Teaching about infinitesimals and integration

[bcrowell]

I'm a lot less comfortable with nonstandard analysis, which posits that there exists an infinitesimal without any further ado or description thereof.

This doesn't seem to me like an accurate characterization of NSA. In a synthetic treatment of the reals, we posit some axioms about the reals, and we simply assume that there exists a systems that obeys those axioms. In a constructive description of the reals, we build them up using Dedekind cuts or whatever. The situation is exactly the same for the hyperreals. One can approach the hyperreals in either a synthetic style or a constructive style.

I guess this is a matter of taste.

It's certainly a matter of taste in terms of your own work. However, when we educate other people in mathematics, IMO it's irresponsible not to help them become literate in the common practices of science and engineering, which include the manipulation of infinitesimals.

 

[zinq]

I guess 99.9% of mathematicians are irresponsible, since we mostly don't use infinitesimals per se, despite having successfully taught legions of math, physics, engineering, and other students how to excel in calculus and calculate everything they ever need to calculate.

Funny, no one ever called me irresponsible before.

 

[bcrowell]

I guess 99.9% of mathematicians are irresponsible, since we mostly don't use infinitesimals per se, despite having successfully taught legions of math, physics, engineering, and other students how to excel in calculus and calculate everything they ever need to calculate.

Yes, I am criticizing current common practices in math education. The fact that something is a common practice in education doesn't mean that it's right. If it did, then education would never change and we'd probably all still be parroting Aristotelianism, discussing the differing mental abilities of the negroid, caucasoid, and asiatic races, or assigning our students essays on how clitoral orgasm is a sign of arrested development.

Your description of the excellent abilities of freshman calc students also seems wildly wrong in my experience. I teach both freshman physics and freshman calc, so I see this from both sides of the fence. When students show up in my physics classes after having taken the calculus prerequisite, I find that the vast majority of them lack any understanding of what they were actually doing in their calculus course or how to apply it to reality. What they do in their calc course consists of solving long lists of stylized textbook problems (almost never any word problems) in which y is always a function of x. There is a yawning chasm between their math education and real-world practices in science and engineering, and anything that helps to perpetuate this gap is certainly a problem. One such problem is the tendency of mathematicians to pretend that scientists and engineers don't use infinitesimals.

 

[micromass]

Yes, I am criticizing current common practices in math education. The fact that something is a common practice in education doesn't mean that it's right. If it did, then education would never change and we'd probably all still be parroting Aristotelianism, discussing the differing mental abilities of the negroid, caucasoid, and asiatic races, or assigning our students essays on how clitoral orgasm is a sign of arrested development.

Your description of the excellent abilities of freshman calc students also seems wildly wrong in my experience. I teach both freshman physics and freshman calc, so I see this from both sides of the fence. When students show up in my physics classes after having taken the calculus prerequisite, I find that the vast majority of them lack any understanding of what they were actually doing in their calculus course or how to apply it to reality. What they do in their calc course consists of solving long lists of stylized textbook problems (almost never any word problems) in which y is always a function of x. There is a yawning chasm between their math education and real-world practices in science and engineering, and anything that helps to perpetuate this gap is certainly a problem. One such problem is the tendency of mathematicians to pretend that scientists and engineers don't use infinitesimals.

Ben, I think you are correct in that infinitesimals do not get enough attention in basically all of math courses. I wish to see this different as well. But (aside from the major problem: tradition), I feel there are some other problems:

1) The infinitesimals I always encounter in physics are the ones of the sort $\epsilon^2 = 0$ (or similar). This doesn't seem to match at all with the hyperreal number system. So while Keisler does a really neat job explaining infinitesimals, it is actually not that kind of infinitesimals you'll actually meet later on. It seems like synthetic differential geometry and other synthetic approaches (including the infinitesimals in schemes of algebraic geometry) are much more closer in spirit to the infinitesimals as used in science and engineering. This is problematic since these approaches are a lot more difficult.

2) For mathematics students, it is very important that everything they see must be based on first principles. This must be the case eventually. This is problematic since NSA requires some very deep logical constructs to really do it properly. On the other hand, constructing the real numbers is (compared to this) a piece of cake. I do not think they should construct the reals in calculus classes of course. I am perfectly ok with calculus classes which assume some results of the reals/hyperreals without proof. But eventually, a construction of the real numbers must be given to the students. Everything must be based on first principles, and accordingly, a construction of the hyperreals should also be given. But this seems just too difficult for the standard undergrad student.

 

[bcrowell]

1) The infinitesimals I always encounter in physics are the ones of the sort $\epsilon^2 = 0$ (or similar). This doesn't seem to match at all with the hyperreal number system.

This is actually pretty easy to take care of, with several centuries of hindsight. In fact, it turns out that Leibniz himself actually got this right, but later workers got sloppy with some of the distinctions he made. For the history, see Blaszczyk, Katz, and Sherry, Ten Misconceptions from the History of Analysis and Their Debunking, http://arxiv.org/abs/1202.4153 . In the hyperreal system, the idea is that the derivative is not dy/dx but the standard part of dy/dx. The squares of infinitesimals do not actually equal zero, but they are eliminated when we take standard parts. Keisler's book has a clear presentation of this at greater length.

So while Keisler does a really neat job explaining infinitesimals, it is actually not that kind of infinitesimals you'll actually meet later on. It seems like synthetic differential geometry and other synthetic approaches (including the infinitesimals in schemes of algebraic geometry) are much more closer in spirit to the infinitesimals as used in science and engineering. This is problematic since these approaches are a lot more difficult.

I would suggest a different way of looking at this. The overarching idea is that we have a common body of practices that have been used for 300 years, and these practices work, regardless of whether you justify them through SIA or NSA.

2) For mathematics students, it is very important that everything they see must be based on first principles. This must be the case eventually. This is problematic since NSA requires some very deep logical constructs to really do it properly. On the other hand, constructing the real numbers is (compared to this) a piece of cake. I do not think they should construct the reals in calculus classes of course. I am perfectly ok with calculus classes which assume some results of the reals/hyperreals without proof. But eventually, a construction of the real numbers must be given to the students. Everything must be based on first principles, and accordingly, a construction of the hyperreals should also be given. But this seems just too difficult for the standard undergrad student.

Mathematics is basically a service department, and freshman calculus is a service course. Math majors are a tiny fraction of the students in such a course. IMO there is a much bigger problem in math education, which is that freshman calc is expected to be a one-size-fits-all course that serves the needs of math majors at the same time that it serves the needs of people who want to be dentists. The first priority should be fixing that massive dysfunction. We really should have different math courses for (1) math majors, (2) engineers and physical scientists, (3) biology majors, and (4) business majors. Nothing is going to get fixed until that happens. Math departments seem to have been asleep at the switch on these issues as the population of biology majors in their freshman calc courses has more than tripled in the last 20 years.

 

[PeroK]

We really should have different math courses for (1) math majors, (2) engineers and physical scientists, (3) biology majors, and (4) business majors. Nothing is going to get fixed until that happens. Math departments seem to have been asleep at the switch on these issues as the population of biology majors in their freshman calc courses has more than tripled in the last 20 years.

I went to a small university (Heriot-Watt in Edinburgh) in the 1980's. We didn't have the same concept of "majoring", but the maths courses for maths, physics & engineering and biology were different. The physics and engineering maths courses were fairly solid; whereas, the maths course for biology and pharmacy were watered down! I did maths, so those were different courses again - just for the maths students.

 

[Andy Resnick]

<snip>
Your description of the excellent abilities of freshman calc students also seems wildly wrong in my experience. I teach both freshman physics and freshman calc, so I see this from both sides of the fence. When students show up in my physics classes after having taken the calculus prerequisite, I find that the vast majority of them lack any understanding of what they were actually doing in their calculus course or how to apply it to reality. What they do in their calc course consists of solving long lists of stylized textbook problems (almost never any word problems) in which y is always a function of x. There is a yawning chasm between their math education and real-world practices in science and engineering, and anything that helps to perpetuate this gap is certainly a problem. One such problem is the tendency of mathematicians to pretend that scientists and engineers don't use infinitesimals.

I agree with this. It doesn't help that these students have to face a barrage of confusing notation: dx, Δx, δx, ∂x, sometimes all combined into a single expression. Similarly, sometimes dx is expressed as an operation: d/dx, sometimes all by itself. Sometimes the concept refers to a mathematical limit, sometimes to an physical quantity (an area or volume, a short length or duration).

The best solution I can think of is to have science and engineering faculty work closely with math faculty in the intro pre-calc and calc sequences, both during course design and student evaluations- we implemented problem-based learning modules a few years ago and are now starting to analyze the data (student grades in subsequent courses) to gauge effectiveness. Honestly, our preliminary results are mixed.

 

[Andy Resnick]

<snip>We really should have different math courses for (1) math majors, (2) engineers and physical scientists, (3) biology majors, and (4) business majors.

I think you have to be very careful here- that approach is used here ("business math", "Statistics in biology", etc.) and it leads to (IMO) two sets of significant problems:

1) Those courses are sometimes offered by the Math department, sometimes not- the claim is that the course content is tailored to the interests of the students, but in practice the content is simply weaker, especially when the course is not offered through the math department. The underlying issue is the difference between education and training- is the purpose of a BS degree to train/qualify a student to work in a particular class of jobs, or is the purpose of a BS to provide a broad intellectual platform? If you believe the function of a BS degree is to provide job training, then you would indeed be justified in controlling as much of the curriculum as possible. Which brings me to problem #2:

2) the battle for credit hours. Department and College budgets are set by the number of credit hours, creating an incentive for (say) the College of Engineering to create its own intro Physics sequence 'that specifically targets the employment needs of our majors". In practice this does not often happen, but there is a constant low-level threat. To be fair, we have a "Intro to theoretical physics" course that is really a math class, and we developed it to address what we perceive is shortcomings in the mathematical preparation of our majors.