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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 agr

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?
  - jerf 17y ago
    "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?' Ah, and therein lies the rub. In math terminology, I think that that itself is an axiom, in the sense that you can choose it or not choose it, and it will strongly affect the rest of the system. If you accept the axiom, you end up going down the constructivist path. If you do not, you end up in "conventional mathematics" (i.e., what you get for a real math degree in college). Personally, as a programmer I am also sympathetic to the constructivist viewpoint, but I am happy to settle for everyone involved understanding when they've left the physical universe (real numbers, axiom of choice, etc) and when they haven't. If some people want to screw around in the unreal universe, more power to them, doesn't have to bother me. And they often produce useful approximations for real-life use. As RiderOfGiraffes said, there are proofs that work on reals much better than any actual number, and I observe that in practice the resulting maths work out pretty well in the real world, even if they aren't actually physical. (We have way larger problems dealing with floating point, monsters of our own creation, than we do in dealing with the fact that our equations of motion are defined based on real numbers that don't actually correspond to anything, where the inaccuracies are typically beyond our measuring ability and dominated by general inaccuracies of the input data which are far larger than the granularity of the universe.) But I do think maintaining that distinction is definitely important. I further observe that while I can't speak for all of mathematics, the mathematicians are very careful to distinguish what takes the Axiom of Choice and what does not, just to draw one example. They are pretty good about showing their work. I guess the takeaway is that you certainly aren't alone; many people agree that real numbers are ridiculous in the way you are sensing. But they can be a useful fiction.
RiderOfGiraffes 17y ago
I offer the proposition that all models are inaccurate, some models are useful, and the usefulness of a model is in the balance between the accuracy and the ease of use. Compared with floating point numbers in computing, and finite difference types of calculus, the model offered by the reals is much, much simpler to work with, and then modelling them with floating point nubmers is comparatively simple. Doing the work directly with floats is hard, and detailed analysis and theorems are virtually impossible. For a sound theoretical basis to many applications, it's easier to work with reals and then approximate with floats than it is to work with floats directly. I further offer the observation made in another item that working directly with computer calculations is making science unreproducable http://news.ycombinator.com/item?id=1069808 http://news.ycombinator.com/item?id=1069808 Suggesting that we simply do the calculations and then work with the results has its drawbacks. Finally, I'm doing some huge simulations at the moment, and every now and again they blow up. The results are supremely plausible, but if I didn't have an underlying theory, rigorously developed in the world of pure math, I wouldn't know that.

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?
- jerf 17y ago
  "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?' Ah, and therein lies the rub. In math terminology, I think that that itself is an axiom, in the sense that you can choose it or not choose it, and it will strongly affect the rest of the system. If you accept the axiom, you end up going down the constructivist path. If you do not, you end up in "conventional mathematics" (i.e., what you get for a real math degree in college). Personally, as a programmer I am also sympathetic to the constructivist viewpoint, but I am happy to settle for everyone involved understanding when they've left the physical universe (real numbers, axiom of choice, etc) and when they haven't. If some people want to screw around in the unreal universe, more power to them, doesn't have to bother me. And they often produce useful approximations for real-life use. As RiderOfGiraffes said, there are proofs that work on reals much better than any actual number, and I observe that in practice the resulting maths work out pretty well in the real world, even if they aren't actually physical. (We have way larger problems dealing with floating point, monsters of our own creation, than we do in dealing with the fact that our equations of motion are defined based on real numbers that don't actually correspond to anything, where the inaccuracies are typically beyond our measuring ability and dominated by general inaccuracies of the input data which are far larger than the granularity of the universe.) But I do think maintaining that distinction is definitely important. I further observe that while I can't speak for all of mathematics, the mathematicians are very careful to distinguish what takes the Axiom of Choice and what does not, just to draw one example. They are pretty good about showing their work. I guess the takeaway is that you certainly aren't alone; many people agree that real numbers are ridiculous in the way you are sensing. But they can be a useful fiction.

RiderOfGiraffes 17y ago

I offer the proposition that all models are inaccurate, some models are useful, and the usefulness of a model is in the balance between the accuracy and the ease of use. Compared with floating point numbers in computing, and finite difference types of calculus, the model offered by the reals is much, much simpler to work with, and then modelling them with floating point nubmers is comparatively simple. Doing the work directly with floats is hard, and detailed analysis and theorems are virtually impossible. For a sound theoretical basis to many applications, it's easier to work with reals and then approximate with floats than it is to work with floats directly. I further offer the observation made in another item that working directly with computer calculations is making science unreproducable http://news.ycombinator.com/item?id=1069808 http://news.ycombinator.com/item?id=1069808 Suggesting that we simply do the calculations and then work with the results has its drawbacks. Finally, I'm doing some huge simulations at the moment, and every now and again they blow up. The results are supremely plausible, but if I didn't have an underlying theory, rigorously developed in the world of pure math, I wouldn't know that.