Why Do Irrational Numbers Exist?

[PeterJ]

Hurkyl? Anybody? I'm getting a little bit paranoid at the lack of response. Was that not an appropriate question here? I suppose it's not exactly a mathematical question. Or is it? It wasn't a trap anyway. I was trying to understand how mathematicians see these issues, exactly where they feel that mathematics turns into metaphysics and so draw the line. Personal opinion would be be fine.

Or is it that mathematicians have no time for such idiotic questions? I think the answer to it is clearly yes, but I don't know if this is a commonplace opinion in mathematics or utter heresy, or even of any interest, and I have no idea as to what the mathematical implications of either answer would be. Brown and Peirce would have answered the same way, if I understand them correctly, but don't imagine I can follow even Brown's painfully simple calculus once it becomes a system of symbols.

Perhaps everyone's down the pub.

 

[Hurkyl]

That's how I imagine points are usually defined, as the end point of a never ending process.

Generally, "point" is never defined at all.

A typical version of the theory of Euclidean geometry, for example, takes "point" as one of its "primitive notions" (others tend to include "line", "lies on", "between" and "congruent") and never attempts to define it. Instead, it postulates the properties that points have, and studies the consequences of those properties.

A typical version of the theory of real numbers is similar. It takes "number" as a primitive notion along with 0, 1, +, *, and <, and postulates the complete ordered field axioms.


A definition of "point" only comes when you want to apply Euclidean geometry to some purpose. e.g. a physical theory might assert that there is a notion of "position" in the universe, which obeys all of the axioms of Euclidean geometry describing points.

Another example is that, to better study the arithmetic of real numbers, we might define a model of Euclidean geometry in which "point" is interpreted to mean an ordered pair of real numbers. Conversely, in order to better study Euclidean geometry, we might construct a number line -- a model of the theory of real numbers -- and work with coordinates. In this sense, the two theories are actually the same theory just presented differently.


Locale theory defines "point", but that's simply because, pedagogically, it doesn't seem useful to take it as a primitive notion. I'm sure that would change if it was shown to be useful.


As you might guess I'm a formalist. But only weakly -- I don't make any assertions on whether or not mathematics has meaning, I just assert that the meaning isn't part to the formalism.
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Now, I tend to get a little uneasy when people talk about "never-ending processes". It's a perfectly good idea, but there are a lot of misconceptions often associated with it.

The most common, I think, is of this form. When talking about things in this way, it is the process itself that names the point. (maybe we have defined "point" to literally be the process, or maybe as a limit in some suitable sense, or some other way to associate the name to the point) However, sometimes people get in their heads that the point is supposed to be the result of the process, either taken after some ill-defined number of steps, or after some mythical final step.

I can't actually tell if you have a bad idea in your head, I'm just a little uneasy about it.

I assume that they simply have to be defined in this sort of way. It's the issues this raises that interest me.

It's a typical approach to naming things. You identify objects by some property they have -- in this case, a point is identified by a collection of "regions" that would contain it, but you need infinitely many regions to uniquely pin down the point.

The definition I cited is actually the unrolling of a simpler definition -- there is a locale called "*", and a "point of the locale L" is really just a mapping from * to L. But when you unfold all the complexities of the locale of real numbers, the locale-theoretic meaning of "point" transforms into something similar to and equivalent to the one I stated.

Typically, definitions that apply in most or all cases of interest tend to require infinite amounts of information. Specific cases often require much less -- e.g. for the Euclidean line, I could instead fix an orientation and name points with intervals (with the idea that the point is to be the left endpoint of the interval). But this particular scheme is very specific to the Euclidean line.

The real numbers suffer from this too. General naming schemes (like decimals, or continued fractions) tend to require infinite amounts of data. However, specific numbers can often be named with much simpler methods, such as the positive square root of 2.

I see no possibility of constructing a coherent metaphysic for which points are not a convenient fiction, and I think this is a quite widespread view.

True or not, convenient fictions are, well, convenient. That's why we have them. It's analogous to scaffolding -- in the end, all you care about is building a building, but it's much easier to do so if you build the scaffolding along with the building, then remove the scaffolding at the end.

This is ubiquitous in mathematics. e.g. if we decide to name rational numbers with names of the form x/y where x and y are integers, one of the first things we do is decide which names really do name rational numbers (1/0 does not), and decide when two different names name the same object (e.g. 2/3 and 4/6). This extends to mathematical structures, structures of structures, and so forth.

This is also one of the reasons physicists are so interested in symmetries. e.g. from the fact the laws of classical mechanics are symmetric under rotations and translations of Euclidean space, we deduce things like an absolute notion of "position" or "direction" have no physical meaning.

 

[PeterJ]

H - Many thanks for a really helpful post! Maybe what I'm exploring is to do with the boundary between mathematical formalism and realism. I'm now a little more clear about one or two things and I'll shut up about this now.

Btw - re the primes - I'll stop bothering you about this also. I've managed to track down a retired prof who is prepared to do a bit of tutoring so I'll see how that goes.

You've been very helpful and patient - thanks.

 

[disregardthat]

Now, I tend to get a little uneasy when people talk about "never-ending processes". It's a perfectly good idea, but there are a lot of misconceptions often associated with it.

The most common, I think, is of this form. When talking about things in this way, it is the process itself that names the point. (maybe we have defined "point" to literally be the process, or maybe as a limit in some suitable sense, or some other way to associate the name to the point) However, sometimes people get in their heads that the point is supposed to be the result of the process, either taken after some ill-defined number of steps, or after some mythical final step.

I think Hurkyl is pointing out the main difference between those who tend to be skeptical towards irrational numbers and those who are not.

Hurkyl, what is your opinion about treating the real numbers as a primitive notion; rather than constructing them from the naturals? I know this is an essentialist issue, but even as a formalist, the essentialist aspect need not be ignored.