Is there some “real-world,” more or less “practical” application of the fact that there are more real numbers than natural numbers?
Any way in which this fact has made a difference to physics or anything?
I’m in a discussion with someone who concedes that Cantor’s diagonalization argument is mathematically valid in a real sense, but who doesn’t buy its conclusion because he thinks talking about infinities is just talking about a sort of “mythology.” I tried reformulating the proof without reference to a commitment to the existence of actual infinities, but he didn’t find this satisfying–he said something like “sure, counting by natural numbers and counting by real numbers would lead to the differing kinds of results you mention, but there’s nothing we can do with that fact. It doesn’t, and can’t, ever make a difference in what we would ever actually do in any situation (other than what we would say if we were mathematicians talking about these mythical entities called ‘infinities.’”
I mentioned all of that, not to introduce a debate, but just to explain why I asked my question, to help people understand exactly what my question is.
I would tend to side with your friend there, I think. Mathematical theory is uber-cool of course, but this kind of mathe is theoretical enough to become a semi-philosophical game. I’m not sure that there are any real-world applications of it.
One does pop into mind as a potential application, though even this one is based on sci-fi type technology as the premise. If there are an infinite number of parallel universes ranging through a spacial dimension perpendicular to our three (four?) dimensions, and transport between them was possible ala ‘sliders’ or ‘stargate’ (Was stargate parallel universes?) then it could be very important to determine if they were countably infinite or uncountable infinite in nature. Countably infinite suggests that it’s practical and possible to establish a categorization scheme that can never be complete, (like mapping a flat world that just keeps going on and on,) but would be sufficient to keep travellers from getting lost so long as they do not accidentally stray into ‘uncharted dimensions.’ Uncountably infinte universes suggest that even the smallest misalignment of the equipment would get you lost in another world very close to the one you were trying to reach, and that it might be better just to avoid the whole mess and stay home.
Two direct applications of Cantor’s diagonalization method are Goedel’s Incompleteness Theorem and Turing’s proof of the Halting Problem.
The latter (and its variations) is extremely important in CS. There will never be an automated method to check correctness of computer programs. The number of related uses is huge. E.g., there can never be a “perfect” computer virus detector. And on and on.
Method aside, there is still the following simple observation: There are a countably infinite number of programs (solutions) but an uncountably infinite number of problems. Ergo, there are a lot more problems that cannot be solved than can be solved. (But ut’s nice to have simple to understand ones like the halting problem at hand.)
With Math, Goedel’s is quite important (to say the least) as well. A lot of interesting problems can be phrased as relatively short Math Theorems. But there is not only no easy way, there is no way at all, to prove such Theorems in general. There is an inherent limitation on human knowledge.
I don’t see how there could be an uncountable infinity of problems since a problem has to be stated as a sentence. Since a sentence is a finite string of words, there are only countably many of them.
I see that Goedel’s theorem is important in computer science, where it is equivalent to the halting problem and the fact that some programs are impossible to prove correct. However, I see no place in math outside of logic in which Goedel’s theorem makes a particle of difference. So there are statements that cannot be decided. There are many more that I cannot decide and as far as I am concerned there is no practical difference between them. Anyway, it is almost never possible to decide if a question is undecidable. The one case that I know is possible–a wide elaboration of Ramsey’s theorem (this is the one that for n=3 states that in any set of 6 people, there will either be at least three who are all acquainted with each other or else at least three none of whom is scquaitned with the other two)–is proved by stating an infinite generalization and proving it using the axiom of choice. Thus it is removed from the domain of arithmetic.
As for the original post, well it is an interesting question. The continuum is used in the development of calculus and the continuum (the real line) is uncountable. The real world may or may not be discrete and therefore calculus maybe should not be used; rather finite differences should replace calculus. A real world example of this is radioactive decay where we get an equation describing the decay over time by pretending the decay is continuous and use calculus, even though we know it is not. Maybe all our real world applications are like that; how would we know. It is not generally known that the reason Cantor developed his theory of cardinal (and ordinal) numbers was to be able to describe the sets of discontinuities of Fourier series. So the first example of uncountable sets arose arose directly from this real world application. Still, it is true that it is hard to see how the distinction could be important in any practical sense.
I don’t agree with this statement. It all depends on what one means by the subjective term “useful”, but I think prime numbers are pretty damn useful. Pretty much all of classical algebraic number theory has to do with primes. The Fundamentla Theorem of Arithmetic would not exist without primes. Primes have so many applications, you couldn’t even begin to list them all.
I was using “problem” in the computability theory sense, which predates complexity theory.
A problem is a mapping from strings to strings. The simplest type of mappings are from strings to 1/0 (or true/false or in/not in a set). There are an uncountable number of mappings of such mappings. Note that (0+1)* -> {0,1} is just one such mapping but is trivially seen to have the same cardinality as the Reals.
Goedel’s Incompleteness Theorem is actually not such a big deal at all in CS outside of being an inspiration to Turing (and Church). Cantor’s diagonalization technique, OTOH, is used extensively even within the realm of the computable. E.g., we know that the class of linear time problems is a proper subset of quadratic time problems.
All fields of Science use Math and quite interested in doing things that are either extremely difficult or impossible. A lot of non-Mathematicians would love reasonable closed form solutions to find polynomial roots or solving multi-body problems. But knowing that a lot of problems have been proven difficult to impossible saves people from wasting their time and get them to focus on more tractable methods.
From one perspective, none of it is “useful.” The whole notion of an infinite set is pretty much an abstract, mathematical concept: there is no such thing in real life. You can’t show me a set of paperclips that is countably infinite. The number of atoms in the known universe is finite (very large, but still finite.)
The concept of the real line having an infinite number of points (of no width or length, mind you) is also only theoretic. You can’t draw a line of no width or length. And if we get down to subatomic levels, you still don’t have an infinite number of points – there are only so many atoms in the line, and so many particles within each atom, and the number of points in the line you draw with your pencil is finite (large, but fininte.)
The concept of an infinite decimal (like pi) is also only theoretic. You cannot draw a line that is pi-inches long, not with any real implement. You can come close, maybe a few hundred decimal places, say, but you’ll never be “exact.” (By the way, you can’t draw a line exactly one inch long, either.)
The notion of infinity is a theoretic one. It creates a nice model of a Real number system, and that model is useful for lots of practical applications, but it’s only a theoretic model.
Dex, I hear arguments like yours all the time from students in courses like Intro to Theoretical CS. They especially like to point out that all computers are Finite State Automata and therefore there is no point in studying any other models.
Needless to say, their argument gets squashed quite easily. And that’s in Computer Science where practicality is in fact quite important.
True, you did prefix your post with “From one perspective…”, but it is a very naive and overly simplistic perspective.
One thing to note is that a lot of Math appears to having nothing to do with “reality” but is tremendously useful. E.g., imaginary numbers.
One of the most theoretical and therefore least “useful” fields in Math is Algebraic Topology. But the 2004 Goedel prize went to two groups who figured out how to apply Algebraic Topology to solve a very difficult open problem in Computer Science concerning decision making in concurrent (parallel) systems.
And infinity most definitely exists in real life. E.g., there are an (uncountably) infinite number of speeds, and directions a moving particle can have.
Speaking as a professional mathematician, I agree completely with this. What happens as I explained in my previous post is that many problems are easier when extended to infinity. This was the intent of my explanation of the fact that finite difference calculus is much harder than ordinary differential calculus. From a practical point of view you may as well assume that all the numbers you deal with are rational and with small denominator (say less than 10^100).
Well, one way it comes up is that mathematical models of physics which assume actual existence of continua can’t be accurate if actual ontologies are countable. That is, if there’s only a countable number of “things” in the universe, then calculus, strictly speaking, doesn’t apply.
ftg, please note, I did use the word “useful” in quotes. And I wound up by remarking that it’s a model that can give very practical results. However, mathematics is just a model, it is not the real world. Measurements in the real world are not precise (although they may be accurate); measurements in the mathematical model are precise, even if it means an infinite number of decimal places. What’s the old story about being off by one inch every mile means that your rocket will miss Mars?
I don’t think we disagree. I’m not denying the practical applications, but I find tons of questions when I sort through Cecil’s mail from people who don’t understand the diff between the theoretical model and realilty.
Simple hijack example: there’s some spot in the bible, I forget where, that a circular tub is described with measurements and ratio of circumference to radius is 3. Some people scoff that the biblical writer thought pi was 3; but of course, the biblical writer was just rounding, as he must do. Otherwise, if he tried to write out the infinite decimal, the bible would be very long and tedious.
My naive way of thinking about countable infinity vs. uncountable infinity is this:
Starting at some point on a number line, and choosing any other point on the number line, can I get from one to the other by counting the points in between?
For the integers, the answer is yes - no matter how large a number I choose, I could count from zero to that number.
For the reals, the answer is no. I cannot even count from 0 to 1 given infinite time.
IMO, saying that a branch of mathematics is useless unless there’s an application of it in the sciences is like saying that a discovery in physics is useless unless there’s an application in engineering. Just because we don’t know how to apply it yet doesn’t mean that there will never be an application of it. Wasn’t the General Theory of Relativity the first use of tensor calculus? Would Einstein have come up with it had he not had the mathematics to express it?
I also recall reading an argument that rejecting the notion of an actual infinity basically implies rejecting the notion of a real number. Unfortunately, I don’t have the book on hand, but I believe that the argument was that you need the actual infinity to construct the real numbers from the integers.
You’re on the right track, but on the other hand, it’s not always naively obvious if you can count from a number to another. For example, given infinite time, you can count from 0 to 1 jumping from rational number to rational number, but not from real number to real number. Naively speaking, what is the difference?
Moreover, your method only deals with ordered sets of numbers. Many examples of sets don’t fall into this class.
