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I enjoyed your comment and agree with all your well put points except one: on the rationals tie with reality. Having implemented exact real computation to bett

by Vetch 5y ago

I enjoyed your comment and agree with all your well put points except one: on the rationals tie with reality. Having implemented exact real computation to better understand reals, I think of a real number as a kind of machine that generates infinite streams. Operations on them instantiate new machines which query their real operands, computating until there's sufficient information to emit a next term of the stream. When the next term needs an infinite amount of information to decide what to spit out next, it results in an "unproductive" infinite loop. Rational numbers are interesting, more realistic, because they always terminate. In the real world, measurement tolerances and physical limits means at some point having to extract a rational. When we work with reals we are really only working with rational approximations or symbols with associated properties and relations. Reals are a powerful and elegant tool to rigorously reason about mathematical spaces and operations on algebraic objects but trying to work with them in reality in their exact form is a fun and visceral lesson on the nature of undecidability. It's hard to go two steps without tripping over a non-terminating loop (such as any operation that starts with an irrational and results in a rational or equality testing in general). This is an observation on our interface with reality and not on its true nature, which may or may not admit reals (although my non-serious guess is that black holes form whenever you try to do something that requires a proper real number).

User23 5y ago
Or, put tersely, most reals aren’t computable. The implications of that observation are not entirely clear or distinct.
- bopbeepboop 5y ago
katmannthree 5y ago
> on the rationals tie with reality I feel like this ties back into the distinction between theoretical and applied math. The basis in reality for the integers is counting discrete objects with fingers, for the rationals it's (likely) an attempt to fill in the spaces between integers using known concepts (ratios / fractions). Rationals are great if you stick to numerical work where discontinuities below epsilon can be ignored, but the rationals don't actually map to what we think of when we consider a philosophically real number system -- a discontinuous set does not match our observed experience which is that you can have any number you want between two you already have. The construction of the reals varies depending on how you want to approach it but each is equivalent: you fill in the all the holes everywhere but at the infinities so that you have a continuous closed set, just like one would intuitively expect from an infinite set of numbers representing segments of reality. There's nothing special about the rationals which ties them more closely to reality than the reals, the rationals are just our first attempt to rigorously define all of the numbers between other numbers using the tools we had at the time. One could just as easily construct the set $ = {x#y for all x, y in Z+} and where a#b === a + the Riemann sum of 1/(a^n) from n = 0 ... b. This also fills in some of the gaps between integers, just not enough to be interesting or particularly useful. The rationals are interesting and stuck around because they fill in almost enough gaps to allow you to conveniently construct useful things. They're not quite there though, which is why we eventually developed the reals. And then the imaginary numbers, because despite the name physical phenomena which can be modeled using square roots of negative numbers end up presenting a compelling use case for adoption. We don't have complex numbers because some math nerd thought they were cool, we have complex numbers because they are useful in describing observed physical phenomena succinctly and as such there's enormous utility in hacking an extension onto the reals to add them. Pulling this back around, from a theoretical perspective real & complex numbers are as real as anything else in math and are very useful to boot. You only run into issues in applied circumstances where nothing is exact and half of the things end up nondeterministic for one reason or another. Applied math requires countless shortcuts and discretionary tactics to convert things with a guarantee of correctness on the theoretical side into things which can actually be computed albeit with a correctness only within specified bounds. Mapping between theoretical and applied math is a decent example of a pseudo one-way function, all of applied math draws from the theoretical but insights from applied math don't really map back into anything useful on the theoretical side. Which is why when we do theory and build models, we use theoretical techniques since the ability to prove correctness is the entire point. If you need numerical computation you must in exchange give up absolute correctness, which is why it is only appropriate to use during numerical computation. > This is an observation on our interface with reality and not on its true nature, which may or may not admit reals (although my non-serious guess is that black holes form whenever you try to do something that requires a proper real number). Well, let us know if you're able to develop a falsifiable experiment one way or another. That is definitely an interesting theory, unfortunately nobody has been able to figure out a way to poke that particular are-the-numbers-real-or-just-made-up bear.
- Pyramus 5y ago
  I think you misunderstood what parent was saying. There is no evidence the real numbers are based in physical reality. As parent was saying, it doesn't make sense to be able to store infinite information in a single number, or even, say, store all of human knowledge in a single number. Generations of physicists have come to the same conclusion [1] and most professional physicists agree. It's just that (a) the real numbers work incredibly well as a "tool" or "model", with negligible shortcomings, and it's (b) extremely tedious to think of alternative number systems that are remotely as convenient as the real numbers. So it's not clear if alternative approaches are a waste of time, but that does not mean the reals are real! If you want to learn more, check out the references in [1]. [1] https://news.ycombinator.com/item?id=18256455 https://news.ycombinator.com/item?id=18256455
  - katmannthree 5y ago
    I do understand that argument, I just remain unmoved by it. Watch this, I'm about to show you a complete finite representation of an irrational transcendental number: π. That took literally three lines to represent and then an additional half page's worth that I'll skip explaining how to calculate a numerical value to however much precision you have time and space for. Now granted, there is an underlying assumption that when you need to use that number you'll select an appropriate algorithm to compute it to the degree of precision you need, much like how if you were instead considering the rational number 22/7 you would need an algorithm to numerically evaluate it. We don't quibble about whether or not the universe has enough space to hold that one though because we have a simple abstraction which lets us refer to it with infinite precision and evaluate it with arbitrary precision. Just. Like. π. Yes, literally none of the reals would fit in the universe no matter how small you wrote them if you want to represent them with full precision. That is literally the point of the reals, that they are an infinitely dense field. It doesn't matter, we wield the same tools we used to construct them and refer to them by their names or by their construction. If your definition of "based in reality" means "can be explicitly written out with full precision" then literally none of the reals or rationals are "based in reality" because for otherwise finite numbers you can keep padding zeros to the right of the decimal place and a finite universe doesn't have enough space to hold infinite objects. Taking a definition of reality that provides actual utility, the reals are clearly based in physical reality by virtue of their construction being explicitly guided by the objective of modeling reality. Just like the rationals and integers before them and the complex numbers after. They were literally created to model reality. Imaginary numbers are based in reality too, despite it being equally impossible to own sqrt(-2) and π melons. At best I will concede that there is an additional layer of abstraction between whatever "reality" is and what the real numbers are, but that's not a very interesting distinction given that humans are already running a dozen intermediate layers of abstraction in order to process their surrounding reality and then overlay math on top of it.

by Vetch 5y ago

User23 5y ago

Or, put tersely, most reals aren’t computable. The implications of that observation are not entirely clear or distinct.

bopbeepboop 5y ago

katmannthree 5y ago

> on the rationals tie with reality I feel like this ties back into the distinction between theoretical and applied math. The basis in reality for the integers is counting discrete objects with fingers, for the rationals it's (likely) an attempt to fill in the spaces between integers using known concepts (ratios / fractions). Rationals are great if you stick to numerical work where discontinuities below epsilon can be ignored, but the rationals don't actually map to what we think of when we consider a philosophically real number system -- a discontinuous set does not match our observed experience which is that you can have any number you want between two you already have. The construction of the reals varies depending on how you want to approach it but each is equivalent: you fill in the all the holes everywhere but at the infinities so that you have a continuous closed set, just like one would intuitively expect from an infinite set of numbers representing segments of reality. There's nothing special about the rationals which ties them more closely to reality than the reals, the rationals are just our first attempt to rigorously define all of the numbers between other numbers using the tools we had at the time. One could just as easily construct the set $ = {x#y for all x, y in Z+} and where a#b === a + the Riemann sum of 1/(a^n) from n = 0 ... b. This also fills in some of the gaps between integers, just not enough to be interesting or particularly useful. The rationals are interesting and stuck around because they fill in almost enough gaps to allow you to conveniently construct useful things. They're not quite there though, which is why we eventually developed the reals. And then the imaginary numbers, because despite the name physical phenomena which can be modeled using square roots of negative numbers end up presenting a compelling use case for adoption. We don't have complex numbers because some math nerd thought they were cool, we have complex numbers because they are useful in describing observed physical phenomena succinctly and as such there's enormous utility in hacking an extension onto the reals to add them. Pulling this back around, from a theoretical perspective real & complex numbers are as real as anything else in math and are very useful to boot. You only run into issues in applied circumstances where nothing is exact and half of the things end up nondeterministic for one reason or another. Applied math requires countless shortcuts and discretionary tactics to convert things with a guarantee of correctness on the theoretical side into things which can actually be computed albeit with a correctness only within specified bounds. Mapping between theoretical and applied math is a decent example of a pseudo one-way function, all of applied math draws from the theoretical but insights from applied math don't really map back into anything useful on the theoretical side. Which is why when we do theory and build models, we use theoretical techniques since the ability to prove correctness is the entire point. If you need numerical computation you must in exchange give up absolute correctness, which is why it is only appropriate to use during numerical computation. > This is an observation on our interface with reality and not on its true nature, which may or may not admit reals (although my non-serious guess is that black holes form whenever you try to do something that requires a proper real number). Well, let us know if you're able to develop a falsifiable experiment one way or another. That is definitely an interesting theory, unfortunately nobody has been able to figure out a way to poke that particular are-the-numbers-real-or-just-made-up bear.

Pyramus 5y ago
I think you misunderstood what parent was saying. There is no evidence the real numbers are based in physical reality. As parent was saying, it doesn't make sense to be able to store infinite information in a single number, or even, say, store all of human knowledge in a single number. Generations of physicists have come to the same conclusion [1] and most professional physicists agree. It's just that (a) the real numbers work incredibly well as a "tool" or "model", with negligible shortcomings, and it's (b) extremely tedious to think of alternative number systems that are remotely as convenient as the real numbers. So it's not clear if alternative approaches are a waste of time, but that does not mean the reals are real! If you want to learn more, check out the references in [1]. [1] https://news.ycombinator.com/item?id=18256455 https://news.ycombinator.com/item?id=18256455
- katmannthree 5y ago
  I do understand that argument, I just remain unmoved by it. Watch this, I'm about to show you a complete finite representation of an irrational transcendental number: π. That took literally three lines to represent and then an additional half page's worth that I'll skip explaining how to calculate a numerical value to however much precision you have time and space for. Now granted, there is an underlying assumption that when you need to use that number you'll select an appropriate algorithm to compute it to the degree of precision you need, much like how if you were instead considering the rational number 22/7 you would need an algorithm to numerically evaluate it. We don't quibble about whether or not the universe has enough space to hold that one though because we have a simple abstraction which lets us refer to it with infinite precision and evaluate it with arbitrary precision. Just. Like. π. Yes, literally none of the reals would fit in the universe no matter how small you wrote them if you want to represent them with full precision. That is literally the point of the reals, that they are an infinitely dense field. It doesn't matter, we wield the same tools we used to construct them and refer to them by their names or by their construction. If your definition of "based in reality" means "can be explicitly written out with full precision" then literally none of the reals or rationals are "based in reality" because for otherwise finite numbers you can keep padding zeros to the right of the decimal place and a finite universe doesn't have enough space to hold infinite objects. Taking a definition of reality that provides actual utility, the reals are clearly based in physical reality by virtue of their construction being explicitly guided by the objective of modeling reality. Just like the rationals and integers before them and the complex numbers after. They were literally created to model reality. Imaginary numbers are based in reality too, despite it being equally impossible to own sqrt(-2) and π melons. At best I will concede that there is an additional layer of abstraction between whatever "reality" is and what the real numbers are, but that's not a very interesting distinction given that humans are already running a dozen intermediate layers of abstraction in order to process their surrounding reality and then overlay math on top of it.