Wow, vector multiplication suddenly makes sense. I had never seen it described with polar coordinates.
It's wonderful to have this little insight now. It's unfortunate that my math knowledge is so filled with holes.
What are the 'real numbers', really?
31–40 of 98 posts
Re: What are the 'real numbers', really?
#32" It seems that any proper theory of real numbers presupposes some kind of prior theory of algorithms; what they are, how to specify them, how to tell when two of them are the same.
Unfortunately there is no such theory."
http://njwildberger.wordpress.com/2012/12/02/difficulties-wi...
Re: What are the 'real numbers', really?
#33Earlier quoted context omitted.
"My understanding of a line is that it is delimited by two points" That is not how Euclid defined it and how it is still seen in geometry today. What you describe is called a (line) segment ( http://en.wikipedia.org/wiki/Line_segment ) "but does not contain any points" Lines extend indefinitely in two directions (if you go past Euclidean geometry, that 'indefinitely' changes meaning a bit) One talks of a point being…
Excuse me. Of course. I was using line and line segment interchangeably there. Which I should have not been doing if I am aiming for clarity but I think my point (ahem) applies to line segments and lines that extend indefinitely in one or two directions. Presumably people will contend that even a line segment "contains" an infinite number of points. But if points have zero extension then even an infinity of them cann…
Perhaps your intuition changes if you think about the line (in a plane) coinciding with the x-axis. Don't you think it is reasonable to define that line as the set of points (in the plane) having y=0? Also, should that line not be identical (isomorphic) to the set of real numbers - which is just a set of numbers? And should not all other lines in the plane be identical (isomorphic) to the first line?
Re: What are the 'real numbers', really?
#34Earlier quoted context omitted.
Added in edit to emphasise a point: If all you want to do is differentiate and integrate, then non-standard analysis is probably, for most people, a faster way to be able to do just that. Now read on ... Non-standard analysis has been put on a firm, formal footing. Theorems have been proven showing that (largely) it's equivalent to the regular form of analysis. Some things are easier to prove in standard analysis, so…
I disagree. The vast majority of students take math classes for the practical applications - science and engineering - not to continue theoretical pure math study. Therefore the focus should be on effective teaching of applied math. I am sure that if a student wishes to explore their studies in pure mathematics they will be clever enough to learn whatever they need in specialized classes.
Actually, you are agreeing with me. You are saying that doing calculus was, for you, much easier using the infinitesimal approach. I'm not disagreeing with you. In fact, you'll find that advanced mathematicians think in that way, although they can drop back to epsilon-delta work if they need to (which they often do).
So we are in agreement. My point is that if you teach calculus that way you have immediately ham-strung anyone who might go on and do anything other than engineering or physics. In fact, there are deep theoretical arguments in physics where you need to use the standard approach, and the non-standard approaches are much more difficult.
My point is that if all you want is calculus then it's very likely that the non-standard approach is fine. I'm also arguing that this is limited thinking. Clearly you were never going to go further in these sorts of subjects - does that mean that everyone else should also be taught in a similarly limited way?
I also observe that limiting arguments are essential in anything other than the most direct and practical versions of engineering, so again, the point isn't in the calculus, the point is learning about limits.
Many people don't need any math at all beyond arithmetic, and I know a lot of people who proudly announce that they can't even do that. And to some extent it's true - most people don't need any math at all. Why were you bothering to take calculus? I'm sure you've never needed it.
But let me add that if all you want to do is arithmetic, why bother? Just use a calculator. If all you want to be able to do is differentiate, why bother? Feed it to Wolfram Alpha. If all you want to do is program, why bother? Hire someone to do it.
But yes, if all you want to do is high-school calculus, there are easier ways to learn the processes to jump through the hoops, pass the exam, and get the piece of paper. For most people that's all they care about. We probably agree on that.
Re: What are the 'real numbers', really?
#35Earlier quoted context omitted.
I agree. It would be truer to say that infinitesimals are studiously ignored by modern mainstream mathematicians because they feel that Dedekind and co. have put the calculus on a firm footing way back when. Anybody with a small bit of curiosity or a dashing of non-conformity will be suspicious of this narrative. If anything, infinitesimals in their various guises carry a certain explanatory heft, and are quite begui…
I loathed limit-based calculus in High School and College. Later I read Elementary Calculus: An Infinitesimal Approach http://www.math.wisc.edu/~keisler/calc.html and it all came clear in a fraction of the pages. It's infuriating that most math curricula won't drop those old, bloated, overly formal calculus tomes to improve the clarity and effectiveness of the instruction method.
That's funny. I despised my college calculus courses because they were so informal. Much like the author of this post said, my calculus education focused entirely on boring rote computation and not at all on proofs, the,a tater of which is. The only part I really consider to be mathematics.
Re: What are the 'real numbers', really?
#36a real number is "a point on the number line" These posts are always stimulating. My understanding of a line is that it is delimited by two points, but does not contain any points. To elaborate, no point could be "on" a line because a point has no extension, whereas a line does. This is the crux of the matter. Therefore a line is not "made up of" points. (By analogy a plane could not be made up of lines.) This begs t…
In geometry it is quite common to define a line as an infinite set of points (namely, those which satisfy a linear equation). Likewise with other figures, like a circle.
Re: What are the 'real numbers', really?
#37Earlier quoted context omitted.
"My understanding of a line is that it is delimited by two points" That is not how Euclid defined it and how it is still seen in geometry today. What you describe is called a (line) segment ( http://en.wikipedia.org/wiki/Line_segment ) "but does not contain any points" Lines extend indefinitely in two directions (if you go past Euclidean geometry, that 'indefinitely' changes meaning a bit) One talks of a point being…
Excuse me. Of course. I was using line and line segment interchangeably there. Which I should have not been doing if I am aiming for clarity but I think my point (ahem) applies to line segments and lines that extend indefinitely in one or two directions. Presumably people will contend that even a line segment "contains" an infinite number of points. But if points have zero extension then even an infinity of them cann…
"But if points have zero extension then even an infinity of them cannot sum to anything greater than zero."
Infinities are weird; anybody who wants to learn math has to accept that. 0.99999… does equal 1, there are as many even numbers as integers, etc. these things are 'true' not because they make sense initially, but because they make the most sense of all the other things we have thought of so far. Similarly, a set of Aleph-0 points can completely cover a line.
Re: What are the 'real numbers', really?
#38What are "real numbers"? A horribly misnamed fiction. Nearly all of them cannot be represented with a finite amount of information. I strenuously object to naming an uncountable set "real" when only a countable subset (measure 0 of the full set) can be worked with in any way at all. We need to stop venerating the "real" numbers and start focusing on sets that are actually usable.
What do you mean by "represented with a finite amount of information"? Are you referring to their representation in a positional notation like decimal or binary? Or are you referring to the much subtler and more advanced fact that almost all reals are uncomputable? The former isn't really true, and the latter, while true, is subtle enough that it doesn't matter for the vast majority of mathematics (and to replace the reals with the computable numbers would make most of mathematics messy).
Re: What are the 'real numbers', really?
#39I do have an issue with this line "Ultimately, infinitesimals were discredited and discarded by mathematicians (though they continued to be mentioned in some physics books many decades later)" Infinitesimals have been made rigorous with modern mathematics.
Indeed, many mathematicians think in terms of non-standard analysis, and then translate their proofs into standard arguments, even if the non-standard ones can be made just as rigorous as the standard ones.
Terry Tao has a wonderful series of posts about hard and soft analysis, ultrafilters, and non-standard analysis. He writes
I feel that one of the reasons that non-standard analysis is
not embraced more widely is because the transfer principle,
and the ultrafilter that powers it, is often regarded as some
sort of “black box” which mysteriously bestows some
certificate of rigour on non-standard arguments used to prove
standard theorems, while conveying no information whatsoever
on what the quantitative bounds for such theorems should
be. Without a proper understanding of this black box, a
mathematician may then feel uncomfortable with any
non-standard argument, no matter how impressive and powerful
the result.
The main drawbacks to use of non-standard notation (apart
from the fact that it tends to scare away some of your
audience) is that a certain amount of notational setup is
required at the beginning, and that the bounds one obtains at
the end are rather ineffective (though, of course, one can
always, after painful effort, translate a non-standard
argument back into a messy but quantitative standard argument
if one desires)
Re: What are the 'real numbers', really?
#40"Since (a,0)+(c,0)=(a+c,0) and (a,0)×(c,0)=(ac,0), the points along the horizontal axis have an arithmetic just like "ordinary" numbers"
Holy hell that is clear, concise and compelling. If only my professors would have explained it like this more often in my freshman calc class which was so much more abstract and proof based than anything I had encountered before. The only thing I remember form that time is hellishly long study groups late into the night with my classmates.