Calculus WITHOUT limits! 

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.

we must see how to integ8 with infinitesimal approach

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 must have missed something... Who is Stephanie?

@@writerightmathnation9481 Stephanie is the editor for Michael's videos.

@@titan1235813 Thanks. I missed that somehow. :)

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 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

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

Be careful what you wish for...

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.

It is literally just the same thing as 'standard' calculus. The only difference is the justification of the math that is being done, and the 'intuition' behind .t

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.

I think that would probably require the more formal definition of surreal numbers though.