Why Do We Need Limits and Infinitesimals?(betterexplained.com)
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Why Do We Need Limits and Infinitesimals?
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18 comments
Great point. I think one of the issues with calculus is that it can be studied at so many levels. Most high school classes give lip service to limits (somehow they're important), and then get into symbol manipulation (move the exponent down, subtract 1, etc.).
I think the most important thing in the beginning is to learn to think conceptually about calculus, about how to break a shape into a simpler model and solve it (not just manipulate models that are given to you). Later on (in analysis, which I've never formally taken :) ) the details about the transfer principle, epsilon-delta proofs, etc. can emerge, but realistically only a very, very small fraction of students need that. That fraction can be increased if they are able to use calculus regularly (with infinitesimals, say) and then wonder the details of how it all works.
I like to ask: Would it be such a bad thing if everyone understood calculus as Newton did, at a baseline level? Many people taking calculus leave with no understanding whatsoever, but the same isn't said for geometry or even algebra (you can make references to parallel lines, equations and feel that people will 'get it', but mention a derivative or limit to someone 10 years after calc class and you'll get a blank stare). I advocate having rough mental models that are successively refined over time (vs. presenting the most rigorous version first).
I think the most important thing in the beginning is to learn to think conceptually about calculus, about how to break a shape into a simpler model and solve it (not just manipulate models that are given to you). Later on (in analysis, which I've never formally taken :) ) the details about the transfer principle, epsilon-delta proofs, etc. can emerge, but realistically only a very, very small fraction of students need that. That fraction can be increased if they are able to use calculus regularly (with infinitesimals, say) and then wonder the details of how it all works.
I like to ask: Would it be such a bad thing if everyone understood calculus as Newton did, at a baseline level? Many people taking calculus leave with no understanding whatsoever, but the same isn't said for geometry or even algebra (you can make references to parallel lines, equations and feel that people will 'get it', but mention a derivative or limit to someone 10 years after calc class and you'll get a blank stare). I advocate having rough mental models that are successively refined over time (vs. presenting the most rigorous version first).
History note for you. Newton understood calculus through the idea of fluxions. It is often claimed that an insistence on teaching calculus with Newton's fluxions instead of Leibniz' infinitesmals held back research in England for a long time. (Another random note. Bishop Berkeley's famous comment about "ghosts of departed quantities" was a criticism of fluxions, not infinitesmals. In fact he had positive things to say about infinitesmals.) So yes, I do think it would be bad for students to understand calculus like Newton did. :-)
That said, to my eyes the primary problem with how we teach calculus is that we introduce too many big concepts at the same time. The key picture that people should come away with is the tangent line. But when you say that the derivative is a function defined at each point as the slope of the tangent line at that point, most people experience brain freeze.
Instead I would like to see students get to the point where they can calculate tangent lines. Then stop. Work out lots of tangent lines for lots of functions. Introduce them to how the tangent line can be used to estimate error bars. (Typical word problem, if a rock take 5 +- 0.5s to fall off a cliff, how tall is the cliff?) And then only after the picture and mechanics of the tangent line is well established should we move on to defining the derivative.
This approach, if done right, would be rigorous but informal. It should avoid the problem of causing brain freeze by presenting too many layers of generality at once. And it would really cement the key image behind half of calculus.
That said, to my eyes the primary problem with how we teach calculus is that we introduce too many big concepts at the same time. The key picture that people should come away with is the tangent line. But when you say that the derivative is a function defined at each point as the slope of the tangent line at that point, most people experience brain freeze.
Instead I would like to see students get to the point where they can calculate tangent lines. Then stop. Work out lots of tangent lines for lots of functions. Introduce them to how the tangent line can be used to estimate error bars. (Typical word problem, if a rock take 5 +- 0.5s to fall off a cliff, how tall is the cliff?) And then only after the picture and mechanics of the tangent line is well established should we move on to defining the derivative.
This approach, if done right, would be rigorous but informal. It should avoid the problem of causing brain freeze by presenting too many layers of generality at once. And it would really cement the key image behind half of calculus.
Ah, thank you for the clarification -- yes, Newton loved his fluxions, didn't he? In future examples I'll s/Newton/Leibniz/ :).
I think the general point is true about the brain freeze and too many big topics -- it's way too easy to just throw a definition out there, say "this represents slope which is the rate of change" and move on to the next idea.
I think the general point is true about the brain freeze and too many big topics -- it's way too easy to just throw a definition out there, say "this represents slope which is the rate of change" and move on to the next idea.
There is also this:
http://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis
It uses intuitionistic logic, so no axiom of choice.
http://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis
It uses intuitionistic logic, so no axiom of choice.
No axiom of choice, but most working mathematicians reject constructivism.
Plus there are some pretty big differences. Do you really want to explain to students why you don't consider f(x) = 1 if x >= 1 and 0 otherwise to be a valid definition of a function? And (in some ways more importantly) why most of the rest of the world does? Furthermore the intermediate value theorem is pretty useful and I wouldn't want to lose it.
Plus there are some pretty big differences. Do you really want to explain to students why you don't consider f(x) = 1 if x >= 1 and 0 otherwise to be a valid definition of a function? And (in some ways more importantly) why most of the rest of the world does? Furthermore the intermediate value theorem is pretty useful and I wouldn't want to lose it.
Just mentioning another approach. I don't really understand it myself, but I kind of find it intriguing because of its relation to computability aspects.
Admirable intent, and it seems pretty good, but I've seen far too many people follow advice like this and end up producing confusing nonsense while thinking they're doing the right thing.
By all means take the concepts, but don't try to apply them like that. There are so many pitfalls that this doesn't help you identify, let alone avoid.
If it helps you grok stuff, great. I think I'm pretty clear on how to do a quadruple coronary bypass - I'll just figure out the details as I go along, shall I?
By all means take the concepts, but don't try to apply them like that. There are so many pitfalls that this doesn't help you identify, let alone avoid.
If it helps you grok stuff, great. I think I'm pretty clear on how to do a quadruple coronary bypass - I'll just figure out the details as I go along, shall I?
Hey, thanks for the comment.
Yep, there is a fundamental tension between intuition and rigor. Unfortunately, I feel that rigor (and not even rigor -- rote details) are overemphasized, without giving any context, and more importantly without giving any inspiration to the student. It's like focusing on the spelling of a poem, not the content.
I do want to jump into the rigorous definitions in later posts (such as when limits exist, when they don't, why it's important) but I find intuition -> rigor is an easier path to navigate than rigor -> intuition [which often becomes rigor -> memorization]. If someone had just said "Hey, limits and infinitesimals help us solve the problem of 'How do we make an approximation which is still accurate enough?'" calculus would have clicked a lot faster for me :).
Yep, there is a fundamental tension between intuition and rigor. Unfortunately, I feel that rigor (and not even rigor -- rote details) are overemphasized, without giving any context, and more importantly without giving any inspiration to the student. It's like focusing on the spelling of a poem, not the content.
I do want to jump into the rigorous definitions in later posts (such as when limits exist, when they don't, why it's important) but I find intuition -> rigor is an easier path to navigate than rigor -> intuition [which often becomes rigor -> memorization]. If someone had just said "Hey, limits and infinitesimals help us solve the problem of 'How do we make an approximation which is still accurate enough?'" calculus would have clicked a lot faster for me :).
I think rigor is imposing to people when you try and combine it with high-level theory. The concepts in that article are built on top of so much theory, that to explain it all would require the equivalent of a course on analysis.
But if you supply rigor in an appropriate dose, it can really be eye-opening. For me, calculus was a nightmare because none of the terms used were well defined. I was just presented with all this highly developed theory and had to take much of it on faith. And I couldn't enjoy nor care about something I felt was just forced memorization of computational techniques for solving contrived problems. But if they'd done it rigorously, the classes would have begun with questions like "what is distance?", then answering that question would lead into "what is a space?", etc. This is rigor and I feel like exploring these types of questions is a lot more intuitive than forcing kids to memorize how to compute the volume of some cone.
But if you supply rigor in an appropriate dose, it can really be eye-opening. For me, calculus was a nightmare because none of the terms used were well defined. I was just presented with all this highly developed theory and had to take much of it on faith. And I couldn't enjoy nor care about something I felt was just forced memorization of computational techniques for solving contrived problems. But if they'd done it rigorously, the classes would have begun with questions like "what is distance?", then answering that question would lead into "what is a space?", etc. This is rigor and I feel like exploring these types of questions is a lot more intuitive than forcing kids to memorize how to compute the volume of some cone.
I think we're pretty much in agreement. I would rather, though, see a real example with proper rigor, but introduced and explained starting with intuition and then developed forwards into the detail.
The problem is, as I'm sure you're aware, just how many traps and pitfalls there are for the insufficiently paranoid. Working "intuitively" has a tendency not to instill that paranoia.
And, as you say, working just on the rigor tends to result in rote work with no understanding.
I think only time and practice is the real solution. The article, though, is good for someone who already has the "rote" understanding, and needs to "grok" the whole thing.
Similar criticisms can be aimed at my stuff: http://news.ycombinator.com/item?id=940169
The problem is, as I'm sure you're aware, just how many traps and pitfalls there are for the insufficiently paranoid. Working "intuitively" has a tendency not to instill that paranoia.
And, as you say, working just on the rigor tends to result in rote work with no understanding.
I think only time and practice is the real solution. The article, though, is good for someone who already has the "rote" understanding, and needs to "grok" the whole thing.
Similar criticisms can be aimed at my stuff: http://news.ycombinator.com/item?id=940169
Yep, I agree with your points about pure intuition leading you down blind alleys unless you have some structure in place.
For the examples, my approach is definitely tailored for people who need to grok the subject after getting a rote understanding. It's not so much to become an expert but to realize what insights would help on the path to becoming an expert.
I figure there are thousands of other sites for the formal definition and practice problems :). I'll check out the article now, looks interesting.
For the examples, my approach is definitely tailored for people who need to grok the subject after getting a rote understanding. It's not so much to become an expert but to realize what insights would help on the path to becoming an expert.
I figure there are thousands of other sites for the formal definition and practice problems :). I'll check out the article now, looks interesting.
I studied calculus on my own in my twenties, and I think the textbooks' focus on rigor led me down a lot of blind alleys before I started trying to come to an intuitive, often graphical, understanding of the problem before attacking the rigorous view. (There is a good textbook from 1977 that was reprinted by Dover in 1998, Morris Kline's "Calculus: An Intuitive and Physical Approach", that I wish I had available when I was studying.)
Just curious, what exactly do you consider the necessary rigor in this case?
Well, there's nothing in this article that really requires rigor, because there aren't really any problems. It's talking about the concepts of limiting processes and infintesimals, but not how to use them.
In the one example given, sin(x)/x, the difficult bit is glossed over when it talks about sin(x) being more-or-less equal to x when x is "close to zero." The rigor there is missing entirely, because that's not the point.
In the one example given, sin(x)/x, the difficult bit is glossed over when it talks about sin(x) being more-or-less equal to x when x is "close to zero." The rigor there is missing entirely, because that's not the point.
Thanks for the comment. I did contemplate mentioning the definition of sin(x) = x - x^3/3! + ...
but thought it would be too leading & tangential -- I wanted to show visually that most curves can in fact be modeled by a straight line over a small distance, without the sledgehammer of showing sine's expansion.
but thought it would be too leading & tangential -- I wanted to show visually that most curves can in fact be modeled by a straight line over a small distance, without the sledgehammer of showing sine's expansion.
Better would simply be to say that sin is opposite over hypotenuse, and when the angle is really, really small, the opposite is effectively the same as the arc length. Since the acr length is the angle in degrees, that means that sin(x) ~ x when x is very small. Then you can be more rigorous with the epsilon/delta, and say "you pick how close, and I'll get it closer by making x small enough."
Similar arguments can show, for example, that acceleration in a circle is v^2/R. "Calculus" was, or course, done for decades, if not centuries, before "calculus proper" was (somewhat) formalised by Leibniz and Newton.
This whole discussion should probably be taken to email if it's to be taken further. The question is, how can we have this sort of discussion, and produce something worth publishing back onto HN?
Similar arguments can show, for example, that acceleration in a circle is v^2/R. "Calculus" was, or course, done for decades, if not centuries, before "calculus proper" was (somewhat) formalised by Leibniz and Newton.
This whole discussion should probably be taken to email if it's to be taken further. The question is, how can we have this sort of discussion, and produce something worth publishing back onto HN?
Yep, definitely -- actually I have an article on radians which does just that (comparing height on the circle, sin, to distance traveled around, x).
Always happy to discuss on email ([email protected]) -- I'm not sure if there's something to produce out of it necessarily, but I enjoy the discussion / investigation of personal philosophies & teaching approaches.
Always happy to discuss on email ([email protected]) -- I'm not sure if there's something to produce out of it necessarily, but I enjoy the discussion / investigation of personal philosophies & teaching approaches.
Of those approaches infinitesmals look the simplest but require the most complex mathematical machinery to really do right. Yes, you can construct a non-standard model of the real number system with a transfer principle that lets you transfer between proofs done in the one to the other which lets you formulate calculus using infinitesmals. But using a construction using the axiom of choice to prove that the derivative of x^2 is 2x is kind of like using a sledgehammer to push a tack into a cork board.