Sunday, January 4, 2009

The reality of sets

I checked the blog of the Maverick Philosopher, aka Dr. William Vallicella, over the course of the last week, and once again find he has provided material to which I would respond. In fact, in writing two posts about the existence of sets, he has touched on a subject near to my heart. In fact, I discuss this very topic, in a much less formal manner, in every single Geometry class. There is a very real distinction between a formal system, a construct of the mind that we use to generalize reality and draw deductive conclusions, and the underlying reality itself. Mathematics is first and foremost a formal system. I think Vallicella provides an excellent example of how the eagerness that believers have to create these non-materials things of real existence spills over into assigning reality to all sorts of fanciful concoctions, much as we see supporters of one type of woo are often more likely to pickup or support other sorts of woo. Vallicella's comments are italicized.

Set theory can be done either naively or axiomatically. In the standard axiomatic approach to the subject, that of Zermelo-Fraenkel, the existence of the null set is posited in a special axiom.

Two errors in two sentences. One of the primary reasons that set theory was put into an axiomatic format is that, when doing set theory naively, you come across both logical and semantic contradictions that are avoided by the axiomatic process. Also, while Zermelo's original approach had a special axiom for the empty, the standard approach to the Zermelo-Fraenkel axiom schema contains no such axiom. Rather, the empty set is the derived result of the axiom schema of replacement, which in the short form says that functions defined on classes will map sets into sets.

Intuitively, the existence of a set depends on the existence of its members. The set consisting of me and my cat cannot exist unless both man and cat exist: if either of us should cease to exist, the set would cease to exist. It exists because we exist, not vice versa. (This is of course not a causal use of 'because.') In the case of the null set, however, there is nothing on which the null set can depend for its existence.

The truth is that sets don't exist, at all. The set of Vallicella and his cat is a mental shortcut that we use, but it has no reality unto itself. When I talk about the red books on my bookshelf, they are a set of books for only as long as any person considers them to be a set of books. There is no ontological status the property of their being a collection. After all, how is the nature of the books changed if I say that I have carefully designed them to be a set, or if I say they are just a bunch of book I have in that place? I can tell Person A the one and Person B the other, they can't compare notes, examine the books, and thereby discern which account is more truthful. So, it seems the nature of being a set has no reality behind it. In particular, why should the empty set have any more reality?

No working mathematician is likely to lose any sleep over this, however. He will tell us that the null set is convenient, computationally useful and ought to be judged by its practical fruits.

Working mathematicians generally seem aware that almost everything they do is based upon assumption that are help arbitrarily, because we have found them useful. The very grammar of mathematics, predicate logic, is based upon the highly arbitrary notion notion that every statement is true or false. The arbitrariness of this decision is shown by the success of the intuitionists and the various alternate logical evaluation schemata in creating perfectly functional mathematical systems. They know that sets exists only to the degree that we think of the collection of various objects as a set, and so one more useful, non-real entity is not going to cause consternation.

From the first post:

So on the one hand, the null set is useful and well-motivated from within the circle of set-theoretical ideas, but on the other hand, it appears philosophically to be a creature of darkness. Is there a way to get rid of the darkness?

By my lights, the philosopher aims at a degree and a type of clarity that the mathematician qua mathematician -- not to mention other nonphilosophers -- does not care about. Of course, I am not saying that he should care about it. He is within his rights in simply dismissing concerns like the one raised in this post as irrelevant to his concerns or unimportant given his goals and priorities. What is intolerable, though, is the mathematician who gives a lousy philosophical answer to a philosophical question, especially if he is only half-aware that it is a philosophical question.

Of course, philosophy itself is also a formal system. When the mathematician chooses to ignore the needs and desires of one formal system in favor of another, they do so 2with the understanding that all formal systems are arbitrarily chosen based upon their perceived usefulness.

He is then like the neuroscientist who, refusing to stick to his subject-matter, says silly things about mind and consciousness, all the while oblivious to the philosophical problems to which he gives silly answers.

Naturally, we see one sort of woo dissolving into another.

From the second post:

A set in the mathematical (as opposed to commonsense) sense is a single item 'over and above' its members. If the six shoes in my closet form a mathematical set, and it is not obvious that they do, then that set is a one-over-many: it is one single item despite its having six distinct members each of which is distinct from the set, and all of which, taken collectively, are distinct from the set.

Or, since the very notion of set is in and of itself a mental construct, and not a reality, it is certainly different from the shoes themselves, which presumably are real enough that Vallicella puts them on his feet once in a while.

Vallicella goes on for quite a while being very clear that sets are created as a product of thinking. Yet, somehow he does not seem to understand this means they are just as unreal as Romeo or a modern-day Casanova. Just as we can use the conventions and shortcuts of our understanding of Shakespeare's play or a historical personage to create mental constructs of what it means to be Romeo of a moder-day Casanova, we use conventions and shortcuts to understand what a set of size six means, and assign contents to sets at need.

Your objection does not show that sets cannot be mental constructions; it shows that sets cannot be mental constructions of a finite mind. If there were an infinite, necessarily existent mind, then sets could be the constructs of such a mind. If you maintain that there is no such mind, then you should also maintain that there are no sets. If, however, you hold that there are mathematical sets, and that nothing contradictory can exist, then you should hold that they are the mental constructs of an infinite mind. If you deny that there is an infinite mind, but hold that there are sets, then you owe us an alternative explanation of how a set can be both one and many.

Left unaddressed is the reason we should think sets exist at all.


Joshua said...

A few remarks:

First, whether sets exist in any sense is a hopelessly ill-defined question. Does one mean exist in a Platonic sense that they exist in some metaphysical other realm? Does one mean exist in some pseudo-Platonic sense? Does one mean exist as agreed upon social constructions?
This entire discussion assumes that the term is well-defined when it isn't necessarily so.

Second, your comment about ZFC not needing a separate axiom for the null-set is nitpicking. One sees the existence of a the null set as a separate axiom slightly over half the time. When I talk about this topic I generally make it a separate axiom for pedagogical reasons. There is arguably an issue in that Vallicela's comment can be misleading since it might imply that one needs such a separate axiom but that's a much weaker objection.

Overall, Vallicela's piece is a good example of why mathematicians don't like philosophers that much. There's serious imprecision and poorly defined terms and then he claims that somehow he is being more precise than what mathematicians do.

Anonymous said...

How to stump any philosopher.

Ask for a definition of philosophy.

One Brow said...


Thank you for your comments, and I am sorry I have not looked at them before.

I can see making the existence of the empty an axiom for pedagogical reasons (I teach a course in elementary Geometry, and probably more than half of the Postulates we list are for the sake of convenience). I evev acknowledge that my post was a bit of nit-picking (in response to Vallicella's own nit-pciking). I suppose I have bought into the notion that a proper aqxiomatic system does not use axioms that can be derived from other axioms, but that is probably more an asthetic preference than anything else.