Wednesday, February 3, 2010

Balaguer responds to my email

I emailed Mark Balaguer about my observation of his entry on Mathematical Fictionalism. In the response email, Balaguer admits that it is one possible way to interpret the fact that both CH and ~CH, e.g., may be consistent with our notion of set and that this would entail that mathematics is, some sense, in trouble.

But he went on to say that mathematicians will likely have recurse to fall back on to render this possibility benign (that is, they will precisify our notions of set in a way that will not entail contradictions). In fact, he used a really interesting analogy that I once had used for a paper on feminist epistemology and realism in the philosophy of science (planet-hood and the case of Pluto).

This was one approach I had envisioned it going and the likely rout taken by mathematicians if no obvious additional axioms are found that would entail the possibility of a proof of either CH or ~CH. But further questions remain. What if we find this additional "obviously true" axiom regarding our precisified notions of our set and then we find another "obviously true axiom" which we may also add to ZFC which renders a proof of ~CH possible (or some alternative theorem and its negation)? There's no way to guarantee this happening and so the possibility of our set theoretic notions being ineluctably imprecise (i.e., any precisifications would just end up pushing the problem back to another level), in a robust and foundation-shaking way will always remain. I feel that this possibility is at least a blow to Platonists by showing that their belief in the complete objectivity of mathematics is an article of faith for at least some of the most interesting problems. There just might not be a fact of the matter about them.

In my paper, the reason I introduced the problem of planet-hood was to give a way for the possibility of objectivity together with some degree of relativism.

Consider the (ex) planet Pluto. It was classified as a planet by Astronomers once but now it is not. The criteria for planet-hood had changed in 2005 and set (arbitrarily) to standards that would disallow classifying Pluto as a planet. Astronomers did this because of the recent discovery of several large objects in our solar system that would, under the old criteria, be classified as planets. Faced with the choice of either adding all of them as planets, they decided by a vote (I think it was about 80% "yea") to reset the criteria so that they would not be classified as planets. One upshot of doing this however, was the now Pluto would fall short of those new standards. This implication was something astronomers were willing to accept.

But let's say that some Martian civilization had astronomers that were willing to accept these other solar planet-like objects as planets, and hence, still able to ascribe Pluto's planet status under their conception. Would we say that they are wrong or that they simply don't have a notion of planet in general?

I don't think we would. I think we'd say that they do have a notion of planet, just that it is slighty different from the astronomer's precisified notion but that they do understand what planets are, etc. Our notion of planet is vague and I argued that this was because planet-hood, like many other natural kinds of things, are vague (I was arguing for vague objects). But this would not mean that anything can be a planet. Floating space debris the size of basketballs are not planets. Anyone that insists they can classify them as such simply do not understand what planets are or have in mind some other concept (though they just use "is a planet" to denote such a concept, it wouldn't actually be one). So I maintained that planets are natural kinds but that planets are vague objects.

Now vague objects are quite controversial, as I understand it. Many philosophers don't believe in them and dismiss them outright (such as Ted Sider in his Four Dimensionalism, if I remember correctly). They think it a "fallacy of verbalism." But at least two very outstanding metaphysicians have defended the thesis: David Lewis and Gareth Evans (making Sider's almost flippant dismissal rather puzzling). I'm not an expert in vagueness so I'll leave it be and hope that it may be defended to save my argument!

But getting back to vagueness, consider this analogy. Hilary Putnam once argued that we, by our linguistic-conceptual faculties, "cut" the world up into conceptually coherent kinds of objects much like a cookie-cutter cuts cookie-dough into cookies. The cookies don't exist before our conceptual renderings (and thus scientific and many commonsense "truths" are not really "mind independent"). In other words, nature has no "joints" but we make them using concepts to "cut" the world (arbitrarily).

This has huge problems. One, it runs against our realist intuitions. Two, isn't the pre-conceptual "dough" of the world a conceptual rendering thereby undermining this kind of theory? By using an understanding of vagueness like I have in my paper, that is, as inherent in the actual world, thus justifying our vague concepts, we can bypass this problem and have a degree of objectivity plus a degree of socio-cultural relativism I argued was desirable (and this would have political implications for feminist and cultural studies).

On my view, cookies already exist in the cookie-dough mind independently, but their borders are not clear-cut as they are when we use the cookie-cutter to cut them free of the surrounding dough. Rather, there are smooth gradients separating the surrounding dough from which the cookies are encompassed therein. We only precisify when we use the metaphorical cookie-cutter such as when the astronomers use precise criteria to classify and reclassify planets. Nature has "joints" but they are smooth, not clear-cut.

What does this have to do with mathematical fictionalism discussed at the beginning of the post? Well, I argued that nature has a way of constraining our beliefs about the world, that is, we can't just completely arbitrarily cut the world up however we like but that doesn't mean we don't have at least some flexibility. But my point was that the objective, physical world constrains us. What in the world could possibly constrain us in one precisification over another in mathematics even when they are dramatically different precisifications? Saying that the abstract "mathematical universe" does so simply begs the question! Besides, wasn't the whole logicist and set-theoretic foundational project started by Cantor and Frege and that carries on today for the purpose of finding mathematics a precise foundation that will eliminate possibilities of ambiguity and thus possible contradictions?


  1. If “Pluto” falls under the concept of “planet” it depends who you are asking. It depends on a community of speakers. We can consider two immediately from the same language group, but with different backgrounds. One is a leading NASA astronomer, and the other is an average person who has not “heard” Pluto is “no longer a planet.” The man asks, “Can you see the planet Pluto from Earth?” and the scientist answers, “Pluto is not a planet.” The man replies, “You are a damned poor NASA scientist if you don't know Pluto is a planet.” Then the NASA scientist says, “Oh, it is no longer a planet” and explains the change.
    The scientist will be using a criterion of size to determine correct usage, but that may not form a critical feature in the use of the word outside the speaking community of scientists except as it translates into “convincing others” Pluto is not a planet.
    In Sanskrit it is “graha” which means “planet.” Of course, they were not aware of Pluto. Does this decision determine my use of “planet” within Sanskrit, perhaps as I talk about astronomy?
    Part of the revision of “planet” will be the role in plays in formal talk about heavenly bodies—the talk of the expert. They have arbitrarily eliminated vagueness and its application in the tightening of their theoretical jargon. Lots of people understandably talk about stars and planets, though, without knowing sizes and compositions. There are two speaking communities with different needs . . . one needs precision in this term while the other needs something loose enough to convey information without getting specific beyond our knowledge of the object/concept. These differences, though, all apply primarily to semantic vagueness, don't they? A "metaphysical" transformation of an object from a non-planet to a planet doesn't happen if I somehow add a few tonnes of mass, does it?

  2. That brings up an interesting point. I think that different linguistic communities can exist within the same language and even within the same individual's history of use. If an argument like that ever happened between the average person and the astronomer I think that the astronomer would admit that the average person isn't necessarily wrong in his usage but just that they are working with two different notions of planet (one would be a layman's understanding and the other is a more scientific one). There is no strict perfect overlap but one is as legitimate as the other in some sense of linguistic legitimacy.