4 ms·

You don't. He basically defines numbers like pi and e not as numbers, but as iterative functions, which you can run to whatever level of accuracy that you want

You don't. He basically defines numbers like pi and e not as numbers, but as iterative functions, which you can run to whatever level of accuracy that you want. It's sort of a silly argument, because _all_ numbers can be treated like the output of a function, including the real numbers, so he has basically smuggled in all reals through the back door, because any real number can just be thought of as a function with increasingly precise return values with an infinitely long description, just like pi is.

MarkusQ 1y ago
You can't get all the reals that way. The reals that can be produced by an algorithm make up a vanishingly small (e.g. countable) subset. Almost all of the reals are inexpressible.
- LPisGood 1y ago
  To an ultrafinitist, there is no such thing as a number that is inexpressible.
  - kragen 1y ago
    Right, but to be clear, it's not that ultrafinitists like Wildberger believe that they can express all the real numbers; rather, they believe that those inexpressible real numbers don't actually exist.
    - vlovich123 1y ago
      How does that work for calculus which regularly looks at the limits of functions as x approaches infinity and has very real real world applications that stem from such algorithms?
      - drdec 1y ago
        Math in general needs to have a big blinking "don't confuse the map for the territory" label on it. E.g. when you calculate the area of a plot of land do you take into account the curvature of the Earth? You have to make a bunch of compromises in the first place to even talk about what the area of a plot land means. Math is a bunch of useful systems that we humans have devised. We tend to gravitate towards the ones that help us describe and predict things in the real world. But there is plenty of math which doesn't do either. It's just as real as the math that does.
        vlovich123 1y ago
        I agree. I’m just trying to speak to the “realist” argument that Wilderberg presents claiming that some numbers aren’t real and there’s no point talking about them when they come from very practical mathematics let alone the ones that aren’t. In no way was I trying to claim that some part of maths aren’t real - I was trying to understand the consistency of what to me seems like a confusing argument to make.
        kragen 1y ago
        I don't think uncomputable numbers "come from very practical mathematics"! Rather, they come from Gödel, Church, and Turing demolishing Hilbert's program of solving the Entscheidungsproblem once and for all. Possibly, if Hilbert had succeeded, that would have made it "very practical mathematics", or possibly not, but that counterfactual is reasoning from a logical contradiction. https://plato.stanford.edu/entries/church-turing/decision-problem.html https://plato.stanford.edu/entries/church-turing/decision-pr...
      - LPisGood 1y ago
        Here is a paper on just how a serious ultrafinitist copes with that https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/... The short answer is that they deal with such things symbolically.
- empath75 1y ago
  What I described isn't really an algorithm, it's just taking the digits of a number, let's say: foo=3.14159265... Where after 5 is some continuing sequence of decimals. The series of functions is literally just: foo(0) = 3 foo(1) = 3.1 foo(2) = 3.14... And to be clear, it's not just like, an algorithm that estimates pi, it's literally just a list of return values that is infinitely long that return more and more digits of whatever the number is. That is actually how he defines pi. https://youtu.be/lcIbCZR0HbU?si=3YxcHfPlCFrlr5h3&t=2080 https://youtu.be/lcIbCZR0HbU?si=3YxcHfPlCFrlr5h3&t=2080 pi _happens_ to be computable, and there are more efficient functions that will produce those numbers, but you could do the same thing with an incomputable number, you just need a definition for the number which is infinitely long. To be clear, I don't think any of this is a good idea, just pointing out that if he's going to allow that kind of definition of pi (ie, admit a definition that is just an infinite list of decimal representations), you can just do the same thing with any real number you like. He of course will say that he's _not_ allowing any _infinite list_, only an arbitrary long one.
  - MarkusQ 1y ago
    That's the key point though, this list isn't infinitely long, and all the numbers in it are rational. And it is an algorithm (specifically, a lookup table). All the numbers you get this way are going to be rational, and if you require them to be finite, you can't even identify them with any irrational numbers. At least with the computable numbers you get an infinite set of irrational numbers along with the rationals, while still never touching the vast majority of all numbers (the remaining, incomputable irrationals).

MarkusQ 1y ago

You can't get all the reals that way. The reals that can be produced by an algorithm make up a vanishingly small (e.g. countable) subset. Almost all of the reals are inexpressible.

LPisGood 1y ago
To an ultrafinitist, there is no such thing as a number that is inexpressible.
- kragen 1y ago
  Right, but to be clear, it's not that ultrafinitists like Wildberger believe that they can express all the real numbers; rather, they believe that those inexpressible real numbers don't actually exist.
  - vlovich123 1y ago
    How does that work for calculus which regularly looks at the limits of functions as x approaches infinity and has very real real world applications that stem from such algorithms?
    - drdec 1y ago
      Math in general needs to have a big blinking "don't confuse the map for the territory" label on it. E.g. when you calculate the area of a plot of land do you take into account the curvature of the Earth? You have to make a bunch of compromises in the first place to even talk about what the area of a plot land means. Math is a bunch of useful systems that we humans have devised. We tend to gravitate towards the ones that help us describe and predict things in the real world. But there is plenty of math which doesn't do either. It's just as real as the math that does.
      - vlovich123 1y ago
        I agree. I’m just trying to speak to the “realist” argument that Wilderberg presents claiming that some numbers aren’t real and there’s no point talking about them when they come from very practical mathematics let alone the ones that aren’t. In no way was I trying to claim that some part of maths aren’t real - I was trying to understand the consistency of what to me seems like a confusing argument to make.
        kragen 1y ago
        I don't think uncomputable numbers "come from very practical mathematics"! Rather, they come from Gödel, Church, and Turing demolishing Hilbert's program of solving the Entscheidungsproblem once and for all. Possibly, if Hilbert had succeeded, that would have made it "very practical mathematics", or possibly not, but that counterfactual is reasoning from a logical contradiction. https://plato.stanford.edu/entries/church-turing/decision-problem.html https://plato.stanford.edu/entries/church-turing/decision-pr...
    - LPisGood 1y ago
      Here is a paper on just how a serious ultrafinitist copes with that https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/... The short answer is that they deal with such things symbolically.
empath75 1y ago
What I described isn't really an algorithm, it's just taking the digits of a number, let's say: foo=3.14159265... Where after 5 is some continuing sequence of decimals. The series of functions is literally just: foo(0) = 3 foo(1) = 3.1 foo(2) = 3.14... And to be clear, it's not just like, an algorithm that estimates pi, it's literally just a list of return values that is infinitely long that return more and more digits of whatever the number is. That is actually how he defines pi. https://youtu.be/lcIbCZR0HbU?si=3YxcHfPlCFrlr5h3&t=2080 https://youtu.be/lcIbCZR0HbU?si=3YxcHfPlCFrlr5h3&t=2080 pi _happens_ to be computable, and there are more efficient functions that will produce those numbers, but you could do the same thing with an incomputable number, you just need a definition for the number which is infinitely long. To be clear, I don't think any of this is a good idea, just pointing out that if he's going to allow that kind of definition of pi (ie, admit a definition that is just an infinite list of decimal representations), you can just do the same thing with any real number you like. He of course will say that he's _not_ allowing any _infinite list_, only an arbitrary long one.
- MarkusQ 1y ago
  That's the key point though, this list isn't infinitely long, and all the numbers in it are rational. And it is an algorithm (specifically, a lookup table). All the numbers you get this way are going to be rational, and if you require them to be finite, you can't even identify them with any irrational numbers. At least with the computable numbers you get an infinite set of irrational numbers along with the rationals, while still never touching the vast majority of all numbers (the remaining, incomputable irrationals).