January 18, 2002
Wittgenstein on Mathematical Necessity and Convention
Pablo Kalmanovitz
Wittgenstein Workshop
University of Chicago
The difficult thing here is not, to dig down to the ground;
no, it is to recognize the ground that lies before us as the ground.
Our disease is one of wanting to explain. (RFM, VI, 31)
The present paper advances some preliminary steps towards a better understanding of Wittgenstein’s ideas on mathematical necessity. Mathematical necessity is one of the main themes in Wittgenstein’s writings about mathematics; the present work will focus on the connection he repeatedly makes between necessity and convention. The connection is an instance of Wittgenstein’s broader interest in going “back to the rough ground”. Within this broader context, there is a permanent underlying tension. If mathematical necessity is to be understood in what Wittgenstein proposes as its proper ground, are we not going to lose it? Wittgenstein was aware of the tension for the case of logic. In PI, §108 he writes: “But what becomes of logic now? Its rigor seems to be giving way here.—But in that case doesn’t logic altogether disappear—For how can it lose its rigor? Of course not by our bargaining any of its rigor out of it.” Even though it is clear that philosophy cannot disintegrate mathematical necessity, how it may not is a main concern of the present work.
Section 1 will present Euler’s proof on the impossibility of walking all the bridges in old Könisberg without crossing any one of them more than once; the purpose is to set the conditions—the feeling of mathematical necessity, the phenomenology of following a proof—to elicit a natural resistance to Wittgenstein’s ideas. Section 2 will make this resistance explicit and will establish some conditions that Wittgenstein’s account should be expected to meet. Whereas sections 1 and 2 present the problem of mathematical necessity and its contrast with convention as clearly as I found possible, Section 3 takes some preliminary, very loose steps towards a solution. While a solution will remain mainly a task for the future, the fact that these steps guided the construction of the problem in sections 1 and 2 is enough reason for their appearance in this paper.
1
Figure 1
The River Pregel traversed the city of Könisberg, in Eastern Prussia. There was an island on the river, called Kneiphof, past which the river forked into two branches. There were seven bridges joining the shores of the river with the island and the land inside the bifurcation (see Figures 1 and 2)[1]. Citizens of Könisberg, it is said, had entertained themselves trying to walk (unsuccessfully) all the bridges without crossing any one of them more than once. Euler was asked for a solution: if the walk was possible, find the path; if it was not, find a proof. Euler submitted his solution to the Academy of Sciences in St. Petersburg in 1736 in an article called “Solutio Problematis ad Geometriam Situs Pertinentis.”[2] This article has been referred to as the birth of Graph Theory.[3] It is not, as one might think, a mere piece of didactic mathematics or an amusing puzzle.[4] The article contains the seed of mathematical questions that are still studied by mathematicians.[5]
Besides the importance of Euler’s article for mathematics, the nature of its subject matter makes it a good “case study” for the purposes of this paper.[6] A book collecting fundamental contributions to graph theory begins as follows: “The origins of graph theory are humble”,[7] as humble as the problem of walking bridges in a city, drawing figures in one single stroke, or coloring maps. Almost every person can understand what is involved in these problems, at least anyone able to walk, draw, and think. One may need a lot of ingenuity to solve them, one may even need some arithmetic and basic number theory to understand some proofs (one might need some higher level mathematics to understand some other proofs), but what is involved in the problems is rather simple; they are related to activities in our ordinary lives. Euler’s solution is part of the humble origin of an important mathematical field—it is, one could hope, a good starting point to think about how we might follow (and also to understand how Wittgenstein followed), within philosophy of mathematics, Wittgenstein’s recurrent calls to guide philosophy towards the ordinary. At least two inter-related questions arise: “how can we give an account of the necessity of mathematics in face of the ordinary?” and “how can we give an account of the ordinary in face of the most technical and specialized results of mathematics?” Mathematical necessity seems to entail that mathematics cannot simply be invented; if its results have to be no matter what, then they are above human activities, decisions and inventions—therefore above the ordinary. On the other hand, the most technical aspects of mathematics seem to be far off from our ordinary experience. If it is this experience that Wittgenstein wants philosophy to be fed with, esoteric results of mathematics can hardly provide a soil for philosophy; we should then ask how far a philosophy of the ordinary can, or needs to, advance in mathematics.[8]
*
In his article on the KB problem, Euler states the problem in §1 as follows:
In Könisberg in Prussia there is an island A, called the Kneiphof; the river which surrounds it is divided into two branches, as can be seen in Figure 2, and these branches are crossed by seven bridges a, b, c, d, e, f and g. Concerning these bridges, it was asked whether anyone could arrange a route in such a way that he would cross each bridge once and only once.
Figure 2
Euler uses capital letters A, B, C, D for each of the land areas separated by the river and represents a walk over the bridges as an array of letters in a natural way: if someone departs from A, goes to B, and then to D, for example, the array is ABD, regardless of which bridge is used to go from A to B.The problem of crossing KB, each bridge once and only once,[9] is represented using this method as that of constructing a specific array of eight letters. In §6 Euler writes:
If a journey of the seven bridges can be arranged in such a way that each bridge is crossed once, but none twice, then the route can be represented by eight letters which are arranged so that the letters A and B are next to each other twice, since there are two bridges, a and b, connecting areas A and B; similarly, A and C must be adjacent twice in the series of eight letters, and the pairs A and D, B and D and C and D must occur together once each.
In §5, Euler states the more general proposition of which the quoted paragraph is a consequence. In modern terms we have it as a trivial Lemma 1: The crossing of n bridges can be represented by an array of n+1 letters. For each bridge you have the area where you enter the bridge (there are n of these, not necessarily different, areas because there are n bridges). These plus the area at the end of the walk compose the array.
There is a second, central and less trivial lemma in §8; in modern terms: Lemma 2: If k bridges lead to A and k is an odd number, the number of occurrences of A in the array of letters is (k+1)/2. In §8 Euler proves this lemma considering an area A and treating the exterior of A as a single region B (see Figure 3). He considers cases k=1 and k=3. The case k=1 is fairly simple: there is only one bridge. You can either start your walk at A (in that case you will never be back, so A appears only once in the array) or start outside A and go there (in that case the walk ends at A—you must cross the bridge once and only once!—and therefore A only appears once in the array, as the final letter). Consider the case k=3. If you start at A you will have to leave A, come back, and then leave to never return (the array will be AX1AX2… with no more A’s); if you don’t start at A, you will get there, leave and then come back to never leave again (the array will be in this case ….X1AX2…A). The reader should now see that it is true for every odd number.[10]
Figure 3
If you accept Euler’s representation of the problem, these two lemmas are enough to conclude the impossibility of crossing KB. By Lemma 1 you need to construct an array of 8 letters; by Lemma 2, given that 5 bridges lead to A and 3 to B, C and D, A must appear 3 times in the array and B, C and D 2 times. 3+2+2+2=9, thus you cannot construct an 8 letters array as you wished. Hence, there is no Euler Circuit in Könisberg.[11]
*
What has this bit of mathematics accomplished? It seems to have hindered our movements in some way; even more perplexing, it seems to predict our future.[12] But how can these mathematical representations have such power on us? We are experiencing the “hardness of the logical must”, the hardness that obstructs our way to what we were trying to accomplish. How can mathematics dictate what is possible and what is not? A metaphysical answer is: mathematics presents the deep, underlying, unmoved, order of all possible facts, an order we have to follow even if we are not aware of it. And hence the high implausibility of a “cultural” account of mathematics, for the proof shows us that any person, belonging to any cultural tradition, whatever conventions he happens to follow, will not be able to perform an Euler Circuit on KB. The strength of the metaphysical picture is proportional to the effort required to see the point of some of Wittgenstein’s Remarks on mathematical necessity. For many of his attacks are directed against it.[13]
It is useful to anticipate at this point a place towards which this paper is directed. The perplexity will begin to recede if we ask what is it that one was trying to accomplish when trying to perform an Euler Circuit in Könisberg.[14] (It seems not very hard to accomplish, as would be the case of Kaliningrad up on the Everest, but even harder—as Wittgenstein likes to put it: ultra-hard.) In a sense, what we were trying to accomplish was an illusion of our language, a consequence of the combinatorics of our grammar; Euler’s result excludes the combination of two concepts: “crossing bridges without repetition” and “the bridges of Konisberg”.[15] It is a discovery in grammar, analogue to the discovery that it is impossible to check in chess with a certain limited set of pieces. For Wittgenstein, the proof manifests our unwillingness to call anything a crossing of KB, given the way we already use the concepts. This unwillingness is not mathematical; rather, no extension will be natural for us.[16]
(It is necessary to remark that the problem solved by Euler is not empirical; in fact, the bridges of Könisberg, as Prussia, do not exist anymore. Könisberg is now called Kaliningrad, belongs to Russia and has lost two of the old bridges (it is possible to find an Euler Circuit in Kaliningrad). The problem of finding a path in Figure 2 is mathematical; it is not concerned with the temporal action of walking Könisberg but with the (non-temporal) question of finding a path in a graph, or an array of letters. The mathematical solution to the problem does not depend upon our availability of time, our willingness, experience, skills or strength.
Kaliningrad)
2
There are several passages in RFM where Wittgenstein’s view on mathematics seems very close to what one may be tempted to call “conventionalism.” Consider RFM, I, 74:
I say, however: if you talk about essence—you are merely noting a convention [Übereinkunft]. But there one would like to retort: there is no greater difference than that between a proposition about the depth of the essence and one about—a mere convention. But what if I reply: to the depth that we see in the essence there corresponds the deep need for a convention [Übereinkunft].[17]
However the inadequacy, or emptiness, of applying any label to Wittgenstein’s thought about mathematics, the question about the relation between convention and necessity demands an answer. How can we understand the sense “convention” has for Wittgenstein in this passage? More precisely: How can we account for a convention (or of what kind) when we are trying to understand the necessity we experience when we follow a proof? We seem to be required by the proof to decline any attempt—it is not by means of a convention, at least in its familiar sense, that we don’t do it; it is because we know that it cannot be done. The impossibility seems to be part of the essence of the KB structure(if it were just a convention, it would make sense to try to cross the bridges). Is the familiar sense of convention not radically opposed to what we perceive in a proof, to the “hardness of the logical must”? For it seems we cannot convene on taking something as necessary. When we convene on something, we fix one (or some) among many possible alternatives; but a thing being necessary implies that there is no alternative.
As a first attempt to understand the RFM passage quoted above, let us say that we convene in the axioms (or rules) wepick. Let us say also that it is true that the axioms can be chosen in several ways (though not in any way), but then what follows from them is not subject to any convention. According to this view, when we choose a set of axioms (in a conventional way), we are committed by their meaning, in some yet unclear way, to what can be derived from them. This is an analogue view to what McDowell refers to as thinking about meaning and understanding in “contractual terms”: “To learn the meaning of a word is to acquire an understanding that obliges us subsequently – if we have occasion to deploy the concept in question – to judge and speak in certain determinate ways, on pain of failure to obey the dictates of the meaning we have grasped.”[18]
This cannot be what Wittgenstein refers to as “convention” for he repeatedly attacks such views of meaning and understanding in the Philosophical Investigations. The picture of the “rails invisibly laid to infinity” is brought up in PI §218 as an image corresponding to the idea that our rules (or axioms) have all their consequences derived in advance.[19] Such idea is called a “mythological description of the use of a rule” in PI §221 and therefore cannot support the sense that “convention” has for Wittgenstein in the passage of the RFM.[20] Besides, we miss the whole strength of Wittgenstein’s idea if we interpret the passage as a convention on our (initial) rules, for Wittgenstein is referring to a convention that is effective at the moment of following a mathematical proof (insofar as essence, or internal properties, are manifested in proofs, not in the axioms chosen). “Convention” in RFM, I, 74 has a different meaning—although probably a related one—to our convening, say, on traffic rules or to the case of two countries agreeing to establish free trade. However, note that Wittgenstein writes, “you are merely noting a convention” and not: “you are noting a mere convention”; we seem to be in danger of seeing too much and getting lost when we talk about the essence, and this is at least part of what Wittgenstein is trying to prevent, but we should not see too little either.
The question is, then, why does Wittgenstein talk about convention when we talk about essence. If it is a metaphor, it is hard to know how to take it. But if we take necessity (and essence) to be in fact a part, or a consequence, of a contract in any sense (though, as we have just seen, it cannot be that of agreeing on a given set of axioms), we face the difficulties presented by Hume in his objection to the Social Contract.[21] It is hard to imagine how this contract could have been signed. We are not aware of having committed to a contract in which, if that is what it is, we promised to follow proofs (What are its clauses? What are its limits? What are the obligations involved?). We can hardly imagine what would count as breaking the contract: maybe to stop following proofs (trying to cross KB in an Euler Circuit or to trisect the angle with ruler and compass); or making wrong calculations, for instance when we spend our money; or not following a map, and therefore getting lost in an unknown city; or writing wrong answers in a mathematical exam. It is not meaningful to stop doing these things, for that would imply stop thinking as we do. In the case of the Social Contract, one might think that leaving society, breaking its laws, or harming fellow citizens is a way of breaking the contract. But in the case of mathematics, breaking the contract seems to mean giving up reason, and how can anyone decide or commit to that? Besides, a contract equivalent to “be rational” does not shed much light on the nature of mathematical necessity—to say that mathematical necessity is an expression of our being rational does not take our understanding any further. There has to be more in Wittgenstein’s picture.
*
Several related concepts surround the idea of convention in RFM; a natural resistance arises from all of them. Take for instance RFM, III, 27, where Wittgenstein writes: “I am trying to say something like this: even if the proved mathematical proposition seems to point to a reality outside itself, still it is only the expression of acceptance of a new measure (of reality)”. As in RFM, I, 74, there is a note of caution: “it is only…” i.e. we must not take the proposition for more than what it is (and, presumably, we usually do). But if we can accept a proved mathematical proposition, then we might also decline it, for the possibility of acceptance implies that of decline, and decline does not seem possible. Besides, does Euler’s proof set a new “measure of reality”? It seems to be just incorporating what one has always known to be called “walking on bridges” and then discovering a property of a specific configuration of bridges. In RFM, I, 33 Wittgenstein writes:
When I say “This proposition follows from that one”, that is to accept a rule. The acceptance is based on the proof. That is to say, I find this chain (this figure) acceptable as a proof. —“But could I do otherwise? Don’t I have to find it acceptable?”—Why do you say you have to? Because at the end of the proof you say, e.g.: “Yes – I have to accept this conclusion”. But that is after all the expression of your unconditional acceptance.”
If my acceptance is indeed unconditional, then why call it acceptance. Is there a situation in which there are conditions that allow for a rejection of a correct proof, only it is not the present one (and what is the present one)? It seems there is none—at least, it is not easy to think about such an alternative situation.
In RFM, III, 27 Wittgenstein writes: “Why should I not say: in the proof I have won through a decision? The proof places this decision in a system of decisions” and in RFM VI, 7: “I decide to see things like this. And so, to act in such-and-such a way” One feels the same resistance again: but do I really have a choice? After I see the proof and see it is correct, what I realize is that it must be like that, that there is no alternative or any decision to make. The only available decision seems to be whether or not to follow the proof because, once you follow it, you are trapped. (And Wittgenstein is not referring to this other kind of decision; for him, the decision comes after seeing the proof.)