Van Fraassen on Peirce’s “Scholastic Realism”

Because I need the practice, this will be in the mould of those short summaries one writes about one’s course readings.

The arguments of the “scholastic realists” van Fraassen attacks in Laws and Symmetry can be broken into two parts: the first being that there exist laws of nature, the second being that we must believe there exist laws of nature (or else sink into an abyss of skepticism).

Van Fraassen quotes a lecture demonstration by C. S. Peirce:

Here is a stone. Now I place that stone where there will be no obstacle between it and the floor, and I will predict with confidence that as soon as I let go my hold upon the stone it will fall to the floor. I will prove that I can make a correct prediction by actual trial if you like. But I see by your faces that you all think it will be a very silly experiment.

This is supposed to demonstrate that there are some things that we know will happen without having to have that demonstrated before our eyes.

Peirce argues that the fact that we can believe that the stone will fall without doing the experiment is proof that the assumed ‘law’ that the stone will fall to the floor corresponds to reality. The idea is that either the fact that the stone will fall to the floor is a matter of chance — it could have failed to fall, but it just didn’t happen to have failed to fall that one time. Or, more plausibly, the fact that the stone will fall is dictated by a law of nature, which is what justifies us in believing that it will fall even before we see it do so. After all, if it were merely a matter of chance, we wouldn’t feel justified in believing it. Van Fraassen points out that this corresponds to the second part of the scholastic realists’ argument: that given our other beliefs, we must believe there exist laws of nature.

To recap Peirce’s argument: IF we are know that certain regularities in nature will occur without observing them to, THEN we must believe there exist laws of nature.

Van Fraassen argues that the dichotomy Peirce draws between events that happen due to ’sheer chance’ and events that happen due to a law of nature is a false one. What, he asks, does ‘by chance’ mean? In the most common interpretations of that phrase, it could mean ‘not due to any law’, or it could mean ‘no more probable than the other possibilities’.

If Peirce means to take the latter interpretation, then it is not true that we know that certain regularities in nature will occur. So the premise of Peirce’s argument is false already, and we can’t argue from that to the truth of its conclusion.

What if Peirce means to take the former interpretation, that ‘by chance’ means ‘not due to any law’? Van Fraassen simply says that that would be a strange use of the phrase ‘by chance’. (I’m not sure I agree with him on this.)

Van Fraassen then goes on to consider if Peirce had perhaps accepted the tacit premise that whatever happens either does so for a reason or else is no more likely to happen than its contraries. Van Fraassen rejects this premise because it would mean that if the universe contained no reasons for regularities, then it would have to be completely chaotic — there wouldn’t even be room for highly probable regularities. In fact, this premise is exactly the first part of the scholastic realist argument — not just that we must believe that laws of nature exist, but that there actually exist laws of nature. It is not clear, though, why we should accept the premise that events must either have a reason behind them or be instances of completely random outcomes.

Yet it is hard to deny the strong attraction of the Peircean intuition that laws of nature have a flavour of necessity to them that mere continuation of a regularity does not. As van Fraassen writes, “A law must be conceived as the reason which accounts for uniformity in nature, not the mere uniformity or regularity itself.” But how do we reconcile this intuitive notion of ‘law’ with our repeated inability (from Hume onwards) to prove that such reasons exist?

Frankly, I don’t have much of a problem with giving up the intuition that laws dictate necessity. It’s true that it’s really convenient, for scientists and even most ordinary people, to think of regularities like falling objects as due to natural laws. And when a mode of thinking becomes convenient enough, people start treating its objects as real existent things. In philosophical parlance, they start inventing an ontology to go with their mode of thinking, which may have started out as a metaphysically innocent heuristic. I tend to think, for example, that scientific realists have unwittingly bought into what started out as a heuristic. So I’m perfectly comfortable with the idea that the regularities we know of now are just there, free of metaphysical baggage. It may please scientists to think of them as caused by laws of nature, but the burden of proof is on them to show that they have to accept the ontology of laws of nature. Seems to me the language of laws of nature is near-indispensable in much of modern day science, but I think it’s quite possible to shift to a more metaphysically conservative language (although I don’t see a point in doing so). In other words, bugger our intuitions. They are often products of extended cultural marination that need not push our intuitions any closer to the truth.

Determinants and Oriented Volume

This was one of the more delightful aspects of linear algebra, when I first learned it properly. It could be because my initial introduction to linear algebra was purely computational, with no discussion of the geometric meaning of the procedures we learned. Sure, we learned how to calculate determinants, eigenvectors, eigenvalues, reduced row echelon forms, and such, but it was a mere drilling of procedures till we could do them blindfolded.

So here’s an ‘intuitive’ explanation of why the determinant of a matrix represents the ‘oriented volume’ of the parallelpiped spanned by the column vectors that constitute the matrix. The rigorous proofs can be found in any good linear algebra text — I am more interested in offering an explanation of why we might want to relate determinants to oriented volume.

The standard definition of the determinant, taught to most students of subjects where mathematics is used as a tool and not understood for its own sake, is as follows:

Here, is the entry of the matrix in the ith and 1st column, and is the matrix obtained by striking out the ith row and jth column of .

We can start with the simplest example: the identity matrix . The determinant of this matrix is 1, and the column vectors constituting it are , , and . The volume that these three vectors span is the cube with vertices at (0, 0, 0), (1, 1, 0), (1, 0, 1), (0, 1, 1), (1, 0, 0), (0, 1, 0), (0, 0, 1), and (1, 1, 1). It is, in short, the cube with a corner at the origin and unit lengths stretching along the x, y and z axes. This cube has side of length 1 each, so it has a volume of 1, the same value as the determinant.

Now consider a similar unit cube, but spanned instead by the vectors , , and . This cube has vertices (0, 0, 0), (1, 1, 0), (1, 0, -1), (0, 1, -1), (1, 0, 0), (0, 1, 0), (0, 0, -1), and (1, 1, -1). If we consider volume to have the standard property of being a positive number, then clearly the volume of this cube is 1, just like the previous cube. Oriented volume, though, as its name suggests, is a kind of volume that is dependent on the ‘directions’ possessed by the object with the volume in question. In this case, since we have fliipped one vector to its ‘negative’, the oriented volume is also ‘negated’, so although the cube has conventional volume, its oriented volume is -1. -1 also ‘happens’ to be the determinant of the matrix composed of , , and .

We can say the same for the matrices composed of , , and of , , . Both are something like the unit cube of , , and “flipped” (“negated”) once; both have an oriented volume of -1, and both have determinants of -1.

We can abstract this property of the determinant as such: when one of the column vectors of the matrix is negated, the entire determinant is negated. The absolute value of the determinant is unaltered. This can be interpreted as showing that when you ‘flip’ a vector about the origin to point in the ‘opposite’ direction, it together with its two other companions then ’span’ a parallelpiped ‘pointing’ in the opposite direction from the original but having the same absolute volume. The change in the ‘direction’ of the parallelpiped is manifested as a change in the sign of its oriented volume, just as the change in the sign of one of the matrix’s column vectors is manifested as a change in the sign of its determinant.

To summarise, we have seen that negating one of the column vectors of a matrix also negates our intuitive notion of ‘oriented volume’, and negates its determinant. This is just a special case of a property shared by both oriented volumes and determinants: multiplying a vector constituent of either by a scalar results in a multiplication of either (oriented volume or determinant) by a factor of .

Another common feature shared by both the oriented volume and the determinant is that both are invariant under the addition of a scalar multiple of a constituent vector to any of the other vectors. For example, if we transform the matrix to by adding to the original middle vector , the determinant remains the same: the extra “1″ in the x-direction is irrelevant somehow.

This irrelevance makes sense in the ‘oriented volume’ interpretation. There, the volume represented by the second matrix is simply the first volume, the unit cube along all the positive axes, with four faces ’skewed’ from squares to parallelograms so that it’s a parallelpiped with a 45^o skew in one direction but with two square faces in the ‘unskewed’ directions. However, since the y-value of the middle vector is unchanged at 1, the ‘component’ that matters for calculating parallelpiped volume is unchanged — this ’skewed’ parallelpiped still has volume 1, in the same way that a right angled triangle skewed to a scalene triangle of the same base length and height retains its original volume. Parallelograms are really just triangles with twice the volume, so for them as well, it doesn’t matter how we angle their sides as long as they retain the same ‘base’ and ‘height’. And parallelpipeds are just higher dimension analogues of parallelograms. So it should be intuitively plausible that we should expect their volume to remain the same under certain ’skew’ transformations in which the ‘components’ added to one of their ‘edges’ only result in them becoming ‘more skewed’ but essentially the same ’size’.