41–50 of 98 posts
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.
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This is a similar argument to sqrt(2) being "not a number", back in the BC's, because it was not rational. And yet, you can construct it in a straightforward manner by making a right angled triangle with catheti of length 1, giving a hypotenuse of length sqrt(2). I suppose this would have made you equally uncomfortable back then. One can definitely "work with" numbers that aren't easy to write. a + (-a) = 0, and this…
The 'problem' with the reals is that there are numbers that cannot be constructed. 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 t…
Then apply the diagonal argument. Take the computable numbers between 0 and 1, including 0, not including 1. These are countable, so we can write them in a list, taking a mapping k from the natural numbers: { 1, 2, 3, 4, ... } to the set of computable numbers in [0,1).
Now let's construct a new number. In the first decimal place we put 1 if the first decimal place of k(1) is 0, and 0 otherwise. In the second place we put 1 if the second decimal place of k(2) is 0, and 0 otherwise. And so on.
This results in a number that's not on the list, and is between 0 and 1. So it must, by our assumption, not be computable.
Things become tricky.
So there's a choice to be made, and most mainstream mathematicians have decided to talk about, use, study, and otherwise accept the existence of the real numbers because it's convenient.
Feel free to choose otherwise.
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 t…
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.
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I'm interested! I think Dedekind cuts are reasonably understandable, but infinitesimals are on the surface of much of our calculus syntax, so I'd be glad to understand where they become so tricky formally.
I wrote a short paper on the topic once upon a time[1] which you may find interesting. It's part history of math, part philosophy of math. It's not a great paper and most of the insights in it come from others but here is some of the arithmetic of nilpotent[1] infinitesimals as shown in the appendix. Imagine an entity which is not equal to zero but that when raised to the power of 2 or higher is equal to zero! Sounds…
(ϵ + 1)(ϵ−1) = −1, or alternately (1 + ϵ)(1 − ϵ) = −1
(1 + ϵ)(1 − ϵ) = 1
(1 + ϵ)(1 − ϵ) = 1 - ϵ^2 = 1Along 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 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!
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One possible replacement is the computable numbers [1]; this includes the algebraic numbers and some common transcendentals (e, pi), and you can even build up something akin to standard analysis (computable analysis [2]). [1] http://en.wikipedia.org/wiki/Computable_number [2] http://en.wikipedia.org/wiki/Computable_analysis
Unfourtuantly, there exist numbers which are definable but not computable.
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…
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.