25 ms·

I would just like to note that this is a moot point, as the computable reals are a subset of the reals, thus by extension equality of the reals is also undecida

by orlp 4y ago

I would just like to note that this is a moot point, as the computable reals are a subset of the reals, thus by extension equality of the reals is also undecidable. This fact is always given as an argument against computable analysis, while completely ignoring the reals suffer from the exact same issue. It bothers me greatly that the mathematical objects we generally refer to as "numbers" have no constructive basis. I really wish the reals were phased out as the default "number" object as a mathematical curiosity (like the surreals are), and that the default interpretation of "number" is replaced by a computational foundation. The idea of an "non-computable number" seems completely silly to me, as the fundamental property for something to be called a number, to me, is the ability to do arithmetic with it. If I can't add with it, multiply by it, or even know its digits, why are we calling it a number (e.g. https://en.wikipedia.org/wiki/Chaitin%27s_constant https://en.wikipedia.org/wiki/Chaitin%27s_constant)? Note that I have no problems with undecidability in general, I am well familiar with it and the above does not stem from ignorance. I am not rejecting the reals as a mathematical concept entirely, not at all. They are perfectly valid mathematical objects to study. I simply reject the current definition we have for a "number" as the correct mathematical object for the job.

jostylr 4y ago
If you have not heard of Norm Wildberger, you may enjoy some of what he has delved into. Here is one video in which he explains some of his difficulties with current mathematics: https://cosmolearning.org/video-lectures/difficulties-with-real-numbers-infinite-decimals-i/ https://cosmolearning.org/video-lectures/difficulties-with-r... He dislikes the idea of presenting infinity as a complete thing. And that one objection leads to a lot of different new directions to pursue. His rational trigonometry does trigonometry entirely without transcendental functions. It can involve square roots at times, but it is kept to a minimum. Everything else is entirely rational based. He has several videos that delve into the problems of each of the standard formulations of real numbers. He also argues for a more practical and computational version of the Fundamental Theorem of Algebra. An interesting demonstration of the difficulty of real number arithmetic, relevant to some other comments here, is multiplying 1/9 by itself. For fractions, it is trivial as it is 1/81 and this can be converted into a repeating decimal, of course. But try multiplying the decimal form of 1/9 by itself. It is all 1's in the multiplication so it should be easy, right? If you write it down, essentially, the n+1th place is generated by summing n 1s. That is, it is .0123456789(10)(11)(12).... where I put in parentheses the sum of that columns digits. So one has to carry and as it goes further out, one is carrying over many digits; when out a trillion places, we are carrying across 12 places, which is larger than the repeating pattern. Just carrying that first 10 leads to .01234567900(11)(12)... And .012345679 is the basic pattern of 1/81 but it is hard to see feeling confident about that if one only had the infinite decimal to work with. The point is that something with a non-repeating pattern such as computing sqrt(2)pie seems difficult enough that it verges on the vacuous. He does point out the difference for his criticism applying to Pure Mathematics rather than Applied Mathematics. Approximations are fine for applications and what Wildberger is really saying is he wants a Pure Mathematics that really supports that explicitly by focusing on rational numbers as much as possible. For example, he introduces differential calculus with polynomials by considering transforming p(x) to q(x) = p(x+r), collecting powers of x, and then translating back to p(x) = q(x-r) which is just replacing x with x-r. If expanded out, all the r's cancel, but if one leaves them and then truncates the different powers, one gets the different polynomial approximations. While neat in avoiding limts, the real nice thing is applying this technique to algebriac curves. For example, we can view the unit circle as the solution to 0= p(x,y) = x^2 + y^2 - 1. We can do the same trick above computing p(x+r, y+s), expand, and then retranslate and truncate. This can give us the approximations to the unit circle at a given point on the circle. This sidesteps having to compute the derivative of the square root function to get the tangent lines to the unit circle. An example of an alternate work flow is multiplying two complex numbers on the unit circle. The traditional approach is to say "compute the angles and then add the angles". But the computing of the angles is impossibly hard to do in a precise fashion (approximate is fine, of course). But there is a perfectly fine accurate procedure. Take the points z and w on the unit circle and draw a line through them. Draw a parallel line through 1. The line will intersect the circle at z*w. As a quick example of this, if you multiply a+bi and -a+bi, this becomes -a^2 -b^2 = -1. Geometrically, the line through these two points is horizontal and the horizontal line through 1 intersects at -1. You can see that with angles, but it feels less intuitive to me that that is how it will work out. Even the set of Natural Numbers being called infinite is something he questions. He used the term "unending" which I like as well. And by understanding that "most" natural numbers cannot be represented in this universe (assuming it is a finite universe), then it leads to questions such as what numbers can be represented? We have islands of simplicity such as 10^10^10^10^10^10^10 + 23. How dense are they in the larger numbers? Can we do anything useful with those islands? These questions are less prompted when we simply think of the natural numbers as this one big set of sameness. But if we demand that being able to do the computations is actually an important requirement, then we can investigate many more interesting ideas. And Wildberger's point is that this should be in the domain of Pure Mathematics with it being taught to future mathematicians instead of it being relegated to Applied Mathematics.
- orlp 4y ago
  I have seen some of Wildberger's work, yes. My stance isn't as strong as him on some points - I certainly don't reject the reals outright or take issue with them. I think they're fascinating mathematical objects, I just reject choosing this particular mathematical object as what we mean by a "number".
quantified 4y ago
Would you include the integers and the rationals in the mathematical objects that have no constructive basis?
- orlp 4y ago
  No, they are absolutely constructive and have well-defined computable addition and multiplication rules.
  - quantified 4y ago
    Thank you, it’s not an area I’ve gone deep on. It seems to me the casual observer that there is not a single thing called “number”, rather there is a family of somewhat related concepts each of which ties in to various semantics and systems of units for relevant operations and interpretations. So “number” needs at least one adjective in front of it, otherwise we don’t know what we’re discussing.
tromp 4y ago
But you can do arithmetic with uncomputable numbers. The probability of a the universal machine not halting on a random input is 1 - Ω. There, I just subtracted an uncomputable number. As you note, you cannot "know its digits", which is just rephrasing the "uncomputable" aspect of these numbers.
- orlp 4y ago
  I disagree that you have done arithmetic here. You have constructed an expression that describes arithmetic. Would you be satisfied if you entered 3 + 8 into a calculator it would simply give you back 3 + 8? Would you say this calculator has performed arithmetic? No. You described to it the arithmetic it must perform, and it failed to do so. In mathematics we almost exclusively work with the descriptions (expressions), using the rules for simplifying and manipulating those descriptions (algebra). But the foundation underlying the algebra is (currently) the reals, which contains uncomputable objects. Ultimately we want every description of a number to map to an actual number. Why are we allowing descriptions that are not computable, and then call the result a number anyway?
  - auggierose 4y ago
    How about instead of number you say "computable number" and call it a day? There, solved it for you. Oh no, I didn't, because the concept of computability is also not absolutely clear. Helpful reading: "The Nature of Mathematics", in "The Fabric of Reality" by David Deutsch.
octachron 4y ago
One issue with computable numbers is that computable analysis is a far more complex subject than real or complex analysis since limits or upper bounds do not preserve computability. Similarly, the continuity of computable functions means that one need a more complex framework to even speak about discontinuity.
orra 4y ago
> the mathematical objects we generally refer to as "numbers" have no constructive basis. Eh? I’m confused by your terminology here. You can construct the set of Real numbers using Dedekind cuts. Less conventionally, you can construct the Surreal numbers, and find the Reals as a subset.
- orlp 4y ago
  I should say computable basis to be precise. Our fundamental object of computation - the number - generally does not have a computable basis in modern mainstream mathematics (where we generally assume the real numbers by default). The standard Dedekind cut construction does not give computable arithmetic because it involves cartesian products of arbitrary (not necessarily enumerable) infinite sets.
  - orra 4y ago
    I can understand some of that discomfort. However, ultimately, uncountable infinities and (the axiom of) choice leads to other interesting scenarios that, IMO, it's best to make peace with. Besides, mathematicians rarely study arbitrary, probabilisticly-common, objects. We focus on objects with “nice” algebraic properties. Similarly, it doesn't matter that almost all numbers are uncomputable when we don't use them in practice.

by orlp 4y ago

jostylr 4y ago

If you have not heard of Norm Wildberger, you may enjoy some of what he has delved into. Here is one video in which he explains some of his difficulties with current mathematics: https://cosmolearning.org/video-lectures/difficulties-with-real-numbers-infinite-decimals-i/ https://cosmolearning.org/video-lectures/difficulties-with-r... He dislikes the idea of presenting infinity as a complete thing. And that one objection leads to a lot of different new directions to pursue. His rational trigonometry does trigonometry entirely without transcendental functions. It can involve square roots at times, but it is kept to a minimum. Everything else is entirely rational based. He has several videos that delve into the problems of each of the standard formulations of real numbers. He also argues for a more practical and computational version of the Fundamental Theorem of Algebra. An interesting demonstration of the difficulty of real number arithmetic, relevant to some other comments here, is multiplying 1/9 by itself. For fractions, it is trivial as it is 1/81 and this can be converted into a repeating decimal, of course. But try multiplying the decimal form of 1/9 by itself. It is all 1's in the multiplication so it should be easy, right? If you write it down, essentially, the n+1th place is generated by summing n 1s. That is, it is .0123456789(10)(11)(12).... where I put in parentheses the sum of that columns digits. So one has to carry and as it goes further out, one is carrying over many digits; when out a trillion places, we are carrying across 12 places, which is larger than the repeating pattern. Just carrying that first 10 leads to .01234567900(11)(12)... And .012345679 is the basic pattern of 1/81 but it is hard to see feeling confident about that if one only had the infinite decimal to work with. The point is that something with a non-repeating pattern such as computing sqrt(2)pie seems difficult enough that it verges on the vacuous. He does point out the difference for his criticism applying to Pure Mathematics rather than Applied Mathematics. Approximations are fine for applications and what Wildberger is really saying is he wants a Pure Mathematics that really supports that explicitly by focusing on rational numbers as much as possible. For example, he introduces differential calculus with polynomials by considering transforming p(x) to q(x) = p(x+r), collecting powers of x, and then translating back to p(x) = q(x-r) which is just replacing x with x-r. If expanded out, all the r's cancel, but if one leaves them and then truncates the different powers, one gets the different polynomial approximations. While neat in avoiding limts, the real nice thing is applying this technique to algebriac curves. For example, we can view the unit circle as the solution to 0= p(x,y) = x^2 + y^2 - 1. We can do the same trick above computing p(x+r, y+s), expand, and then retranslate and truncate. This can give us the approximations to the unit circle at a given point on the circle. This sidesteps having to compute the derivative of the square root function to get the tangent lines to the unit circle. An example of an alternate work flow is multiplying two complex numbers on the unit circle. The traditional approach is to say "compute the angles and then add the angles". But the computing of the angles is impossibly hard to do in a precise fashion (approximate is fine, of course). But there is a perfectly fine accurate procedure. Take the points z and w on the unit circle and draw a line through them. Draw a parallel line through 1. The line will intersect the circle at z*w. As a quick example of this, if you multiply a+bi and -a+bi, this becomes -a^2 -b^2 = -1. Geometrically, the line through these two points is horizontal and the horizontal line through 1 intersects at -1. You can see that with angles, but it feels less intuitive to me that that is how it will work out. Even the set of Natural Numbers being called infinite is something he questions. He used the term "unending" which I like as well. And by understanding that "most" natural numbers cannot be represented in this universe (assuming it is a finite universe), then it leads to questions such as what numbers can be represented? We have islands of simplicity such as 10^10^10^10^10^10^10 + 23. How dense are they in the larger numbers? Can we do anything useful with those islands? These questions are less prompted when we simply think of the natural numbers as this one big set of sameness. But if we demand that being able to do the computations is actually an important requirement, then we can investigate many more interesting ideas. And Wildberger's point is that this should be in the domain of Pure Mathematics with it being taught to future mathematicians instead of it being relegated to Applied Mathematics.

orlp 4y ago
I have seen some of Wildberger's work, yes. My stance isn't as strong as him on some points - I certainly don't reject the reals outright or take issue with them. I think they're fascinating mathematical objects, I just reject choosing this particular mathematical object as what we mean by a "number".

quantified 4y ago

Would you include the integers and the rationals in the mathematical objects that have no constructive basis?

orlp 4y ago
No, they are absolutely constructive and have well-defined computable addition and multiplication rules.
- quantified 4y ago
  Thank you, it’s not an area I’ve gone deep on. It seems to me the casual observer that there is not a single thing called “number”, rather there is a family of somewhat related concepts each of which ties in to various semantics and systems of units for relevant operations and interpretations. So “number” needs at least one adjective in front of it, otherwise we don’t know what we’re discussing.

tromp 4y ago

But you can do arithmetic with uncomputable numbers. The probability of a the universal machine not halting on a random input is 1 - Ω. There, I just subtracted an uncomputable number. As you note, you cannot "know its digits", which is just rephrasing the "uncomputable" aspect of these numbers.

orlp 4y ago
I disagree that you have done arithmetic here. You have constructed an expression that describes arithmetic. Would you be satisfied if you entered 3 + 8 into a calculator it would simply give you back 3 + 8? Would you say this calculator has performed arithmetic? No. You described to it the arithmetic it must perform, and it failed to do so. In mathematics we almost exclusively work with the descriptions (expressions), using the rules for simplifying and manipulating those descriptions (algebra). But the foundation underlying the algebra is (currently) the reals, which contains uncomputable objects. Ultimately we want every description of a number to map to an actual number. Why are we allowing descriptions that are not computable, and then call the result a number anyway?
- auggierose 4y ago
  How about instead of number you say "computable number" and call it a day? There, solved it for you. Oh no, I didn't, because the concept of computability is also not absolutely clear. Helpful reading: "The Nature of Mathematics", in "The Fabric of Reality" by David Deutsch.

octachron 4y ago

One issue with computable numbers is that computable analysis is a far more complex subject than real or complex analysis since limits or upper bounds do not preserve computability. Similarly, the continuity of computable functions means that one need a more complex framework to even speak about discontinuity.

orra 4y ago

> the mathematical objects we generally refer to as "numbers" have no constructive basis. Eh? I’m confused by your terminology here. You can construct the set of Real numbers using Dedekind cuts. Less conventionally, you can construct the Surreal numbers, and find the Reals as a subset.

orlp 4y ago
I should say computable basis to be precise. Our fundamental object of computation - the number - generally does not have a computable basis in modern mainstream mathematics (where we generally assume the real numbers by default). The standard Dedekind cut construction does not give computable arithmetic because it involves cartesian products of arbitrary (not necessarily enumerable) infinite sets.
- orra 4y ago
  I can understand some of that discomfort. However, ultimately, uncountable infinities and (the axiom of) choice leads to other interesting scenarios that, IMO, it's best to make peace with. Besides, mathematicians rarely study arbitrary, probabilisticly-common, objects. We focus on objects with “nice” algebraic properties. Similarly, it doesn't matter that almost all numbers are uncomputable when we don't use them in practice.