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I do not want to poison this forum with politics. But I want to understand, precisely, what is meant by the bolded statement. It is made by a physicist who used to work at Harvard regarding the relationship between pure math and physics.

A physics-oriented question appears at the end of Chapter 3: Are we really talking about continuous objects themselves or about finite sequences of symbols that talk about continuum?

That's a good question and I am inclined to say the latter. If we talk about specific things, these specific things are always countable because they must be describable by a finite sequence of symbols. Even when we talk about intervals of real numbers that arguably contain an uncountably infinite number of real numbers, we must still specify the endpoints of the interval by a finite sequence of words or symbols – and such sequences are "discrete" i.e. "countable".

This is why I tend to consider the very fact that real numbers are uncountable to be nothing else than a linguistic curiosity: the actual, well-defined real numbers you may ever encounter form a countable set! This is why the uncountability of the real numbers – and the whole discipline of maths based on this formally provable claim and similar claims – doesn't have implications for "talking about physics".

Here is the link to the full blog. If you want more context, I would suggest to start reading from the "Chapter 2 talks about sets, their elements..." part.

It appears he is talking about the symbols and notation we write down to describe the real numbers. But the bolded part is explicitly referring to the set of real numbers.

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Obligatory XKCD: xkcd.com/435 –  BlueRaja - Danny Pflughoeft Apr 22 '13 at 23:19
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The blog's author is active on Stack Exchange, particularly on Physics, if you want to ask him directly... –  AakashM Apr 23 '13 at 8:24
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It seems strange for people to respond to this question without noting that the physicist is Lubos Motl; and his article is a wild ad hominem attack on complexity theorist Scott Aaronson. Motl is a smart guy, but well known for having no manners online and for making these kinds of shoot-from-the-hip attacks against anyone who doesn't think exactly like he does. I do hope people clicked on the link and read the entertaining dialog between Motl and Aaronson in the comments section. Warning, f-bombs and non sequiturs abound. –  user76014 May 4 '13 at 19:20
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I am perplexed by the wording "a physicist who used to work at Harvard" and the fact that the answers similarly play pronoun games. This is something that Lubos Motl posted on his blog; it can't be a secret. Why not just identify Motl by name and give the link to his blog? And what is the relevance of the fact that he "used to work at Harvard"? –  Pete L. Clark May 4 '13 at 20:15
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I also have to ask: if you have a question about something that was written on someone's blog, why not just post your question to the blog? –  Pete L. Clark May 4 '13 at 20:30

6 Answers 6

He's saying that the definable numbers are countable. In some sense we cannot talk directly about a real number that isn't definable. (However, we can still talk about the set of real numbers even without being able to talk about each of its elements individually.)

The quote seems to be advocating some form of finitism.

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Without passing judgement, I believe what the physicist is calling attention to is the set of Computable numbers, which is countable. Such numbers are the ones that can be computed, to arbitrary precision, using a computer. His point is that most real numbers are not computable, and is advocating restricting attention to such numbers.

A comparable example is that of a non-measurable set or non-principal ultrafilter, objects that can be proved to exist with the use of the Axiom of Choice, but cannot be explicitly constructed (since without the Axiom of Choice they would not exist).

This point of view is related to mathematical constructivism, a philosophy advocated by a minority of mathematicians.

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There are definable numbers that aren't computable, though, and "finite sequences of symbols that talk about the continuum" smells to me more like definable numbers than computable numbers. –  Qiaochu Yuan Apr 22 '13 at 21:24
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It may be worth adding that the definable numbers are countable, but cannot be counted, i.e. there is no "finite sequence of words or symbols" that enumerates them - so that the author should admit by his very argument that the set of (definable) reals is not (definably) countable. –  Hagen von Eitzen Apr 22 '13 at 21:33

There is a direct rebuttal: show me the method by which one would count the "actual, well-defined real numbers". If you have an actual well-defined method for enumerating real numbers, then I can invoke Cantor's diagonal argument to show you one that you missed. :)


Although I don't think it is widely known, there is another interpretation of the notion of cardinality: that it talks about complexity rather than size.

In a finitist context I have a little familiarity in (based on Turing machines and computability), it's a nearly trivial to find a partial surjection from the "set" of natural numbers to the "set" of real numbers.

However, in this setting, one cannot always tell@ if a particular natural numbers gets turned into a real number, or whether two different natural numbers get mapped to the same real number. Thus, the "set" of real numbers still cannot be counted -- the set remains uncountable in the only sense that is meaningful within this setting.

@: I don't just mean it's hard: I mean impossible. e.g. look at things like the "halting problem".


However! In the example above, the computable numbers can be "revealed" to be countable by looking at it from some external perspective -- e.g. if we use ZFC as a foundation for the theory of computation, then the computable reals are countable, but only in the sense of countability that makes sense in ZFC@.

This reveals a subtle aspect of this issue: there are a lot of ways one might mean words like "countable", and it's very easy to mix them up. In fact, it's very easy to not even be aware that there are multiple things one could mean.

My general advice is to take such statements with a grain of salt. While I believe there is usually some angle by which such statements are meaningful (and sometimes the speaker is even aware of it!), it is rarely clear just what that angle is. If you believe a writer's views are worthwhile to understand, it's better to ignore the shocking comments and instead focus on trying to gain knowledge from the way he actually uses his views to solve problems. (which may possibly even require translating examples out of the language the writer thinks in, and into the way you see things)

@: This gets especially confusing when there are multiple models of ZFC under discussion. e.g. look up "Skolem's paradox".

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+1 for this answer. I think though that the state of affairs could be made more clear: yes, the actual well-defined real numbers are countable, and provably so from a purely mathematical/logical/linguistic point of view, but no, there is no actual well-defined method to enumerate all those numbers in a bijective way. The main difficulty is that one cannot avoid repetition in the enumeration, which means one cannot decide whether two well-defined real numbers are equal; they may be well-defined, but that does not make them well-known. –  Marc van Leeuwen Apr 23 '13 at 9:14

As others have said, in this extract, it looks like he means something like:

  • The set of computable numbers is countable
  • The set of definable numbers is countable
  • The set of numbers which can be 'experienced' in any sense by a human equipped with the usual senses is finite

However, he goes on to say [my emphasis]

[...] the fact [...] that we only talk about finite sequences of symbols does not mean at all that discrete maths and its axioms should be the foundation of physics. Even when I say that we are talking about finite sequences from a discrete set, it's still important what we're saying about them. And to do physics, we should organize these sequences and "discrete objects" in such a way that they talk about properties of continuous objects because those are ultimately fundamental in physics, as we know because of many general reasons.

So talking uses finite sequences from a countable set but in physics, it still matters what we are saying, and if we're not saying things about intrinsically continuous structures and properties that continuous structures may have, it will be either invalid or non-fundamental as chatter about physics!

Having looked at this, I get the impression that instead he is really saying that propositional logic and the language of mathematics (and natural language) involves writing down and working with countable sequences of symbols. He seems OK with talking about continua using such a 'discrete' framework. (Edited to avoid a bone of contention with André - see comments!)


By the way, as an aside,

This is why the uncountability of the real numbers [...] doesn't have implications for "talking about physics".

seems like a very odd thing to say. The entire language of physics is set in the language of $\mathbb R$, and were we to decide to solve physical problems talking about only a countable subset we would be pretty insane.

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This is definitely not maths; it's completely dependent upon what "thinking about" means. The neurons in your brain are storing an essentially countable number of bits of information; your brain is processing a countable amount of data. On the other hand, abstractly, you might wish to give a definition which is somehow independent of the fact that humans are not processing to infinite accuracy; but that's difficult to pin down IMHO. –  Sharkos Apr 22 '13 at 22:14
    
Also, even in this abstract sense, we may not be thinking about the phonemes, but we are thinking in 'lumps of idea', not continua thereof. Our conception of the reals is not of simultaneously perceiving all of them, but rather of individuals found amongst a continuous block. –  Sharkos Apr 22 '13 at 22:16
    
I've edited that paragraph to remove this issue, since it's really not what I was trying to say anyhow. –  Sharkos Apr 22 '13 at 22:18

This problem is closely related to mathematical logic and the notion of definability and truth.

One view is through the Löwenheim–Skolem theorem in mathematical logic, which states that

if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ.

This has somewhat counter-intuitive consequences, for example that any theory of real numbers (or even set theory) has a countable model. Or that any theory of natural numbers has an uncountable model.

Let's say we have a countable model of a theory of $\mathbb{R}$. Does it mean that all subsets of $\mathbb{R}$ are countable? No. We're viewing only one possible model of the theory. All subsets are countable only when this particular model is viewed from outside. But when viewed from "inside" of this model, we still can't construct a bijection between $\mathbb{N}\subset\mathbb{R}$ and $[0,1]\subset\mathbb{R}$. This is because the model doesn't contain such a bijection (even though such bijection exists when viewed from outside - it's not available inside the model). The existence of objects or their definability is different when viewed from outside of some particular model form when viewed from inside the theory.

Therefore it is a bit imprecise to say just that there is only a countable number of real numbers. What is "countable" depends on if work inside a theory or if we take a step "outside" on the meta level.


Such situations or paradoxes occur often in logic. For example Gödel's second incompleteness theorem states that a (reasonable) theory can't prove its own consistency. But of course we can prove its consistency from outside, for example by embedding the theory in some other, stronger theory.

See also Berry's paradox: the smallest positive integer not definable in fewer than twelve words. The key is that the notion of "definable" is different when observed from outside and when constructed inside the theory (for example using Gödel numbering).

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I honestly don't see the relevance of Lowenheim-Skolem and countable models here. A countable model of $\mathbb{R}$ is simply a countable real-closed field, e.g. the real algebraic numbers. None of these models possess the important completeness properties of $\mathbb{R}$ that makes $\mathbb{R}$ a central object in geometry and physics: they are first-order models only. I don't see why this is counterintuitive, and talking about countability of subsets within a model of $\mathbb{R}$ doesn't make sense to me: the language of fields cannot express such things. –  Pete L. Clark May 4 '13 at 20:22

Let's start the clock 4 billion years ago, just for the sake of starting somewhere. The first time someone ever wrote down or even thought about some specific number, that was term $a_1$ in a sequence. Every time since then someone mentioned or thought of or wrote down a number, that added to this sequence. By now, the number $\sqrt{2}$ I just wrote might be $a_{10^{100}}$. In this way, the author means that every number which has ever had relevance to a human being insofar as they crystalized it into form in some way, has been part of a sequence.

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