Hyper real numbers are truly weird and can you prove that they actually exist? Is this definition sound?
They are very attractive, because they simplify calculus a lot. Physicists often use this idea in explanations without spelling it out. Therefore, I am not surprised but I remain puzzled as I have been for a long time.
Maybe, some day someone will makes sense of these numbers like Gauss and Euler made sense of the imaginary numbers by placing them on the y axis in the 2D vector space.
@@kilianklaiber6367 You provably cannot construct an "explicit" model of the hyperreal numbers. To be technical, to do so, you need a free ultrafilter of the set of natural numbers and to construct that, you need the axiom of choice. So simplifying a bit, the situation is "hyperreal numbers exists only if we make some strong assumptions about the foundations of mathematics and even then, nobody knows (and cannot ever know) what they really look like" 🙂
Frankly, I do not understand your comment and what this all has to do with the axiom of choice. The axiom violated by the hyperreal numbers is the axiom of Archimedes, according to my understanding...
@@kilianklaiber6367 That is correct, the hyperreal numbers are what is called non-Archimedean, but that is not in itself a problem. There are plenty of number systems which are non-Archimedean, with infinities and infinitesimals, and which we can explicitly construct, one example being the Levi-Civita field (has an article on Wikipedia). But those systems don't have properties which are essential if want to use them for calculus. The hyperreal field has those properties, we can consistently assume that hyperreal numbers exists, but if we want to *prove* they exists and that all our results we proved using them are valid, we need the axiom of choice. And because of this, we'll only ever know they exists, without ever having a description of their complete structure.
@@jakubledl1602 o. K. Interesting. I would be interested in the proof of their existence using the axiom of choice.
Oh! You could extend this into talking about the surreal numbers, which I think hyperreals are a subset of I think.
I think that would probably require the more formal definition of surreal numbers though.
Indeed the def with limit in the real number is cleaner but using the hyperreal number is a great way to simplify the notations as soon as we know the link to the the real numbers.
Are there hyperimaginary numbers, too? I think "The Imaginary Infinitesimals" would make a good band name.
You can make nonstandard extensions of complex number as well
So cool to see this after so many years. Dr. Greg Foley taught Calc 2 with hyperreals at App State.
Yes. I would definitely like to see how to use hyperreals to do integrals!
If omega is the "smallest" infinite ordinal, then what's omega - 1, for example?
I don’t think the \omega used here necessarily is the ordinal \omega ? I know he says it is, but at least if arriving via nonstandard models of the natural numbers, the infinite values are *not* ordinals.
The system of hyperreal numbers and the calculus that descents from them is more or less formal then the standard one?
*YES, PINKY! THE HYPERREALS!* I've been living for this moment ever since I've read an obscure intro on the subject in some murky corner of the internet 😛
I'd love a video on integrals with infinitesimals
I loved this video! I tried to find it again to share with a friend, but it was hard to find, as hyperreal/infinitesimal numbers weren't mentioned in the title or description. 😅
Would love to see integrals, but more than that, how do hyperreals relate to dual numbers, if at all?
How does 0 minus an infinitesimal work? For example:
(0 minus epsilon) times 3
Is the standard part still just 0? Or is there a concept of negative 0 in this space?
Similarly, what about division by 0 +/- epsilon? Are we essentially able to calculate a result as if we were taking a limit from one side in this space?
0-epsilon is some infenitesimal
non zero real number* infenitesimal
is infenitesinal so standard part of this is 0.
Not there is no concept of negative 0. Here same first order logic sentences works so sentences like (for any x)
x≠0=> 0/x=0, x*0=0, x≠0=>x^0=1, x+0 =x, etc. Holds for all hyperreal numbers
Video on integrals please! Seems to me this way of understanding calculus is much more intuitive
Yes, I hope you do one on Integrals.
Nice. Never thought espilon or omega had any value outside number theory!
This is how I imagine calculus every time I work with it. This is more natural than limits😆
I've read about this about 17 years ago while starting the math career (as a curiosity), the Spanish Carlos Ivorra Castillo has a really nice book on this stuff (I'm from Argentina). Didn't get in contact with this nonstandard analysis anymore for some years but it's nice to revisit it!
Can you get the same differential truck by just defining *R as a+be where e^2=0
I assume that there is either an Isomorphism between the Limit Derivative and the Hyperreal Derivative or some sort of equivalent notion?
I don't think there is much about any Isomorphism about anything in this case (Isomorphism is between sets, not functions. And R and R* aren't isomorphic).
But in general (via transfer principle) the things that work in standard Analysis will work as same in non-standard analysis
Sir can we prove that there does not exist any real number between an infinitesimal and zero in the non Archimedean field and hence infinitesimals exists?.