4 ms·

As a programmer, the thing that bothers me about Cantor's diagonalization (and the whole concept of Real numbers) is that it presumes an infinite number of bits

As a programmer, the thing that bothers me about Cantor's diagonalization (and the whole concept of Real numbers) is that it presumes an infinite number of bits of information can be stored in each number. So it's kind of like saying that you can't enumerate all the values of an infinitely large floating point number using any arbitrarily large, but finite, integer. Of COURSE you can't. Which is kind of Cantor's point, but at least from the perspective of a computer scientist and a mathematical constructivist, it seems a little cheap because I consider the idea of real numbers themselves to be somewhat "overpowered", leading to contradictions inherent in the very idea. I'd make an analogy to the question of can God make a rock so big he can't lift it? Whether you believe in God or not, you can probably recognize that as a faintly ridiculous question; I feel real numbers as a concept are ridiculous in the same way.

RiderOfGiraffes 17y ago
Without the reals you can't assume that all limits exist. The reals are the only totally ordered complete field. As such, whether they're "practical" or "useful" is irrelevant. They are a mathematical object worthy of study in their own right. Calculus is easier with the reals than without. The computing you do is an approximation to the reals - trying to do numerical analysis without the reals is horrendous. I've seen so-called "constructivists" attempting it, and I'd rather work with the reals any day.
- phaedrus 17y ago
  Quote: "Without the reals you can't assume that all limits exist." Oh, I agree completely. I can't assume that all limits exist. Modded you up because I agree with your premises, but I disagree with your conclusion. Both constructivism and "regular" mathematics are systems, but rather than views computable numbers as approximations of reals, I prefer to think of real valued calculus as a degenerate case numerical analysis where delta goes to zero. Here's the thing: most of mathematics was developed prior to computers, and I question how much of what's in mainstream mathematics is influenced by the requirement that ultimately the math had to be tractable to humans without the aid of a computer.
  - RiderOfGiraffes 17y ago
    Interesting observations. I'd like to write this up - I might later - I don't have time now - here's some material from Feynman ... Take successive square roots of 10. As you get further and further you spot a pattern, that when e is very small, 10^e ~ 1+c.e for some constant c. Now take 10^(i.e) where e is very small and i=sqrt(-1). Then 10^(i.e) ~ 1+c.i.e. Take repeated powers and you end up moving in a circle. Following that we end up with the formula e^(i.t) = cos(t)+i.sin(t) where t is a real angle. None of that works if you don't basically take the reals for granted. Or at least, it's very hard.
  - jerf 17y ago
    Are you offended at the idea that a mathematical thing called a "real" exists, or offended at the idea that somebody thinks real numbers actually exist in the real (no pun intended) universe? Because nobody thinks real numbers actually exist in a usable manner; whether the base of the universe is "ultimately real" or "ultimately discrete" is an open problem, but also irrelevant since we can't get to real values (if they exist) with infinite precision thanks to the various uncertainty principles. Ultimately, there really isn't anything to "agree" or "disagree" with when it comes to the reals; they are just a definition, and everyone who knows what they are talking about know they don't actually exist. See also the Axiom of Choice; really there isn't anything to "believe" or "not believe" about it. (I say "offended" and "believe" but I am not trying to put words into your mouth. I am not satisfied that either of them describe your tone, but I needed something. Please do clarify if those are wrong.)
    - phaedrus 17y ago
      To answer your first question: Yes. I mean, both. (And no, I'm not offended by your choice of words.) Since you've restated the discussion on more neutral terms, allow me to rephrase my position in your terms: The article is about how people have trouble swallowing Cantor's argument. Yet clearly Cantor's argument is correct within a framework that includes the reals. And my position is that the existence of Cantor's argument leads me to question the framework itself. You say that there really isn't anything to agree or disagree with, that we are just talking about definitions. Absolutely, I agree. It's unfortunate that the only words we have to talk about these things also have extra connotations that might be more forceful than what we really mean. We're really just talking about the consequences of manipulating symbols under certain rules. HOWEVER... Isn't the whole point of math that you expect at some point to analyze the result of the symbol manipulation and "read off" an interpretable meaning from the answer? At that point may you not make a judgement that the combination of the system you choose and your interpretation of it lead to "believable" results? Is it entirely correct to say that there is no such thing as "agreeing" or "disagreeing" with a definition of a mathematical object, if putting it into the system leads to results you don't consider believable?
WilliamLP 17y ago
Reals numbers give you the property that if a continuous curve is bigger than zero somewhere, and smaller somewhere else, then it must cross the line at a certain point. You probably want to say that sqrt(2) has a solution. Without reals you don't have this.
- phaedrus 17y ago
  I would say that the solution to sqrt(2) is the expansion (to however many finite number of iterations you run it for) of the bits produced by the Turing machine that computes sqrt(2). Sqrt(2) is a computable number, so it exists both in my system and in yours.
lmkg 17y ago
Can you say more specifically in what way you feel the reals are 'overpowered' or contradictory? As a mathematician, I'm interested in why people think the concept is odd, because it's not intuitively odd to me and I want to understand this alternative viewpoint. Would I be right in thinking it's related to reals being 'infinitely precise?'
- phaedrus 17y ago
  By overpowered, I mean that a model that allows the idea of something having infinite precision leads to apparent paradoxes. For instance you get constructs like Gabriel's Horn that have greater surface area than volume, or the case of choosing a random real number between 0 and 1: the probability of picking any particular real number is (approaches?) zero, yet some number will be picked. But in the case of the Cantor argument specifically he seems to be encoding an infinite amount of information in one number simply by extending the digits after the decimal point. It's almost like - cheating, or more specifically an "exploit" allowed by what I call reals being overpowered. (Here I have to make an aside: the value Pi, which is actually a computable number, can also encode any amount of information because any number will appear at some point in the expansion. But you'd have to indicate where in Pi your number starts, and your index values would (on avergage) be so large as to consume at least as many bits of information as it would to simply state the number you want.) In the physical world, when you do something like measure the voltage on a capacitor you can only measure it to within a certain accuracy & precision. If you wanted to make an analog computer that uses different voltages to represent different values, you'd be limited by the accuracy and precision which you could measure it. Note that, even if the voltage is infinitely variable, your ability to differentiate those values is still limited, so you still do not have infinite precision. (In Turing's paper on computable numbers, he makes the case that even if a human brain can have infinite states, there would at some point be a threshold of similarity at which the brain couldn't distinguish two similar states from each other.) So my claim is that real numbers imply the ability to measure something to infinite precision, and that that is contradictory to physical experience.
  - lmkg 17y ago
    Reality and measurable reality were separate concepts, until quantum mechanics came along. If the intricacies of the Reals demonstrate the philosophy they come out of, it has as much to do with predating the atom in chemistry and the Heisenberg Principle in quantum mechanics as it does with predating float-point numbers. I feel that math is intended to appeal to logical intuition rather than physical intuition, so my belief is that adding limits to precision is an unnecessary complication to the number system. By extension, I don't believe that math or numbers reveal 'fundamental truths' of the physical universe, nor are they intended to. I view math as a disembodied exercise in logic, and if we happen to find a physical system that satisfies the hypotheses of a mathematical system, then the results will apply as well. The result of this is that I view the reals as 'purer' math because they have less hypotheses, and numerical approximations as a set of additional hypotheses that are more commonly seen in practice. The natural result is that there's a lot of mathematical ideals with cool properties that are ruled out in practical terms. I have a few questions for you and your viewpoint (for my own curiosity, not antagonistic). 1. Do you believe that Pi actually does have an infinite number of decimal places, or that it's limited by the precision with which we can measure a circle? I think I read somewhere that 50 decimal places is enough to calculate the radius of the visible universe from the radius to within an atom's width of accuracy. 2. What do you think of the Random Oracle Model in cryptography[1]? I think there's an anology between the ROM vs real hashes, and how I conceive of the reals vs finite-precision mathematics. In particular, simple and (logically) intuitive idealized models have properties that are not held by any possible real approximation. Feel free to ask me more if you're totally confused =P. 3. Do you accept infinite-precision rational numbers? Why or why not? My motivation in asking is that there's a difference between computability and measurability, and you appeal to both as a source of intuition. [1] http://en.wikipedia.org/wiki/Random_oracle_model http://en.wikipedia.org/wiki/Random_oracle_model

RiderOfGiraffes 17y ago

Without the reals you can't assume that all limits exist. The reals are the only totally ordered complete field. As such, whether they're "practical" or "useful" is irrelevant. They are a mathematical object worthy of study in their own right. Calculus is easier with the reals than without. The computing you do is an approximation to the reals - trying to do numerical analysis without the reals is horrendous. I've seen so-called "constructivists" attempting it, and I'd rather work with the reals any day.

phaedrus 17y ago
Quote: "Without the reals you can't assume that all limits exist." Oh, I agree completely. I can't assume that all limits exist. Modded you up because I agree with your premises, but I disagree with your conclusion. Both constructivism and "regular" mathematics are systems, but rather than views computable numbers as approximations of reals, I prefer to think of real valued calculus as a degenerate case numerical analysis where delta goes to zero. Here's the thing: most of mathematics was developed prior to computers, and I question how much of what's in mainstream mathematics is influenced by the requirement that ultimately the math had to be tractable to humans without the aid of a computer.
- RiderOfGiraffes 17y ago
  Interesting observations. I'd like to write this up - I might later - I don't have time now - here's some material from Feynman ... Take successive square roots of 10. As you get further and further you spot a pattern, that when e is very small, 10^e ~ 1+c.e for some constant c. Now take 10^(i.e) where e is very small and i=sqrt(-1). Then 10^(i.e) ~ 1+c.i.e. Take repeated powers and you end up moving in a circle. Following that we end up with the formula e^(i.t) = cos(t)+i.sin(t) where t is a real angle. None of that works if you don't basically take the reals for granted. Or at least, it's very hard.
- jerf 17y ago
  Are you offended at the idea that a mathematical thing called a "real" exists, or offended at the idea that somebody thinks real numbers actually exist in the real (no pun intended) universe? Because nobody thinks real numbers actually exist in a usable manner; whether the base of the universe is "ultimately real" or "ultimately discrete" is an open problem, but also irrelevant since we can't get to real values (if they exist) with infinite precision thanks to the various uncertainty principles. Ultimately, there really isn't anything to "agree" or "disagree" with when it comes to the reals; they are just a definition, and everyone who knows what they are talking about know they don't actually exist. See also the Axiom of Choice; really there isn't anything to "believe" or "not believe" about it. (I say "offended" and "believe" but I am not trying to put words into your mouth. I am not satisfied that either of them describe your tone, but I needed something. Please do clarify if those are wrong.)
  - phaedrus 17y ago
    To answer your first question: Yes. I mean, both. (And no, I'm not offended by your choice of words.) Since you've restated the discussion on more neutral terms, allow me to rephrase my position in your terms: The article is about how people have trouble swallowing Cantor's argument. Yet clearly Cantor's argument is correct within a framework that includes the reals. And my position is that the existence of Cantor's argument leads me to question the framework itself. You say that there really isn't anything to agree or disagree with, that we are just talking about definitions. Absolutely, I agree. It's unfortunate that the only words we have to talk about these things also have extra connotations that might be more forceful than what we really mean. We're really just talking about the consequences of manipulating symbols under certain rules. HOWEVER... Isn't the whole point of math that you expect at some point to analyze the result of the symbol manipulation and "read off" an interpretable meaning from the answer? At that point may you not make a judgement that the combination of the system you choose and your interpretation of it lead to "believable" results? Is it entirely correct to say that there is no such thing as "agreeing" or "disagreeing" with a definition of a mathematical object, if putting it into the system leads to results you don't consider believable?

WilliamLP 17y ago

Reals numbers give you the property that if a continuous curve is bigger than zero somewhere, and smaller somewhere else, then it must cross the line at a certain point. You probably want to say that sqrt(2) has a solution. Without reals you don't have this.

phaedrus 17y ago
I would say that the solution to sqrt(2) is the expansion (to however many finite number of iterations you run it for) of the bits produced by the Turing machine that computes sqrt(2). Sqrt(2) is a computable number, so it exists both in my system and in yours.

lmkg 17y ago

Can you say more specifically in what way you feel the reals are 'overpowered' or contradictory? As a mathematician, I'm interested in why people think the concept is odd, because it's not intuitively odd to me and I want to understand this alternative viewpoint. Would I be right in thinking it's related to reals being 'infinitely precise?'

phaedrus 17y ago
By overpowered, I mean that a model that allows the idea of something having infinite precision leads to apparent paradoxes. For instance you get constructs like Gabriel's Horn that have greater surface area than volume, or the case of choosing a random real number between 0 and 1: the probability of picking any particular real number is (approaches?) zero, yet some number will be picked. But in the case of the Cantor argument specifically he seems to be encoding an infinite amount of information in one number simply by extending the digits after the decimal point. It's almost like - cheating, or more specifically an "exploit" allowed by what I call reals being overpowered. (Here I have to make an aside: the value Pi, which is actually a computable number, can also encode any amount of information because any number will appear at some point in the expansion. But you'd have to indicate where in Pi your number starts, and your index values would (on avergage) be so large as to consume at least as many bits of information as it would to simply state the number you want.) In the physical world, when you do something like measure the voltage on a capacitor you can only measure it to within a certain accuracy & precision. If you wanted to make an analog computer that uses different voltages to represent different values, you'd be limited by the accuracy and precision which you could measure it. Note that, even if the voltage is infinitely variable, your ability to differentiate those values is still limited, so you still do not have infinite precision. (In Turing's paper on computable numbers, he makes the case that even if a human brain can have infinite states, there would at some point be a threshold of similarity at which the brain couldn't distinguish two similar states from each other.) So my claim is that real numbers imply the ability to measure something to infinite precision, and that that is contradictory to physical experience.
- lmkg 17y ago
  Reality and measurable reality were separate concepts, until quantum mechanics came along. If the intricacies of the Reals demonstrate the philosophy they come out of, it has as much to do with predating the atom in chemistry and the Heisenberg Principle in quantum mechanics as it does with predating float-point numbers. I feel that math is intended to appeal to logical intuition rather than physical intuition, so my belief is that adding limits to precision is an unnecessary complication to the number system. By extension, I don't believe that math or numbers reveal 'fundamental truths' of the physical universe, nor are they intended to. I view math as a disembodied exercise in logic, and if we happen to find a physical system that satisfies the hypotheses of a mathematical system, then the results will apply as well. The result of this is that I view the reals as 'purer' math because they have less hypotheses, and numerical approximations as a set of additional hypotheses that are more commonly seen in practice. The natural result is that there's a lot of mathematical ideals with cool properties that are ruled out in practical terms. I have a few questions for you and your viewpoint (for my own curiosity, not antagonistic). 1. Do you believe that Pi actually does have an infinite number of decimal places, or that it's limited by the precision with which we can measure a circle? I think I read somewhere that 50 decimal places is enough to calculate the radius of the visible universe from the radius to within an atom's width of accuracy. 2. What do you think of the Random Oracle Model in cryptography[1]? I think there's an anology between the ROM vs real hashes, and how I conceive of the reals vs finite-precision mathematics. In particular, simple and (logically) intuitive idealized models have properties that are not held by any possible real approximation. Feel free to ask me more if you're totally confused =P. 3. Do you accept infinite-precision rational numbers? Why or why not? My motivation in asking is that there's a difference between computability and measurability, and you appeal to both as a source of intuition. [1] http://en.wikipedia.org/wiki/Random_oracle_model http://en.wikipedia.org/wiki/Random_oracle_model