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Correction, tons of physicists use real values (or complex real values) in their calculations. That doesn't imply that the corresponding values in the world are

Correction, tons of physicists use real values (or complex real values) in their calculations. That doesn't imply that the corresponding values in the world are irrational, transcendental, infinitely-precise, etc. There are a few ways we can avoid the reals in physics. Abstractly, there are results such as Skolem's "paradox" ( https://en.wikipedia.org/wiki/Skolem%27s_paradox https://en.wikipedia.org/wiki/Skolem%27s_paradox ) which tell us that theories involving uncountable sets (such as the reals) can be satisfied by countable models. More concretely, physicists never manipulate real numbers directly; only symbols which may represent real numbers. Such symbolic manipulation could, in principle, be performed by a turing machine, and hence any value obtained from such theories (e.g. experimental predictions) must be in the set of computable numbers. Hence the reals are just an 'implementation detail' of any theory, and their purported presence cannot have any influence on the theory's predictions. As I said previously, I have nothing against the reals as a useful mathematical tool; I just disagree with the name, and claims that they are physically real. Others go further though, arguing that discrete mathematics is more general and useful than continuous, e.g. http://www.math.rutgers.edu/~zeilberg/OPINIONS.html http://www.math.rutgers.edu/~zeilberg/OPINIONS.html

wyager 10y ago
Nothing you said in your post has actually demonstrated that it's unlikely that reals underly physical reality. It sounds like you just have some sort of aesthetic agenda. Your entire premise (that we might replace reals in current physical theory with some other structure and get the same result) is trivially debunked. Pi shows up all over the place, and e.g. rationals are not closed under exponentiation by e. The schroedinger equation applied to physical hilbert spaces is inextricably real-valued.
- chriswarbo 10y ago
  > Nothing you said in your post has actually demonstrated that it's unlikely that reals underly physical reality. Sorry, I was taking it for granted that the "reals" are problematic and unphysical (based on the sentiment of the grandparent). For clarification, the real numbers are a superset of the "computable" numbers (those numbers which can be approximated, to any given precision, by a universal turing machine). The computable numbers are countable (since there are countably-many turing machine tapes), but the reals are uncountable. That means "almost all" of the reals are uncomputable. Since all the laws of physics are computable, all physical processes must be based on computable numbers; even if some initial condition of the universe resulted in an uncomputable number, e.g. if the digits of the muon's magnetic moment are the same as the digits of the 100th busy beaver number, we can never know since we'll never compute what that busy beaver number is (we could test such predictions empirically, but it could always just be a coincidence). > Pi shows up all over the place, and e.g. rationals are not closed under exponentiation by e. Pi and e are computable numbers (quite trivially so, there are loads of known algorithms to do it). Exponentiation is a computable operation (again, the algorithm is pretty trivial; it's just repeated multiplication, multiplication is repeated addition and addition is repeated increment, so you only need 3 loops) https://en.wikipedia.org/wiki/Computable_number#Can_computable_numbers_be_used_instead_of_the_reals.3F https://en.wikipedia.org/wiki/Computable_number#Can_computab... > The schroedinger equation applied to physical hilbert spaces is inextricably real-valued. It's been a while since my physics master's, so I wouldn't want to refute this face-on; however, it's trivial to notice that (by definition) we can only use computable numbers in our calculations, and hence even if there are any non-computable reals in a theory, they cannot affect the predictions of that theory. If some prediction depended on a value outside of the computable numbers, e.g. the 100th digit of Chaitin's constant, or the 100th busy beaver number modulo 7, then that prediction would not be calculable and could only be given in symbolic form, e.g. 1.8 * Chaitin(100) ^ 3.45. This is not the case for the schroedinger equation, which gives results that are analytically and/or numerically computable. > Your entire premise (that we might replace reals in current physical theory with some other structure and get the same result) is trivially debunked. It's not my premise to "replace reals in current physical theory" (although it seems that Doron Zeilberger might be pleased if that happened). I was pointing out that physicists, mathematicians and many others claim to use the reals, when in fact they're only using the computable numbers. This leads to confusion when unintuitive results about the reals are applied to the world. For example, when choosing a real number randomly from the unit interval, the probability of each point is 0, but the sum of their probabilities is 1. That's an example of the reals being unintuitive, and that's fine. However, trying to apply such statements to the world (e.g. "I can prove that the chance you've measured exactly 1 litre of water is zero") leads to confusion. Even worse, the confused person (the one making the claim, confusing uncomputable numbers with reality) may then "prove wrong" some perfectly justified refutations. In case you can't see the problem, the "proof" that an amount other than 1 litre was poured relies on checking more and more decimal places until a difference is found between the measured amount and 1 litre. However, we have no way of knowing that the "keep checking decimals" algorithm will halt, so the claim might be an unsound counterfactual based on the "result" of an infinite computation. Any claim that the algorithm will halt is pre-supposing that there is a difference, and hence is not a proof. At any finite time, we can only ever compare the amounts to within some epsilon > 0, hence we can only ever claim the probability is arbitrarily small. Dubious non-constructive, uncomputable results like this appear all over the place when dealing with reals. This makes the reals an interesting mathematical curiosity worthy of study, like the surreals, the hyperreals, transfinite cardinals, etc. and their properties can certainly be useful when required. However, their status as the "default" set of (non-whole) numbers should have been removed after Turing's "On Computable Numbers" paper. > It sounds like you just have some sort of aesthetic agenda. When it comes to the reals I'm very serious. When it comes to renaming the "negatives" to the "other imaginaries", replacing spurious use of "integer" with "natural" and outright banning the awful phrase "unsigned integer", then that's an aesthetic agenda :)
  - wyager 10y ago
    Assuming the universe is computable using a UTM, I agree. Why do you think the universe is computable? It would be nice if it were, but I have seen no proof.
    - chriswarbo 10y ago
      Physical theories can't be "proven" in the same way as mathematical conjectures, so we have to rely on agreement with experiment, self-consistency, "simplicity" (e.g. Kolmogorov complexity or some approximation), lack of counterexamples/disproof, etc. In this sense, I agree with Robert Harper (e.g. Practical Foundations for Programming Languages, section 22.2) that unprovable claims regarding computability may be merely "theses" to logicians, but to scientists they should be considered "laws" just as much as Newton's law of universal gravitation. In fact, other than purported claims of "hypercomputation" ( https://en.wikipedia.org/wiki/Hypercomputation#Criticism https://en.wikipedia.org/wiki/Hypercomputation#Criticism ) or the anthropocentric "microtubule"-type theories popularised by Penrose ( https://en.wikipedia.org/wiki/Orchestrated_objective_reduction#Criticism https://en.wikipedia.org/wiki/Orchestrated_objective_reducti... ), it seems that the computability of the universe is commonly accepted 'background knowledge' by most researchers in the field. Even those making extreme claims in this area, like those refuted in http://www.scottaaronson.com/papers/npcomplete.pdf http://www.scottaaronson.com/papers/npcomplete.pdf are presumably assuming that the universe is computable; there doesn't seem to be much point debating computational efficiency of soap bubbles if I thought (for example) that halting oracles existed!
      - wyager 10y ago
        Unless I'm mistaken, the computable numbers are not complete in the Cauchy Sequence sense. Therefore, there can be no hilbert space based on the computable numbers, because a hilbert space has to be complete in the Cauchy sense. It would only be a pre-hilbert space. I believe this means one can no longer correctly use limits in QM; are there other consequences of this?
        10y ago

wyager 10y ago

Nothing you said in your post has actually demonstrated that it's unlikely that reals underly physical reality. It sounds like you just have some sort of aesthetic agenda. Your entire premise (that we might replace reals in current physical theory with some other structure and get the same result) is trivially debunked. Pi shows up all over the place, and e.g. rationals are not closed under exponentiation by e. The schroedinger equation applied to physical hilbert spaces is inextricably real-valued.

chriswarbo 10y ago
> Nothing you said in your post has actually demonstrated that it's unlikely that reals underly physical reality. Sorry, I was taking it for granted that the "reals" are problematic and unphysical (based on the sentiment of the grandparent). For clarification, the real numbers are a superset of the "computable" numbers (those numbers which can be approximated, to any given precision, by a universal turing machine). The computable numbers are countable (since there are countably-many turing machine tapes), but the reals are uncountable. That means "almost all" of the reals are uncomputable. Since all the laws of physics are computable, all physical processes must be based on computable numbers; even if some initial condition of the universe resulted in an uncomputable number, e.g. if the digits of the muon's magnetic moment are the same as the digits of the 100th busy beaver number, we can never know since we'll never compute what that busy beaver number is (we could test such predictions empirically, but it could always just be a coincidence). > Pi shows up all over the place, and e.g. rationals are not closed under exponentiation by e. Pi and e are computable numbers (quite trivially so, there are loads of known algorithms to do it). Exponentiation is a computable operation (again, the algorithm is pretty trivial; it's just repeated multiplication, multiplication is repeated addition and addition is repeated increment, so you only need 3 loops) https://en.wikipedia.org/wiki/Computable_number#Can_computable_numbers_be_used_instead_of_the_reals.3F https://en.wikipedia.org/wiki/Computable_number#Can_computab... > The schroedinger equation applied to physical hilbert spaces is inextricably real-valued. It's been a while since my physics master's, so I wouldn't want to refute this face-on; however, it's trivial to notice that (by definition) we can only use computable numbers in our calculations, and hence even if there are any non-computable reals in a theory, they cannot affect the predictions of that theory. If some prediction depended on a value outside of the computable numbers, e.g. the 100th digit of Chaitin's constant, or the 100th busy beaver number modulo 7, then that prediction would not be calculable and could only be given in symbolic form, e.g. 1.8 * Chaitin(100) ^ 3.45. This is not the case for the schroedinger equation, which gives results that are analytically and/or numerically computable. > Your entire premise (that we might replace reals in current physical theory with some other structure and get the same result) is trivially debunked. It's not my premise to "replace reals in current physical theory" (although it seems that Doron Zeilberger might be pleased if that happened). I was pointing out that physicists, mathematicians and many others claim to use the reals, when in fact they're only using the computable numbers. This leads to confusion when unintuitive results about the reals are applied to the world. For example, when choosing a real number randomly from the unit interval, the probability of each point is 0, but the sum of their probabilities is 1. That's an example of the reals being unintuitive, and that's fine. However, trying to apply such statements to the world (e.g. "I can prove that the chance you've measured exactly 1 litre of water is zero") leads to confusion. Even worse, the confused person (the one making the claim, confusing uncomputable numbers with reality) may then "prove wrong" some perfectly justified refutations. In case you can't see the problem, the "proof" that an amount other than 1 litre was poured relies on checking more and more decimal places until a difference is found between the measured amount and 1 litre. However, we have no way of knowing that the "keep checking decimals" algorithm will halt, so the claim might be an unsound counterfactual based on the "result" of an infinite computation. Any claim that the algorithm will halt is pre-supposing that there is a difference, and hence is not a proof. At any finite time, we can only ever compare the amounts to within some epsilon > 0, hence we can only ever claim the probability is arbitrarily small. Dubious non-constructive, uncomputable results like this appear all over the place when dealing with reals. This makes the reals an interesting mathematical curiosity worthy of study, like the surreals, the hyperreals, transfinite cardinals, etc. and their properties can certainly be useful when required. However, their status as the "default" set of (non-whole) numbers should have been removed after Turing's "On Computable Numbers" paper. > It sounds like you just have some sort of aesthetic agenda. When it comes to the reals I'm very serious. When it comes to renaming the "negatives" to the "other imaginaries", replacing spurious use of "integer" with "natural" and outright banning the awful phrase "unsigned integer", then that's an aesthetic agenda :)
- wyager 10y ago
  Assuming the universe is computable using a UTM, I agree. Why do you think the universe is computable? It would be nice if it were, but I have seen no proof.
  - chriswarbo 10y ago
    Physical theories can't be "proven" in the same way as mathematical conjectures, so we have to rely on agreement with experiment, self-consistency, "simplicity" (e.g. Kolmogorov complexity or some approximation), lack of counterexamples/disproof, etc. In this sense, I agree with Robert Harper (e.g. Practical Foundations for Programming Languages, section 22.2) that unprovable claims regarding computability may be merely "theses" to logicians, but to scientists they should be considered "laws" just as much as Newton's law of universal gravitation. In fact, other than purported claims of "hypercomputation" ( https://en.wikipedia.org/wiki/Hypercomputation#Criticism https://en.wikipedia.org/wiki/Hypercomputation#Criticism ) or the anthropocentric "microtubule"-type theories popularised by Penrose ( https://en.wikipedia.org/wiki/Orchestrated_objective_reduction#Criticism https://en.wikipedia.org/wiki/Orchestrated_objective_reducti... ), it seems that the computability of the universe is commonly accepted 'background knowledge' by most researchers in the field. Even those making extreme claims in this area, like those refuted in http://www.scottaaronson.com/papers/npcomplete.pdf http://www.scottaaronson.com/papers/npcomplete.pdf are presumably assuming that the universe is computable; there doesn't seem to be much point debating computational efficiency of soap bubbles if I thought (for example) that halting oracles existed!
    - wyager 10y ago
      Unless I'm mistaken, the computable numbers are not complete in the Cauchy Sequence sense. Therefore, there can be no hilbert space based on the computable numbers, because a hilbert space has to be complete in the Cauchy sense. It would only be a pre-hilbert space. I believe this means one can no longer correctly use limits in QM; are there other consequences of this?
      - 10y ago