Michael Murray and Jan Cover (Purdue) Take on Evil

My colleague Jan Cover appears in the latest edition of Percontations, a Bloggingheads series which has in the past tackled other philosophical topics like the nature of time. This time the nature of evil is discussed, with special reference to God and Leibniz.

The Honeycomb Conjecture (Cont.)

Following up my earlier post, and in line with Kenny’s perceptive comment, I wanted to raise two sorts of objections to the explanatory power of the Honeycomb Conjecture. I call them the problem of weaker alternatives and the bad company problem (in line with similar objections to neo-Fregeanism).

(i) Weaker alternatives: When a mathematical result is used to explain, there will often be a weaker mathematical result that seems to explain just as well. Often this weaker result will only contribute to the explanation if the non-mathematical assumptions are adjusted as well, but it is hard to know what is wrong with this. If this weaker alternative can be articulated, then it complicates the claim that a given mathematical explanation is the best explanation.

This is not just a vague possibility for the Honeycomb Conjecture case. As Hales relates

It was known to the Pythagoreans that only three regular polygons tile the plane: the triangle, the square, and the hexagon. Pappus states that if the same quantity of material is used for the constructions of these figures, it is the hexagon that will be able to hold more honey (Hales 2000, 448).

This suggests the following explanation of the hexagonal structure of the honeycomb:

(1) Biological constraints require that the bees tile their honeycomb with regular polygons without leaving gaps so that a given area is covered using the least perimeter.

(2) Pappus’ theorem: Any partition of the plane into regions of equal area using regular polygons has perimeter at least that of the regular hexagonal honeycomb tiling.

This theorem is much easier to prove and was known for a long time.

If this is a genuine problem, then it suggests an even weaker alternative which arguably deprives the explanation of its mathematical content:

(1) Biological constraints require that the bees tile their honeycomb with regular polygons without leaving gaps so that a given area is covered using the least perimeter.

(2’) Any honeycomb built using regular polygons has perimeter at least that of the regular hexagonal honeycomb tiling.

We could imagine supporting this claim using experiments with bees and careful measurements.

(ii) Bad company: If we accept the explanatory power of the Honeycomb Conjecture despite our uncertainty about its truth, then we should also accept the following explanation of the three-dimensional structure of the honeycomb. The honeycomb is built on the two-dimensional hexagonal pattern by placing the polyhedron given on the left of the picture both above and below the hexagon. The resulting polyhedron is called a rhombic dodecahedron.



So it seems like we can explain this by a parallel argument to the explanation of the two-dimensional case:

(1*) Biological constraints require that the bees build their honeycomb with polyhedra without leaving gaps so that a given volume is covered using the least surface area.

(2*) Claim: Any partition of a three-dimensional volume into regions of equal volume using polyhedra has surface area at least that of the rhombic dodecahedron pattern.

The problem is that claim (2*) is false. Hales points out that Toth showed that the figure on the right above is a counterexample, although “The most economical form has never been determined” (Hales 2000, 447).

This poses a serious problem to anyone who thinks that the explanatory power of the Honeycomb Conjecture is evidence for its truth. For in the closely analogous three-dimensional case, (2*) plays the same role, and yet is false.

My tentative conclusion is that both problems show that the bar should be set quite high before we either accept the explanatory power of a particular mathematical theorem or take this explanatory power to be evidence for its mathematical truth.

Schupbach Crushes Pincock!

Over at Choice and Inference, Jonah Schupbach has initiated a discussion of my PSA 2008 paper on mathematics, science and confirmation theory. Readers of this blog may be interested in how it is going ...

What Follows From the Explanatory Power of the Honeycomb Conjecture?

Following up the intense discussion of an earlier post on Colyvan and mathematical explanation, I would like to discuss in more detail another example that has cropped up in two recent papers (Lyon and Colyvan 2008, Baker 2009). This is the Honeycomb Conjecture:

Any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling (Hales 2000, 449).

The tiling in question is just (Hales 2001, 1)



The Honeycomb Conjecture can be used to explain the way in which bees construct the honeycombs that they use to store honey. The basic idea of this explanation is that the bees which waste the minimum amount of material on the perimeters of the cells which cover a maximum surface area will be favored by natural selection. As Lyon and Colyvan put it:

Start with the question of why hive-bee honeycomb has a hexagonal structure. What needs explaining here is why the honeycomb is always divided up into hexagons and not some other polygon (such as triangles or squares), or any combination of different (concave or convex) polygons. Biologists assume that hivebees minimise the amount of wax they use to build their combs, since there is an evolutionary advantage in doing so. ... the biological part of the explanation is that those bees which minimise the amount of wax they use to build their combs tend to be selected over bees that waste energy by building combs with excessive amounts of wax. The mathematical part of the explanation then comes from what is known as the honeycomb conjecture: a hexagonal grid represents the best way to divide a surface into regions of equal area with the least total perimeter. … So the honeycomb conjecture (now the honeycomb theorem), coupled with the evolutionary part of the explanation, explains why the hive-bee divides the honeycomb up into hexagons rather than some other shape, and it is arguably our best explanation for this phenomenon (Lyon and Colyvan 2008, 228-229).

Lyon and Colyvan do not offer an account of how this conjecture explains, but we can see its explanatory power as deriving from its ability to link the biological goal of minimizing the use of wax with the mathematical feature of tiling a given surface area. It is thus very similar to Baker's periodic cicada case where the biological goal of minimizing encounters with predators and competing species is linked to the mathematical feature of being prime.

Baker uses the example to undermine Steiner’s account of mathematical explanation. For Steiner, a mathematical explanation of a physical phenomenon must become a mathematical explanation of a mathematical theorem when the physical interpretation is removed. But Baker notes that the Honeycome Conjecture wasn’t proven until 1999 and this failed to undermine the explanation of the structure of the bees’ hive (Baker 2009, 14).

So far, so good. But there are two interpretations of this case, only one of which fits with the use of this case in the service of an explanatory indispensability argument for mathematical platonism.

Scenario A: the biologists believe that the Honeycomb Conjecture is true and this is why it can appear as part of a biological explanation.

Scenario B: the biologists are uncertain if the Honeycomb Conjecture is true, but they nevertheless deploy it as part of a biological explanation.

It seems to me that advocates of explanatory indispensability arguments must settle on Scenario B. To see why, suppose that Scenario A is true. Then the truth of the Conjecture is presupposed when we give the explanation, and so the explanation cannot give us a reason to believe that the Conjecture is true. A related point concerns the evidence that the existence of the explanation is supposed to confer on the Conjecture according to Scenario B. Does anybody really think that the place of this conjecture in this explanation gave biologists or mathematicians a new reason to believe that the Conjecture is true? The worry seems even more pressing if we put the issue in terms of the existence of entities: who would conclude from the existence of this explanation that hexagons exist?

Hales, T. C. (2000). "Cannonballs and Honeycombs." Notices Amer. Math. Soc. 47: 440-449.

Hales, T. C. (2001). "The Honeycomb Conjecture." Disc. Comp. Geom. 25: 1-22.

Two New Drafts: Surveys on "Philosophy of Mathematics" and "The Applicability of Mathematics"

I have posted preliminary drafts of two survey articles that are hopefully of interest to readers of this blog. The first is for the Continuum Companion to the Philosophy of Science, edited by French and Saatsi, on "Philosophy of Mathematics":

In this introductory survey I aim to equip the interested philosopher of science with a roadmap that can guide her through the often intimidating terrain of contemporary philosophy of mathematics. I hope that such a survey will make clear how fruitful a more sustained interaction between philosophy of science and philosophy of mathematics could be.

The second is for the Internet Encyclopedia of Philosophy on "The Applicability of Mathematics":

In section 1 I consider one version of the problem of applicability tied to what is often called "Frege's Constraint". This is the view that an adequate account of a mathematical domain must explain the applicability of this domain outside of mathematics. Then, in section 2, I turn to the role of mathematics in the formulation and discovery of new theories. This leaves out several different potential contributions that mathematics might make to science such as unification, explanation and confirmation. These are discussed in section 3 where I suggest that a piecemeal approach to understanding the applicability of mathematics is the most promising strategy for philosophers to pursue.

In line with the aims of the IEP, my article is more introductory, but hopefully points students to the best current literature.

Both surveys are of course somewhat selective, but comments and suggestions are more than welcome!

Leitgeb Offers an "Untimely Review" of the Aufbau

Topoi has a fun series of "untimely reviews" of classic works in philosophy commissioned with the following aim: "We take a classic of philosophy and ask an outstanding scholar in the same field to review it as if it had just been published. This implies that the classical work must be contrasted with both past and current literature and must be framed in the wider cultural context of the present day."

Hannes Leitgeb has carried this off with great panache with the Aufbau. The opening paragraph sets the tone:

Philosophy is facing a serious crisis, but no one cares. When
German Idealism, Existentialism, and Marxism allied with
Sociology, Psychoanalysis, Cultural History, and Literature
Studies in the early 20th century, all attempts at conducting
philosophy in a style similar to that of the scientists got
expelled from the High Church of Philosophy. The creation
of the Wykeham Professorship in Hermeneutics (formerly:
Logic) at Oxford and the Stanford Chair of Textual Non-
Presence (formerly: Methodology of Science) are wellknown
indicators of these, by now, historical developments.
The best philosophical work since then is to be found in the
history of philosophy—if one is lucky. One cannot help but
wondering what turn philosophy would have taken if
someone had picked up the revolutionary developments in
logic and mathematics in the 1920s and directed them
towards philosophy. Maybe there would still be logic
courses in philosophy departments? Who knows?

Here's hoping that some more classics of analytic philosophy get similar treatments soon!

El Niño Has Arrived. But What is El Niño?

According to Nature the lastest El Niño has begun in the Pacific. I got interested in this meteorological phenomenon back when I was living in California and coincidentally read Mike Davis' polemic Late Victorian Holocausts: El Niño Famines and the Making of the Third World . While a bit over the top, it contains a great section on the history of large-scale meteorology including the discovery of El Niño. As I discuss in this article, El Niño is a multi-year cyclical phenomenon over the Pacific that affects sea-surface temperature and pressure from India to Argentina. What I think is so interesting about it from a philosophy of science perspective is that scientists can predict its evolution once a given cycle has formed, but a detailed causal understanding of what triggers a cycle or what ends it remains a subject of intense debate. See, for example, this page for an introduction to ths science and here for a 2002 article by Kessler which asks if El Niño is even a cycle. This case provides yet one more case where causal ignorance is overcome by sophisticated science and mathematics.

Scientists Wonder If Philosophy Makes You a Better Scientist

Over at Cosmic Variance Sean Carroll has initiated an ongoing discussion of the following passage from Feyerabend:

The withdrawal of philosophy into a “professional” shell of its own has had disastrous consequences. The younger generation of physicists, the Feynmans, the Schwingers, etc., may be very bright; they may be more intelligent than their predecessors, than Bohr, Einstein, Schrodinger, Boltzmann, Mach and so on. But they are uncivilized savages, they lack in philosophical depth — and this is the fault of the very same idea of professionalism which you are now defending.

With some hesitation Carroll concludes that "I tend to think that knowing something about philosophy — or for that matter literature or music or history — will make someone a more interesting person, but not necessarily a better physicist." (See comment 56 by Lee Smolin and comment 64 by Craig Callender for some useful replies.)

Beyond that debate, it's worth wondering how knowing some science and mathematics helps the philosopher of science and mathematics. Pretty much everyone in these areas of philosophy would agree that it does help, but exactly how is probably a controversial issue.

Colyvan Blocks the "Easy Road" to Nominalism

In a paper posted on his webpage listed as forthcoming in Mind, Mark Colyvan launches a new offensive against fictionalists like Azzouni, Melia and Yablo. They present a non-platonist interpetation of the language of mathematics and science that, they argue, does not require the "hard road" that Field took. Recall that Field tried to present non-mathematical versions of our best scientific theories. As Colyvan describes the current situation, though, "There are substantial technical obstacles facing Field's project and thse obstacles have prompted some to explore other, easier options" (p. 2). Colyvan goes on to argue that, in fact, these fictionalists do require the success of Field's project if their interpretations are to be successful.

I like this conclusion a lot, and it is actually superficially similar to what I argued for in my 2007 paper "A Role for Mathematics in the Physical Sciences". But what I argued is that Field's project is needed to specify a determinate content to mixed mathematical statements (p. 269). Colyvan takes a different and perhaps more promising route. He argues that without Field's project in hand, the fictionalist is unable to convincingly argue that apparent reference to mathematical entities is ontologically innocent. This is especially difficult given the prima facie role of mathematics in scientific explanation:

The response [by Melia] under consideration depends on mathematics playing no explanatory role in science, for it is hard to see how non-existent entities can legitimately enter into explanations (p. 11, see also p. 14 for Yablo).

I have noted this explanatory turn in debates about indispensability before, but here we see Colyvan moving things forward in a new and interesting direction. Still, I continue to worry that we need a better positive proposal for the source of the explanatory contributions from mathematics, especially if it is to bear the weight of defending platonism.

Mancosu on Mathematical Style

Paolo Mancosu continues his innovative work in the philosophy of mathematics with a thought-provoking survey article on Mathematical Style for the Stanford Encyclopedia of Philosophy. From the introductory paragraph:

The essay begins with a taxonomy of the major contexts in which the notion of ‘style’ in mathematics has been appealed to since the early twentieth century. These include the use of the notion of style in comparative cultural histories of mathematics, in characterizing national styles, and in describing mathematical practice. These developments are then related to the more familiar treatment of style in history and philosophy of the natural sciences where one distinguishes ‘local’ and ‘methodological’ styles. It is argued that the natural locus of ‘style’ in mathematics falls between the ‘local’ and the ‘methodological’ styles described by historians and philosophers of science. Finally, the last part of the essay reviews some of the major accounts of style in mathematics, due to Hacking and Granger, and probes their epistemological and ontological implications.

As Mancosu says later in the article "this entry is the first attempt to encompass in a single paper the multifarious contributions to this topic". So it is wide-open for further philosophical investigation!