Lessons from Feyerabend

A standard account of the Copernican revolution might run like this: as the body of celestial observations grew (aided by the invention of the telescope in the early 17th century) anomalies were recorded which revealed inadequacies in the Ptolemaic system, motivating astronomers to consider rival theories. Among these was the heliocentric system of Copernicus, which ultimately turned out to be the best (it explained the data in the simplest, most elegant way, adopted the least auxiliary hypotheses, etc.). Despite the efforts of the Church it gradually came to be accepted. According to this story observation and theory are independent of each other — observation is the neutral party which can topple old theories and adjudicate between new ones.

This presumed independence of theory and observation is one of the main targets in Paul Feyerabend’s Against Method [1]. Feyerabend illustrates his point by considering the tower argument against the motion of the earth, which was for a time one of the main sources of resistance to the heliocentric model of the solar system. He quotes Galileo [2]:

[H]eavy bodies. . . falling down from on high, go by a straight and vertical line to the surface of the earth. This is considered an irrefutable argument for the earth being motionless. For, if it made a diurnal rotation, a tower from whose top a rock was let fall, being carried by the whirling of the earth, would travel many hundreds of yards to the east in the time the rock would consume in its fall, and the rock ought to strike the earth that distance away from the base of the tower.

Needless to say, this argument eventually lost its purchase. Nowadays we understand that the observation that rocks dropped from towers fall perpendicular to the earth is not evidence that the earth is stationary. The force of the reasoning derived from something we have lost [3]:

[Galileo] tells us that the everyday thinking of the time assumes the ‘operative’ character of all motion, or, to use well-known philosophical terms, it assumes a naive realism with respect to motion: except for occasional and unavoidable illusions, apparent motion is identical with real (absolute) motion.

Without absolute motion the tower argument makes little sense, and a crucial component of the Copernican revolution was the adoption by astronomers of Galilean relativity. As the standard story tells it this shift in attitudes toward motion occurred at the level of theory, while the observations — e.g. that falling rocks fall straight to earth — remained stable. Feyerabend disputes this. Naive realism about motion cannot be considered part of the old Ptolemaic theory; one reason is that it is not required, but the more important is that it couldn’t even be stated until Galileo had provided an alternative. Naive realism about motion was not part of the explicit theory, it was an implicit natural interpretation of the observations [4].

Natural interpretations, unlike explicit hypotheses or postulates, cannot be said to be distinct from the data. On the contrary: they permeate the observation language itself. To put it another way, they are part of what the terms used to state and record observations mean to the people using them. According to Feyerabend, it is not correct to say that in championing Copernicus Galileo was offering a new theory that did a better job of accounting for the data. Quite the opposite: it was inconsistent with the data as then understood. As he writes [5]:

[A] theory may clash with the evidence not because it is not correct, but because the evidence is contaminated.

Galileo did not just argue for a new theory, he provided a new observation language. From this Feyerabend concludes that scientists must from time to time proceed by what he calls ‘counterinduction’, that is by actively developing theories which contradict the current body of observation. By doing so they can unearth (and perhaps then discard) the natural interpretations it contains.

While others have made similar points about holism and the theory-ladenness of observation, Feyerabend is more suspicious than most of overly prescriptive views of scientific method. Taking all this a step further, we can note that his idea that new theories provide not just new explanations of the data, but in effect change the very data to be explained is one that applies to rational discourse in general.

Take materialism and idealism. Idealists have on occasion been accused of denying the existence of their own hands [6]. This criticism likely rings hollow for them — while the idealist does deny that the hand exists in any mind-independent sense, they need not deny the existence of the hand as such, because on their view the things they refer to as their hands were just perceptual entities all along. In virtue of their idealism they mean something different by ‘my hand’.

Similarly, materialism often gets criticised on grounds like “it is the nonsensical idea that mindless physical stuff can crash together to create minds”. The materialist will be unmoved by this, because what is materialism if not the thesis that physical systems can indeed be mindful? As with the above comment on idealism it is at best a cynical rephrasing of the position, and at worst a question-begging criticism of it.

The pattern crops up all over the place, particularly in relation to the big questions of morality, truth, and the existence of God. Beliefs on these topics have deep semantic ripples, and the opportunities for those with differing views to talk past one another are manifold. Feyerabend’s philosophy of science can help us see that this needn’t be because of malice or disinterest on either side, but may simply come down to the fact that charitable interpretation is really hard — it requires more than paying attention to someone’s definitions, it requires stepping into their entire paradigm. The general point can be put somewhat crudely: it is not just the case that the beliefs we hold depend on what we mean by the terms that compose them, but also that what we mean by those terms will depend on (and change with) the beliefs we hold that involve them.

Notes:

1. Paul Feyerabend, Against Method (4th edition). First published in 1975. (This really was a pleasure to read.)

2. Galileo Galilei, Trattato della sfera, quoted AM p50.

3. AM p54.

4. ‘natural interpretation’ is a phrase Feyerabend borrows from Francis Bacon. See AM p55.

5. AM p15.

6. This argument sometimes gets accredited to GE Moore. Presumably this comes from his essay A Defence of Common Sense, though in it he seems to be making a subtler point.

Some thoughts on mathematics

Any successful philosophy of mathematics will have to face up to these two facts:

1. Mathematical truths are, or at least seem to be, necessary truths.
2. Mathematics is immensely useful in physics and other sciences.

At first glance these appear to be in conflict. One way we might try to account for necessity is by holding that mathematical truths are analytic truths, i.e. that their truth is derived from the meaning of the terms that occur in them, and nothing else. The problem here is that if mathematics is analytic, its utility becomes mysterious. Analytic truths are by definition not about the world, they are about the relationships between meanings.

In the opposite direction we face a similar dilemma. Because mathematics is so helpful in our theories of the world, we would expect it to be in some sense about the world. But the world is a shifty place, full of contingent structure and beings that might not have been. If mathematics describes a contingent world, how can its truths be necessary?

Platonists try to resolve this tension by positing that there is some ideal, non-contingent portion of reality which mathematics describes. If true this deals with necessity, and then utility is handled by saying that the physical world is somehow derivative from or dependent on this ideal world. This approach raises many questions, arguably more than it answers. 

But perhaps more tellingly, Platonism offers accounts of things it doesn’t need to. In positing that an ideal portion of the world accessible via mathematics is necessary and that the physical world is dependent on it, Platonism implies that the mathematics which is useful in this world will be useful in all other possible worlds. In other words, it provides an account not just of the necessity and utility of mathematics, but also of the necessity of its utility.

But this necessity needn’t be accounted for, because while a truth being necessary and useful does imply that it couldn’t have been false, it does not imply that it couldn’t have been useless. This shows us that Platonism is stronger than required, suggesting that a weaker theory might be able to do the same explanatory work with less of the metaphysical baggage. Here’s a thought about how this might work.

If we stick to our guns and maintain that mathematical truths are indeed analytic truths (contra Platonism), we’re left with troublesome questions of utility. We might contend that theorems of group theory, say, just follow from the definitions of a group, and are necessarily true in this stipulative, almost trivial sense. But then why is group theory so handy for doing quantum mechanics? Perhaps we can say that it’s useful because some bits of the physical world exhibit group-like structure. Ah ha! says the Platonist, but the physical world is contingent, so it might not have exhibited group-like structure. How can you square that with the necessity of group theoretic theorems?

But at this point we might wonder what exactly the problem is. We have two things: a theory which consists of some definitions plus theorems derived from them – i.e. a system of analytically true propositions – and a mapping between the theory and a portion of the physical world. But there’s no reason to suppose the mapping between theory and world is anything but contingent. In another world group theory might be irrelevant to physics (or to any other descriptive aspect of it). But this doesn’t mean that group theory might have been false, it just means that group theory might not have been useful. Mathematics that is true and useful in one world might be true and useless in a world with different physics. 

What I’m advocating here is something like a mix between mathematical formalism and mathematical nominalism. On their own these theories face objections – nominalism can’t cope with necessity, and formalism seems at a loss when it comes to utility. But once we understand these questions as decoupled (that coupling them was a mistake in the first place) we can see that there’s scope for tethering the good ideas in formalism to the good ideas in nominalism. This hybridisation seems to me to provide a framework in which an account of both the necessity and utility of mathematics can be approached, and without taking on ad hoc metaphysical commitments.

A revision (09/06/14):

I’ve realised that I was being unfair to Platonism when I said that it implies that the utility of mathematics is necessary utility. I didn’t offer any justification for this because it seemed obvious to me, but having received a question about it which I didn’t have an answer for it now seems likely to me that, like many things which seem obvious at first, it is false. I had in mind the notion that were Platonism true, the abstract entities referred to by mathematical terms would be the same in any given possible world, and then that since the physical world is derived from them the physics of each would appeal to the same mathematics. Unfortunately that looks like a non sequitur. I suppose the Platonic world could contain all the abstract objects referred to in (any) mathematical statements, with the utility of each object (or class of objects) being contingent on the physical world. (I’m not sure whether a Platonist would want to argue in that manner, but perhaps it’s an option open to them.)

Hopefully this doesn’t obscure the main point of the post, which is that despite appearances questions about the necessity and utility of mathematics don’t cause problems for each other, and that their decoupling can help us gain a sense of what a successful philosophy of mathematics might look like. I still think that a formalism/nominalism combo can do this better than Platonism, though in light of this revision I can’t claim the reasons I’ve given here are especially good ones. Still, should any errant Platonist wander this way I’d welcome any comments about how the utility of mathematics might be worked out in that sort of framework, and whether what is mathematically useful in describing our world might not have been.