4 ms·

I've made the argument many times before that the reals are certainly an interesting set of objects, but a poor abstraction for the concept of a number. We shou

by orlp 3y ago

I've made the argument many times before that the reals are certainly an interesting set of objects, but a poor abstraction for the concept of a number. We should work towards replacing them with a more well-behaved set without eldritch monsters such as uncomputable 'numbers' that you can't add or multiply with. E.g. https://en.wikipedia.org/wiki/Computable_number https://en.wikipedia.org/wiki/Computable_number.

zeroonetwothree 3y ago
Computable functions do also have some weird behavior. For example: a) all functions are continuous, b) equality is not computable (maybe not so weird for anyone used to floating point), c) differentiation is not computable unless you work over complex numbers (but integration is!)
- orlp 3y ago
  a) All functions from computable reals to computable reals are continuous. This makes sense, because an arbitrarily small input difference leading to an arbitrarily large output difference is not a good basis for computation. You can still have discontinuities for discrete computable maps. b) Neither is equality computable for the reals. Nothing is lost here. c) Differentiation is computable for the functions of computable reals if you have a computable rate of convergence.
  - chromoblob 3y ago
    How to do (c)? I tried to model the reals as lazily computed sequences of intervals with rational bounds where every next interval is subset of previous one, and couldn't write a higher-order differentiator function because on every step for a given input interval there is output interval with non-zero (in most cases) size and the differentiated function may contain quick enough oscillations that the derivative is unbounded, and thus the derivative will stay unbounded no matter what you do. What do you mean by rate of convergence?
    - orlp 3y ago
      https://en.wikipedia.org/wiki/Modulus_of_convergence https://en.wikipedia.org/wiki/Modulus_of_convergence
      - chromoblob 3y ago
        (I assume you refer to the sequence of reals that is used to compute the limit in the definition of derivative) But is there a way to somehow construct this modulus automatically for some functions? And if there isn't, then isn't requiring manual input of the modulus like just requiring manual input of the derivative itself? Can you typically find the modulus manually more easily than the derivative?
        orlp 3y ago
        > But is there a way to somehow construct this modulus automatically for some functions? In a sense. You would end up with https://en.wikipedia.org/wiki/Automatic_differentiation https://en.wikipedia.org/wiki/Automatic_differentiation. The amount of functions you can construct in this manner is massive, along with their derivatives, but it is not an operator that given an arbitrary computable function gives you its derivative.
        chromoblob 3y ago
        Thank you.
mananaysiempre 3y ago
There’s a tradeoff[1] throughout mathematics, not unlike the expressiveness—analyzability one in programming languages: either the set of things you’re working with has nice properties (e.g., for the reals, completeness) and supports nice constructions with no caveats (e.g., for the reals, limits); or each of the things in the set is individually nice and tractable (e.g. computable). There isn’t a single universally appropriate choice here, and sometimes even the direction of the axis is not obvious (real numbers lack solutions for algebraic equations; complex numbers lack smooth functions nonzero only in a finite region), but usually a simple family of eldritch objects is easier to deal with than an byzantine clan of cuddly objects. Could you do simple physics and more generally ordinary differential equations over the computable numbers? You probably could to some extent, but it’s unlikely you’d get the nice intuitive geometry you get over the reals (see e.g. Arnold’s ODE textbook). The (basic) geometry of surfaces parametrized by real coordinates is fairly straightforward (differential geometry); the geometry of surfaces parametrized by rational coordinates is famously mind-blowingly abstract (algebraic geometry). The (basic) theory of real or complex functions is mostly understandable (analysis); the theory of natural or rational functions is horrifying (number theory). Also, as a sibling reply mentions, it’s not like computable numbers will solve every problem: even though arithmetic over them is decidable, order still isn’t, and computable transcendental functions aren’t exactly a cakewalk either. It doesn’t mean the computable reals aren’t worth knowing about, but, well, we live in a world where mandatory education has violently rejected[2] every advance in mathematical thought since Otto von Bismarck left office, and reduced the ones it accepted to a desiccated husk[3], so there are a great many things in that category. (I cannot help but remember the SMBC patriotism vs nationalism comic[4]: “Now I am mathematics.”) [1] https://ncatlab.org/nlab/show/dichotomy+between+nice+objects+and+nice+categories https://ncatlab.org/nlab/show/dichotomy+between+nice+objects... [2] http://www.ams.org/notices/201201/rtx120100031p.pdf http://www.ams.org/notices/201201/rtx120100031p.pdf [3] https://www.maa.org/external_archive/devlin/devlin_03_08.html https://www.maa.org/external_archive/devlin/devlin_03_08.htm... [4] https://www.smbc-comics.com/comic/an-important-distinction https://www.smbc-comics.com/comic/an-important-distinction

by orlp 3y ago

zeroonetwothree 3y ago

Computable functions do also have some weird behavior. For example: a) all functions are continuous, b) equality is not computable (maybe not so weird for anyone used to floating point), c) differentiation is not computable unless you work over complex numbers (but integration is!)

orlp 3y ago
a) All functions from computable reals to computable reals are continuous. This makes sense, because an arbitrarily small input difference leading to an arbitrarily large output difference is not a good basis for computation. You can still have discontinuities for discrete computable maps. b) Neither is equality computable for the reals. Nothing is lost here. c) Differentiation is computable for the functions of computable reals if you have a computable rate of convergence.
- chromoblob 3y ago
  How to do (c)? I tried to model the reals as lazily computed sequences of intervals with rational bounds where every next interval is subset of previous one, and couldn't write a higher-order differentiator function because on every step for a given input interval there is output interval with non-zero (in most cases) size and the differentiated function may contain quick enough oscillations that the derivative is unbounded, and thus the derivative will stay unbounded no matter what you do. What do you mean by rate of convergence?
  - orlp 3y ago
    https://en.wikipedia.org/wiki/Modulus_of_convergence https://en.wikipedia.org/wiki/Modulus_of_convergence
    - chromoblob 3y ago
      (I assume you refer to the sequence of reals that is used to compute the limit in the definition of derivative) But is there a way to somehow construct this modulus automatically for some functions? And if there isn't, then isn't requiring manual input of the modulus like just requiring manual input of the derivative itself? Can you typically find the modulus manually more easily than the derivative?
      - orlp 3y ago
        > But is there a way to somehow construct this modulus automatically for some functions? In a sense. You would end up with https://en.wikipedia.org/wiki/Automatic_differentiation https://en.wikipedia.org/wiki/Automatic_differentiation. The amount of functions you can construct in this manner is massive, along with their derivatives, but it is not an operator that given an arbitrary computable function gives you its derivative.
        chromoblob 3y ago
        Thank you.

mananaysiempre 3y ago

There’s a tradeoff[1] throughout mathematics, not unlike the expressiveness—analyzability one in programming languages: either the set of things you’re working with has nice properties (e.g., for the reals, completeness) and supports nice constructions with no caveats (e.g., for the reals, limits); or each of the things in the set is individually nice and tractable (e.g. computable). There isn’t a single universally appropriate choice here, and sometimes even the direction of the axis is not obvious (real numbers lack solutions for algebraic equations; complex numbers lack smooth functions nonzero only in a finite region), but usually a simple family of eldritch objects is easier to deal with than an byzantine clan of cuddly objects. Could you do simple physics and more generally ordinary differential equations over the computable numbers? You probably could to some extent, but it’s unlikely you’d get the nice intuitive geometry you get over the reals (see e.g. Arnold’s ODE textbook). The (basic) geometry of surfaces parametrized by real coordinates is fairly straightforward (differential geometry); the geometry of surfaces parametrized by rational coordinates is famously mind-blowingly abstract (algebraic geometry). The (basic) theory of real or complex functions is mostly understandable (analysis); the theory of natural or rational functions is horrifying (number theory). Also, as a sibling reply mentions, it’s not like computable numbers will solve every problem: even though arithmetic over them is decidable, order still isn’t, and computable transcendental functions aren’t exactly a cakewalk either. It doesn’t mean the computable reals aren’t worth knowing about, but, well, we live in a world where mandatory education has violently rejected[2] every advance in mathematical thought since Otto von Bismarck left office, and reduced the ones it accepted to a desiccated husk[3], so there are a great many things in that category. (I cannot help but remember the SMBC patriotism vs nationalism comic[4]: “Now I am mathematics.”) [1] https://ncatlab.org/nlab/show/dichotomy+between+nice+objects+and+nice+categories https://ncatlab.org/nlab/show/dichotomy+between+nice+objects... [2] http://www.ams.org/notices/201201/rtx120100031p.pdf http://www.ams.org/notices/201201/rtx120100031p.pdf [3] https://www.maa.org/external_archive/devlin/devlin_03_08.html https://www.maa.org/external_archive/devlin/devlin_03_08.htm... [4] https://www.smbc-comics.com/comic/an-important-distinction https://www.smbc-comics.com/comic/an-important-distinction