Monday, December 19, 2016
Of Foreign Lands and Maths
by Carl Pierer
In 1786, Goethe began his famous journey to Italy, of which he kept a diary to be published as The Italian Journey in 1816. Even though his main interest lies elsewhere, he finds time to write about Italians. Early on his journey, he writes, for example (Goether, 1982):
So far I have seen only two Italian cities and only spoken to a few persons, but already I know my Italians well. They are like courtiers and consider themselves the finest people in the world, an opinion which, thanks to certain excellent qualities which they undeniably possess, they can hold with impunity.
22. September 1786
The generalisation at hand is striking. Whilst admitting a very limited experience, Goethe feels in a position to talk about Italians, as a whole. Or does he?
Who are the Italians Goethe is talking about? It seems unlikely that he is talking about all of them, at all times. Yet even restricted to his contemporaries, it would be bold to assert that this is a necessary feature of being Italian. Such a reading most surely would miss the point. It seems more appropriate to suggest that he is talking about some sort of Mentalität, a commonality or stereotypical property of Italians. Even if this were so, it would be a confusion to suggest, anachronistically, that Goethe means by "Italians" the citizens of Italy. The "Italians", it seems, have a rather different, curious status.
In mathematics, it is most common to form sets by taking a property (or properties, but let us stick with a single property for now) and collecting all objects that satisfy this property. For example, when we talk about even numbers we mean all numbers that are an integer multiple of 2. It is fairly clear that this property is not, or at least not necessarily, a property of the set – what would it even mean for the set to be an integer multiple of 2? Yet, the property is not part of the set either, in the strict sense. "being an integer multiple of 2" is not an element of the set of even numbers. What happens if we talk about a general element? A general element, distinct from a specific element, captures the defining property and only the defining property of any element of the set. For even numbers, a general element is of the form 2k, where k is an integer. This is subtle: the most general element captures the defining property, but has the form of an element. It is neither a particular element of the set nor the property itself. No even number actually is 2k, but any even number is of this form. Even more confusingly, we know that 2k is an even number (it is obviously an integer multiple of 2), but if we look at the set of even numbers, we will not be able to find 2k. It is some integer, but we do not know which one. In fact, if we knew which one it was, it would become a particular element and cease to be general. Yet it is precisely this ambiguous structure that allows us to proceed in mathematical proofs. Suppose we want to show something about even numbers. A typical proof would get started by saying: "let x be an even number, then x = 2k for some integer k", or something along those lines. Obviously, it would be fallacious to begin by saying: "let x be an even number, so x = 4, say", and then deduce that the statement arrived at holds for all even numbers. The general element ensures, through capturing the defining property and only the defining property, that we are really talking about what we are interested in and do not get side-tracked by any accidental properties (as we would, if we said x = 4 = 22 and then talk about squares, for example).
So what is the role of k in the above example? We might want to say it is a variable in the integers. Variables have a certain notoriety[i]. This would explain k, but not quite the situation of the general element. We could treat it as a function going from the integers to the even integers. However, when we talk about the general element, we really do want to say something about the even integers, not the function. This suggest we should take the general element to be the image of the function. This won't do either, as the image of the function is just the entire set. So the general element is neither a variable, nor a function, nor the image of a function. Is it then a purely formal, if hugely important construction?
The construction will look even more suspicious if we turn our attention to ‘real world' examples. If we consider the set of books, for instance, the general element would have to capture all and only the defining properties of being a book. It appears that it could not possibly be any actual book in the world, as that would host some features nonessential to its being a book (the number of pages, for instance, or the material of the cover). Rather, it would need to be something that, if instantiated, would yield a book. Is the general element in mathematics then a platonic residue, some lingering commitment to platonic forms build into the very fabric of set theory? Without entering into this debate here, we see that these consideration lead to well-known, hotly debated questions in the philosophy of mathematics.
Outside of mathematics, we likewise feel tempted to assert statements about sets of things or people. "They [the Italians] are like courtiers (…)", or, more contemporary on the subject of the Trump victory: "People voted for him because they didn't believe him. They wanted change but they also had confidence in the basic durability and decency of America's political institutions to protect them from the worst effects of that change. They wanted Trump to shake up a system that they also expected to shield them from the recklessness of a man like Trump"[ii]. Criticising the statements by referring to the classical gap of inductive reasoning is trivial. We are not convinced of the inaccuracy of such an analysis if someone were to point out a Trump voter who didn't fit the bill. Conversely, it is not statistic verification that does convince us of the meaningfulness, the utility of the analysis. An appeal to the empirical, actually existing voter alone will not do to dispose of the analysis. It misses the point. Rather, it might prove more interesting to read these statements in a different way, and neither attempt to justify (or, for that matter, attack) them on merely empirical grounds. What if, instead, we tread the voter, the Frenchman, the Italian, as something akin to a general element? This, precisely, would fall prey to the statistician, for if the Trump voter in the above sentence is identified with the general element of the set of all Trump voters, any statement about them can be discredited by finding a single person who voted for Trump but does not fit the bill. More subtle, perhaps, would be to suggest that the ‘Trump voter' is akin to a general element in that they sit just as uncomfortably in between being a formalisation of a relation (‘voting for Trump') and actually being an element of the set (‘a person who voted for Trump'). An appeal to an actual voter, then, would appear like a person asking about 2k ‘Which even number is it now?' In this way, we might understand why reference to the empirical seems like a misunderstanding of the point of the sentence.
Of course, this only goes so far. As the discussion of the general element showed, it is structurally uncomfortable and its status hasn't been sorted out. The more difficult question, by far beyond the humble scope of this essay, is what the general element really is. In turn, this might illuminate who the Italians, the Trump voters really are.
[i] Karl Menger, for instance, writes: "But what is a variable? Russell, who investigated all aspects of this notion, called it ‘certainly ... one of the most difficult to understand' and in 1903 stated that a satisfactory theory of variables was not to be found in his Principles of Mathematics. In fifty years this situation has not been improved." (Menger, 1954)
[ii] See the intriguing analysis provided in (Runciman, 2016)
Goether, J. (1982). Italian Journey (1786-1788). (W. Auden, & E. Mayer, Trans.) San Francisco: North Point Press.
Menger, K. (1954). On Variables in Mathematics and in Natural Sciences. The British Journal for the Philosophy of Science, 134-142.
Runciman, D. (2016, December 1). Is this how democracy ends? London Review of Books, pp. 5-6.
Posted by Carl Pierer at 12:20 AM | Permalink