Friday, May 10, 2013

Scientific Truth and the Nature of Mathematical Inquiry

Today I heard that proving the efficacy of the technique called "forcing" in set theory is made much more elegant and clear by the assumption of Platonism.

So what? I hear you ask.

So... so everything.
Scientific truth is determined by a variety of criteria, but some quite important ones are practical utility and elegant descriptive power. I think everyone who understands any scientific field can agree that mathematics is of significant practical utility, and set theory is immensely useful within mathematics. In some scientific sense then, the philosophical assumptions underlying set theory are true.

I feel I need to justify the reliance on elegant descriptive power more than simply citing it (paraphrased) as one of the criteria by which scientific theories should be judged mentioned in The Grand Design by Stephen Hawking and Leonard Mlodinow, so here goes. Occam's razor warns us to not multiply entities beyond necessity, but this principle naturally extends itself to a broader view of complexity, namely that one should generally choose "simpler" explanations in place of more "complicated" ones, for some reasonable definitions of those terms. The reason I attribute this to elegance, particularly in the metamathematical use of the term, is exemplified by the original "revolution":
Copernicus did not actually "prove" that the motions of the planets are those of massive spheroids orbiting the sun, he did not demonstrate that the preceding system of calculation using epicycles was actually incorrect per se, what he proved was that the heliocentric model provides a much more mathematically elegant description of their motion. Technically everything which can be calculated using the Copernican system can also be calculated using epicycles, but it is much more intricate and difficult to do so, leading to the worldwide scientifically-minded community naturally adopting the view that Copernicus' model of the motion of the planets is in some important sense closer to the truth.

Apparently certain important proofs in set theory (at least the efficacy of forcing) are made more elegant by assuming a Platonic view of first-order set theory - by this I mean assuming that there is some part of reality which set theory describes to some degree of accuracy, and that the practice of set theory is in some way the apprehension of this part of the external reality. As I take a materialist view of reality this assumption is quite incompatible with the way I have so far dealt with mathematics on a philosophical level.

Anybody who is familiar with the dialogue of mathematicians will know that we all tend to speak as though the objects we are discussing are in some sense external to ourselves, that there are "real truths" to be discovered. I have thought of this as largely a reasonable way to speak about it for three reasons.
1) It is a very useful shorthand.
2) In a sense it is true no matter which view of the subject the speaker takes.
3) The culture of mathematicians has formed with this mode of speech built into it, so to participate in the mathematical culture one must first adopt some of its basic mannerisms.
All of these reasons are perfectly valid for a materialist and do not at all rely on the objects of study existing in some actual realm of Platonic ideal forms. In particular my own view is that mathematics is a description of the subjective experience of precise comprehension shared by all human(-like) minds (roughly speaking - it's an idea I find difficult to put into words).
However when Kurt Gödel proved his incompleteness theorems the philosophical question of what mathematics is actually describing (if anything) was dropped squarely into the core of the foundations of math, of which set theory is an important part. To describe why this philosophical question suddenly became significant in philosophy of math would likely be redundant to mathematicians (and philosophers of math and logicians) and merely confusing to everyone else, so I will forego a review of the history of this area here.

Now while it is arguable whether or not mathematics is a science at all (I would argue against that proposition) it is certainly a discipline with immense practical utility. I hope I don't need to say anything here to justify that statement, so I will continue without doing so. Set theory is important within the practice of mathematics, in part because I have yet to encounter a mathematical subdiscipline which does not occasionally make use of some of its more basic results. Suffice it to say that if set theory were to crumble it would seriously threaten to take most if not all of mathematics with it. Set theory, I have now heard, is noticeably more complicated and nuanced when working without the assumption of Platonic reality. Obviously (to me, anyway) this assumption, being a philosophical rather than a mathematical one, does not affect the actual mathematical content of the results of set theory, but it may still affect their phrasing and thereby their apparent applicability outside of math itself. To me this is reminiscent of the row between constructive and classical logics; that ultimately the mathematical community decided that since results could be equivalently phrased in either logical system constructive logic, being somewhat unwieldy, was to be considered redundant, effectively granting victory to the side of classical logic. This conclusion of the fight between constructive and classical logics is another instance of the more elegant theory being considered closer to the truth in virtue of its elegance.

So it seems that some metascientific methodologies tend to be applied within philosophical questions of metamathematics at least, and the results can then be widely accepted as "true" in some real but non-mathematical sense, and their use throughout the scientific disciplines lends the philosophical assumptions underlying mathematics seemingly as much claim to "truth" in the external world as the physical sciences themselves can claim. Directly applying this approach to the assumption of Platonism in set theory (as it results in a more elegant set theory, it can be assumed to be true in a scientific sense), we reach the conclusion that a platonic realm of ideal forms (consisting at least of sets) provisionally likely exists as a part of objective reality. This obviously contradicts philosophical naturalism, but more interestingly (and confusingly) it also can be seen that the existence of this nonmaterial realm of Platonic ideal forms has been arrived at using scientific principles of decision-making, which also contradicts the idea that science is methodologically naturalistic.

So has methodological naturalism reached a decidedly non-naturalistic conclusion, even provisionally?

I suppose we add this to Lawrence Krauss' paradoxical conclusion that empirical rationalism tells us that at a future time empirical rationalism will correctly reach a false conclusion.