The problem with "points on a number line" as a definition for real numbers is that it's not clear how you can tell if you have all of them. You can populate a number line as densely as you care to using just rational numbers, but that's not all of them, you're missing out on numbers like the square root of two. You can toss in the non-intergral powers of rational numbers, but you still won't have all of them, you're missing out on col numbers like pi (or tau, if you prefer). Even after you toss in every solution to every differential equation you can name, and every number you can generate using well defined finite or infinite serieses, there's probably some horrible diagonaliztion proof that says you still don't have all of them.
I have an issue with this (albeit parenthesised) line: "It turns out that, in some sense, the real numbers would still look like a line under infinite magnification, but the rational numbers would be dots separated by spaces."
In-between any two rational numbers there's an infinite number of other rational numbers. So, in any reasonable sense and at any level of "magnification", if you can "see" two dots representing two rational numbers then they are connected by a line of other little dots (just like the reals). Perhaps you could argue though that at "infinite magnification" there are no rational numbers to be seen, it's just empty space, whereas the reals of course still make a nice line.
"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.
What 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.
"Points on the line" is fine for the first, second, ..., tenth cut at a definition. Sure, completeness is the biggie for the reals compared with the rationals, algebraics, etc.
Still, as in the OP, mentioning Dedekind cuts is okay since it is one way to establish completeness, but there is much more, e.g., as in
John C. Oxtoby, Measure and Category.
and even that doesn't fathom all that is special about the reals. E.g., for just a little more, there is the continuum hypothesis, that little thing!
The OP wants to say that by mentioning Dedekind and completeness he is getting at what the reals really are; no, instead he is just cutting one layer deeper of something that has likely some infinitely many layers available.
Yes, yes, yes, I know; I know; the reals are the only complete, Archimedean ordered field, okay, after we have defined completeness, Archimedean ordered, and field and explained why these are important.
So, back to "points on the line" -- it's actually pretty good for a first cut.
I have a Master's in Applied Math.
The comments about how "few students take [Real Analysis]" doesn't square with my experience and survey of an undergraduate mathematics education. Such a course is often called "Advanced Calculus", and is a required course for a Bachelors-level education in Math. I also understand in the European-style approach to teaching Math, students start off with a foundational approach to Calculus through Real Analysis, and not the hand-wavy & computation-driven Calculus course.
The equivalence class approach attributed to Cantor is more generalizable in discussing sets. The theoretical foundation of Fourier Transforms lies in a similar completion of functions.
Along a similar vein you may also enjoy http://arxiv.org/pdf/1303.6576
The foundations of analysis by Larry Clifton. I always enjoy checking out the references in his papers as they are often hundreds of years old or more.
This is a great article but unfortunately has one thing horribly wrong: Democracy far preceded the Age of Enlightenment. A form of democracy was already in place in ancient Greece at around 500 BC. Newton and the Age of Enlightenment were much later, at 1600+ AD. See Wikipedia: http://en.wikipedia.org/wiki/Democracy#History, http://en.wikipedia.org/wiki/Age_of_enlightenment, http://en.wikipedia.org/wiki/Isaac_Newton.
Other than that, a great article!
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.
" 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...
a 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 the question, what are lines made up of? Are they made up of anything? Is a point really where two (or more) lines would intersect if they could intersect. Is this what is meant by a Dedekind cut?
The contents of the linked page were the first lecture I had in my undergraduate calculus course. At the end of the lecture, we all looked around at each other wondering what we had just signed up for.
Can someone explain the setup of the 0=1 exercise? It's poorly worded. Is it saying find: (Y,1,+,1,×) or is it saying find what "1" has to be to make it a valid field?
I 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.