Infinitely impaired: an appeal

1. Sep 5, 2006

loom91

I was wondering about something. Has anyone noticed the sheer volume of meaningless math questions that tend to be inspired by the concept of infinite in calculus? These range from the intelligent but misguided to the utterly illegible. One recent example was a poster who considered every point to contain more points, apparently because he was taught that points are what you get when you make straight lines smaller.

I can also see this state of education first-hand. When our physics teacher introduced the concept of limit by saying things like "very small" and "unimaginably small", I didn't mind thinking that he is just seeking to convey a working idea so that we can utilize the tools of calculus. But I was shocked when our maths teacher used the same language! I opened my textbook and found an explanation of limit saying that as the limiting variable deviates from actual value by a very small amount, we can neglect this difference. The actual definition was hidden in a corner with very little importance, and our teacher didn't even mention the epsilon-deltas, immediately moving over to computing limits of indeterminate forms by factorisation and standard limits.

Such a handling of one of the most fundamentally important and useful branches of mathematics and an overwhelming emphasis on getting the right answer with 'infinite' disregard of the underlying theoretical basis is destroying the future of students and contributing to the general resentment of mathematics by students. As someone particularly fond of calculus, I find this to be heart-breaking. What can be done to improve matters?

Molu

2. Sep 5, 2006

Nothing can be done. There is a difference between calculus courses which can be taken in engineering courses, and between courses for students of mathematics, which, for obvious reasons, contain far more mathematical consistency. Engeneering math is 'packaged' in order to serve as a tool which is to be applyed while solving routine basic problems and understanding general physical concepts. Furthermore, people don't like math in general, so it's reasonable to assume that the more math you give them, the worse they'll do. Also, people like to think math is hard, which it isn't, at least not to some point. If you ask me, I'd blame the whole educational system for teaching math in a wrong way. But if you'd ask me which the right one is, I wouldn't have a clue. :tongue2:

3. Sep 5, 2006

MeJennifer

It is not completely fair to blame it on education IMHO.

Infinity is simply not an intuitive concept (we have a few hundred years of proof with "struggling smart mathematicians" on the topic).
Yes, when you know the right approach you can say it is easy and simple but that is really hindsight.

4. Sep 5, 2006

matt grime

would you mind rewriting that so it makes sense as a sentence, and perhaps then you might illustrate it with some examples as to what you mean? It doesn't seem to me like a reasonable statement. Mathematicians don't appear to have any problem with 'infinity', whatever it may be.

5. Sep 5, 2006

MeJennifer

I simply meant that in the history of mathemathics infinity was not as straightforward even by established mathematicians of their time.
Only after Cantor was it relatively straightforward. You can see a similarity with imaginary numbers.

6. Sep 5, 2006

matt grime

Actually, no I can't see the similarity.

I can't see what cardinals have to do with a failure to understand that something is not finite, which is all infinity alludes to. After all, we figured out why Xeno's (or Zeno's) paradox was nonsensical well before Cantor came along.

It wasn't as though people were attempting to do arithmetic with 'infinity' in the same way that the need to deal with 'imaginary' numbers came up.

7. Sep 5, 2006

franznietzsche

No, but historically the development of the concept of imaginary numbers was muddled and confused, right up through hamilton's quaternions.

8. Sep 6, 2006

matt grime

Who is denying that? It was the similarity with the alleged muddled notion of infinity that I am asking about.

9. Sep 6, 2006

loom91

But I'm talking about calculus that is taught at level before you specialise in engineering or math, the high-school level. Is it right to convey completely false ideas in the name of simplification (I have the same grudge against teaching the Bohr model)? Also, how much does it really simplify to be doing the thing before you know what you are supposed to be doing?

I'd say a system more right than the current one would be one which emphasised clear and unambiguous conceptions about the fundamentals before the calculations begun.

10. Sep 6, 2006

MeJennifer

Well then Matt let me ask you this: Are you of the opinion that in the history of mathemathics at each era the top mathematicians had a solid grasp of infinity?

11. Sep 6, 2006

matt grime

I am of the opinion that the description of something as infinite if it is not finite is and always has been well understood by mathematicians. Mathematicians rarely use the word 'infinity', by the way in any definite sense, unless they are geometers when it, again, has never had a 'vague' meaning.

I would certainly agree that there was contention around the time Cantor attempted to introduce the notion that two infinite sets can be 'quantitatively' different, but that is almost entirely a point about philosophy. And that certainly does not imply that Euclid had some misapprehension in his proof that there are 'an infinity' of primes'. There is not a finite number of them, therefore there is an 'infinity' of them. There is no 'wobbly grasp of infinity' there at all.

12. Sep 6, 2006

Hurkyl

Staff Emeritus
I wanted to point out that history is not an indicator of what is "intuitive". Intuition is a wholly subjective thing, and is a product of one's formal and informal education. (remember Einstein's "Common sense is the collection of prejudices acquired by age eighteen" )

Most laypersons are not formally taught about the infinite, and there's a lot of popular junk out there to confuse them... so it should not be surprising that a lot of laypeople find it counter-intuitive.

But when you study infinite things and become used to them, they can become rather intuitive.

And as mathematics progresses, new ways to look at things are discovered. For example, geometrically, there is no essential difference between the entire real line and the open interval (0, 1) -- and forming the extended real numbers by adding in the "numbers" $\pm \infty$ is exactly analogous to adding the endpoints 0 and 1 to the interval (0, 1).

13. Sep 7, 2006

loom91

Those who use infinity in a definite sense, such as null infinity or the extended real number [itex]\infty[/tex] have (I hope) a good grasp of what they are talking about. But a beginner encounters infinity (or the infinitesimal) as a limit in calculus, where it is not at all a definite quantity. Still our teachers write things like 1/infinity = almost 0, whatever that's supposed to mean. Any half-competent highschooler can compute the limit $$lim_{x\rightarrow 1} \frac {x^2 - 1}{x-1}$$ but how many of them know what they are computing?

They are all given vague ideas such as very small, very close, and negligible differences, which are not simplifications but simply a bag of lies invented by witless teachers who are either too ignorant or too lazy to tackle the real thing.

14. Sep 7, 2006

pete5383

Loom, this may be a stupid question, but I don't understand how the answer to that limit is something beyond the grasp of any highschool student (I assume most understand what 2 is), or am I missing something deeper in that...which is likely the case. I guess using L'Hospitals rule involves an indeterminate form, but in my evaluation of that limit, I didn't even have to think the word "infinty". Ooooh, or was that just a statement about the teaching of limits in general?

Hehe. Sorry, I kind of babble in that post.

15. Sep 7, 2006

CPL.Luke

loom your limit is undefined in the above as it simplifies to 0/0

16. Sep 7, 2006

Omega_6

No it isn't. You have to use L'Hospital's Rule (the answer is 2).

0/0 is not considered undefined such as 3/0 is anyways...it is considered an indeterminate quantity.

Last edited: Sep 7, 2006
17. Sep 7, 2006

Hurkyl

Staff Emeritus
There's a much easier way than using L'Hôpital's rule.

18. Sep 7, 2006

Omega_6

Yeah, you could just simplify...:tongue2:

19. Sep 7, 2006

matt grime

what is null infinity?

Eh? The notions that the limit of something is 'infinity' is purely a statement about the finite. It doesn't matter what you think 'infinity' might be in any larger sense at all. 1/x tends to infinity as x tends to zero is purely a statement about the finite - that 1/x can be made larger than any Y for x suitably small. There is nothing tricky about the notion of infinity there at all.

if they are truly even half competent then they know what they are doing. Perhaps we have different ideas about competency. And the answer is 'draw a picture', by the way.

I am not going to defend, support or deny such insults. (If teachers are witless what does that make the student, most of whom are incapable of becoming teachers?)

20. Sep 8, 2006

loom91

I do not know exactly, but it is a tool sometimes employed in general relativity, related to conformal infinity.

I'm not claiming that the limiting infinity is a tricky concept, I'm claiming that the students and teachers I've seen do not understand it.

Perhaps the state of education is better where you live, but here only a select few students will be able to tell you what a limit is. Most of them will regurgitate what they were told by their teacher, which makes limit sound like a numerical approximation.

Because of this perception of limits as approximate rather than exactly defined quantities, I had a hard time convincing a student (who happens to a wizard at solving math problems) that events that are not impossible may nevertheless have exactly 0 probability. He kept claiming that they will have almost 0 probability, and I'm not sure I managed to convince him completely at the end.

Teachers are not witless, only the witless among them are, and unfortunately these seem to constitute a depressingly large proportion among those I've met. I've found that this can not be related to their educational qualifications. There are three teachers holding doctorate in my school, and they are among the poorest teachers in the school. Exams taken to give teaching posts test only knowledge, but extent of knowledge is almost independent of capability to teach that knowledge. As for your question, a witless teacher usually makes a witless student, the point I was trying to make.

21. Sep 8, 2006

loom91

State of education.

22. Sep 8, 2006

matt grime

I am not surprised at this: perhaps you were thinking of different things. Certainly in probability there are non-trivial events with measure zero, such as P(number picked at random from [0,1] is rational) is zero. But if the other person is thinking of the real world then this doesn't make much sense since there is no such thing as the interval [0,1] nor a random generator of its elements in the real world.

Without wishing to get into too much philosophy, the limit in the example you gave is exactly just a definition, in my opinion. It does not have any significance other than a calculation. What you can do with it afterwards is a separate issue, and if they don't explain that then it is a short coming, but not necessary for understanding how to take limits.

Which goes someway to your first post. When you say your 'maths teacher' glossed over some of the detail, who do you mean? Are you at university here? Is this a class for mathematics majors? Is it one for general engineering people? Because that is why they'll omit epsilons and deltas: the answer is more important than understanding the method rigorously. The thing you can do is wait until you do proper courses in this case. Won't help the students who hate maths and won't do more of it, but then I doubt they'd appreciate a proper treatment of calc. Of course that kkind of assumption might be totally wrong and the cause of all the problems. But see the thread started by someone on 'proofs' who railed against mathematics being taught with boring rigor in it. You can't please all of the people all of the time.

23. Sep 9, 2006

loom91

That is where I disagree. I don't think throwing the concepts out of the window and going for the formula memorisation approach helps students who hate math. Such an approach is precisely the cause for the pervasive resentment for math which is a global phenomena. It's quite possible to teach without sacrificing rigour yet not making it boring by filling pages with dry unmotivated derivations. The best textbooks and teachers have done this, but these are rather few.

24. Sep 9, 2006

matt grime

A pervasive resentment of mathematics throughout the world? Really? I know students who found it dull, but 'resentment' is a little strong don't you think? I can't say I know how people in Mali feel about maths, to be honest.

Oh, and it is perfectly possible to teach the entireity of calculus without mentioning infinity at all (getting back to the point of the thread). People's misuse of infinity in mathematics is entirely to do with the use of it outside of mathematics.

Last edited: Sep 9, 2006
25. Sep 11, 2006

loom91

I was not talking specifically about the abue of infinity, but rather the abuse of limit, trying to sell it using grossly inaccurate language because it would take time and thinking to come up with something better.