# Calculus without limitsSeptember 17, 2014 5:23 PM   Subscribe

Hyperreal numbers: infinities and infinitesimals - "In 1976, Jerome Keisler, a student of the famous logician Tarski, published this elementary textbook that teaches calculus using hyperreal numbers. Now it's free, with a Creative Commons copyright!" (pdf—25mb :)

also btw :P
posted by kliuless (34 comments total) 78 users marked this as a favorite

It's simply unfathomable to me that anyone could consider this a sound pedagocical approach. Only a mathematical logician could love this book. Limits, and traditional delta-epsilon proofs are considered "too difficult" for beginners to understand, but they are expected to swallow the heavy dose of mathematical logic (here relegated to the Epilogue) to actually construct these "Hyperreal numbers". Sheer folly. For instance, on p. 907 one finds

DEFINITION:

A real statement is either a nonempty finite set of formulas T or a combination involving two nonempty finite sets of formulas S and T that states that "whenever every formula in S is true, every formula in T is true"

Seriously, that's easier to understand than limits (which are a useful concept on their own)?

The only reason to write a calculus text like this is as a stunt, to show that it can be done. This is no way to teach calc.
posted by crazy_yeti at 6:09 PM on September 17, 2014 [3 favorites]

When I tried reading Kiesler's book a few year ago I found it difficult to make sure that every statement stayed in first-order logic (a requirement for a mathematical statement to automatically transfer over to the hyperreals). Having to be paranoid about every statement I made not accidentally quantifying over predicates made me start to long for some good ol' ε's and δ's that were just arbitrary and > 0. I'd dread the idea of teaching someone calculus for the first time and worrying about formal logic theory. Maybe you'd just move quickly, not mention it, and hope no-one trips over it.

Of course, I'd had years of ε's and δ's in functional analysis, so maybe it's just what I was used to. Has anyone had a better experience?
posted by benito.strauss at 6:23 PM on September 17, 2014 [2 favorites]

Heh, I started writing my previous comment before crazy_yeti had posted. I guess he/she isn't going to provide that "better experience" I asked about.

Still, nice that the .pdf is now available for free over the internet. And that article "The Logic of Real and Complex Numbers" by John Baez ("The only reason I ever go on Google+") is pretty fascinating — I look forward to reading it more deeply. Thanks for the post.
posted by benito.strauss at 6:30 PM on September 17, 2014

Personally, something like the hyperreals was the way I intuitively grasped much of calculus in high school. Discovering the nonstandard approach was a revelation because I realized that my intuitions had a valid formal basis. I started reading Keisler's book for kicks a while ago (I've yet to finish) and for the most part the experience has been one of deja-vu as I recognize approaches I already somewhat knew. Frankly, I never really grokked the epsilon-delta approach until well into college, so count me as one vote in favor of the hyperreals having potential.
posted by Wemmick at 6:30 PM on September 17, 2014 [3 favorites]

This book is free now, assuming values of now greater than 2002.

I used the Kiesler textbook for my Honors Calculus section when I was a freshman. I used the book before it was even the first edition. We received new chapters on Mondays, xeroxes of the galley proofs. Then on the next class, we'd get errata and have to write corrections all over our galleys. We didn't get a printed, bound book until the second semester.

This class was an absolute disaster. The teacher had obvious difficulty understanding how he was supposed to teach the infinitesimal calc concept. The class full of Honors students (back in the days before AP Calc but generally the most mathematically adept students in any calc class section) performed miserably on tests and homework. The teacher seemed infuriated that the students could not understand what he was teaching. Many students (like me) asked for tutoring from grad students, who understood the concepts even less. I remember my tutor complaining that the infinitesimal methods were ridiculous, now watch how you're supposed to do it. So then I'd get back tests with correct answers marked as incorrect, with big red pen marks over my work with notes "that's not the way we taught you to do this!" The math department was infuriated that the class had the lowest test scores and highest dropout rate of any calc class in recent history. Most students transferred to other classes early in the semester, when the early problems became obvious. But I bravely soldiered onward, having been conned into believing this was the latest, greatest concept in teaching calculus and that I would eventually develop superior skills than no other method could achieve. I think there were only like 5 students left in the second semester. The book was abandoned at the end of the year and never ever used again for any calc class.

My strongest memory of this class is the first semester final exam. I had to take a makeup since I had another final scheduled at the same time. The teacher came to the test room, made sure I had no backpack or books, only a couple of pencils, then he gave me the test, and said absolutely do not leave this room, he was leaving and putting a sticker on the door to seal it and he would return in an hour. If I left the room he'd know it and, I'd flunk. Then about halfway through, the fire alarm went off. I continued the test, even after the room began filling with smoke from the ventilators. I was almost done when a security guard broke the seal and entered, and told me to leave. I refused, I had to be there when the teacher came back. I was physically escorted out of the building. I went to get the teacher to tell him what happened, he said I left the room so I flunked. I told him, did you not see the fire trucks and the smoke pouring out of the building? It took a bit of heated argument before he would accept my test for a score. I got the test back on the first day of the next semester, as usual, marked with "that's not the way we taught you."

Well anyway, about the book.. I have seen this PDF of the econd edition, but I have been unable to locate my first edition, I know I still have it in a storage box somewhere. According to the preface of the second edition, it appears that the basic lessons on the theory of infinitesimals, how it applies to calculus, and what the hell is Epsilon anyway, which we labored over for weeks, has been demoted to an epilog in the second edition. That is absolutely infuriating to have my primary calculus lessons demoted to an optional footnote. But I am not surprised. This whole damn calculus instruction method is ass backwards. It doesn't even get to differential equations until the last chapter, according to the new preface, this chapter was added only in the second edition. But I recall covering this differentials in the first semester, in a haphazard manner, scattered around the other chapters. I clearly remember this utterly ridiculous diagram that is now at the beginning of the chapter on differentiation. Yes, that is Mr. Infinitesimal riding a little unicycle with a steering wheel, which confounds the whole concept since the wheel would rotate around the Y axis, making it move in the Z axis (not pictured in the diagram).

Recently I have been reviewing several Calc 101 courses online, because somehow I have become a professional mathematician as part of the capacity of my job. I came upon another one of these "New Method" calc classes, it reminded me a lot of Kiesler. The course starts Chapter 1 with a common explanation of a function, there's this little box with a funnel on top, you drop in a number, turn the crank, and a number comes out the side. I remember my High School math teacher drew this same diagram. Then in Chapter 2 it says, let's zoom ahead to show you what the final goal of this course is, let's look at e, the exponential function, and let's expand it as a Taylor function and learn the ultimate theory of all functions. And then it goes off into the wild blue yonder and never comes back to earth. I questioned the value of this method, and showed it to our resident Math PhD ABD, he was baffled by this method just as much as I was. He asked me if I was sure this wasn't 2nd year calc. I found the MIT Open Courseware Calc 101 section, it is legendary. I watched the video lectures, the final exam review was almost identical to Chapter 2 of the New Method course.

I have not gone back to work through Kiesler's second edition in detail, I want to locate my first edition to compare. But I have examined it and it contains the same ridiculous methods I remember learning so many years ago. It seemed just as baffling, just as disorganized, and just as terrible a method of instruction. If you have any regard for your mathematical abilities, absolutely do not use this textbook. It is likely to do more damage than good.
posted by charlie don't surf at 6:48 PM on September 17, 2014 [28 favorites]

My first semester college calculus course used the second edition of this book and it was a fantastic experience. The professor was a great teacher who worked with non-standard analysis. I wish we could have continued with it for the rest of calculus. As Wemmick says, it's a very intuitive approach. You aren't supposed to need the Epilogue, that's why all the heavy-duty higher math is relegated in there -- just learn the rules for hyperreal numbers and the math will all work out.
posted by Harvey Kilobit at 6:52 PM on September 17, 2014 [3 favorites]

The hyperreals are incredibly fun to learn about, but I'm not convinced that it's a particularly useful approach to teach early on. I think I agree with the early posters that most of the fun of Robinson analysis (the standard non-standard analysis) is in the logic and modern algebra needed to construct the hyperreals. Without the actual formalization of the hyperreals, the calculus that follows from it is basically just old school (pre-epsilon-delta) Liebniz in fancy clothes.

Tangent time: What I think would be really interesting to see is a formalization of Newton's calculus. Both the Weierstrauss calculus (and it's accompanying standard Analysis) and the Robinson analysis (and its accompanying calculus) are really based in Leibniz. And that makes sense. The Leibniz calculus was designed to do math, and mathematicians developed analysis.

Newton is different. Newton developed his calculus to do physics. And a lot of physics and applied mathematicians use Newtonian notation instead of Leibnitz because Newton's way of thinking was well suited to do mathematical science.*

But to my knowledge, Newton's approach really hasn't be formalized the way Leibniz' has. Which is a real pity, because so many scientists think like Newton. If someone were to build a rigorous version of Newtonian calculus (the method of fluents and fluxions). That would be really interesting to see, and quite possibly useful for teaching.

*One of the big differences is that Leibnizian approaches focus on functions from x to y, while Newtonion approaches are parametric, with y and x both being implicit functions of t, rather than functions of each other. This way of thinking seems to be particularly useful in building equations like Ohm's law or the ideal gas law, where the relationship is more important than the (input/output) direction of calculation. But there are other differences as well; including huge differences in style and values.
posted by yeolcoatl at 7:01 PM on September 17, 2014 [3 favorites]

yeocoatl, you ought to watch this video: The Birth of Calculus. It's very basic, but shows both Newton and Leibniz's actual notebooks and examines how they developed their methods and notation.

I remember discovering this video at the end of a long week of arguing calculus methods with a room full of mathematicians. I was doing some research at home and this video popped up in a web search. I was watching it and then they started explaining how Newton was trying to calculate cardoids. I was dumbfounded by two thoughts. First, why the hell didn't anyone explain calculus to me using these practical methods instead of Mr Infinitesimal on his unicycle? And secondly, why the hell am I sitting here alone on a Friday evening watching a documentary on calculus?

This documentary is so good, I can forgive him for touching Newton and Leibniz's handwritten notebooks with his bare fingers. I can even forgive him for calling it The Calculus.
posted by charlie don't surf at 7:21 PM on September 17, 2014 [9 favorites]

You aren't supposed to need the Epilogue, that's why all the heavy-duty higher math is relegated in there -- just learn the rules for hyperreal numbers and the math will all work out.

Call me old-school, but I think that when learning/teaching math, it's best to build up progressively on what's already been learned - not taking things on faith.
posted by crazy_yeti at 7:30 PM on September 17, 2014 [1 favorite]

"Call me old-school, but I think that when learning/teaching math, it's best to build up progressively on what's already been learned - not taking things on faith."

So the curriculum should start with category theory, followed by group theory and set theory, and after all those are covered we get to arithmetic?
posted by idiopath at 7:34 PM on September 17, 2014 [8 favorites]

So based on the first link, it looks like ε essentially means, a number that is basically zero, but we can divide by it, and we'll use it as an unknown value when we do symbolic algebra, but when we're reducing/simplifying, we treat it like zero. That sort of makes sense to me.
posted by CheeseDigestsAll at 7:43 PM on September 17, 2014

> yeocoatl, you ought to watch this video: The Birth of Calculus. ..

I love that video, and have been urging people watch it for a few years now. The presenter doesn't just say "Here are Newton's and Liebniz's notebooks. Be impressed.". He's read them and points out many very specific passages/figures that are key developments in both of their work. He also does a very good job narrating their progress and contrasting their two approaches. I guarantee that if you are one of those people who actually has an opinion on calculus, you'll get a lot out of watching it. If you don't, I'll buy you a beer.

The weird thing for me is that I think Leibniz "reducing" infinitesimal analysis to a formal symbolic calculus made it available to many more people than Newton's methods alone would have. Whenever I see Newton's work I think "That's amazing! There's no way I can do that." Because, let's face it, dividing zero by zero is dangerous stuff and, while a virtuoso like Newton can do it without a safety net or padding, most of the rest of us benefit from having a set of provably reliable rules to work with.
posted by benito.strauss at 7:58 PM on September 17, 2014 [4 favorites]

My six year old son drew an infinity symbol onto a chain restaurant kid menu "Mad Lib" field reserved for "a number." I almost went into the spiel my dad went into when I first insisted to my dad as a 14 year old that x/infinity equals zero while 0/0 simultaneously exists as oneness (x/x), zeroness (0/x), nothingness (a null set, non answerable), and infinity (1/x), in which I am explained what limits are and why you can't just treat infinity like a number. I knew that division by zero was not technically allowed but like all things we can get quite close enough, and smiled and did not intrude.
posted by aydeejones at 8:14 PM on September 17, 2014 [2 favorites]

For maximum madness read The Mystery of the Aleph, which touches on all this and more, like the different orders of infinity such as where the set of positive integers is greater than the set of all integers, but smaller than the number of transcendental numbers or the infinitesimal continuum between zero and one. All sets are infinite yet some are larger still. It's enough to make mathematicians obsess about Shakespeare vs Bacon or Leibniz vs Newton
posted by aydeejones at 8:21 PM on September 17, 2014

So the curriculum should start with category theory, followed by group theory and set theory, and after all those are covered we get to arithmetic?

LOL I remember doing something vaguely like this, when I was in maybe 2nd grade learning arithmetic. They introduced Base 6 numbers and demonstrated formal proofs like 1 + 1 = 2 using group theory. What can I say, it was the sixties and New Math. But maybe this teacher was just weird. Maybe this is why I object to weird teaching methods.
posted by charlie don't surf at 8:26 PM on September 17, 2014 [1 favorite]

> For maximum madness read The Mystery of the Aleph, which touches on all this and more, like the different orders of infinity ...

Or just watch Vi Hart.
posted by benito.strauss at 8:35 PM on September 17, 2014

I'd dread the idea of teaching someone calculus for the first time and worrying about formal logic theory.

Heh. Sounds like trying to teach someone practical statistics the Bayesian way...
posted by Jimbob at 8:49 PM on September 17, 2014 [1 favorite]

This class was an absolute disaster. The teacher had obvious difficulty understanding how he was supposed to teach the infinitesimal calc concept. . . . Many students (like me) asked for tutoring from grad students, who understood the concepts even less.

Whether or not the standard method of teaching calculus is better or not in some abstract sense, it certainly has the huge advantage of 300+ years of pedagogical development behind it, a hefty 'math-industrial complex' consisting of everything from textbooks to study notes to teachers and grad students who all understand and teach the same basic concepts in the same general way to a large supply of potentially helpful fellow students who took the same course a year or two ago and understand same concepts, and so on.

It's hard to overcome all the inertia no matter how good your new method might be, in theory.
posted by flug at 9:00 PM on September 17, 2014 [5 favorites]

they are expected to swallow the heavy dose of mathematical logic (here relegated to the Epilogue) to actually construct these "Hyperreal numbers"

I'm not sure this is a very valid criticism. This stuff would be relegated to the Epilogue precisely because the author did NOT expect ordinary students to plow through it and understand it.

Similarly, no standard calc course starts out by wading through a few months of the technical definition of what real numbers are.

In both cases, you hand-wave a few of the main characteristics and properties that students will need to know, and then move along. Among other things, students won't even have the beginnings of a foundation to understand these technical definitions and constructions until they've worked their way through a bunch of the concepts they'll learn in the standard calculus courses.
posted by flug at 9:09 PM on September 17, 2014 [1 favorite]

This stuff would be relegated to the Epilogue precisely because the author did NOT expect ordinary students to plow through it and understand it.

As I said, the first weeks of my 1st edition class was demoted to the optional epilog of the 2nd edition. Keisler absolutely did expect every ordinary student to understand this material, before doing anything else. Apparently he was persuaded this was an unrealistic expectation, even a diversion from the primary goal. Some of the things he considered prerequisites, still stun me. Seriously now, just how much hyperreal math do you need to master, in order to understand the mere concept of integration and differentiation?
posted by charlie don't surf at 9:49 PM on September 17, 2014

It's a choice between an intuitive idea with inaccessible foundations and a less intuitive idea with accessible foundations. From a pedagogical point of view, and in a tradition that teaches geometry before algebra for a reason, I think the choice is obvious.
posted by sjswitzer at 11:33 PM on September 17, 2014

My fifth grade daughter is having trouble with some math. I'm not sure if this is the best or worst idea I've ever had, but I'm going to sit down with her and playtest chapter 1 of this edition of the rules, and see how it goes.
posted by mikelieman at 2:37 AM on September 18, 2014 [1 favorite]

I actually really love these weird textbooks and I think that more approaches to intuitive understanding of a subject can only be better. Sure, maybe these things aren't useful across an entire system, but someone, somewhere could pick this up and unlock something wonderful.

I have a big soft spot for Keisler's Book too.
posted by angusiguess at 5:45 AM on September 18, 2014

For maximum madness read The Mystery of the Aleph, which touches on all this and more, like the different orders of infinity such as where the set of positive integers is greater than the set of all integers, but smaller than the number of transcendental numbers or the infinitesimal continuum between zero and one. All sets are infinite yet some are larger still.

The set of positive integers is not greater than the set of all integers. There's a 1:1 mapping from positive integers to integers*, so there are exactly as many positive integers as there are integers!

*Take the integer, double it, and if it's negative, take the absolute value and subtract one.
posted by a snickering nuthatch at 7:42 AM on September 18, 2014 [1 favorite]

So the curriculum should start with category theory, followed by group theory and set theory, and after all those are covered we get to arithmetic?

Clearly not, as nobody would have any context for understanding categories. Start with arithmetic, fill in the foundations with some naive set theory (axiomatic treatment can come later), then algebra, calculus, etc. the time-honored way. Students can work with integers without having the Peano axioms and set theory at their disposal - their naive understanding of what an integer is is pretty good. But I think bringing in anything so sophisticated as "hyperreal numbers" at this stage of education is a terrible idea. The gap between "intuitive understanding of numbers" (integers, rationals, and reals) and "axiomatic understanding of numbers" is nowhere near as great as the gap for hyperreals. You just simply cannot explain what a hyperreal number is without having a lot of mathematical logic at your disposal. If a student came to me and asked "What's really up with the real numbers", I could explain Dedekind cuts. But if they asked about hyperreals - there's a lot of strange principles you have to accept, such as, giving up the principle that two numbers are either equal, or else one of them is greater (the hyperreals do not share this property). Hard to see that as a pedagocical improvement over deltas and epsilons. And "we're going to use these hyperreal numbers without really going into what they are" is an intellectual disservice to students. And, finally, they are going to have to get in touch with limits, deltas and epsilons sooner or later, why try to "hide" these real and important concepts?
posted by crazy_yeti at 8:42 AM on September 18, 2014

Tangent time: What I think would be really interesting to see is a formalization of Newton's calculus.

The book you want to read is Huygens and Barrow, Newton and Hooke by the great Russian mathematician V.I. Arnold.

Basically, 'formalizing Newton's mathematics' summarizes about 300 years of mathematics after Newton. It wouldn't go over any better with the undergraduates.
posted by ennui.bz at 9:43 AM on September 18, 2014 [1 favorite]

The book you want to read is Huygens and Barrow, Newton and Hooke by the great Russian mathematician V.I. Arnold.

yeocoatl, you ought to watch this video: The Birth of Calculus. ..

This are both really cool. I particularly like that Arnold covers Descartes, who I don't think gets enough credit for calculus, but neither is what I was talking about. I'm not looking for a history, I'm looking for a rigorous mathematical construction. I'm looking for the completion of this analogy: Weierstraus:Leibniz::????:Newton; although given how weird Newton is, it's probably closer to the completion of Robinson:Leibniz::????:Newton
posted by yeolcoatl at 10:36 AM on September 18, 2014

I went through the epilog a few years a go; it's not so bad, for someone with a bit of Engineering math done a long time ago.
posted by Monday, stony Monday at 10:40 AM on September 18, 2014

Even though I've sworn off commenting on MeFi, I had to log in to reply to the completely BS review given above by charlie don't surf.

Read the first chapter. If you already know calculus you can skip most of it, since it's just basic pre-calc (functions, etc). It's a completely clean, sane presentation of calculus. I have no idea why anyone would be confused by it, even if you'd been teaching the limits-based way for 30 years. Maybe the confusion was what to do with all their free time, once they didn't have to worry about all that limits baggage.

As the book says, infinitesimals is how everyone (other than Cauchy himself, I suppose) actually does calculus. For 100 years, we are taught with limits and then after the final we just mentally use infinitesimals. This textbook just skips that confusing first step and teaches the infinitesimals to begin with. The Epilogue (which is short, not difficult and optional) justifies this simple approach with more rigorous arguments.

It is not at all confused or disorganized, it is perfectly straightforward, contains a profusion of examples (both mathematical and physical, thank god) and omits the confusing side issues of limits to say nothing of series expansions.

I can't find that diagram "at the beginning of the chapter on differentiation" but if Keisler's worst crime is drawing a steering wheel on a unicycle then this book is still 100x better than how calculus is taught today. I'd never heard of this but it's how I'm about to teach my teens calculus since it's so easy and clear.
posted by DU at 5:16 AM on September 23, 2014 [1 favorite]

You have inadvertently identified the problem, DU. You are skilled at mathematics, you already learned this subject so this approach seems like a good one. But you have "The Curse of Knowledge." You do not know what students do not know. But I can tell you what precalc students do know: limits, sums and series. You think these are "confusing side issues" but I assure you that the SAT, ACT, and other exams will test high school kids for their knowledge of these subjects.

The diagram I posted is clearly labeled 14.1.3 which you might have guessed was in Chapter 14. I searched and found a similar diagrams in Chapter 8, labeled Figure 8.3.1, 8.6.2 etc. And remember, I studied from the first edition so read the preface to the second edition, where he describes a complete reorganization of the book. Even Keisler had trouble organizing this material.

I cannot see any way to make this method substantially easier, it does everything the hard way. That is why this book exists, Keisler thinks the hard way is the best, most correct way. Perhaps he is right, but his book does not demonstrate this to me.
posted by charlie don't surf at 11:17 AM on September 23, 2014

No, I'm sorry, that's ridiculous.

1) Precalc students do NOT know limits, sums and series. I didn't when I was one and my current pre-calc children don't. We got this stuff in calculus itself.

2) "They'll need it on the ACT" is not an argument for teaching things a certain way.

3) Nothing here is done the hard way. In the Chapter 1 he writes out the intuitive version of differentiation that's handwavey and then formalizes it with absolutely no change in notation. If that's not the easy way, I don't know what is.

I did find that diagram in chapter 14, you are right. And from the text it is perfectly clear that the steer is going up and down the slope of the line.

I dunno why you hate this book so much, but you shouldn't be projecting your problems onto people who want to learn and use calculus in a simple way.
posted by DU at 12:12 PM on September 23, 2014 [1 favorite]

There is only so much I can say about this subject, since I actually work in educational testing and have signed nondisclosure agreements covering what I know about the questions on future math tests.

But I assure you there is plenty of publicly available information on the content that students are expected to master in high school precalc. You may not consider it important for students to understand this material in a way that they can demonstrate it on a standardized test. But there is a high probability that some college admissions bureaucrat will think your children should be able to do that.

We can differ in opinion on Keisler's methods, but I think it is fair to say I have more detailed knowledge about how a student will deal with this method, since I spent my freshman year learning calculus from Keisler's book and you are looking back at it from the perspective of someone who already knows the material. If you want your kids to learn it Keisler's way, you better get them all the way to college level Calc II before they have to take their college entrance exams. If you're going to homeschool them in Calc, they better be able to test out of taking Calc at all.
posted by charlie don't surf at 1:14 PM on September 23, 2014

Once again, standardized tests should not dictate pedagogy. I'm not interested in how well kids pass the SATs, I'm interested in how well they understand how to use the mathematical tools.

As for the rest: I spent MY freshman AND sophmore year learning from a standard book that used limits and I had zero clue what I was doing. This was after a pre-calc course that I *did* understand and did *not* mention limits.

I eventually learned on my own by doing some physics, where they didn't bother with any of that junk and just treated Δx and Δy as variables. That's basically what the hyperreals method does, although the physics book didn't formalize anything.

Since one data point damns Keisler in your view, I rest assured that one data point will also damn limits in your view.

Anyway, the main point is that anyone who wants to try to learn calculus, or get a refresher or just see what this hyperreals thing is about should read the first chapter. It's very short, clear, simple and will not "do more damage than good".
posted by DU at 4:52 PM on September 23, 2014 [1 favorite]

I eventually learned on my own by doing some physics, where they didn't bother with any of that junk and just treated Δx and Δy as variables.

I never considered that part of the Freshman Year Bullshit is the different ways Physics 101 and Calc 101 approached the same thing from different ways. 30 years too late for me, but maybe someone can benefit from the insight.
posted by mikelieman at 4:14 AM on September 24, 2014

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