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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 n

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.

gizmo686 13y ago
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.
- moyix 13y ago
  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 http://en.wikipedia.org/wiki/Computable_number [2] http://en.wikipedia.org/wiki/Computable_analysis http://en.wikipedia.org/wiki/Computable_analysis
  - gizmo686 13y ago
    Unfourtuantly, there exist numbers which are definable but not computable.
    - moyix 13y ago
      Sure. Chaitin's Omega is a good example. The question is whether such numbers occur in the real world.
      - deleted 13y ago
        [deleted]
      - jfarmer 13y ago
        The trick here is defining "occur" and "real world" precisely. Are you saying the act of me writing down those symbols and expressing the idea does not count as "occurring in the real world?" ;)
        moyix 13y ago
        This reminds me of the self-defeating property of an "uninteresting" number -- a reasonable definition might be "any number that does not have any property of human interest", but then of course there is a smallest such number, and so that has the interesting property of being the first uninteresting number, a contradiction! You're right though that a more precise definition of "real-world numbers" is needed, but I confess that my attempts to think of one in the past few minutes have been essentially circular (coming down to "the ones we know how to compute")!
        jfarmer 13y ago
        Well, we can and have made the idea of a computable number precise: http://en.wikipedia.org/wiki/Computable_number http://en.wikipedia.org/wiki/Computable_number It's not clear whether the universe is computable, however, in the sense that we only find computable numbers in nature. This is kind of an epistemological catch-22, though. How would we know whether this were the case or not?
dnautics 13y ago
imaginary numbers are also a horribly misnamed fiction. For decades, my dad was mesmerised by how if you plugged in a bigger than c value into the lorentz transformations (he had long since forgotten the form of the transformation) you would become "imaginary". To set him straight, I asked him, if instead we named them "Green" numbers, would you become "green" if you went faster than the speed of light?
flebron 13y ago
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 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.
- gizmo686 13y ago
  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 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.
  - ColinWright 13y ago
    Indeed, and the constructable numbers are studied as a subset of the reals, as are the algebraics, and the computables. You can make a choice as to the domain of discourse. If you like, feel free to restrict it to the computables (or the constructables). 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.
acjohnson55 13y ago
Unfortunately, the set of computable numbers, while countable, is more difficult to work with than the much more conceptually reasonable set of reals. Luckily, we have pretty great theoretical tools for dealing with uncountability, so I don't think the fact that the vast, vast majority of real numbers are unidentifiable is really that big of a problem. That, and the continuum is a really useful concept, even if it very well might not have any basis in physical reality.
baddox 13y ago
What do you mean by "represented with a finite amount of information"? Are you referring to their representation in a positional notation like decimal or binary? Or are you referring to the much subtler and more advanced fact that almost all reals are uncomputable? The former isn't really true, and the latter, while true, is subtle enough that it doesn't matter for the vast majority of mathematics (and to replace the reals with the computable numbers would make most of mathematics messy).
- gizmo686 13y ago
  I don't think "represented with a finite amount of information" means computable. For example, consider BusyBeaver(n). We have shown that their exists an n such that BusyBeaver(n) is uncomputable. However, "BusyBeaver(n)" still contains enough information to describe this number. However, because all descriptions are a finite string from a finite alphabet, we can show that only a countable infinity of descriptions exist. However there exist an uncountable infinity of real numbers. Therefore, most real numbers cannot be unambiguously described.
  - baddox 13y ago
    Again, it just comes down to what we mean by "information" and "description." We can certainly construct the real numbers using a finite amount of precise language, so it's reasonable to claim that we have described all real numbers. Heck, even the existence of the English phrase "all undescribable real numbers" evokes an interesting linguistic and philosophical debate, similar to http://en.wikipedia.org/wiki/Interesting_number_paradox http://en.wikipedia.org/wiki/Interesting_number_paradox.
    - gizmo686 13y ago
      We can construct the set of all real numbers with a finite amount of information. However, that set contains elements which we cannot precisely describe with a finite amount of information. The phrase "all undescribable real numbers" does not introduce any problems, because we have still not described any specific undescribable number. We would run into a problem with a phrase such as "the smallest undescribable real number", as that would be a description of a specific undescribable real number. Fourtuantly, that particular phrase does not raise any problems because we can simply conclude that their is no smallest undescribable real number, in the same way that there is no smallest real number in general.
scotty79 13y ago
IMHO real numbers are anything but. I believe there isn't a single thing in the universe that is represented by real number. Any physical law that involves pi should be considered as statistical in nature. There are no perfect circles. Only the things that are really well approximated by them.
- twobits 13y ago
  Amen to that. The real world is discrete. The real numbers in our equations are just approximations.
lmkg 13y ago
The Reals are venerated because they're actually usable. Other numbers systems tend to be a gigantic pain in the ass to get any work done with. 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.
GFK_of_xmaspast 13y ago
I don't see why we need to be shackled to the bounds of countability.

gizmo686 13y ago

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.

moyix 13y ago
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 http://en.wikipedia.org/wiki/Computable_number [2] http://en.wikipedia.org/wiki/Computable_analysis http://en.wikipedia.org/wiki/Computable_analysis
- gizmo686 13y ago
  Unfourtuantly, there exist numbers which are definable but not computable.
  - moyix 13y ago
    Sure. Chaitin's Omega is a good example. The question is whether such numbers occur in the real world.
    - deleted 13y ago
      [deleted]
    - jfarmer 13y ago
      The trick here is defining "occur" and "real world" precisely. Are you saying the act of me writing down those symbols and expressing the idea does not count as "occurring in the real world?" ;)
      - moyix 13y ago
        This reminds me of the self-defeating property of an "uninteresting" number -- a reasonable definition might be "any number that does not have any property of human interest", but then of course there is a smallest such number, and so that has the interesting property of being the first uninteresting number, a contradiction! You're right though that a more precise definition of "real-world numbers" is needed, but I confess that my attempts to think of one in the past few minutes have been essentially circular (coming down to "the ones we know how to compute")!
        jfarmer 13y ago
        Well, we can and have made the idea of a computable number precise: http://en.wikipedia.org/wiki/Computable_number http://en.wikipedia.org/wiki/Computable_number It's not clear whether the universe is computable, however, in the sense that we only find computable numbers in nature. This is kind of an epistemological catch-22, though. How would we know whether this were the case or not?

dnautics 13y ago

imaginary numbers are also a horribly misnamed fiction. For decades, my dad was mesmerised by how if you plugged in a bigger than c value into the lorentz transformations (he had long since forgotten the form of the transformation) you would become "imaginary". To set him straight, I asked him, if instead we named them "Green" numbers, would you become "green" if you went faster than the speed of light?

flebron 13y ago

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 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.

gizmo686 13y ago
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 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.
- ColinWright 13y ago
  Indeed, and the constructable numbers are studied as a subset of the reals, as are the algebraics, and the computables. You can make a choice as to the domain of discourse. If you like, feel free to restrict it to the computables (or the constructables). 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.

acjohnson55 13y ago

Unfortunately, the set of computable numbers, while countable, is more difficult to work with than the much more conceptually reasonable set of reals. Luckily, we have pretty great theoretical tools for dealing with uncountability, so I don't think the fact that the vast, vast majority of real numbers are unidentifiable is really that big of a problem. That, and the continuum is a really useful concept, even if it very well might not have any basis in physical reality.

baddox 13y ago

What do you mean by "represented with a finite amount of information"? Are you referring to their representation in a positional notation like decimal or binary? Or are you referring to the much subtler and more advanced fact that almost all reals are uncomputable? The former isn't really true, and the latter, while true, is subtle enough that it doesn't matter for the vast majority of mathematics (and to replace the reals with the computable numbers would make most of mathematics messy).

gizmo686 13y ago
I don't think "represented with a finite amount of information" means computable. For example, consider BusyBeaver(n). We have shown that their exists an n such that BusyBeaver(n) is uncomputable. However, "BusyBeaver(n)" still contains enough information to describe this number. However, because all descriptions are a finite string from a finite alphabet, we can show that only a countable infinity of descriptions exist. However there exist an uncountable infinity of real numbers. Therefore, most real numbers cannot be unambiguously described.
- baddox 13y ago
  Again, it just comes down to what we mean by "information" and "description." We can certainly construct the real numbers using a finite amount of precise language, so it's reasonable to claim that we have described all real numbers. Heck, even the existence of the English phrase "all undescribable real numbers" evokes an interesting linguistic and philosophical debate, similar to http://en.wikipedia.org/wiki/Interesting_number_paradox http://en.wikipedia.org/wiki/Interesting_number_paradox.
  - gizmo686 13y ago
    We can construct the set of all real numbers with a finite amount of information. However, that set contains elements which we cannot precisely describe with a finite amount of information. The phrase "all undescribable real numbers" does not introduce any problems, because we have still not described any specific undescribable number. We would run into a problem with a phrase such as "the smallest undescribable real number", as that would be a description of a specific undescribable real number. Fourtuantly, that particular phrase does not raise any problems because we can simply conclude that their is no smallest undescribable real number, in the same way that there is no smallest real number in general.

scotty79 13y ago

IMHO real numbers are anything but. I believe there isn't a single thing in the universe that is represented by real number. Any physical law that involves pi should be considered as statistical in nature. There are no perfect circles. Only the things that are really well approximated by them.

twobits 13y ago
Amen to that. The real world is discrete. The real numbers in our equations are just approximations.

lmkg 13y ago

The Reals are venerated because they're actually usable. Other numbers systems tend to be a gigantic pain in the ass to get any work done with. 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.

GFK_of_xmaspast 13y ago

I don't see why we need to be shackled to the bounds of countability.