January 18, 2002
Wittgenstein on Mathematical Necessity and Convention
Pablo Kalmanovitz
[email protected]
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
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
1
Figures 1, 2 and 3, as well as the story, are taken from Biggs, Lloyd & Wilson [1976], pp. 2, 3, 5.
I will use the translation in Biggs, Lloyd & Wilson [1976]: “The Solution to a Problem Relating to the
Geometry of Position” pp. 3-8.
3
Journal of Graph Theory, Vol. 10, No. 3, dedicated to the anniversary of Graph Theory (1736-1986), especially
Wilson [1986]. See also the Preface of Biggs, Lloyd & Wilson [1976].
4
However, in 1736 Euler wrote to the man who proposed him the Könisberg problem: “Thus you see, most noble
Sir, how this type of solution bears little relationship to mathematics, and I do not understand why you expect a
mathematician to produce it, rather than anyone else, for the solution is based on reason alone, and its discovery
does not depend on any mathematical principle” (quoted in Sachs, Stieblitz & Wilson, p. 136). Since he changed
his mind, he must have seen what mathematics had to do with a simple crossing of bridges. He saw at least two
things: (1) the opportunity to generalize the result to any conformation of islands, land and rivers and (2) as he
wrote to the Italian mathematician Marinoni: “this question is so banal, but seemed to me worthy of attention in
2
2
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:
that neither geometry, nor algebra, nor even the art of counting was sufficient to solve it. In view of this, it
occurred to me to wonder whether it belonged to the geometry of position, which Leibniz has once so much
longed for” (Ibid). The geometry of position later came to be known as topology.
5
Euler’s result is covered in graduate introductory texts on graph theory. Cf. for instance Bollobás [1979], pp 1315.
6
It should be observed that the problem is akin to some of Wittgenstein’s examples, e.g. RFM, VII, 20.
7
Biggs, Keith & Wilson [1976], p. 1.
8
Dummett [1959] criticizes Wittgenstein’s scarcity and quality of examples in RFM. “The examples given in
Wittgenstein book are—amazingly for him—thin and unconvincing. I think that this is a fairly sure sign that there
is something wrong with Wittgenstein account” (p. 498). We can be sure that it is not for lack of expertise that
Wittgenstein’s examples are so elementary, but all the same the question seems fair. If we are not going to deal
with the technical results of mathematics, in what sense is this a philosophy of mathematics?
3
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
9
In graph theory, this kind of “walk” is now called an Euler Circuit.
4
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
10
A diagram might be useful; the following shows the case when you start at A (if you start at B you only have to
reverse the orientation of the arrows) for k=5. You can group the bridges as shown by the arrows—for k=7, one
couple of arrows has to be added, for k=9 two couples, and so on. It should be clear now that the number of times
A appears in the array is 1+(k-1)/2.
It must be said that the case of KB is mathematically trivial, as Euler himself states in his article in §3, because
you might list all possible 7-bridge paths without repetitions (there are 7! (=5040) bridge crossings without
repetitions) and then check that none of them is feasible in Könisberg. However, its mathematical triviality does
not make it less relevant for what we are dealing with. The fact that the set of cases is exhaustive is an element of
this alternative method worth considering under the light of Wittgenstein Remarks. In connection with this, there
are several issues beyond the boundaries of the present work which are interesting enough to be stated here: the
rigor of a proof; the triviality of some proofs, the changing history of the moment in which one can be satisfied
with a proof, when one can finally say “it must be like that”; in the case of Euler’s proof, for instance, of how one
can decide if mathematical induction is needed, or if it is already clear enough. There is a history of rigor that
seems to conflict with the non-temporal status of the results attained under those very (historical) standards of
rigor.
5
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
11
The result can be further generalized by means of additional considerations for the case of an even number of
bridges. It is not in the scope of this project to get into the details of the results, but here they are, for the sake of
completeness. At the end of the article (§21) Euler claims he has proved:
If there are more than two areas to which an odd number of bridges lead, then such a journey is
impossible.
If, however, the number of bridges is odd for exactly two areas, then the journey is possible if it starts in
either of these areas.
If, finally, there are no areas to which an odd number of bridges leads, then the required journey can be
accomplished starting from any area.
In fact, Euler only proved the first proposition; he gave a rule for finding the array corresponding to the cases in
the other two propositions, but it was not intended as a proof. That the cases in the three propositions are
exhaustive follows form the so-called Handshaking Lemma, first stated, according to Wilson [1986], in Euler’s
KB article.
12
Wittgenstein acknowledged the predictive character of impossibility proofs. The proof must be “a forcible
reason” for giving up the search (RFM, I, AIII, 14). For him, the character of the prediction is, we may say,
grammatical (cf. RFM, III, 66.)
13
On this, see also Floyd [1991], especially pp. 147, 157.
14
For this question, cf. Floyd [1995]’s excellent treatment of the impossibility of certain mathematical
constructions, especially section II. There is a critical point in p. 383. Floyd asserts that it is fair to maintain that
the search for a construction that turns out to be impossible (the trisection of the angle, crossing the bridges) is a
way of entertaining or believing a contradiction. This seems to imply that the contradiction was always there,
independently of how we manage to do our mathematics (which in turn contradicts Wittgenstein’s ideas about
6
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:
mathematical contradiction, cf. for instance RFM, III, 81). It is true that now we know that the belief that the
construction was possible cannot be fulfilled, that it implies a contradiction. But there is a whole reframing of the
question behind this (both in the case of the trisection and in the much simpler case of KB) and, even more
important, our willingness to accept the re-framing (or re-presentation.) In any case, there still is a tension here
between the temporality of the activity of doing mathematics and the a-temporality of its results. We now know
that the trisection of the angle cannot ever be made.
15
This is connected—in some way I have not yet got a clear grasp of—with PI, §125.
16
Cf. Cavell [1989], pp. 41-42. Cavell emphasizes the importance of the connection between “the natural” and
the idea of “conventions” in Wittgenstein’s philosophy. In his writings, Wittgenstein never seems to explain or
talk much about the natural—in this paper, it will remain as one of the major questions (Is it an end of the road
for philosophy?).
7
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) we pick. 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
17
For Wittgenstein, the idea of “essence” (or of “internal property”) is closely connected with that of
mathematical proof (cf. RFM, I, 32; RFM, I, 99). Mathematical proofs construct and show essential features; in
this sense, mathematical necessity manifests what is essential (hence the connection of this passage with what we
are dealing with).
18
McDowell, p. 221. This is not McDowell’s point of view, he wants to criticize it as a misreading of
Wittgenstein.
8
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
19
There are passages in RFM related to this part of the Investigations. For instance RFM, I, 3: “My having no
doubt in face of the question does not mean that it has been answered in advance.”
20
At this point I only want to show the implausibility of an interpretation of convention as a contract in the sense
of McDowell for RFM, I, 74. I won’t go further into the discussion of following a rule.
21
I rely heavily in this paragraph on Cavell’s discussion of the social contract and Wittgenstein’s sense of
“agreement” in his The Claim of Reason, pp 22-28.
9
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
10
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.)
At this point, it is possible to begin to suspect that seeing the construction as a proof
involves a great deal (a system of decisions, a set of interconnected actions at the bottom of
those decisions, the unconditional acceptance of practices and systems of decisions) and that
the natural resistance is an expression of how much we count on it as a matter of fact. Parts of
Wittgenstein’s work aims to uncover this expression; by uncovering it, mathematical necessity
is clarified.
*
Michael Dummett’s attitude towards Wittgenstein’s RFM in his paper “Wittgenstein’s
Philosophy of Mathematics” (first published in 1959, only three years after the first edition of
RFM) is well summarized in the following sentence:
I must emphasize that I am not proposing an alternative account of the necessity of mathematical
theorems, and I do not know what account should be given. I have merely attempted to give
reasons for the natural resistance one feels to Wittgenstein’s account, reasons for thinking that it
must be wrong.22
Even though several passages of his paper are arguable (for instance, that the reasons for the
natural resistance to Wittgenstein’s account are reasons for thinking he “must be wrong”), the
problems Dummett raises, when confronted with Wittgenstein’s texts, certainly open a way to
understand what is involved in some of his Remarks on the Foundations of Mathematics.
According to Dummett, “Wittgenstein goes for a full-blooded conventionalism.”23 At least
at the beginning of his paper, conventionalism seems advantageous to Dummett, since it
accomplishes “a liberation” for philosophy.24 In the conventionalist account, “all necessity is
imposed by us not on reality, but upon our language; a statement is necessary by virtue of our having
chosen not to count anything as falsifying it. Our recognition of logical necessity thus becomes a
particular case of our knowledge of our own intentions.”25 We can naturally resist this account as
above, and even in some new ways (how can we decide whether or not something falsifies a
statement? It is from the statement, not from our will, that we can know nothing will falsify it;
if necessity is imposed by us, then we might as well not impose it and try to cross the bridges;
22
Dummett [1959] p. 504.
Ibid, p. 495.
24
Ibid, p. 494.
25
Ibid
23
11
how can our intentions be involved, when we face a statement, given that…). But
Wittgenstein’s extreme version of conventionalism, according to Dummett’s interpretation, is
even more implausible, as it “seems to hold that it is up to us to decide to regard any statement we
happen to pick on as holding necessarily, if we chose to do so.”26 Dummett takes this to be so
attributing to Wittgenstein the idea that “we have the right to attach what sense we choose to the
words we employ.”27 Dummett’s Wittgenstein is then an anarchist of meaning, and it is natural
to react and declare, even if one is a conventionalist, “I wish […] to set the conventional view that
in deciding to regard a form of words as necessary, or to count such-and-such as a criterion for making
a statement of a certain kind, we have a responsibility to the sense we have already given to the words
of which the statement is composed”28 or equivalently, if we are going to be conventionalists, we
must be responsible conventionalists.
Dummett’s position as a conventionalist is ambivalent, and the point around which he
cannot find equilibrium shows what is the problem at stake. Dummett explicitly states that he
wants to set, as his own view, what may be called Responsible Conventionalism. However, at
the beginning of his paper, in the midst of his enthusiasm about conventionalism, he criticizes
what he calls “Modified Conventionalism,” according to which
the axioms of a mathematical theory are necessary in virtue of their being direct registers of
certain conventions we have adopted about the use of the terms of the theory; it is the job of
the mathematician to discover the more or less remote consequences of our having adopted
these conventions, which consequences are epitomized in the theorems.29
This is precisely the contract-on-axioms version of conventionalism that was addressed and
criticized above. Dummett has a similar criticism for it: “this account is entirely superficial and
throws away all the advantages of conventionalism, since it leaves unexplained the status of the
assertion that certain conventions have certain consequences.”30 However, Dummett himself asserts
that the mathematician must be responsible for the meaning of the words, but the status of this
responsibility—and of the alleged determination of the words used by the mathematician—is
left unexplained. In other words, the assertion according to which “the senses of the words in the
statement may have been already fully determined, so that there is no room for any determination ”31 is
left unexplained by Dummett (i.e. Dummett’s Responsible Conventionalism is a form of
26
Ibid, p. 500.
Ibid.
28
Ibid, p. 503.
29
Ibid, p. 494.
30
Ibid.
31
Ibid, p. 500.
27
12
Modified Conventionalism, and therefore suffers from the very philosophical weaknesses
Dummett points out).
It is not easy to detect a possible source for Dummett’s claim that Wittgenstein’s idea
“appears to be that […] we have the right to attach what sense we choose to the words we employ.”32
If by “sense” Dummett means “our understanding of a word”, he may be referring to
something analogous to PI, §§139-142, where Wittgenstein states that our understanding of a
word—if one sees it as a picture, or as a “method of projection that comes before our
mind”33—can be applied in several different ways and hence does not determine its use (i.e.
images corresponding to our understanding can be interpreted as to correspond to any use). 34
This does not mean that we can use words as we please; rather, it means that we can
“understand” words as we please, as long as we use them properly. On the other hand, if by
“sense” Dummett means “use”, then why should Wittgenstein bother explaining how our
pictures of meaning might be applied in different ways—showing thus how application, use, is
crucial, unlike our mental pictures—if he thought that we can use words as we please.
Therefore, if “sense” is taken, as Wittgenstein takes it, as “use”, it cannot be attributed to him
the idea that we can attach any sense we choose to the words we employ; consequently, the
idea that it is up to us to take any statement we pick as necessary cannot be attributed to
Wittgenstein on this ground, if at all.35
In any case, the question raised by Dummett is central. That mathematical necessity is
conventional seems to imply that the results of mathematics are in some way arbitrary, that
they just depend on the will of mathematicians and on the way they happen to agree and set
their conventions.36 This is the very center of Dummett’s ambivalence between his
Responsible Conventionalism and the Modified Conventionalism, and also the center of the
tension between necessity and conventions that was anticipated in the first paragraph of this
32
Dummett remits to RFM V,23 and last par. on p. 179. to sustain his claim about Wittgenstein’s alleged
meaning anarchism, but those texts are not connected with the issue. In his Truth and Other Enigmas the article is
reprinted with the same references.
33
PI, §140.
34
Cf. also RFM, VI, 38: “How can I follow a rule, when after all whatever I do can be interpreted as following
it?”
35
In RFM, I, 116 Wittgenstein writes: “ ‘Then according to you everybody could continue the series as he likes;
and so infer anyhow!’ In that case we shan’t call it ‘continuing the series’ and also presumably not ‘inference’.
And thinking and inferring (like counting) is of course bounded for us, not by an arbitrary definition, but by
natural limits corresponding to the body of what can be called the role of thinking and inferring in our life.”
36
For arbitrariness, cf. RFM, I, 167. Also PI, 372.
13
paper. If an account in terms of collective agreement is given for mathematical necessity—an
agreement in which every one can decide whether to join or not, in which every one can
accept or reject what is proposed—then necessity disappears. Furthermore, we would not call
a proposition necessary if we must ask: “do we, the members of this community, agree to take
this proposition as necessary?” Necessity needs to be given from more than an aggregate of
individuals—there has to be something more from which necessity is conceded, from which
necessity arises and can be extended.37 The community Wittgenstein invokes when he talks
about agreement, or convention, has to be more than a mere aggregate of individuals. If we
can make that clear, maybe we will be able to find what bounds the mathematician’s
inventions, what is it that a mathematician must obey, or limit himself to, or be responsible for.
3
3.1. To better understand Wittgenstein’s idea of mathematical necessity, it is useful to sketch
the metaphysical idea he is committed to attack. Consider RFM, I, 8:
But still, I must only infer what follows!—Is this supposed to mean: only what follows, going by
such rules of inference as somehow agree with some (sort of) reality? Here what is before our
minds in a vague way is that this reality is something very abstract, very general, and very rigid.
Logic is a kind of ultra-physics, the description of the “logical structure” of the world, which we
perceive through a kind of ultra-experience (with the understanding e.g.). Here perhaps
inferences like the following come to mind: The stove is smoking, so the chimney is out of order
again”. (And that is how the conclusion is drawn! Not like this: “The stove is smoking, and
whenever the stove smokes the chimney is out of order; and so…”)
The idea that Wittgenstein is attacking is twofold. On one hand, in several passages of RFM
Wittgenstein dreads the logicist idea of mathematics, what he calls ‘“the disastrous invasion”
of mathematics by logic’.38 For him, logic obscures mathematics’ “motley of techniques of
proof.”39 On the other hand, there is what Wittgenstein calls the “sublimation of logic” in the
Philosophical Investigations (mainly in the context of the discussion of §§89-108.)40 These
two are combined to yield a picture of mathematical necessity Wittgenstein is committed to
unmake.
37
The problem is stated in a different context by McDowell [1984], specially pp. 225, 234, 252-253. I address his
way out of the problem briefly in 3.4 below.
38
RFM, V, 24. Cf. additionally RFM, V, 25; V, 48; III, 53 and RFM, III, 44.
39
RFM, III, 46.
40
Wittgenstein writes in PI, §89: “In what sense is logic something sublime? For there seemed to pertain to logic
a peculiar depth—a universal significance. Logic lay, it seemed, at the bottom of all sciences. For logical
investigation explores the nature of all things. It seeks to see the bottom of things and is not meant to concern
itself whether what happens is this or that—it takes it rise, not from an interest in the facts of nature, nor from a
need to grasp causal connexions: but from an urge to understand the basis, or essence, of everything empirical”.
14
In PI, §106 Wittgenstein sets forth what is unsatisfactory about the sublime image of
logic: “Here it is difficult as it were to keep our heads up—to see that we must stick to the subject of
our every-day thinking, and not go astray and imagine that we have to describe extreme subtleties,
which in turn we are after all quite unable to describe with the means at our disposal. ”41 According to
Wittgenstein, there is a sharp conflict between our actual language and our requirements for it,
a conflict that becomes intolerable: “We have got on to slippery ice where there is no friction”,
where the conditions are ideal but, because of that, “we are unable to walk.”42 At this moment, it
is imperative to turn back; “We are talking”, Wittgenstein reminds, “about the spatial and
temporal phenomenon of language, not about some non-spatial, non-temporal phantasm.”43 “Keeping
our heads up”, not “going astray”, using the “means at our disposal” are imperatives for all
philosophical investigation; then, of course, it cannot be different in the case of mathematics.
However, part of the required work is to unveil the proper ground for the investigation in
mathematics, while being able to preserve mathematical necessity; philosophy has to describe
how the spatial, temporal phenomenon of language is the ground for the non-spatial, nontemporal results of mathematics.
The example of the smoking stove in RFM, I, 8 is a sample of the rough ground that
constitutes the means of philosophy. Logic, as is remarked in the parenthesis in the passage,
does not necessarily underlie the smoking stove inference—it is more likely that a picture of
the mechanism is used, or that the experience of previous cases is applied. By the same token,
the inexorability of the results of arithmetic is only a consequence of our inexorable way of
counting and adding and multiplying.44 And then it seems as if it were not the strength of the
proof that forces us to give up any attempt to cross the bridges but it is instead our giving up
any attempt that has given the proof all its strength. And again, this seems to imply that after
all we should try, and that maybe we will succeed.
41
PI, §106.
PI, §107.
43
Ibid.
44
Cf. RFM, I, 4: “But then what does the peculiar inexorability of mathematics consist in?—Would not the
inexorability with which two follows one and three two be a good example? […] Counting (and that means:
counting like this) is a technique that is employed daily in the most operations of our lives. And that is way we
count as we do: with endless practice, with merciless exactitude.”
42
15
3.2. The preceding prescription for philosophy is condensed in a maxim that resembles a
method: philosophy must “do away all explanation and description alone must take its place.”45 In
the case of mathematical necessity, it is e.g. the inexorability in the way we count, or derive,
or multiply that constitutes the description. However, there seems to be something missing; a
description here seems superficial, as if it were unable to penetrate the problem. The case is
analogous to the discussion on mental processes in PI, §§305-306, where Wittgenstein
explicitly confronts the accusation that he “wants to deny something”. Wittgenstein replies:
“What we deny is that the picture of the inner process gives us the correct idea of the use of the word
“to remember”.”46 But in the discussion on mental processes we did not want to know anything
about mere words, we wanted to know about the process of remembering (it is similar with
mathematical necessity: we do not want to know something about just the way we calculate, or
about the sense of the word “must”, but about the necessity of mathematics).
Wittgenstein briefly considers the issue in PI, §§370-374.47 The way in which essence
is expressed in grammar may be hinted at when considering the way mathematical
representations, or pictures, are applied. We set our practices, actions and ways of thinking
according to pictures we share and apply in agreement, in a uniform manner. In the case of
mathematics, some of these pictures can even be drawn on paper. Take for instance the
diagram in footnote 10; it stipulates what counts as “crossing”. The essence of crossing is
captured, or created, in it. It must be noted, however, that the diagram can be applied in many
ways, and that the fact that we know how to apply it, and that we do apply it in consonance, is
not determined by it—there is no reason for us to follow it as we do. This is why it can be said
that in mathematics we are closer to the bottom, where we cannot explain why we act as we
do. We cannot give reasons for how we apply a rule, or follow a proof—there are no reasons
left. The proof is the end in the chain of reasons; in it, we are left with just a “this is what I do”
or “this is how it has to be done.”
3.3. Mathematics provides pictures for us to understand and judge our experiences. In the case
of KB, the water and the bridges pose limits in the way we walk. They set and determine a
domain for our possible activities. Euler’s representation captures and is based on this
45
PI, §109. Cf. also RFM, V, 52 :“It is not that a new building has to be erected, or that a new bridge has to be
built, but that the geography, as it now is, has to be described.”
46
PI, §305. Cf. also PI, 370.
47
PI, 371: “Essence is expressed in grammar”. PI, 373: “Grammar tells what kind of object anything is.”
16
determination. The hardness of our practice gives the proof its consistency. The prophecy
made by the proof is based on a grammatical stipulation registered in it: “The prophecy does not
run, that a man will get this result when he follows this rule in making a transformation—but that he
will get this result, when we say that he is following the rule.”48 If someone says he made the Euler
Circuit, we will say he is lying because what we call an Euler Circuit is something that cannot
be done in KB.
3.4. To belong to a community means, among other things, to act and react in determinate and
“tuned” ways vis-à-vis the pictures and rules we share.49 Our unhesitant action when following
a rule is an expression of our being trained in our actions and reactions as members of a
community, and hence the analogy made by Wittgenstein between mathematical propositions
and orders (“You must cross the bridge, and only once!”).50 The agreement is not so much one
of acceptance but of initiation. For McDowell “a linguistic community is conceived as bound
together not by a match in mere externals (facts accessible to just anyone), but by a capacity of a
meeting of minds.”51 It might very well be “a meeting of minds”, but what we can see is our
being conditioned by our training to follow rules as we do, to apply pictures as we do and to
act and see things in agreement. We were taught to act like that, and in this sense there is no
decision to make. However, we could have been taught differently, we might have had a
different life52 and we may very well act differently now if we are willing to pay the very high
price. The necessity of mathematics is an expression of regularities and coincidences in our
life and in nature. Coincidence must exist not only in our actions, also in our appreciations and
inclinations. The agreement involved in these considerations is called “agreement in forms of
life.”53 Wittgenstein refers to our natural inclinations to explain the shape of certain extensions
of language games (for instance in PI, §48), including some of mathematics.54
48
RFM, III, 66.
Cf. Cavell, p. 32: “The idea of agreement here is not that of coming to or arriving at an agreement on a given
occasion, but in being in agreement throughout, being in harmony, like pitches or tones, or clocks, or weighing
scales, or columns or figures.”
50
Cf. for instance RFM, VII, 39.
51
McDowell [1984], p. 253.
52
Cf. RFM, I, 148-150.
53
PI, §241.
54
Consider this dialogue between Wittgenstein and Turing in LFM, pp. 186-187:
Wittgenstein: “The point is: Is it or is it not the case of one continuation being natural for us? Or ought one to say
that there is something more to it than that? Ought one to give a reason why one continuation is natural for us?
Ought one to say this, for example: “If we learn to use orders of the form ‘p’, ‘q’, ‘p and q’, ‘p and not-q’ etc,
then so long as we give the phrase ‘p and not-p’ the sense which is determined by the previous rules of training, it
is clear that this cannot be a sensible order and cannot be obeyed. If the rules for obeying these orders—for
49
17
3.5. To make sense, a problem needs to be nested within a web of problems, in the same way
as a single rule, or a single game, needs to be part of a web of rules, or of other games. In
RFM, VI, 32, Wittgenstein calls what surrounds a rule and makes it possible “an institution”.
Teaching the problem of crossing KB requires the introduction of (at least part of) the web of
our customs. Someone who has understood the problem has to have understood many
additional things; a criterion for his understanding is that, when presented the proof, he says:
“it cannot be done.” Wittgenstein’s use of the word “institution” shows the dimension of the
community he is thinking of when he asks the question “how do we use a word?” or when he
invokes a convention to explain mathematical necessity. A simple aggregate lacks institutions.
In this sense of convention, Wittgenstein’s remarks do not amount to saying that mathematics
can be different for us; they are deeply entrenched in our practices and in our needs.
We are inclined to say that an alien to our culture will not be able to find an Euler
Circuit in KB, and then that our mathematics are universal. It should be noted, however, that
for an alien to mathematics there is no KB problem; he might jump and cross the Kneiphof
swimming without seeing a violation of any stipulation. He would not know how to follow the
order “you must cross the bridge, and only once!” But if what he does falls under our
description of walking the bridges, then we know he will not make the Euler Circuit, because
Euler’s proof stipulates our description (not being able to make an Euler Circuit is conforming
to what we call walking the bridges.)
logical product and negation–are laid down, then if we stick to these rules and don’t in some arbitrary way
deviate from them of course ‘p and not-p’ can’t make sense and we can’t obey it”. Isn’t that the sort of thing you
would consider not cheating?
Turing: I would say that it is another kind of cheating. I should say that if one teaches people to carry our orders
of the form ‘p and not-q’ then the most natural thing to do when ordered ‘p and not-p’ is to be dissatisfied with
anything which is done.
Wittgenstein: I entirely agree. But there is just one point: does “natural” mean “mathematically natural”?
Turing” No.
Wittgenstein: […]It is not mathematically determined what is the natural thing to do.”
18
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