Why Study Analytic Philosophy?

So you don't wind up saying something like this:

Broken into 37 meditations, Being and Event is centrally an intervention in what Badiou calls the “Cantor-event.” It goes something like this: Georg Cantor’s work in set theory circa 1874 shatters the distinction between the finite and the infinite by proposing that in any given set of numbers, say [a,b,c], the one, a, is merely a count and not oneness in and of itself. Rather, it is an effect of the presentation of the multiple, a, b, and c; all three take place in the particular situation of the set. Such a presentation allows for the members of the set, and not vice versa. Badiou uses set theory to revise the Heideggerean being-as-one: “Ontology, if it exists,” he says, “is a situation,” that is, one in which beings-as-multiples are presented. It is this structure of which a representation of oneness is an effect.

And his startling proposition: ontology, if it exists, is mathematics.

There's definitely something startling here, but it's not that proposition.  Rather, it's the fact that Pythagoreanism is being cited with tacit approval.  But the point of this post is not to dismiss the notion that the only fundamental kinds of things which exist are mathematical entities.  Oft times those of pseudo-intellectual persuasion invoke scientific principles or mathematical theorems with the mistaken belief that these principles provide them with novel arguments for some thesis.  This is almost always not the case.  The Heisenberg Principle is likely the most abused scientific proposition, but here set theory is the whipping boy.

Cantor did not shatter the distinction "between the finite and the infinite".  In fact, under any natural understanding of what it might mean to "shatter" such a distinction, there is no longer any distinction between the finite and the infinite.  That, of course, is false.  This post contains a finite number of letters.   

What Cantor did was to introduce the notion of equicardinality between sets, such that set A and set B have the same cardinality if and only if there is a 1-1 function from A onto B.  A function is a set of ordered pairs R such that, for all x, y and z, if <x,y> and <x,z> are members of R, then y = z.  (Think of a set of points.  The requirement for a set of points constituting a function is that for any x value of the function, there is only one y value.  For the intuitive graphical examples, note that a vertical line is not a function, a horizontal line is a function, and a diagonal line is a one-one function.)  Finally, and to fully define all the terms used in the definition of "equicardinality", a function with domain A is "onto B" just in case the domain of the function is identical with B.

Here's a nifty way of thinking of equicardinality that was recently suggested to me.  Imagine a table at your favorite fine restaurant with dinner service set for eight.  Suppose you wonder whether the table setting is complete, or perhaps, whether everybody had a wine glass.  There are two obvious ways to figure out whether or not everybody has a wine glass.  The slowest way would be to count all the chairs first, count all the wine glasses second, and then see if your numbers match.  Most of us, however, would probably instead check to see if every chair is paired with a single unique wine glass.  If you can match every chair to a single unique wine glass, then there is a one-one function from the set of chairs onto the set of wine glasses.  That is, the sets have the same cardinality.  For any finite number of chairs and wine glasses, these two methods of counting will return the same answer, but matters become tricky if we move to infinitely long tables.    

There was a distinction between the finite and the infinite both before and after Cantor.  But after Cantor, we could draw further distinctions between infinite sets on the basis of their cardinality.  (It should be remembered that cardinality is a semi-technical notion, stipulatively defined.)  Consider two infinite sets: the set of natural numbers {0, 1, 2, 3, 4...} and the set of even numbers {0, 2, 4, 6, 8...}.  Intuitively, the set of even numbers contains exactly half plus one as many members as the set of natural numbers.  But the two sets have the same cardinality.  Each member of the evens can be paired with exactly one unique member of the naturals.  That is, there is a one-one function from the evens onto the naturals.  To see this, pair zero with zero, one with two, two with four, three with six, etc...

The notion of cardinality is, pretty obviously, not the same as the commonsense notion of size.  Thus, the fact that two infinite sets have the same cardinality does not entail that they have the same number of members, though the fact that two finite sets have the same cardinality does entail this.  (The former claim is mildly controversial.)  But in any case, cardinality is a technical notion, and while applying it to infinite sets yields amazing mathematics, the results are not really surprising since "cardinality" is a notion introduced by stipulative definition.  You shouldn't be shocked to hear that the set of the evens has the same cardinality as the set of the naturals, though you might well be shocked to hear that these sets are the same size, or have the same number of members, etc...  With the notion of cardinality defined above, it can be proven that some infinite sets have greater cardinality than others.  So  while Cantor opened up the field of transfinite mathematics, but, not to belabor the point,  shattered no distinction between the finite and the infinite.

I have no idea whether Badiou makes such a ridiculous claim or whether Alexandra Heifetz invented it, but either way, some training in analytic philosophy will prevent one from making such bizarre assertions.  I leave to my readers the task of attempting to come to grips with the rest of that strange business involving sets not being "oneness in and of themselves".

Comments

[everett]

just for clarification: you should be shocked to hear that two sets have the same size only because size is a physical property that can only be meaningfully applied to physical objects. cardinality is the set-theoretic analogue of size and is, as you rightly point out, introduced by stipulation.

however, i have to disagree with your claim that two infinite sets having the same cardinality does not entail that these sets have the same number of members on the grounds that it is not clear to me what you mean by the phrase 'same number of members'. you also point out that the failure of this entaliment is a midly controversial claim. so, if you would indulge me, please explain.

thanks.
Everett

[Scott]

I'm not sure what you mean by "physical property", Everett. Are physical properties just those properties that will be used in future (ideal) physics? If they are, then are physical properties had only by entities in the actual world, and perhaps, worlds with sufficiently similar physical laws? Other possible worlds, for example, may be populated with alien entities and governed by very different laws, but I see no reason to think that they couldn't come in sizes. Immatierial ghosts and blobs of ectoplasm may also come in different sizes. I take your point to be that something must have spatial dimensions in order to have a size, but I'm not sure this is right either. One might wish to speak of the size of a mathematical object like a perfect sphere. And the universe may have some size, though you might not think it's either spatially located or a physical object. There are other worries as well, but suffice it to say that it's not obviously ridiculous to assert that the size of a set supervenes on the number of its members. Offer me an analysis of "size" though and I might have to agree with you.

Regarding your second point, I am simply employing the intuitive notion of "same number of members". You might be worried about this notion - some are - but I'm not.

[everett]

hmmm. after having thought about this for some time, i find that your response is very compelling. the (or perhaps i should say 'my') intuitive concept of size seems circular.

there does, however, seem to be a difference between the size of a set and the members of that set. in one sense you might feel compelled to say that the size of the set of human beings on the planet is approaching six billion. this is also the number of entities or members in this set.

but this seems a little strange. what about the size of a table? normal tables, even if we grant that they have well-defined edges are typically between, say, 3 and 20 feet long. if we consider tables as sets of all molecules located within some well-defined boundaries, then we might say that the size of the table is 10^25 molecules of some specified type. but this doesn't seem to be what we mean when we ask about the size of a table. we expect a number accompanied by a unit; a particular *kind* of unit. it seems like the relevant units cannot be the things that make up or compose the object in question. units are more abstract things and whatever they are their meaning is fixed by some convention or reference. molecules and atoms might be abstractions, too, but i am more inclined to say that they are more concrete than collections or sets or extensions of predicates.

the relevant difference is that the size of physical things is given in units where units are abstract entities, words or meanings from a previously given list. physical objects are not composed of these kinds of entities.

sets and other non-physical things are "measured" in terms of the things that compose them. their units of measurements are the things themselves, if that can make any sense.

so maybe an analysis of 'size' is the following. x has size y if y is the concatenation of a number followed by an abstract object called a unit and these units do not compose x and cannot be reduced to other units that compose x. if there is a y satisfying these conditions, then we say x has a size.

ad hoc, maybe?

the concept 'the size of the (a) set of natural numbers' is then incoherent and we defer to the notion of cardinality.

[everett]

Also, what is the size of red? Intuitively, this seems like an incoherent question. But to ask about the cardinality of the set of red objects seems to make sense, doesn't it?

[Noumena]

Well, I have no idea what `shatter[ing] the distinction between the finite and the infinite' and `ontology, if it exists, is mathematics' are supposed to mean. But there's an interesting point in the second half of the first quoted paragraph. In more Analytic language, the claim is some sort of contextualism: identifying something as something is only done through its context, situation, and/or relation to other things. In particular, a is only an element because it's located (in some sense) in the context of the set.

Then the author (or Badiou) seems to want to push this down -- maybe, the essence of a is contextual or situational or relational. If this is what Badiou's up to, then it's arguably wrong (it's confusing properties with relations, might be the objection), but it's not clearly crazy or incoherent.

It also doesn't turn too heavily on the mathematical details of Cantor's work. Set theory is used as a jumping-off point, not premisses in an argument.