The most natural way of understanding the role of mathematics in science is as a representational one. When we say that ‘water boils at 100 Celsius’ we do not tend to think that the number 100, in itself, explains the boiling. Rather, number 100 *represents *the physical processes that explain it. This can clearly be seen if we change the units to Fahrenheit. In that case, it would be 212, and not 100, the number that would pick out the relevant physical processes. What is more, *in principle* we can express these numbers in first order logic, thus removing any trace of mathematics. As Joseph Melia puts it:

“[M]athematics is the necessary scaffolding upon which the bridge [of science] must be built. But once the bridge has been built, the scaffolding can be removed”

In recent years, some philosophers have claimed that mathematics is capable of doing more than this; that mathematics can in fact explain the occurrence of a physical phenomenon. In fact, many of these philosophers argue that this is a reason to be a mathematical realist (a believer in the existence of mathematical objects). One of the most prominent authors defending this view is Mark Colyvan. For Colyvan, mathematics plays a genuinely explanatory role when the mathematical component in a scientific explanation is indispensable to the success of the explanation. If a case is found where mathematics plays such a role, we would have a genuinely mathematical explanation of a physical phenomenon (MEPP). One of the alleged cases of MEPPs proposed by Colyvan is the honeycomb case.

Honeycombs divide the space into hexagonal shapes instead of other possible shapes like triangles, squares, etc. The explanation of this comes in two parts:

1) It is evolutionarily advantageous for bees to minimize the amount of wax they use in building their honeycombs.

2) Following the honeycomb theorem (known as the ‘honeycomb conjecture’ before it was proved by Thomas Hales in 2001), the best way of dividing a surface into regions of equal area while minimizing the total perimeter is by a hexagonal grid; and this is why bees build their combs in hexagonal shapes.

This evolutionary explanation depends on a mathematical fact (the honeycomb theorem). It supposedly shows why, no matter which shapes the bees tried throughout their evolutionary history, once they tried hexagons this trait passed on, because hexagons are evolutionarily advantageous for any bee that builds combs.

For Colyvan, the upshot is that, unlike the temperature case, in this one mathematics itself would be playing an explanatory role, in the sense that, if you remove the mathematics the explanation would fall apart. And if you were a scientific realist who justifies your realism by the principle of inference to the best explanation, then you would have to accept mathematical realism (unless you feel comfortable engaging in double-think).

Now, despite the widespread attention devoted to this case, it has recently been discovered that the honeycomb theorem does not explain the hexagonal shape. An explanation that does not appeal to the honeycomb theorem has been recently advanced by engineer Bhushan Karihaloo, and reported by Philip Ball in a recent volume of *Nature *(2013).

According to Karihaloo, the hexagonal shape is due to some properties of wax, as well as the particular procedure bees follow for building their combs. In an experiment, Karihalloo interrupted the bees in the process of making their combs and found that, whereas the older cells were hexagonal, the newer cells were circular. He concluded that bees simply make circular cells packed together like a layer of bubbles. Because the bees heat the wax with their bodies as they build the circular cells, once they finish and move to another cell the wax starts to harden into hexagons (that hot wax retracts into hexagons as it cools and hardens had already been proven in 2004). Thus, as Ball reports, the hexagonal array of honeycombs “owes more to simple physical forces than to the skill of bees”. This explanation sketches the causal history of the explanandum, showing that the hexagonal shape of the combs was due to specific causal processes undergone by the wax. The explanation suggests that the hexagonal honeycombs occur, not because they are evolutionarily advantageous, but because of the relevant forces produced by the interaction between wax and heat.

Karihaloo’s account undermines the explanation that appeals to the honeycomb theorem. Since the bees do not build hexagonal combs (they build circular combs that later become hexagons), the explanation that appeals to the honeycomb theorem relies on a false assumption. Therefore, the purported MEPP is not an explanation (let alone the *best *explanation) of the hexagonal shape. This example should no longer be used as a MEPP.

***

This case has taught me many lessons about the philosophy of science. I want to share two. Just like other branches of philosophy, philosophy of science also deals with perennial questions (existence, knowledge, duty, etc.), but our subject matter is constantly moving. In 2012, Colyvan had a good case (albeit not conclusive) to the effect that there are MEPPs in science. Some philosophers not convinced by other alleged cases were in fact on board on this one. But a new scientific study shows up and the example pretty much becomes useless. This of course doesn’t only affect philosophy of science (think, for example, of the ethics of technology), but I believe that it does affect us more prominently.

Another lesson I learned from the honeycomb case is that this fluidity in the philosophy of science sometimes requires you to act quickly. By 2014, I was already aware of both Colyvan’s views and the Karihaloo experiment, but at that time I was very busy with grad school and did not have time to publish my criticisms to Colyvan. But a few years later, when I did have the time, I found a 2017 paper making the *exact *point I was going to make…

Hi Manuel, very interesting piece. I have two questions.

ReplyDelete1. You seem to suggest that the mathematical explanation provided by Colyvan is not an explanation at all, and that strikes me as incorrect. Based on what you say, I would only draw the conclusion that it is not the best available one.

2. This is not your problem, but I really do not understand why Colyvan's argument is interesting. It seems to me that there are countless explanations that depend critically on mathematics in the way his does. Randy and Manuel began at the same point and walked to the Philosophy office at exactly the same speed. Yet Manuel arrived long before Randy. Why? Because Manuel walked in a straight line and Randy did not. This explanation relies on the mathematical fact that in Euclidean space the shortest distance between two points is a straight line.

If I am right that there are lots of such explanations, then it seems to me that his point still stands, but it just does not seem like a very interesting point to me. In my example, it is not the mathematics alone that explains why Manuel arrived first. It explains it, given a huge set of physical background conditions. I could just as easily make the mathematics a background condition and supply a purely physical explanation for it.

Maybe I am not understanding the significance of the idea that the mathematics can be dispensed with once the explanation has been discovered. That strikes me as saying that once an explanatory framework like evolution has been developed, people who do not know the mathematics of population genetics are still in a position to appreciate why specific species have specific traits.

Hi Randy,

ReplyDeleteThanks for your comments.

1) I am happy to say that a bad explanation is not really an explanation because it doesn’t explain the explanandum. I take explanation to be something like a success term. This makes sense from an ontic perspective. If an explanation ‘must cite the objective relationships of dependence responsible for the occurrence of the explanandum-event’, then any alleged explanation that cites irrelevant structures is not going to explain the explanandum. The astrologer that cites the position of the planets to explain my behavior is not really explaining it. She hasn’t provided an explanation of my behavior.

2) I don’t agree with Colyvan’s mathematical realism either, but I do think that his argument is interesting. Colyvan is replying to Penelope Maddy’s objection against Quine’s version of the indispensability argument (IA). She says that confirmational holism can’t be right because oftentimes scientists use mathematics to describe idealized versions of physical reality (frictionless planes, infinitely deep oceans, etc.); but idealization seems to be indispensable to scientific practice; therefore, scientists are not committed to “all” the entities indispensable to science. Colyvan’s reply is that idealizations are not really indispensable because, in principle, you can get rid of them (you can de-idealize your representation of the slope as frictionless and have a more realistic representation once your computational powers increase). So he’s trying to provide cases where the mathematics cannot be dispensed with. He’s also reacting against Hartry Field’s notion that all science can be nominalized. For Colyvan, genuine mathematical explanations provide great support for mathematical realism because 1. You don’t have to use confirmational holism anymore (you use IBE). 2. (Supposedly) mathematics can’t be nominalized away.

2*) I think that if you put mathematics as a background condition and use the resultant explanation to support the IA, then the argument would become circular, because you would be assuming the truth of mathematics in order to prove the existence of mathematical objects.

2**) If Euclidian space provides a good model of physical space (under a certain level of description), you can perfectly say that the mathematical fact that the shortest distance between two points is a straight line would provide good insight into the properties of physical space. But the role of mathematics would still be representational, not explanatory, so that example wouldn’t count as a genuine MEPP. [Now, as you know, I do believe that mathematics ALWAYS plays this representational role and that the IA, even in its explanatory version, is doomed to fail, but what is interesting is the realist quest to come up with explanations where mathematics is not simply representational. The honeycomb case was a good candidate.]

2***) The point is not that mathematics is dispensed with once you discover the real explanation. Mathematical realists argue that MEPP are the real explanations of the phenomena they describe, and this includes evolutionary phenomena. For example, one of the most discussed candidates of MEPP is the explanation of the life cycles of some periodical cicadas that emerge every 13 or 17 years. Biologists appeal to both evolutionary facts and facts about prime numbers to explain why the cycles are evolutionarily advantageous (they minimize the probabilities of intersecting with predators or cicadas of different life cycles). In this particular case, it is a bit more difficult for the nominalist to show why the mathematical property primeness is not explanatory. [I think I’ve managed to do it in one of my papers, but I don’t think I’ve convinced too many people].

Thanks Manuel.

ReplyDeleteMaybe I just do not understand the difference between mathematics playing an explanatory role and a merely representational role. My example seems to me to preserve what Colyvan claims to achieve with his. "The best way of dividing a surface into regions of equal area while minimizing the total perimeter is a hexagonal grid" and "the best way of connecting two points so as to minimize the total distance is a straight line" strike me as explanatorily isomorphic. Both are mathematical facts about Euclidean space and they seem to me to play the same explanatory role. If one is merely representational, the other one seems to me to be as well.

I wonder if you may be conflating false with irrelevant in your defense of saying that Colyvan has not provided an explanation at all. When people provide irrelevant explanans, we ordinarily are able to detect (or at least reasonably suspect) this a priori. For example, if your wife asked you why you are late for dinner and you reply that it is because no two snowflakes look alike, she would be justified in saying that is not an explanation as the dissimilitude of snowflakes seems irrelevant to the explanandum.

But what you teach us here is that Colyvan turns out to have made a false claim. The false claim is that bees are evolutionarily programmed to build hexagonal cells. At least on the basis of what you say here, if bees were programmed to build hexagonal cells, then we would still be taking the explanation seriously. Hence, I would say that this is definitely an explanation, and quite an interesting one, just one that turns out to be based on a false assertion. I think this way of speaking is consistent with most theories of explanation I am acquainted with, beginning with Hempel.