Its interesting. When I first encountered complex numbers when starting high school it was very difficult to wrap my head around how they could be actual numbers.
I no longer have that problem, ever since I truly understood how all numbers are simply abstract tools for reasoning. In a way, it's interesting that complex numbers seem more "real" than the real numbers themselves.
I remember listening to a radio show where a physicist discussed the link between quantum mechanics and complex numbers, and thus how they were fundamental to reality [1], whereas we don't know whether real numbers actually describe physical reality.
[1] If I remember correctly, one argument was that although a common use of complex numbers is an alternative number system for making trigonometric/polar calculations simpler, they underpin quantum mechanics in a way that cannot be alternatively formulated in terms of real number numbers
A lot of physics equations describe real quantities like E=mc2. We just kind of take it for granted. You can formulate quantum mechanics without complex numbers but they seem kind of fundamental to it in a similar way to how real quantities like energy seem fundamental to reality.
I came up with a different definition that is a kind of inverse of Dedekind cuts. It is the idea that a real number is the set of all rational intervals that contain it. Since this is circular, there are properties that I came up with which say when a set of rational intervals qualifies to be called a real number in my setup. I have an unreviewed paper which creates a version that is a bridge between numerical analysis and the theoretical definition of a real number. Another unreviewed paper shows the equivalence between my definition and Dedekind cuts. You can read both at [1].
There is a long tradition of using intervals for dealing with real numbers. It is often used by constructivists and can be thought of viewing a real number as a measurement.
Hopefully someone better educated than me can answer this - several of the definitions in the link feel constructivist, i.e. they describe constructions of of real numbers. It seems easy to think of methods of constructing non-rational numbers, by e.g. using infinite sequences, by taking roots, or whatever.
It seems harder to prove that every real number can be constructed via such a method.
Is there a construction-based method that can produce ALL real numbers between, say, 0 and 1? This seems unlikely to me, since the method of construction would probably be based on some sort of enumeration, meaning that you would only end up with countably many numbers. But maybe someone else can help me become un-confused.
> several of the definitions in the link feel constructivist, i.e. they describe constructions of of real numbers.
If you are a constructivist, then you will supply direct proofs for your results as you reject indirect proof, proof by contradiction, law of excluded middle, and things of this nature.
The converse does not necessarily hold. Providing a direct construction of an object satisfying the field and completeness axioms (e.g. the Dedekind construction) does not necessarily mean that one is a constructivist. Indeed, one can use the Dedekind construction and still go on to prove many more results on top of it that still do rely on indirect proof and reductio ad absurdum.
Interestingly, constructive mathematics cannot prove that the Cauchy and Dedekind constructions are isomorphic:
"As often happens in an intuitionistic setting, classically equivalent notions fork. Dedekind reals give rise to several demonstrably different collections of reals when only intuitionistic logic is assumed"
The definitions provided appear as though they are constructive, but they are not actually constructive, they are set-theoretic existence claims that quantify over all sequences, in particular over undefinable sets. Specifically, the description that appears constructive doesn't actually define any particular real number, it only defines the universe in which the real numbers live.
Another subtle detail is that while it's true that every real number corresponds to (and can be represented by) a Cauchy sequence of rationals, the very sequence itself might be undefinable.
Constructivist basically means being able to be explicit. Dedekind cuts and Cauchy sequences are not necessarily constructivist though something described by one of them can be explicitly descriptive for some applications. Any approach which produces all real numbers as commonly accepted will fail to be explicit in all cases as such explicitness presumably implies the real number has been expressed uniquely with finite strings and finite alphabets which can describe at most a countable number of them.
The decimal numbers, for example, can be viewed as an infinite converging sum of powers of ten. Theoretically one could produce a description, but only a countable number of those could be written down in finite terms (some kind of finite recipe). So those finite ones could fall in a constructivist camp, but the ones requiring an infinite string to describe would, as far as I understand constructivism, not fall under being constructivist. To be clear, the finite string doesn't have other be explicit about how to produce the numbers, just that it is naming the thing and it can be derived from that. So square root of 2 names a real number and there is a process to compute out the decimals so that exists in a constructivist sense. But "most" real numbers could not be named.
Your original intuition, that only a countable subset of real numbers can be described or used in any way, is correct. The rest are just “there.” They exist, but we can’t really use them for anything.
It gets weirder. What is a set? For finite sets, we know it intuitively. But consider the Axiom of Choice. There is a consistent mathematics in which a choice set is a set, and one in which the same meta-mathematical object is not a set. (Unless, of course, ZF is inconsistent.)
I am no mathematician, but the idea of real numbers as the limit of rational numbers that don't belong to the set of all rational numbers blows my mind. And to top, the set of real numbers is so much bigger than the set of all rational numbers!
Its interesting. When I first encountered complex numbers when starting high school it was very difficult to wrap my head around how they could be actual numbers.
I no longer have that problem, ever since I truly understood how all numbers are simply abstract tools for reasoning. In a way, it's interesting that complex numbers seem more "real" than the real numbers themselves.
I remember listening to a radio show where a physicist discussed the link between quantum mechanics and complex numbers, and thus how they were fundamental to reality [1], whereas we don't know whether real numbers actually describe physical reality.
[1] If I remember correctly, one argument was that although a common use of complex numbers is an alternative number system for making trigonometric/polar calculations simpler, they underpin quantum mechanics in a way that cannot be alternatively formulated in terms of real number numbers
A lot of physics equations describe real quantities like E=mc2. We just kind of take it for granted. You can formulate quantum mechanics without complex numbers but they seem kind of fundamental to it in a similar way to how real quantities like energy seem fundamental to reality.
If everything is actually quantized, then real numbers do not reflect reality
I came up with a different definition that is a kind of inverse of Dedekind cuts. It is the idea that a real number is the set of all rational intervals that contain it. Since this is circular, there are properties that I came up with which say when a set of rational intervals qualifies to be called a real number in my setup. I have an unreviewed paper which creates a version that is a bridge between numerical analysis and the theoretical definition of a real number. Another unreviewed paper shows the equivalence between my definition and Dedekind cuts. You can read both at [1].
There is a long tradition of using intervals for dealing with real numbers. It is often used by constructivists and can be thought of viewing a real number as a measurement.
1: https://github.com/jostylr/Reals-as-Oracles
Hopefully someone better educated than me can answer this - several of the definitions in the link feel constructivist, i.e. they describe constructions of of real numbers. It seems easy to think of methods of constructing non-rational numbers, by e.g. using infinite sequences, by taking roots, or whatever.
It seems harder to prove that every real number can be constructed via such a method.
Is there a construction-based method that can produce ALL real numbers between, say, 0 and 1? This seems unlikely to me, since the method of construction would probably be based on some sort of enumeration, meaning that you would only end up with countably many numbers. But maybe someone else can help me become un-confused.
> several of the definitions in the link feel constructivist, i.e. they describe constructions of of real numbers.
If you are a constructivist, then you will supply direct proofs for your results as you reject indirect proof, proof by contradiction, law of excluded middle, and things of this nature.
The converse does not necessarily hold. Providing a direct construction of an object satisfying the field and completeness axioms (e.g. the Dedekind construction) does not necessarily mean that one is a constructivist. Indeed, one can use the Dedekind construction and still go on to prove many more results on top of it that still do rely on indirect proof and reductio ad absurdum.
Interestingly, constructive mathematics cannot prove that the Cauchy and Dedekind constructions are isomorphic:
"As often happens in an intuitionistic setting, classically equivalent notions fork. Dedekind reals give rise to several demonstrably different collections of reals when only intuitionistic logic is assumed"
https://arxiv.org/pdf/1510.00641
The definitions provided appear as though they are constructive, but they are not actually constructive, they are set-theoretic existence claims that quantify over all sequences, in particular over undefinable sets. Specifically, the description that appears constructive doesn't actually define any particular real number, it only defines the universe in which the real numbers live.
Another subtle detail is that while it's true that every real number corresponds to (and can be represented by) a Cauchy sequence of rationals, the very sequence itself might be undefinable.
Constructivist basically means being able to be explicit. Dedekind cuts and Cauchy sequences are not necessarily constructivist though something described by one of them can be explicitly descriptive for some applications. Any approach which produces all real numbers as commonly accepted will fail to be explicit in all cases as such explicitness presumably implies the real number has been expressed uniquely with finite strings and finite alphabets which can describe at most a countable number of them.
The decimal numbers, for example, can be viewed as an infinite converging sum of powers of ten. Theoretically one could produce a description, but only a countable number of those could be written down in finite terms (some kind of finite recipe). So those finite ones could fall in a constructivist camp, but the ones requiring an infinite string to describe would, as far as I understand constructivism, not fall under being constructivist. To be clear, the finite string doesn't have other be explicit about how to produce the numbers, just that it is naming the thing and it can be derived from that. So square root of 2 names a real number and there is a process to compute out the decimals so that exists in a constructivist sense. But "most" real numbers could not be named.
Your original intuition, that only a countable subset of real numbers can be described or used in any way, is correct. The rest are just “there.” They exist, but we can’t really use them for anything.
It gets weirder. What is a set? For finite sets, we know it intuitively. But consider the Axiom of Choice. There is a consistent mathematics in which a choice set is a set, and one in which the same meta-mathematical object is not a set. (Unless, of course, ZF is inconsistent.)
I may be misunderstanding your concern, but I believe this is what is meant by "Categoricity for the real numbers"
I am no mathematician, but the idea of real numbers as the limit of rational numbers that don't belong to the set of all rational numbers blows my mind. And to top, the set of real numbers is so much bigger than the set of all rational numbers!
Joel's blog in general is an extremely great read. I highly recommend subscribing.
Maybe worth checking out if you want a formal (as in machine checkable) definition of reals:
https://leanprover-community.github.io/mathlib-overview.html
Real numbers are the concept of quantities built up from continuous flows.
Something we made up before we knew Avogadro’s Number and no longer need.
(That was trolling.)