In math and computer science, we have a tendency to talk about “going meta”. It’s actually a
\npretty simple idea, which tends to crop up in other places, as well. It’s also one of my favorite concepts – the idea of going meta is just plain cool. (Not to mention useful. There’s a running joke among computer scientists that the solution to any problem is to add a level of indirection – which is programmer-speak for going meta on constructs inside of a programming language. Object-orientation is, in some sense, just an example of how to go meta on procedures. Haskell type-classes are an example of going meta on types.) <\/p>\n
Going meta basically means taking a step back, and instead of talking about some subject X, you talk about talking about X. <\/p>\n
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For example, we can talk about numbers. We can make statements about specific numbers – for example, “4<6”. We can make general statements about all numbers: “For all numbers x, there are an infinite number of numbers larger than x”. Those are both basic ground-level statements about In a slightly more formal mode, think about standard, first order predicate logic<\/a>. In first order logic, we have a set of objects<\/em> that we can talk about. We make statements about But we can go meta – and jump to second order<\/em> predicate logic. In second order predicate logic, first order predicates can be treated as objects that can be reasoned about. So we can have second-order predicates that talk about first-order predicates: “ The fundamental source of meta in most modern math is set theory. Early efforts to use The problem with Russell’s paradox is pretty obvious when you walk it through. Is R a member of itself? Suppose that R is a member of itself. Then by the definition of R, since R∈R, that means that it’s not a member of itself. Suppose it’s not a member of itself – then by the definition of R, it is<\/em> a member of itself. So no matter what we do, we have a contradiction: if R is in itself, that leads to the inference that it’s not in itself, and vice versa.<\/p>\n The attempts to solve that work by creating a strict separation of levels – that is, separating
\nnumbers. Going meta is shifting from talking about numbers to talking about talking about numbers. So most of this paragraph is actually meta-discussion about numbers: it’s talking about how we can talk about<\/em> numbers. And we can meta many times if we want – for example, the previous sentence went meta on the meta – it talked about how we were talking about talking about numbers; and this sentence went even more meta, by talking about the sentence where we were talking about talking about numbers. <\/p>\n
\nthose objects using predicates<\/em>. We can make statements about specific<\/em> objects by referring to them directly in a predicate – for example, “HasBigNose(MarkCC)<\/code>“. We can make general statements by introducing variables using quantifiers, and making statements in terms of those variables: “∀ x: HasBigNose(x) ⇒ SneezesLoudly(x)<\/code>“. But we can’t talk about predicates<\/em> using first order logic – predicates are fixed things, and there’s no way to make a statement about a predicate<\/em> in first order logic – we can’t say that the predicate “HasBigNose” is a member of the set of really silly predicates – there’s no way to refer to the predicate that would allow us to say that.<\/p>\nIsSillyPredicate(HasBigNose)<\/code>“.<\/p>\n
\nset theory as a unifying basis for mathematics encountered a serious problem: set theory
\nwas discovered to include paradoxical constructs. The most canonical example is the<\/em> classic paradoxical statement: Russell’s paradox<\/a>, which is a statement about the set R={x | x∉x} – that is, the set of all sets that do not<\/em> contain themselves as a member. Is R an element of R?<\/p>\n
\nbase from meta. For example, von Neumann’s version of
\nthe solution<\/a>, called NBG (von Neumann-Bernays-Gödel) set theory, is to take the idea of a
\ncollection of stuff, and separate it into two different kinds of things: classes<\/em> and
\nsets<\/em>. A class is any<\/em> collection of things which can be defined by some property
\nheld by all of its members. A set is a class which is a member of some other class. Thus, a class
\nis a sort of meta-set: it’s a kind of set which can contain other sets – but by making the
\nseparation between classes and sets, we’ve created a sort of fire-break that prevents things like
\nRussell’s paradox from working. In this construct, Russell’s set becomes Russell’s class<\/em>;
\nthere’s no consistent statement of Russell’s class that can allow it to be a member of a set; and
\nso the paradox (seems to) disappear. So von Neumann and friends temporarily defeated Russell’s
\nparadox by introducing one level of meta into set theory.<\/p>\n