Infinitesimals have been made rigorous with modern mathematics.
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)
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 beguiling little creatures if you take the time to get to know them. I'd be happy to elaborate or leave a few links here if anybody is interested.
Suppose there's a location on the line that's somehow missing - call it x. Let A be all the numbers less than x, let B be all the numbers greater than x, and that gives you your Dedekind cut. That Dedekind cut is, in a very real sense, x, and that means x is a real. QED.
That needs tidying up and formalising, but it does work.
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.
Consider that the integral of the ruler function from 0 to 1 is 0 (as is stated in your reference 1). In layman's terms you could express this as "there are infinitely more irrational than rational numbers between 0 and 1". At the same time, "for every two rational numbers there are infinitely many rational numbers in-between them". What sort of "picture" is this compatible with?
I still think that the only picture that really makes any sense is a solid line at any finite magnification, yet empty space at infinite magnification.
(Compare e.g. with the fourier transform of a function. It consists of a sum series which comes "arbitrarily close" to the function, but "at the limit" when the number of terms approaches infinity the function and its fourier transform is one and the same.)
There exists a function of the reals which is continuous at every irrational point but discontinuous at every rational point. However, there is no function of the reals which is discontinuous on the irrationals but continuous on the rationals. In this sense, the irrationals are "more continuous" than the rationals.
That's about the best I can do, though, which I admit is a stretch.
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.
We need to stop venerating the "real" numbers and start focusing on sets that are actually usable.
One can definitely "work with" numbers that aren't easy to write. a + (-a) = 0, and this is valid for every real number a, not just "the ones which I can describe with a finite amount of information", or the ones I've written down at some point in my life.
Every number that we can construct can be constructed in a finite amount of symbols. For example sqrt(2) is an unambiguous description of itself. Without use of the sqrt function, we can also call it the number x such that x*x=2. However, every description is a finite string constructed from a finite alphabet. We can easily show that the set of all such descriptions is countably infinite. However, we can also show that the set of all real numbers is uncountably infinite. Therefore, there is an uncountable infinity of real numbers that cannot be constructed.
The Reals are constructed specifically to be the smallest set that has some nice algebraic properties, like Least Upper Bounds. Sets that model the real world, like the constructables, countables, computables, etc. tend to be subsets of the Reals, and therefore don't have those properties. That absence makes life difficult.
The Real Number system, like almost everything in mathematics, is an approximation of reality that makes a trade-off between faithfulness and tractibility. As it turns out, gaining more of the former loses you quite a bit of the latter. It's generally not worth it.
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.
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.
Yes. Where I graduated, all engineering majors learn the axiomatic definition of the real numbers including the "supremum (least upper bound) axiom" at the beginning of the first calculus class.
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 curious paper. It's a rigorous derivation of (positive) real numbers without the use of 0 or negative numbers anywhere. It isn't very useful, although the fact that this can easily be done is by itself interesting.
I have sometimes thought about the possibility of us encountering an advanced alien civilization and trying to match our math to theirs. Someone told me recently that if aliens were able to get into space, we can take it for granted that they knew negative numbers (in addition to more advanced concepts). I disagreed. Negative numbers are very convenient, but all the math that's needed for modern physics can, I think, be built up without them in a way that's more bulky and awkward, but not an order of magnitude bulky. This paper is weak evidence of my position.
Other than that, a great article!
It also states that you cannot order the field of complex numbers. Whereas I seem to recollect that there are ways to do so. For instance, z1 < z2 if x1 < x2 or x1 = x2 and y1 < y2.
It's wonderful to have this little insight now. It's unfortunate that my math knowledge is so filled with holes.
Unfortunately there is no such theory."
http://njwildberger.wordpress.com/2012/12/02/difficulties-wi...
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 line you are talking about in the rest of your post seems to be an 'unrelated' object that is used in geometry. I am not familiar with the formal definition of line that is used in geometry, but one way of defining a line is as the set of all points which satisfy "y=mx+b", for a given (m,b). A line segment would be the above definition with restrictions on the domain: x_0<x<x_f.
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 _on_ a line in geometry. 'contains' is something from set theory: "the set of all points on line l contains point P" is a perfectly valid expression (but "P is on l" is way shorter)
A line is (or can be viewed as) an infinite set of points.
> To elaborate, no point could be "on" a line because a point has no extension, whereas a line does.
That seems to be a consequence of an unusual definition of "on".
You can deal instead with Euclid's axiomatization of geometry, and there "line" is an abstract thing defined by two points. Different animal, although seldom explained clearly by teachers, who often themselves don't really understand what's going on. (Although some do, and don't get the chance to explore these things because of the pressure of the curriculum, and students who don't care, but need to pass.)
All too often people get confused about this and are told to shut up by their teacher, whereas in fact the student has had an insight, and demonstrated deeper understanding.
You may find the Fano plane (a three-dimensional finite projective space) interesting:
* http://en.wikipedia.org/wiki/Fano_plane (brief description)
* http://math.ucr.edu/home/baez/octonions/node4.html (connections with higher math)I don't remember the precise proof, but if memory serves it derives from the existance of opposites and inverses, and 0 and 1 being unique in the set, due the commutative properties of abelean groups.