Grazer Philosophische Studien
62 (2001) 215-247
WITTGENSTEIN AND THE REGULAR HEPTAGON*
Felix MÜHLHÖLZER
Universität Göttingen
Summary
The later Wittgenstein holds that the sole function of mathematical propositions is to determine the concepts they invoke. In the paper this view is
discussed by means of a single example: Wittgenstein’s investigation of
the concept of a regular heptagon as used in Euclidean geometry (i.e., the
Euclidean construction game with ruler and compass) and in Cartesian analytic geometry. Going on from some well-known passages in Wittgenstein’s Lectures on the Foundations of Mathematics, and completing these
passages, it is shown that Wittgenstein’s view makes perfectly good sense
and can be very well defended.
1. Wittgenstein’s philosophy of mathematics
The later Wittgenstein (in what follows by “Wittgenstein” I will always mean the later Wittgenstein, i.e. the Wittgenstein after BB)
holds that mathematical propositions are ‘grammatical’ prop- ositions1, that is, very roughly speaking, that their sole function is to determine the concepts they invoke. To speak a little less roughly: Ac* I am grateful to Benno Artmann, Bernd Buldt, Marianne Emödy, HeinzJürgen Schmidt and Wolfgang Spohn for valuable remarks on earlier versions of
this paper, to Yonatan Gelblum for correcting my English, and to an anonymous
referee of this journal for protecting me from a serious mistake. Research was
supported by the Deutsche Forschungsgemeinschaft (grants Mu 687/3-1 and
3-2).
1. See RFM, p. 162: “Let us remember that in mathematics we are convinced
of grammatical propositions; so the expression, the result, of our being convinced is that we accept a rule.”
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cording to Wittgenstein’s view the typical use of a mathematical
proposition is much more similar to the use of propositions in order
to determine concepts or to state rules (“The bishop in chess moves
only diagonally”) than to the use of propositions in order to report
facts. The term “mathematical proposition” has to be understood
here always in the sense of a proven mathematical proposition.2
When in Euclidean geometry, for example, we prove that for any triangle the sum of its three interior angles equals two right angles,
then, according to Wittgenstein, our subsequent use of this proposition does not report any facts discovered by our proof, but merely expresses our decision to call any sum of three angles “two right angles” if the three angles are interior angles of a triangle. Wittgenstein’s philosophy of mathematics in its entirety can be regarded as
the attempt to let us see, on the basis of a lot of examples from different mathematical contexts, the plausibility of this seemingly
bold-looking thesis that (roughly speaking) mathematical propositions are nothing but determinations of concepts.
This thesis, of course, is by no means identical to the logical positivist’s thesis that mathematical propositions are analytic, where a
proposition is defined as ‘analytically true’ if it is made true merely
by the meanings of its constituents, quite independently of any facts
of the world. Wittgenstein rejects this notion because he does not accept its underlying assumption of the existence of certain ‘meanings’ which have the capacity to ‘make propositions true’. Nothing
could be further from Wittgenstein’s later philosophy than a semantical mythology of this sort. Grammatical propositions in his sense
are not analytic propositions but, rather, propositions used to determine concepts, as, for example, the propositions in a rule book
which we consult in order to know when we are in accordance with
the rules and when not.
Not only are grammatical propositions in Wittgenstein’s sense
not analytic propositions, what I called Wittgenstein’s ‘thesis’
should also not be regarded, strictly speaking, as a thesis at all but as
2. I will not deal with the ticklish question about Wittgenstein’s view of
mathematical propositions which are not proven, as Goldbach’s conjecture or the
continuum hypothesis, or which are disproved. To this see, for example, Floyd
1995, pp. 382-386 and 394f.
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an invitation to investigate our use of mathematical propositions in
detail. We then should perceive that these propositions are more
similar to determinations of concepts than to statements of facts. In
Wittgenstein’s most mature notes on the philosophy of mathematics, his Remarks on the Philosophy of Mathematics, this investigation partly presents itself as a struggle for the appropriate formulations of his main philosophical point, beginning with relatively
clear-cut statements like:
[T]he proof [of a mathematical proposition] changes the grammar of
language, changes our concepts. It makes new connexions, and it creates the concepts of these connexions. (It does not establish that they
are there; they do not exist until it makes them.) (RFM, p. 166),
over doubtful ones like:
Now ought I to say that whoever teaches us to count etc. gives us new
concepts; and also whoever uses such concepts to teach us pure mathematics? (RFM, p. 166),
up to, temporarily, defeatist ones like:
The word ‘concept’ is too vague by far. (RFM, p. 166).
The delicate way in which mathematical propositions, together with
their proofs, function as ‘determinations of concepts’ cannot be
stated in simple declarations, and they perpetually force Wittgenstein into further and further investigations.
The intention not to advance philosophical theses but to prompt
investigations of our actual use of language is certainly of central
importance to Wittgenstein’s later philosophy and it is expressed by
him in many well-known passages, such as the following:
One of the greatest difficulties I find in explaining what I mean is this:
You are inclined to put our difference in one way, as a difference of
opinion. But I am not trying to persuade you to change your opinion. I
am only trying to recommend a certain sort of investigation. If there is
an opinion involved, my only opinion is that this sort of investigation is
immensely important, and very much against the grain of some of you.
If in these lectures I express any other opinion, I am making a fool of
myself. (LFM, p. 103)
Wittgenstein is right, up to the present day, to claim “that this sort of
investigation is [...] very much against the grain of some of you”. In-
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vestigations of the sort envisaged by him remain on a purely descriptive level; they only aim at a ‘perspicuous representation’ (PI § 122)
of our use of words and are abstinent with respect to any serious theorizing. Outside of Wittgenstein’s own work such investigations are
very rarely found in the literature. Obviously, they contradict our
deep-seated tendency to concoct theoretical constructions and to
search for far-reaching generalizations. There is no reason to discredit this tendency, which is, of course, the main driving force for
science, but in philosophy it brings about serious dangers since it
threatens to make us blind to the actual complexity of our forms of
life. Investigations of the Wittgensteinian sort would be extremely
desirable to counteract this blindness. This is especially important in
the philosophy of mathematics where we tend to flee to the levelling
reconstructions of mathematics, for example set-theoretical ones,
which have been proposed by the mathematicians themselves (logicians included) but which divert us from the actual diversity of our
real mathematical practice. A Wittgensteinian therapy might be
quite healthy in this case.
In this essay I want to carry out, by concentrating on a special
mathematical example, an investigation as intended by Wittgenstein. The example is one which has been dealt with by Wittgenstein
himself in his Lectures on the Foundations of Mathematics: the
failed attempts within Euclidean geometry to construct a regular
7-gon, culminating in the mathematical proposition, proven in the
19th century, that such a construction is impossible.3 In this case
Wittgenstein’s view may appear particularly implausible. Doesn’t
everybody understand what a regular 7-gon is and what a Euclidean
construction consists of? And doesn’t the impossibility theorem in
question simply state that in case of a regular 7-gon such a construction does not exist? The latter seems to be a plain mathematical fact,
and Wittgenstein’s claim that the impossibility theorem does not express such a fact, but merely determines, or at least changes, the concepts it invokes – the concept of a regular 7-gon and the concept of a
Euclidean construction – may appear entirely mistaken. A closer investigation, an investigation Wittgenstein himself sets going in
LFM and which I wish to continue now, will show, however, that
3. The relevant passages in LFM are mainly on pp. 45-91.
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things are not so easy.
It might be objected that passages from LFM are a textual basis
too insecure to be the starting point of an investigation in
Wittgenstein’s style, since LFM is nothing but a piece of handicraft
created by its editor (Cora Diamond), patched together by different
notes taken by students of Wittgenstein’s lectures, the reliability of
which may be doubted. Such an objection does not hold water, however, since one cannot really doubt that LFM breathes the spirit of
Wittgenstein’s later philosophy – even in a particularly fresh and
spontaneous way. This can be easily verified by a comparison with
Wittgenstein’s authentic remarks in RFM which show sufficiently
well at least the sort of investigation he had in mind.4 It is exactly the
same sort of investigation we meet in LFM. Admittedly, Wittgenstein’s way of dealing with the concept of a regular 7-gon is not particularly systematic and contains noticeable argumentative gaps; but
it is clear enough to encourage the attempt at a completion and independent continuation. In what follows I want to embark on such an
attempt. In order not to disturb the flow of the investigation, I will refrain from substantiating my reflections by always meticulously citing corresponding passages in LFM. So the aim of this paper is not
primarily an exegetical one. This, of course, does not mean that I ignore Wittgenstein’s statements, but they are too incomplete and too
imperfect to be content with them. Instead, I try to complete and to
improve them in order to make a good case for them. Critical readers
may reread the relevant sections in LFM, compare them with my assertions and judge in the end whether Wittgenstein’s intentions have
been met. But I hope that the following investigation also contributes to an understanding of mathematics itself, quite independent of
any relation to Wittgenstein.
4. Because of the heavy encroachments of its editors, RFM too must be taken
with grains of salt, of course. But one cannot really doubt that even in its present,
very unsatisfactory version RFM at least shows the Wittgensteinian plan for his
philosophy of mathematics and his philosophical method in a sufficiently clear
way.
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2. How to understand the impossibility of the construction of a regular 7-gon?
Wittgenstein’s reflections on the regular 7-gon start with a consideration of the following two propositions (see LFM, p. 45) in which
mathematical and non-mathematical issues are combined:
(P) Smith drew the construction of a regular 5-gon.
(H) Smith drew the construction of a regular 7-gon.
(Unless otherwise specified, the term “construction” is always to be
understood in the sense of a Euclidean construction, allowing only
the use of ruler and compass in the well-known way described in Euclid’s Elements.) These two propositions are interesting since they
do not differ in form. Reconstructions in the language of formal
logic, for example, may treat them as completely analogous. If one
does, however, what Wittgenstein repeatedly admonishes us to do, if
one cares about the use of propositions5, big differences turn up.
Wittgenstein’s Lectures on the Foundations of Mathematics are
valuable not only because they breathe the spirit of his later philosophy in a particularly fresh way, but also because they were attended
by Alan Turing, who played a relatively active, and critical, role in it.
Thus, Wittgenstein had to grapple with the reactions of an eminent
mathematician and was forced to state his own position more precisely. In LFM, Wittgenstein says that (P) may be either true or false;
what about (H), however? Turing spontaneously exclaims: “That is
undoubtedly false.” (LFM, p. 45. – Remark for non-mathematicians:
As already said, it has been mathematically proven that there is no
Euclidean construction of the regular 7-gon. The regular 5-gon, on
the other hand, can be easily constructed.) Wittgenstein then asks:
“Isn’t it queer that the case of the [5-gon] is so different from the case
of the [7-gon]?”, whereupon Turing, showing real philosophical
sensibility, admits: “There is something queer about saying that
5. See LA, p. 2: “If I had to say what is the main mistake made by philosophers of the present generation [...], I would say that it is that when language is
looked at, what is looked at is the form of words and not the use made of the form
of words.”
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[(H)] is certainly false. For it suggests that it might be true but is certainly false.”
Of course, complementary to what Turing said, one can respond
to (P) too by saying, “That is undoubtedly false”. But what one
means by this is: “Smith is already dead”, or “Smith didn’t have the
necessary drawing instruments”, or similar contingent things. Quite
differently with (H). Calling (H) false for the reason, say, that Smith
did not have the necessary drawing instruments, would be very misleading, to say the least, because the proper reason for the falsity of
(H) is not an empirical but a mathematical one. Whatever Smith may
accomplish, it cannot be the Euclidean construction of a regular
7-gon. This we can claim without knowing anything about Smith at
all, since the impossibility of such a construction has been mathematically proven.
But suppose we meet Smith and he constructs one regular 7-gon
after the other! (See LFM, pp. 46f.) We then would say: “He must
have made a mistake somewhere: the construction is not really a Euclidean one, or the 7-gon is not really regular.” And, of course, it is a
fact that in the majority of cases where someone claims to have
achieved such a construction, we sooner or later find a mistake. But
suppose that in Smith’s case we do not find one. Let no one say that
this is impossible. Of course, it is possible. Our search for a mistake
may fail not only in such obvious cases where Smith’s construction
is extremely complicated such that we loose track of its mechanism;
it may even fail when the construction is easily surveyable. Even
then it could be the case that we simply don’t find a mistake. This
could be a brute empirical fact, and we would overcharge mathematics when demanding that it should demonstrate the impossibility of
facts of this sort. Their possibility or impossibility cannot be the subject matter of mathematics.
How would we react in the situation envisaged, where our search
for a mistake in Smith’s construction failed? Undoubtedly, we
would say: “At the moment, we seem to be struck with blindness, but
Smith somewhere must have made a mistake.” But this reaction is
not the only possible one, and it is not the only sensible one. Other
sensible reactions are entirely conceivable. For example, we also
could say: “It turns out that Smith hasn’t made a specific mistake;
there is no specific fault in his construction; but, as the impossibility
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proof shows, his construction must be wrong ‘on the whole’. There
is no localizable mistake, but on the whole what Smith did is
wrong.” However, it is a fact about ourselves, a characteristic feature
of our actual form of life, that we do not react in such a way.6 We
deem it obvious that, if the construction is wrong ‘on the whole’,
there must exist a specific, localizable fault, even if we don’t find it.
Such is the way we treat wrongness in mathematics.
Quite another, but also sensible reaction would be to distance
oneself from the formerly accepted impossibility proof. This does
not imply that we should have found a mistake in this proof. As before, let us suppose that in the impossibility proof, too, no fault has
been discovered. Nonetheless, we could say that the proof is not relevant to the Euclidean circumstances, for it requires us to go beyond
Euclidean geometry by embedding it in analytic geometry, that is, in
the sort of geometry invented by Descartes which expresses the geometrical relationships arithmetically, as abstract relations between
coordinates. It could be said that by this move we leave the genuinely Euclidean circumstances: The latter ones essentially rely on
intuition, whereas the impossibility proof, by its abstract nature,
from the outset cannot affect issues related to intuition. In this vein
somebody might say that Smith’s constructions actually give us
good reasons to claim that a Euclidean construction of the regular
7-gon is possible after all.
What should we say when confronted with alternatives of this
sort: a seemingly faultless and persuasive proof that the regular
7-gon cannot be constructed, on the one hand, and seemingly faultless and persuasive examples of such constructions, on the other? In
favour of what should we decide? This is difficult to say, and if we
choose the proof – as we presumably are prone to do – this would be
a genuine act of decision indeed. It would be a decision to simply refuse to apply the expression “construction of a regular 7-gon” to
anything whatsoever, or – as Wittgenstein put it – to simply exclude
this expression from our notation (see LFM, pp. 47 and 91). In reality, of course, our situation is considerably less unpleasant because
the alleged counterexamples to our proof typically do not bear closer
scrutiny. That is what we typically experience, and the impossibility
6. This point is stressed by Michael Dummett; see, e.g., Dummett 1994, p. 54.
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proof, therefore, stands out as something special and unassailable.
This is an important empirical fact.
But does this fact mean that we are now relieved from any decision? Not at all. There remains something similar to a decision since
we do not, of course, scrutinize all alleged counterexamples to our
proof; and even if we had so far scrutinized and falsified all of them,
there is no guarantee that no new and more persuasive ones will turn
up in the future. But we decide not to be troubled by insecurities of
this sort and unwaveringly take side with the theorem: After it has
been proven, and as long as no mistakes have been discovered in it,
nothing will be accepted as a legitimate application of the expression “Euclidean construction of a regular 7-gon”. In Wittgenstein’s
own words: “Whether or not we say, ‘There must be a mistake in the
construction’, is a question of decision.”7 In situations like these,
where in our applications of concepts we simply disregard possible
alternatives, Wittgenstein from the beginning of his post-Tractatus
philosophy until its end uses the term “decision” in order to express
the fact that we disregard alternatives without being forced to do so.8
But what about the insight which the proof undoubtedly conveys?
Doesn’t the proof show, or even explain, why there cannot be a Euclidean construction of the regular 7-gon? And doesn’t the proof
force us to admit that? – In a sense, this is obviously so. But what we
call “insight” here is a relative affair, as our thought experiment with
the seemingly faultless and persuasive counterexamples shows. Relative to the empirical fact that we practically never meet such
counterexamples, the theorem appears to give us indubitable insight
into the geometrical facts; but this would be considerably altered if
such counterexamples began to pile up. So simply referring to the
‘insight’ conveyed by the proof of the theorem does not in itself repudiate Wittgenstein’s point that there is an important element of decision in our acceptance of the theorem. The same is true, in a sense,
7. LFM, p. 56. Wittgenstein here actually deals with the trisection of the angle which, however, with respect to the issue at hand, is quite analogous to our
present case. (See, however, note 12, where a disanalogy becomes relevant.)
8. See PR § 149, where in a context of this sort he added a note on the margin
saying: “Act of decision, not insight.” And in C § 368 he writes in an analogous
context: “If someone says that he will recognize no experience as proof of the opposite, that is after all a decision.”
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with respect to the objection that the proof forces us to accept the
theorem. If we begin to think about what this supposed ‘force’really
amounts to, it turns out to be an unclear and evasive affair. Certainly,
we are not causally forced to say “yes” to the theorem: even after
having fully understood the proof, we, of course, are free to say what
we want. But isn’t a certain conviction causally forced on us by the
proof which would be offended by our saying “no” to the theorem?
This may be so, but this force is, so to speak, of the wrong sort. It may
produce any conviction whatsoever, irrespective of the fact that the
conviction is right or wrong, mathematically justified or not. What
we want, however, is a force which in a sense reflects the mathematical domain in that it securely leads us from mathematical truths to
mathematical truths, and this force is not so easy to find. At this point
we tend to take refuge in metaphors like the metaphor of the paths of
mathematically correct thinking which we have to follow and which
lead us to the correct mathematical results, as, for example, the paths
we follow in the proof of our impossibility theorem which lead to the
result that a Euclidean construction of the regular 7-gon does not exist. But this is only metaphorical talk, and the philosophical task,
then, consists in a non-metaphorical description of what is really
happening in such cases.
To put it differently: It is the task of a non-metaphorical description of what is really happening when we are following mathematical rules. Thus we are now amidst the problems concerning rule-following, that is, to use Wittgenstein’s own words in LFM, p. 125, “in
the midst of a large number of queer puzzles”. Of course, I cannot go
into these puzzles here (but see Section 5 below, where the rule-following problem will turn up again). Suffice it to say that deeper investigations would show a characteristic tension between, on the
one hand, an important normative component which – in a sense to
be clarified – in fact may be described as ‘forcing’ us to act in a certain way in order to be in accord with the rule and, on the other hand,
an important freedom in our actions. When looking at this freedom,
together with certain alternative ways in which we could act, our actions indeed appear as the results of decisions. Wittgenstein time and
again presents us such alternatives9 in order to let us see that our ac9. The paradigmatic presentation is in PI § 185, where a pupil reacts to the or-
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tual practice is not the only possible and not the only sensible one.
When confronted with these alternatives, his use of the term “decision” is not unjustified.
It must be admitted, however, and Wittgenstein himself expressly
admits10, that the term “decision” has connotations which are quite
unacceptable in the present context, as it suggests that in the sorts of
situation envisaged – the situation, e.g., when we accept our impossibility theorem – we actually choose one of several alternatives, or
that we even struggle through to this choice. This is wrong on two
counts. Firstly, in reality we scarcely ever think of such alternatives.
They are presented to us only by philosophical reflection. In reality
we normally see only one way, and that’s the way we go. We see the
impossibility theorem and its proof, and we adopt it without much
ado; we are given the instruction to develop the sequence of even
numbers (as in PI § 185), and having reached the number 1000 we
continue with 1002, 1004, 1006 and so on; no alternative occurs to
us. Secondly, even if we accidentally thought of alternatives, we
would rule them out without hesitation. In the sorts of situation envisaged here, we do really not choose at all; we simply act (see LFM,
pp. 237f; PI § 219). In order to do justice to this indubitable fact and
at the same time not lose sight of the possible alternative actions,
which should be recognized, even if they only turn up by philosophical reflection, Wittgenstein sometimes took refuge with the expresder to develop the sequence of even numbers by writing the even numbers from 0
to 1000 and then going on with 1004, 1008, 1012, because this, as we might say,
is his way of understanding the order. Wittgenstein’s later philosophy is replete
with examples of this sort. In PI § 144 he explicitly explains what is their purpose: “But was I trying to draw someone’s attention to the fact that he is capable
of imagining the [alternative behaviour]? – I wanted to put that picture [of alternative behaviour] before him, and his acceptance of this picture consists in his
now being inclined to regard a given case differently: that is, to compare it with
this rather than that set of pictures. I have changed his way of looking at things.”
This new way of looking at things reveals a characteristic space of freedom such
that our actual, definitive actions appear like decisions.
10. See BB, p. 143; LFM, pp. 30f., 237 and 238; and PI § 186, where
Wittgenstein says: “It would almost be correct to say [...] that [in the development of the sequence of even numbers] a new decision was needed at every
stage” (my emphasis).
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sion “spontaneous decision”. In RFM, p. 236, he writes: “‘We decide spontaneously’ (I should say) ‘on an new language game’” (the
new language game, for example, with the expression “Euclidean
construction of the regular 7-gon”, in which we refuse to apply this
expression to anything whatever), and in RFM, p. 326, he explicitly
warns us (or himself) not to fall prey to the wrong connotations:
I have a definite concept of the rule [for example the rule to develop the
sequence of even numbers]. I know what I have to do in any particular
case. I know, that is I am in no doubt: it is obvious to me. I say ‘Of
course’. I can give no reason.
When I say ‘I decide spontaneously’, naturally that does not mean: I
consider which number would really be the best one here and then
plump for …
Why all these contortions? They are characteristic of Wittgenstein’s
endeavour to give sensitive descriptions of our language games
which let us see what is philosophically relevant to them. Very often
our common vocabulary, which typically serves quite different purposes, is quite unfit for such philosophical aims, and when we nevertheless use it for these aims, we must put up with considerable distortions. The term “decision”, in our present context, is a typical example for this. In our philosophical context it has its use, but also its
shortcomings, and in the end, when it has led us to see what Wittgenstein wanted us to see, we might kick it away like the proverbial ladder. But Wittgenstein himself used it, with this philosophical aim,
until the end of his life, and in what follows I will side with him, with
all the caveats needed.
The term “decision” used as a ladder, whether we throw it away in
the end or not, lets us see an important element of concept-determination in the manner in which the proof of the impossibility theorem
leads to our acceptance of this theorem. In Wittgenstein’s own
words (aimed at a mathematically different, but philosophically
analogous situation):
The spectator sees the whole impressive procedure [i.e., the impressive
argumentative steps in the proof]. And he becomes convinced of something [namely, the theorem] […]
[…]
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He tells us: ‘I say that it must be like that’.
[...]
I decide to see things like this. And so, to act in such-and-such a way.
[...]
“This must shows that he has adopted a concept.
[...]
He has read of from the process [i.e. the actual steps carried out in the
proof], not a proposition of natural science [namely: that this process
has happened, or that it is factually possible, or similar empirical
things] but, instead of that, the determination of a concept.” (RFM, pp.
108-110)
In case of our impossibility theorem it is the determination of the
concept “Euclidean construction of a regular 7-gon”: We decided to
use this concept in such a way that nothing should count as an instance of it.
Note that Wittgenstein is not just saying that our mathematical
proposition involves a conceptual decision. This would be true of all
our propositions, whether mathematical or not. All the judgements
we make can be confronted, in the Wittgensteinian way, with alternative ones such that they may appear as ‘decisions’. In fact, Wittgenstein’s rule-following considerations apply to all our applications of concepts, not only to the application of mathematical concepts in purely mathematical contexts. The decisive point is that for
Wittgenstein, mathematical judgements are nothing but conceptual
determinations, they do not possess any factual content. Of course,
as we have seen, our acceptance of the mathematical proposition
that there is no Euclidean construction of the regular 7-gon is backed
up by a lot of empirical facts: that we did not find a mistake in its
proof; that the proof lets us see the geometrical situation in a characteristic sort of light such that our proposition in the end may appear
‘evident’; that all the alleged counter-examples were found to be dubious; that the mathematical experts agree in their judgements concerning our theorem and its proof; and so on. But these empirical
facts certainly do not belong to what is expressed by the mathematical proposition, they do not contribute to its factual content. For
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Wittgenstein, then, no factual content remains and what we really
have is nothing but concept-determination.
This view at first sight appears very unconvincing. It will be objected that Wittgenstein simply was looking for the wrong facts: of
course, our mathematical proposition does not express an empirical
fact but a mathematical one. The following account may seem very
plausible: We all understand what a ‘regular 7-gon’ is and what a
‘Euclidean construction’ consists in; and our proposition simply expresses the mathematical fact that in case of the regular 7-gon such a
construction does not exist. So our mathematical proposition seems
to inform us about a mathematical fact in a way that is quite analogous to the way that an empirical proposition such as “No human being can, without technical aid, jump further than 20 meters”, informs
us about an empirical fact. On a closer look, however, things prove to
be not as easy.
3. The concept of a regular 7-gon within Euclidean geometry
Let us ask, first, whether we really know, within Euclidean geometry, what the expression “regular 7-gon” means. This question is
suggested by another remark of Alan Turing’s, this time an imprudent one. Wittgenstein asks in LFM whether, in case of Euclidean
constructions, the representation of a regular 5-gon, say, is the mathematician’s end, and the construction only the means to this end, or
whether the construction itself is the end. Whereupon Turing replies: “It is the construction, since it would be no good producing a
regular [5-gon] by a fluke.” (LFM, p. 49) What I am interested in
here is not Wittgenstein’s question about means and ends in mathematics (important as it may be), but Turing’s idea of a ‘fluke’. Is it
possible in Euclidean geometry to produce a regular 5-gon, not by a
construction, but rather by a fluke? It seems to me that this does not
make any sense at all. Of course, one can, using a ruler only, draw as
many 5-gons as one pleases, and then, by measuring their sides and
angles, some of them may turn out to be fairly regular; but this process of measurement is not a Euclidean means to guarantee the
equality of the sides and the equality of the angles of the figure.
By “Euclidean geometry” I always mean the geometry as re-
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corded in Euclid’s Elements and as it has been practiced and understood before the advent of more advanced mathematical means, like
its embedding in analytic geometry or the purely formal
axiomatization in Hilbert’s style, accompanied by metamathematical investigations. Of course, the Euclidean sort of geometry, without
embedding in analytic geometry or axiomatization à la Hilbert, is a
respectable piece of mathematics on its own, and what is important,
now, is that in this sort of geometry line segments and angles can
only be shown to be equal by constructing them in the Euclidean
way. In Euclidean geometry, given line segments, and given angles,
can in no way be shown to be equal; this does not make any sense at
all. Rather, line segments, and angles, can only be constructed as
equal. But this means that Turing’s talk about the production of
something ‘by a fluke’ can only make sense in case of the construction of a geometrical figure – a regular n-gon, say – but not in case of
the figure itself, independently of its construction.
But the regular 7-gon cannot be constructed at all. So what sense,
then, could the expression “regular 7-gon” have in Euclidean geometry?11 Measurement has already been excluded as a Euclidean criterion for the equality of the sides and the angles of a 7-gon; and Euclid’s restrictive construction-rules do not provide such criteria via
construction. Furthermore, existence claims of the form “There is a
regular n-gon” also can be proved in Euclidean geometry only by
proving the possibility of a construction of the regular n-gon. So
shouldn’t we say that the expression “regular 7-gon” does not have
any sense at all in Euclidean geometry? This view is not explicitly
expressed by Wittgenstein, but I think it is implicitly contained in his
reflections in LFM and, if provided with the necessary caveats, can
be defended in a Wittgensteinian manner.
To this end, a lot of objections have to be discussed. The most natural objection to the view that in Euclidean geometry “regular
7-gon” lacks sense simply refers to the obvious definition of a regu11. The following reflections are only of a provisional nature. They are questionable because they rely on a mathematical result, the theorem that a Euclidean
construction of the regular 7-gon is impossible, which involves a transgression of
the Euclidean framework. From a Wittgensteinian point of view this is inadmissible. As we will see later, it can be corrected, however.
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lar 7-gon: “A regular 7-gon is a 7-gon with equal sides and equal angles.” Doesn’t everybody understand what that means? – But, as already said, we in fact do not really understand it in Euclidean geometry, because there, in the case of the 7-gon, we do not posses any criterion for the equality of sides and the equality of angles. Of course,
we can, for example, mark seven points on a circle and regard these
as the vertices of a 7-gon; to establish, however, that the distances
between these vertices are equal lies beyond the Euclidean possibilities. The rules of the Euclidean construction game do not allow us to
establish it. So within the system of these rules it simply does not
make sense to talk of ‘equal sides’in the case of a 7-gon. Such talk is
empty.
Perhaps this result should be expressed more carefully by saying
that, with respect to the case in question, no appreciable sense – no
sense worth mentioning – is involved. Of course, even within the restricted system of the Euclidean construction-rules we cannot resist
associating a host of ideas and pictures with expressions like “regular 7-gon”, ideas and pictures stemming from other contexts – contexts of measurement, of other construction-games, of analytic geometry, of Hilbert-style axiomatizations, etc. –; but these associations should not be considered relevant in the Euclidean context.
They are not relevant to the language-game played with the expression “regular 7-gon” within Euclidean geometry. In this game there
is no substantial use for the expression “regular 7-gon”, and when I
say that in Euclidean geometry this expression ‘has no sense’ – or,
more carefully, ‘has no appreciable sense’ – I mean exactly that.12
12. Juliet Floyd recommends a still more cautious attitude. On p. 385f. of
Floyd 1995 she says, with regard to the impossibility of trisecting the angle:
Wittgenstein is not insisting that conjectures in mathematics are meaningless,
or that we do not understand a mathematical proposition until we possess its
proof. […] Whatever shift in understanding takes place as a result of the
proof, it is not that we move from a situation in which there is no concept of
trisecting – that is, a situation in which no meaningful statements may be
made concerning trisection – to a situation in which we now have such a concept, can intelligibly talk. […] Most generally, in the Investigations
Wittgenstein uses the trisection example to try to complicate our idea of what
it is to ‘really’ understand, to fully mean or express, to ‘really’ want to utter, a
particular sentence. […] Wittgenstein is sceptical that there is any systematic
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Admittedly, this sense of “has no appreciable sense” is not very
precise, and it is not part of any ‘semantic theory’. As to be expected
from a Wittgensteinian approach, this sense should become clear
only in the course of a meticulous investigation of our linguistic and
mathematical practice. That is what I try to pursue here, and the discussion of objections to Wittgenstein’s view is part of this investigation. So let us proceed with the objections.
What about Euclidean constructions which approximate the regular 7-gon? Such constructions are possible to an arbitrary degree of
precision: do they not bestow an appreciable sense on the expression
“regular 7-gon” within Euclidean geometry? No, for the simple reason that in Euclidean geometry all approximations are approximations of figures which are constructible. This is quite obvious in the
case of the so-called ‘exhaustions’, where, for example, a circle is
exhausted by a sequence of inscribed polygons. In this case, of
course, the circle has already been constructed (and the inscribed
polygons have to be constructible, too). In Euclidean geometry there
is no exhaustion of non-constructible figures. Furthermore, in the
Euclidean theory of proportions the ratios of two magnitudes a and b
may certainly be seen (anachronistically speaking) as the limit of rational numbers, but for Euclid it does not make sense to speak of
such a limit if the magnitudes a and b are not already given.13 If, for
theoretical account which will informatively distinguish, in particular cases,
between uttering or thinking a sentence with ‘real’ meaning (that is, clearly
and fully or completely expressing a thought, belief, desire or intention) and
uttering or thinking a sentence which does not fully, clearly or completely express a thought, belief, desire or intention. The trisection example serves this
scepticism concerning a general theoretical account of rational (logical) language use – at least if it is unattentive to our applications of logic in particular
circumstances.
I agree with Floyd’s remark about Wittgenstein’s skepticism concerning systematic theoretical accounts of sense, meaning and understanding; and with respect
to the “trisection of the angle” she is certainly right in insisting that this term has
sense within Euclidean geometry; so, for example, there are angles which can be
trisected by Euclidean means (e.g., 90 and 180 degrees). With respect to the regular 7-gon, however, we are in a rather different situation. That is what I am going
to argue.
13. Euclid’s theory, which stems from Eudoxus, is structurally very similar
to the modern theory of Dedekind cuts which leads from the rational to the real
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example, b is the length of the side of a regular 7-gon inscribed in a
circle with radius of length a, then the Euclidean theory of proportions is not applicable to the ratio of b to a, since b is not given, that
is, constructed. A fortiori it does not make sense for Euclid to speak
of an ‘approximation of the regular 7-gon’. Thus, staying within Euclidean geometry, we still haven’t found an appreciable sense of the
expression “regular 7-gon”.
A further objection might be that Wittgenstein’s view makes it incomprehensible how, within Euclidean geometry, it can occur to us
at all to talk about ‘regular 7-gons’ and to try to construct them. If
“regular 7-gon” does not make sense, nobody should be expected to
talk about these things – or better: non-things –, let alone strive for
their construction. – Two replies are in order here. Firstly, Euclid
himself, i.e. the author of the Elements, in fact did not talk about
them! In Euclid’s Elements you neither find any reference to regular
7-gons nor to regular n-gons in general.14 Euclid only deals with
such regular n-gons which he actually can construct. I interpret this
fact as an indication of the appropriateness and realism of Wittgenstein’s view.
Secondly, Wittgenstein can say a good deal about why people,
working in the Euclidean framework, eventually got the idea to attempt a construction of the regular 7-gon. To appropriately understand this, we should realize to what extent our thinking is led, and
misled, by our language, especially by linguistic forms. With regard
to the Euclidean constructibility of regular n-gons, what we in fact
and in a mathematically substantial way find out is that the Euclidean rules permit the construction of regular 3-, 4-, 5- and 6-gons; and
then, by a very superficial and purely formal analogy, we form the
linguistic expressions “regular 7-gon” and “construction of the regular 7-gon” and believe we understand these expressions. But when
we try to give an account of what this alleged understanding really
amounts to, we discover that it does not go beyond our understandnumbers. But there is one decisive difference: (Anachronistically speaking,
again) Euclid presupposes that the cut is already given, whereas Dedekind thinks
that the cut is to be created out of the rational numbers. See Artmann 1999, pp.
123-129.
14. Benno Artmann pointed out this fact to me.
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ing of the expressions for regular 3-, 4-, 5- and 6-gons (including
trivial generalizations thereof) and the aforementioned superficial
linguistic analogy. That is, it is extremely meagre: strictly speaking
it is no understanding at all, at least not in a mathematically substantial sense of “understanding”.
Usually we do not realize this lack of understanding because, as
mentioned earlier, there are many criteria for being a regular 7-gon
which lie outside the domain of Euclidean geometry and, of course,
these criteria, too, are responsible for our having the idea of attempting a construction of the regular 7-gon in Euclidean geometry. There
are, for example, the usual empirical criteria, involving actual measurements, according to which, if we are no fanatics of precision, we
quite loosely may talk about ‘regular 7-gons’. Or, alternatively, we
may rely on criteria belonging to other construction-games which
are richer than the Euclidean one and which allow the construction
of regular 7-gons. Archimedes, for example, by making use of the
marked ruler, was very well in the position to construct the regular
7-gon (see Artman 1999, p. 115). But all these additional criteria do
not belong to Euclidean geometry and, in our present, strictly mathematical context have to be left out of it.
It might be objected that precisely this is the mistake: to leave out
these other criteria, to isolate the Euclidean construction-game from
other language games which involve the expression “regular 7gon”. After all, it certainly is no accident that in all these different
language games we use the same expression: “regular 7-gon”. – But
this is a dubious objection in the philosophy of mathematics. Mathematicians simply regard it as malpractice – as an indication that one
has not understood what a real mathematical approach consists of –
if somebody uses non-Euclidean criteria within Euclidean geometry. This is simply a violation of the Euclidean construction-game.15
15. The term “game” seems quite appropriate in the case of Euclidean geometry, since one cannot discover any stringent reasons for the restrictedness of the
Euclidean construction rules. It is quite unclear, for example, why the use of a
marked ruler is forbidden (see Artmann 1999, pp. 103-107). The Euclidean game
consists in a sort of contest: how far one can get with these rules alone; and contests of this sort are quite characteristic to the agonistic character of the life and
thinking of ancient Greece. (I am grateful to Benno Artmann for stressing this
point to me.)
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And the fact that we (but not Euclid!) nevertheless talk about ‘regular 7-gons’ in the context of Euclidean geometry can be easily explained by the foregoing lines: We are led to this talk (a) by quite superficial linguistic similarities, which do not supply any appreciable
sense, and (b) by inadvertently projecting non-Euclidean uses into
Euclidean geometry, which we actually should refrain from when
practicing real mathematics.
But shouldn’t we recognize the empirical applications of mathematics and doesn’t this, in the end, bestow an appreciable sense upon
“regular 7-gon”? Not in the case of Euclidean geometry. The empirical applications of Euclidean geometry make use of Euclidean constructions – we measure the empirical relations against certain Euclidean constructions – but there is no Euclidean construction of the
regular 7-gon and so the expression “regular 7-gon”, as (superficially) used in Euclidean geometry, cannot be empirically applied at
all. Any empirical application of this expression must rely on
non-Euclidean insights.
So our foregoing result stands: within the restricted framework of
Euclidean geometry the expression “regular 7-gon” does not have an
appreciable sense. A fortiori the same is true, then, of the expression
“construction of a regular 7-gon”, and the search for such constructions has to be understood accordingly. This search is rather vaguely
oriented by the Euclidean constructions already contrived, especially the constructions of the regular 3-, 4-, 5- and 6-gon, but it is not
– and cannot be – guided by any respectable mathematical idea of
what has to be reached. Such an idea cannot exist, since – as we
know today – the sought-for construction is not mathematically conceivable within Euclidean geometry. The ‘content’, as it were, of
this search consists of scarcely more than a blind hope that one day
we might stumble across a construction of which we would say,
then, “This is the construction of a regular 7-gon”. There is scarcely
any mathematical substance in this hope.
4. The impossibility of a Euclidean construction of the regular 7-gon
If the foregoing considerations are correct we must say that we do
not really understand what “regular 7-gon” means when we remain
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within the restrictive framework of Euclidean geometry. Since Descartes, however, we are used to extend this framework almost automatically by embedding Euclidean into analytic geometry, and
thereby our understanding is strongly enriched. The expression
“regular 7-gon” can now be explained and used in mathematically
substantial ways; e.g., we can specify now, by means of an algebraic
formula, the length of the sides of a regular 7-gon inscribed in the
unit circle; etcetera. And we have a tendency, then, to project this
new-gained understanding back into the Euclidean realm, as if in analytic geometry we only had made explicit what was implicitly contained in Euclidean geometry already; as if, for example, the sense
of “regular 7-gon”, as we understand it now, had already been present, though in a hidden way, within Euclidean geometry. This view,
of course, contradicts the foregoing reflections, and we therefore
should turn to the transition from Euclidean to analytic geometry in
order to see more clearly what’s happening there with the concepts
involved.
One of the most important aspects of this transition certainly lies
in the fact that we can now answer questions concerning the possibility and impossibility of Euclidean constructions in truly mathematical ways. As before, let us consider the specific example of the
regular 7-gon, and let us stay for a little while longer within Euclidean geometry. When, after a lot of futile searching, people eventually began to suspect that the Euclidean construction of the regular
7-gon might be impossible, they at first could use the term “impossible” only in a very weak and merely empirical sense: One had made
many attempts at such a construction; more precisely, to accomplish
something which one would say was the construction of a regular
7-gon; all these attempts failed; and sooner or later one said: “Maybe
it’s impossible.” The reason for this claim then is only an empirical
one, and even as such of an extremely weak sort: The construction
simply had not yet succeeded. Usually, our empirical impossibility
claims are of a much stronger nature, since they can be backed up by
additional reasons. If one says, e.g., “No human being can, without
technical aid, jump further than 20 meters”, this claim can be justified not only by the fact that so far nobody succeeded in jumping further than 20 meters, but also by theoretical reasons concerning our
physical capabilities. In our mathematical case, not even that was
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possible. There was only the mere fact that constructions had not
been found so far.
However, people had some sort of ersatz for this shortcoming in
the form of certain platonistic phantasies such as the following: In
Euclid’s Elements the rules of the Euclidean construction-game are
recorded and thereby it is determined once and for all which geometrical figures are constructible according to these rules and which are
not. And the regular 7-gon – whether we know it or not – either belongs to the constructible ones or does not. Tertium non datur.
Through the formulation of the Euclidean construction-rules, at one
blow, as by magic, the entire space of all constructible geometric figures is stretched, and saying that the construction of the regular
7-gon is ‘impossible’ is nothing more than saying that the regular
7-gon does not lie in this space. – Most of us, I think, entertain a
phantasy of this sort. Let us call it the “phantasy of the space of all
constructible figures”.
In the restricted framework of Euclidean geometry, however, this
is merely a phantasy without any serious mathematical substance,
because in this framework we can make sense of the constructibility
of a figure only by actually constructing it, and the non-constructibility is only accessible by a nebulous suspicion. Strictly speaking,
there is no concept of non-constructibility at all, but only the concept
“not yet constructed”.
But if we now make the transition to analytic geometry, things
change dramatically: Our phantasy looses its air of magic and gains
mathematical substance. Now we are able, like 19-year-old Gauss,
to prove for example the constructibility of the regular 17-gon without actually constructing it, simply by showing that, say, the cosine
of 360°/17 can be obtained by certain algebraic operations which
correspond to the Euclidean construction-rules. To express it a bit
more technically: that the cosine of 360°/17 belongs to an iterated
quadratic extension of the field of rational numbers. And we are now
able to prove the non-constructibility of the regular 7-gon by showing that the cosine of 360°/7 does not lie in such an extension of the
field of rational numbers.16 The magical phantasy of the space of all
16. This can be shown by a simple proof by contradiction which is very similar to the classical proof of the irrationality of the square root of 2. See Martin
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constructible figures has now been replaced by mathematics.
But we have a tendency, then, to project this new-gained geometrical substance back into Euclidean geometry, as if, in some mysteriously hidden way, it were already present there. This tendency is unjustified, but easily explained. It stems from the psychological fact
that to most of us the usual embedding of Euclidean into analytic geometry appears completely natural. When acquainted with the normal translations of Euclidean terms and construction-rules into the
language of analytic geometry, we react by saying, “Yes, that’s the
way it should be done”, or “Yes, that’s exactly how it was meant in
Euclidean geometry”. But by this we merely express that we feel the
translations to be natural and are ready to accept them. By saying,
“Yes, that’s exactly how it was meant in Euclidean geometry”, we
certainly do not mean that the Greeks had these translations already
in mind17, nor do we mean that these translations existed in any
other, mysterious, way. Even the staunchest Platonist would not
claim so. Obviously, these translations were created during the development of analytic geometry, and our statement, “Yes, that’s how it
was meant”, can only mean that we now find them natural and acceptable. In other words, these so-called ‘translations’ are not translations in the sense that they would ‘transfer pre-existing meanings’.
Our foregoing conclusion that in Euclidean geometry the expression
“regular 7-gon” has not been given any (appreciable) meaning remains valid even if we homophonically translate this expression – or
1998, pp. 28 and 45. For the following it is by no means necessary to understand
these mathematical technicalities.
17. At least, we should not mean that. From time to time, however, mathematicians tend to talk in exactly that way; for example Joe Shipman who, answering
the question of what he meant by “the structure of R2 in the geometric sense” (i.e.
“the structure of the Euclidean plane” in our present-day understanding of “Euclidean plane”), said the following: “Of course I meant what the mathematicians
in Descartes’s day meant. And the true first-order sentence about this structure
(using Euclid’s primitive terms and a standard Cartesian translation between
terms like ‘right angle’and real arithmetic) are not just an ‘arbitrary’theory. They
are the propositions Euclid, Descartes, Kant, etc. had in mind all along!” This
can be found on the Internet: http://www.math.psu.edu/simpson/fom/postings/9810/msg00062.html. I hope that Shipman does not want to be understood
too literally here.
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better: take it over – into analytic geometry. We should not be misled
by the word “translation”.
This point can be considerably deepened when we imagine – as I
have done before – people, members of a different culture, who do
not find these translations natural and acceptable, whose practices
belong, to speak in Wittgenstein’s way, to a different form of life.
Our mathematical form of life is characterized by the fact that, as
soon as the non-constructibility of the regular 7-gon has been proven
within Cartesian analytic geometry, professional mathematicians
terminate all attempts at constructing regular 7-gons in pre-Cartesian Euclidean geometry as well, and someone who carries on with
such attempts is regarded as a ‘crank’. In this other form of life the
reactions to the impossibility proofs might be quite different. People
there would say, perhaps, “Of course, it’s not so easy to give a Euclidean construction of a regular 7-gon”, but they would continue
with their attempts since, for them, results in analytic geometry simply are not relevant to Euclidean geometry.
This story is, I think, much less fictitious than it may appear, and
if it is well told (and in more detail than I can provide here) we might
be led to saying that those people are by no means ‘less intelligent’
than we are, but that they are simply different from us. Their form of
life is not ours, but it is not ‘wrong’. In the 19th century one can find,
in fact, tendencies of this sort, when certain professional geometers,
for example Poncelet, vehemently tried to defend the independence
of geometry in opposition to arithmetic and algebra. This is also true
of Frege, who, by basing geometry on pure intuition and arithmetic
on logic alone, insisted upon a categorical difference between the
two.18 I do not know whether any mathematician in the 19th century
went so far as to consider the impossibility proofs of analytic geom18. More precisely, it is true of the Frege of the Grundlagen and the
Grundgesetze der Arithmetik. In his later years, after having convinced himself
that his logicist project was shipwrecked, Frege came to the conclusion that
arithmetic, too, must be based on pure intuition; see Dummett 1982. As for
Poncelet’s view, see, e.g., Wilson 1992. Bernd Buldt pointed out to me that the
insistence on a categorical difference between arithmetic and geometry can, of
course, be traced back to Aristotle and the Aristotelians who quite generally emphasized the characteristic, distinguishing features of different disciplines; to
this see, e.g., Funkenstein 1986, pp. 35ff. and 303ff.
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etry irrelevant to pre-Cartesian Euclidean geometry, but there must
have been a temptation for this in certain quarters.
Why, then, does this form of life nevertheless appear so unattractive to us? I think this is simply due to certain fundamental empirical
facts. Remember the earlier-mentioned Smith, who, as it seemed,
was constructing one regular 7-gon after the other and who could not
be shown to be mistaken. If such events occurred rather frequently,
the alternative form of life just imagined might not appear so strange
after all. Without doubt, empirical facts of this sort also might affect
our sense of what is ‘natural’, and, furthermore, in light of such facts
the alternative form of life might even be very successful in its empirical applications. Our reality, however, is different, and so we regard our mathematical form of life as distinguished. Rightly so, but
this right is not based on the ‘mathematical correctness’ of our form
of life – that it, so to speak, ‘correctly reflects the mathematical
facts’ – but rather on the aforementioned empirical facts to which it
is so well adapted.
This is a down-to-earth view of mathematics, which is very different from the way many mathematicians see their science. Let me,
in order to illustrate this, quote a short passage from Hermann
Weyl’s preface, written in 1913, to his book Die Idee der Riemannschen Fläche. (Weyl’s pathos certainly is not representative of mathematicians, but the tendency of his assertion will, I suppose, meet
with agreement by many of them.) Weyl says the following about
Riemann surfaces: “One now and then still comes across the view
that the Riemann surface is nothing but a ‘picture’, a […] means to
bring to mind and to picture the multiple-valuedness of [analytic]
functions. This view is completely wrong. The Riemann surface is
nothing […] which, in an a posteriori and more or less artificial way,
is distilled from the analytic functions, but has to be considered as
something prior, as the topsoil necessary for the functions to grow
and to flourish.” (Weyl 1955, pp. VIIf.; my translation) Analogously, somebody might want to say that analytic geometry is the
topsoil necessary for the Euclidean entities to grow and flourish – as
if Euclidean geometry from the outset were aimed at what analytic
geometry reveals. But this would be a transfiguring myth, and
Wittgenstein’s philosophy fights against such myths and tries to redirect us, beyond the verbal trimmings of some mathematicians, to-
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ward the actual practice in mathematics which leaves us with a much
more down-to-earth impression.
5. Rule-following
It is time, now, to come back to an earlier issue which has to be corrected and deepened. It concerns what I have called the “phantasy of
the space of all constructible figures”, i.e. the idea that through the
formulation of the rules of the Euclidean construction-game is determined once and for all which figures can be produced according
to the rules and which cannot. I called this idea a “phantasy”, and
even said that it involves some sort of magic, since one cannot see in
which way the formulation of these rules manages to establish the
space of all constructible figures. When we write down or read these
rules, and when we do this with all the understanding we may afford
and with as many additional explanations we may deem necessary,
nevertheless only very few of the applications of the rules, i.e. only
very few constructions, are actually present to our mind – and the
rest of all the possible constructions should then be somehow anticipated already. The question then is: exactly how? exactly in which
way? These possible constructions certainly are not ‘contained’, in a
mysterious way, ‘in our minds’. So, how should we conceive of this
anticipation in the formulation of the rules?
This is, again, Wittgenstein’s rule-following problem which, after all, cannot be completely avoided in the present context. It is a
very general problem. It concerns all axiom systems because one
can always ask how the formulation of the axioms and the inference
rules determines the rest of the theorems, and it concerns – much
more generally, still – rules of any sort, because one can always ask
how, through the formulation of a rule, including all understanding
and explanation you may afford, all its applications are determined;
more precisely: how it is determined what has to be counted as correct and as incorrect application of the rule. Wittgenstein’s question
is: How should we conceive of this determinateness?
Of course, I cannot really go into this problem here. Suffice it to
say that, in any case, this determinateness should not be seen as a
causal one. A paradigm for causal determinateness would be a real
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machine wherein, by pushing a button, a mechanism is set in action
which causally produces a certain result. In an analogous way one
might conceive of a human being whose understanding of a rule corresponds with a certain inner state which, when stimulated by a certain query, activates a mechanism which causes a certain answer; in
case of the rule of counting, for example, when asked “What is the
next natural number after 135?” the mechanism would cause the answer “136”. Obviously, this is not the determinateness of rule-following since a purely causal determinateness lacks the normative dimension, which distinguishes between ‘right’ and ‘wrong’ answers,
that is characteristic of rules. Purely causal processes occur as they
occur, and there is no standard of correctness against which they
should be measured. If our causal mechanism, after having been
stimulated by the question ”What is the next natural number after
135?”, caused the answer “137”, or any other answer, we would have
to accept that as the appropriate one if we were oriented by causal
mechanisms only. With respect to rules and their applications this is
obviously different: In this case, according to what we understand
by ”counting", only the answer “136” is the correct one, no matter
what the causal mechanisms lead us to do.19
So how should this determinateness of the applications of rules,
with its characteristic normative component, be conceived of?
Wittgenstein gives a twofold answer, a negative and a positive one.
The negative answer says that it is inappropriate to suppose that
there is, on the one hand, a definite ‘inner state’ constituting our understanding of the rule, which is present in a finished way as soon as
we have understood the rule; and that there is, on the other hand, the
array of our single acts of applying this rule, which in principle is
never finished; and to ask, then, how this inner state determines
these acts in the sense that it fixes which of the acts are in accordance
with the rule and which are not. When we describe the situation in
this way we generate a gulf between the inner state and the acts,
which cannot be closed, such that the determinateness of these acts –
the way these acts are determined as ‘right’ or ‘wrong’ according to
19. See Mühlhölzer 1998 for an attempt to show in detail why the causal
mechanisms typically referred to in cognitive science are not the appropriate
ones for understanding the phenomenon of rule-following.
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the rule – appears completely mysterious.
Wittgenstein’s positive answer is, very roughly speaking, as follows: Our practice of rule-following is not ‘based on’ our understanding the rules, it is not, so to speak, ‘derived from’ this understanding. On the contrary, this practice itself is the basis of our concept of ‘understanding a rule’. Our acts of applying a rule serve as
important criteria for our understanding; our understanding manifests itself in these acts, and without such manifestations the concept
of ‘understanding a rule’ lacks any appreciable content. Furthermore, the way a person applies a rule also indicates which rule it is
that the person has understood. To express it very roughly: It is the
applications of a rule which determine the rule, and not the other
way round. The rule-following problem thus has been dissolved.
This view is highly relevant to all the foregoing considerations,
since it underpins the general insight underlying the whole of
Wittgenstein’s philosophy of mathematics: that it is on our actual
practice, our actual behaviour, especially our verbal behaviour, we
should concentrate when dealing with questions about the sense and
our understanding of mathematical expressions. In order to further
elucidate this insight, let us come back to our former discussion of
the expression “regular 7-gon” in Euclidean geometry, which must
now be corrected in the light of what Wittgenstein says about
rule-following. I was arguing that in the restricted framework of Euclidean geometry the expression “regular 7-gon” does not have any
appreciable sense, and the essential pillar of this argument was the
fact that regular 7-gons cannot be constructed within Euclidean geometry. But this mathematical insight does not belong to Euclidean
geometry: it did not exist (as an insight, rather than as a ‘conjecture’)
in the practice of Euclidean geometers, and so I actually violated the
Wittgensteinian maxim to rely only on the practice in hand and on
nothing else. This was a mistake which, however, can easily be corrected. It is enough to refer to the fact that within Euclidean practice
the regular 7-gon simply has not been constructed, that all attempts
failed, such that the alleged understanding of the expression “regular
7-gon” actually could not manifest itself. Consequently this understanding did not exist, at least not in a mathematically substantive
way, and consequently “regular 7-gon” did not have an appreciable
sense. This insight remains valid even if now we can no longer afford
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to make use of the results of analytic geometry. The factual poorness
of the Euclidean practice alone is sufficient to make this philosophical point.
6. Holism and the necessity of further investigations
One final, and extremely important, objection against the
Wittgensteinian view should still be discussed. It was an essential
trait of the foregoing argumentation that the Euclidean construction-game has been isolated from all alternative geometrical and
other mathematical activities. This, however, contradicts a nowadays popular view of mathematics, a holistic one, which is known
from Quine’s very influential work and which considers the whole
of mathematics to be an integral part of our total theory of the world,
such that it has no need to discuss mathematical subtheories, like Euclidean geometry. The objection then says that our Wittgen- steinian
argumentation, which deals with single subtheories only, simply
contradicts important, unavoidable holistic insights and must be rejected.
There are weighty objections, however, which can be raised
against Quinean holism when applied to the realm of mathematics.
It seems to me that, with respect to mathematics, the Quinean view
can only appear plausible when, from the outset, mathematics as a
whole is presented in a unifying reconstruction, say, the usual settheoretical one. Mathematics in its entirety is regarded, then, as
nothing but set theory, and it is set theory which ‘faces the tribunal of
experience’ 20 in company with all the other theories belonging to our
total theory of the world. According to this view, all mathematical
subtheories – theories from antiquity until today – are already ‘implicitly contained’ in set theory. Of course, this view has hardly anything to do with the practice of real mathematics that Wittgenstein is
20. See Quine 1951, pp. 41-45, especially: “the abstract entities which are
the substance of mathematics – ultimately [!] classes and classes of classes and
so on up – are another posit [alongside with physical objects] in the same spirit.
Epistemologically these are myths on the same footing with physical objects and
gods, neither better nor worse except for differences in the degree to which they
expedite our dealings with sense experiences” (p. 45).
244
interested in. The thesis that mathematics in its entirety is ‘ultimately’ nothing but set theory has been an ideology during a short
spell in this century and may be useful for specific mathematical
purposes. However, it can hardly be taken seriously as an assertion
about the real phenomenon of mathematics in our real world.
So Quinean holism with respect to mathematics is by no means a
plausible position, and it seems to me that, on the contrary, an essential characteristic of mathematics must be seen in the fact that its different subtheories are considered in isolation from each other. Of
course, important mathematical progress very often is made by establishing interesting relations between different subtheories – as
the case of Euclidean and analytic geometry convincingly shows –
but this does not alter the fact that in mathematics we are not allowed
to simply transport criteria belonging to one theory into another. In
this respect, mathematics is deeply different from the empirical sciences. – This point certainly should be elaborated further; however,
this cannot be done here. Suffice it to say that Wittgenstein’s point of
view by no means proves to be untenable when confronted with
Quinean holism.
Let us now take stock. I interpret the foregoing results as confirming Wittgenstein’s view that mathematical propositions are nothing
but concept-determinations. We took one proposition as our example: the proposition that there is no Euclidean construction of the
regular 7-gon, which is proved in the framework of analytic geometry. Wittgenstein’s view is that this proposition does not describe a
mathematical fact but merely reflects our decision to use the concept
“Euclidean construction of the regular 7-gon” in such a way that
nothing should be counted as an instance of it. We confirmed this
view by (a) in fact diagnosing an essential element of ‘decision’ in it
and by (b) rejecting its natural alternative which says that we already
understand this concept sufficiently well in Euclidean geometry and
that the impossibility proof simply reveals the mathematical fact
that the concept so understood doesn’t have instances. This seemingly natural alternative is dubious because a sufficient understanding of the concept “Euclidean construction of the regular 7-gon” is
not to be found in Euclidean geometry. The concept of a regular
7-gon does not have an appreciable sense there.
By its restrictedness to Euclidean and elementary analytic geom-
245
etry, the foregoing discussion had the disadvantage of being a little
antiquated, but it was very well suited for exemplifying, based on
Wittgenstein’s own reflections in LFM, the characteristic philosophical method of the later Wittgenstein. Certainly, our example
could – and should – be discussed further, by taking into account
formalistic reconstructions of Euclidean geometry in the manner of
Hilbert or other sorts of embedding one mathematical structure into
another (a manoeuver omnipresent in mathematics), and, of course,
many other examples from all mathematical domains should be discussed as well. In all these cases conceptual investigations in the
way just exercised are possible, and in all of them Wittgenstein’s
view, that mathematical propositions are more similar to concept-determinations than to reports of facts, should be examined.
This view, of course, concerns not only antiquated mathematics but
the whole of mathematics from Euclid to the present day. So a huge
field of philosophical investigation opens up, which, in particular,
has the task of discussing and rejecting all sorts of objections which
immediately come to mind against such a seemingly strange, and, at
first sight, implausible view as the Wittgensteinian one. In our present discussion there was the objection, for example, that this view
makes it incomprehensible how, within Euclidean geometry, it can
occur to us at all to talk about ‘regular 7-gons’ and to try to construct
them. We have seen, however, that Wittgenstein could answer this
objection. But new contexts will bring along new objections and
Wittgenstein’s view will have to stand the test every time afresh.
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