Comment by a-dub | Hacker News Reader

Comment by a-dub | Hacker News Reader

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i always found it funny how much time was spent drilling techniques for different integral types or doing transforms for derivatives, yet the most important idea: the continuous nature of the reals and why this is the rug that pulls it all together, and the delta-epsilon definition of the limit, only saw about a grand total of 3 minutes of hand waving and an optional problem on one homework.

ug calculus was about having algebraic/trig manipulations memorized along with a table of transforms and a handful of tricks; where the actually beautiful ideas that if we use an infinitely "elastic" representation of numbers, we can solve hard approximation problems both correctly and easily- get totally glossed over.

physics has the same problem. basic physics without calculus is just a bunch of rote memorization. the idea that such a thing is taught and that is somehow "easier" is nuts. they should be taught together, as they were developed, as many of the expressions given to undergrads in physics are simply definitions of integrals and derivatives applied a few times.

no student in calculus should ever be wondering what the constant is for in a computed integral and no student in physics should wonder where the constants come from in equations of motion. (there should be no equations of motion, just definitions in integral/derivative form and definitions of integrals and derivatives)

As someone who has taught intro calculus a few times, one reason for the emphasis on computational techniques is simply that 90% of the students in such a class do not care/are not capable of grasping the epsilon-delta definition of the limit.

Solution: spend more time on epsilon-delta so that students have time to wrap their minds around the idea. But I think the engineering departments would complain that the students who we send on to them cannot do basic computations. Also students would complain that we spend too much time on theory and not enough on application. There are probably other reasons that someone more experienced would know about.

I tutored calc too, but I had an unanswered complaint about the epsilon/delta definition of a limit - I found it to be circular. There was no impact on the computations required by the class, but the definition seemed rather circular, as it presupposes the reality of the infinity implicit in induction. Over time I recognized that my intuition was rebelling against unphysical math concepts. I was a physics major and so I was content to hand-wave my own concerns away as a merely philosophical problem, since you could always pick a sufficiently large number of induction iterations to satisfy any practical need. But yeah, it left a bad taste in my mouth that no-one seemed to care about this issue. (Oddly the use of induction for similar things, like Cantor's diagonal argument, didn't bother me at all. It was specifically epsilon delta that did, and still does, bother me.)

The epsilon delta definition of limit is not circular and there is no infinity anywhere in it. Are you referring to the limit as something goes to +infinity on the real line? That's just bookeeping shorthand for some quantifiers over bounded numbers.

I'm genuinely curious as to what you were getting hung up on, because odds are you learned the wrong definitions or somehow changed the definition to something that is incorrect, and then discovered the error later on. It's a fairly straightforward definition if you can do the proper bookkeeping over the quantifiers, and is designed to not have anything to do with infinities. But many people struggle with quantifiers.

this is the point. they do not stress in beginning calculus that continuity is actually a profound property of a mathematical construction (the reals) that we use for modeling physical phenomena.

it's not the delta-epsilon definition that is slightly weird, it's what it means to be truly continuous that is. (uncountable infinity, or as i like to call it, endless zooming everywhere)

once that is clear, the delta-epsilon definition becomes obvious and clear.

Continuity is just a formal property. I'm not sure if it is "profound", that's more of an emotional response, which is very personal.

Mathematicians use words like "deep" to describe properties that are useful in many branches of math. Is continuity deep? I think what's deep is the notion of compactness and the essence of compactness is open covers and finite subcovers, so the deep idea really is about epsilons and deltas and not handwaving about "what it really means" to be continuous.

For ruminations on what things "truly mean", you are not going to find that in a math book. Perhaps a philosophy book would better scratch that itch for you. Math is self-contained and provides its own meaning in terms of the relationships between formal properties and precise answers to precise questions. Some people truly love exploring the relationships between formal properties and discovering the answers to difficult questions that reveal new logical structures. If that is not interesting enough for you, perhaps you will get more satisfaction studying a different field.

Any statement about infinity is not something you can check directly, by hypothesis. The powerful influence of social proof cannot and should not be ignored. There are too many smart people who don't know anything about the foundations of mathematics, and believe that there aren't any consistency issues.

My point is that as a practical matter, they're right. Calculus works really well. But consider for the moment the question of why you are convinced that induction is valid. There is no proof of induction's validity. The epsilon delta thing is a definition, not a proof, and serves, effectively, to invert infinity into an infinitesimal. I am on firm ground with these statements, like it or not.

I agree there's little to be gained examining what things really mean when it comes to meaning itself - it is always going to be circular. I believe that the foundations of math lay primarily in its utility, and secondarily in the social fabric of experts (which may or may not be related to utility). The eerie, beautiful way in which math describes the world is fundamentally a human phenomena, and it's based on aesthetics, not logic. I suppose my objection is simply that one shouldn't go around thinking that the epsilon delta definition removes the ambiguity and messiness that actually underlies mathematics.

In an effort to understand what you mean by 'induction' and 'infinity', can you explain where induction comes up in the following standard definition of a limit:

"f has a limit L at x" means exactly that

"forAll epsilon>0 thereExists delta>0 such that if |x-y|<delta then |f(y)-L|<epsilon."

"forAll"

Okay, I see a link to 'infinity' there, because there are infinitely many reals. Is that what you're going for?

If so, the link seems irrelevant and in particular I don't see how this leads to circularity in the definition. Can you explain this further?

> But consider for the moment the question of why you are convinced that induction is valid. There is no proof of induction's validity. The epsilon delta thing is a definition, not a proof, and serves, effectively, to invert infinity into an infinitesimal. I am on firm ground with these statements, like it or not.

I'm afraid these statements are a mishmash of some correct and incorrect statements, and a logical argument like that is considered incorrect.

* Yes, the definition of limit is a definition.

* The definition of limit has nothing to do with induction or infinity. I'm honestly baffled why these three distinct concepts are being conflated.

* For well ordered sets, induction is just reductio ad absurdum in which you assume the smallest element does not satisfy a condition and then show this to be incorrect because the next smallest element must meet the condition and it's satisfication means the next larger element must meet it as well. There is a valid question as to whether every set can be well-ordered, which is an axiom, but for countable sets, which is where virtually all induction arguments are used, no axiom of choice is needed to use induction-style arguments.

* The statement "to invert infinity into an infinitesimal." is gibberish.

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