September 11, 2005

Reading blogging -- Russell, "Mathematical logic as based on the theory of types", 1908

In the first decade of the twentieth century, a number of logical paradoxes were known, including several that involved the logicist foundationalist project for mathematics. Russell himself had come up with one, now known as Russell's Paradox, which starts by considering the collection, W, of 'all collections' which are not members of themselves, ie, X is in W if and only if X is not in X. But then, is W in W? By definition, W is in W if and only if W is not in W, contradiction. In this paper, he makes the following diagnosis:

we found that all of [these paradoxes] arise from the fact that an expression referring to all of some collection may itself appear to denote one of the collection (101, his emphasis)

For example, the Russell paradox assumes that there is a collection of all the collections, and W is a member of this 'universal collection'. He continues:

We decided that, where this appears to occur, we are dealing with a false totality....

By Kantian grounds, this is an accurate diagnosis: while we can talk about collections through abstract, conceptual representations, the only way we can be sure that a collection exists is if it can be given in intuition. Logic cannot tell us whether anything exists, so the definition of W is not a 'purely logical' definition; and since we cannot represent the act of 'collecting together' all collections into one universal collection in intuition, we can't use the universal collection to define an object, W, and then act as though we know W exists.

But Russell, who follows Frege in bucking two thousand years of metaphysics and giving logical constructions ontological weight, takes a very different approach: he orders variables into a hierarchy of 'types', 'proceeding upon the principle that any expression with refers to all of some type must, if it denotes anything, denote something of a higher type than that to all of which it refers'. That is, W is not of the same type as any of the collections which it contains (the variable X ranges over one type lower than the type of W), and hence there is no contradiction for W to not contain itself.

This makes a colossal mess of his symbolic logic, and doesn't end up solving anything; a few years later, he has to 'ramify' each of his types, creating a new hierarchy at each level of the old one, to avoid similar paradoxes; and at the same time, these strictures make it impossible to talk 'across types', eg, the predicate 'is a type' becomes incoherent, and has to be replaced with unanalyzable predicates we might write as 'is a type1', 'is a type2', 'is a type3', and so on. 'The hierarchy of types' itself becomes incoherent, as (in an ironic echo of the Burali-Forti paradox Russell proposed the hierarchy of types to address) its type must be higher than that of any type.

This is one reason why I don't think all that highly of Bertrand Russell.


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