What are the 'real numbers', really?
math.vanderbilt.edu
What are the 'real numbers', really?
1–10 of 98 posts
Re: What are the 'real numbers', really?
#2Along 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.
Re: What are the 'real numbers', really?
#3I 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.
Re: What are the 'real numbers', really?
#4a 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?
Re: What are the 'real numbers', really?
#5I 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.
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 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.
Re: What are the 'real numbers', really?
#6What 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.
Re: What are the 'real numbers', really?
#7a 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…
> My understanding of a line is that it is delimited by two points, but does not contain any points.
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".
Re: What are the 'real numbers', really?
#8a 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…
Your argument is more philosophical than mathematical. Lines are traditionally defined as the set of all points which satisfy some critera. In this case, a line is precisely made up of points.
Re: What are the 'real numbers', really?
#9a 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…
The "point on the number line" definition has always been non-rigourous. It is meant to imply the intuition that real numbers are what we typically think of as "numbers", notably that they extend to infinity, are ordered, and are dense (for any two distinct real numbers, there exists a real number between them). Of course from a rigorous perspective, this does not even suggest a difference between the reals and the rationals.
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.
Re: What are the 'real numbers', really?
#10What 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.
Do you have any idea of what set we should use to replace them with? The rational numbers can do a lot, but we have discovered that there are numbers worth talking about (and which can be described) that are not rational. Whatever replacement you propose must be usable where ever we would use real numbers, and must be at least as simple to use.