Alternative Mathematics: The Vague Way

JEAN PAUL VAN BENDEGEM

ALTERNATIVE MATHEMATICS: THE VAGUE WAY

ABSTRACT. Is alternative mathematics possible? More specifically, is it possible to ima-gine that mathematics could have developed in any other than the actual direction? Theanswer defended in this paper is yes, and the proof consists of a direct demonstration. Analternative mathematics that uses vague concepts and predicates is outlined, leading up totheorems such as “Small numbers have few prime factors”.

1. THE CONTINGENCIES OF MATHEMATICS

Philosophers of mathematics on the one hand and mathematicians them-selves on the other are extremely fond of some kind of platonism.

In its strongest form, it involves the existence of a mathematical world(or World?) to be clearly distinguished from the physical world, we,mortals, happen to inhabit. Nevertheless, mathematicians, through somestrange and hard to explain faculty, usually called mathematical intuition,succeed in penetrating into this mysterious realm. Therefore what math-ematicians do, is to describe this ideal world (or World?) in the fullestdetail possible. It follows straightaway that one and only one descriptionwill do, so mathematicians have a clear goal to attain. It has to developin a cumulative fashion – leaving room for occasional historical accidentswhere intuition was badly understood – and it has to do sonecessarily.

In its weakest form, platonism makes a claim about the necessity ofmathematical development. One can leave room for the possibility thatthe description we now have, may not necessarily be the right one, butsomewhere out there the right one truly exists, so, in the long run – whetherhumanly reachable or pointing at omega, is a matter for discussion – wewill arrive at that description.

In short, mathematics and contingency exclude one another. In the verysame way that a statement such as “2 + 2 = 4” appears to us as necessarilytrue, so the development of mathematics cannot be otherwise. One of thefinest examples of this way of thinking is the necessary development fromnatural numbers to integers, to rationals, to reals, to complex numbers and,finally, to quaternions and there the story ends. There is actually a theoremstating that “ifD is a finite-dimensional vector space overR (the reals),

Synthese125: 19–31, 2000.© 2000Kluwer Academic Publishers. Printed in the Netherlands.

20 JEAN PAUL VAN BENDEGEM

then it must be isomorphic to the quaternions, or to the complex numbers,or to the reals themselves” (see MacLane (1986, 121)).

Although there are many platonists, weak and strong, not everyonehappens to be one. Intuitionists (of different kinds), constructivists (ofall kinds), strict finitists (of rare kind) stress the fact that mathematics isa human enterprise. Imre Lakatos and sociologists of mathematics (seeRestivo et al. (1993) for an overview) confirm that even the development ofmathematics is typically human because it demonstrates all the mistakes,blunders, and other characteristics of human doing and thinking. In thiscontext, contingencies are the rule, necessities are the exceptions, probingthe rule. Hence, alternative mathematics has to be seen as a real possibility.The mathematics we have today just happens to be one particular way ofdoing things, but it could very well have been completely different. At thisstage of the discussion, the same question pops up every time (and, let therebe no mistake, rightly so):What on earth could alternative mathematicslook like?

In order to avoid trivialities, let us put aside answers to this question thatare all too easy. An example will make clear what I mean. Take classicalnumber theory. Take any predicate, say “being a prime number”, Pr(n),wheren is a natural number. Now either Pr(n) or ∼Pr(n). Define a newpredicate Comp(n) such that Comp(n) if and only if ∼Pr(n). Then it isthe case that we could develop an arithmetic that does not mention primenumbers except as a derived notion, namely prime numbers are numbersthat are not composite. It is a rather amusing exercise to “translate” clas-sical theorems in this “new” language.1 But the label “alternative” does notreally apply in this case.

A serious candidate or candidates of recent times for alternative math-ematics are no doubt, the proposals for inconsistent and/or paraconsistentmathematics. These approaches differ clearly from existing mathematics.So, from this point of view, the case for the contingency of the devel-opment of mathematics has been made.2 However, one might still havequalms about whether these alternatives are real historical possibilities. Inother words, the additional question is: is it actually possible that math-ematics could have developed in this way? Although it is clear that JohnWallis, Isaac Newton, Gottfried Wilhelm Leibniz, Leonhard Euler, and somany others, struggled with inconsistencies in their mathematical theories(see Grattan-Guinness (1992, 1997) for overviews), they rarely had theattitude – with the possible exception of Georg Cantor (for full details,see Dauben (1979)) – of adopting them and integrating them in theirtheories. In summary, the topic of this paper is the question whether analternative mathematics is possible that is sufficiently different from, yet

ALTERNATIVE MATHEMATICS: THE VAGUE WAY 21

to some degree compatible with mathematics as we know it and how it hasdeveloped?3

The claim that forms the core of this paper is, firstly, that the answeris yes, and, secondly, that the clue concept isvagueness. Even a rathersuperficial look at the history of mathematics indicates that a story can betold that centers around the elimination of vague concepts from mathem-atics and their replacement by sharply defined concepts instead. Thus, thenotion of small and large numbers disappears quite soon from mathematics(see, e.g., the now rarely mentioned, yet famous text of Archimedes,TheSand Reckoner, dealing with large numbers4), and the notion of infinites-imals – perhaps one of the vaguest concepts ever – has been eliminatedand replaced by the notion of limit (see Boyer (1959) for a classic). It is,of course, well known that sharply defined predicates are a lot easier towork with than vague predicates, but that does not entail that it is thereforeimpossible. In this paper I will sketch the outlines of vague mathematics,an alternative form of mathematics.5

2. PROBLEMS WITH VAGUENESS

It is generally known that the idea of vagueness is a deep and difficult oneto handle. As far as I know, there is not one solution to what vagueness isand how it can formally be expressed that satisfies everyone. Neverthelessit must be emphasized that in our daily speech we seem to be able to handlevague predicates without too many difficulties. Whether we are talkingabout beards, heaps, having-a-lot-of-money,. . . weseem to reach agree-ment most of the time. It is not my intention in this paper to present a fulldiscussion of the problem. I will, quite simply, pick my favorite approach.Nevertheless, let me say a few words about why I believe that at least somesolutions can be dismissed at once. If we do not enter into the details – afull treatment of vagueness can be found in Williamson (1994); see alsoSainsbury (1995) – two sets of solutions can be distinguished:

(a)Replacement solutions. A vague predicate can always be replaced bya sharply defined predicate. Thus a predicate such as “n is a small number”should be interpreted as stating that “There is a fixed numberN , such thatall numbersn ≤ N will be called small and all numbersn > N will becalled not small. But this is a solution that eliminates vagueness, becausethe predicate “small” in the redefined sense no longer means small, butclearly something else.6


(b) The fuzzy logic (and many-valued) solutions. In the naive sense, theidea of a fuzzy logic interpretation is that a predicate such as “having abeard” orB(x) for short, is to interpreted in terms of the number of hairson x’s chin. In addition, two numbersL andM have to be specified, afixed lower and a fixed upper bound, respectively. Now, ifI stands for theinterpretation of “having a beard”, then we can stipulate that:

I (B(x)) = 1 if the numbern of hairs onx’s chin> M

I(B(x)) = 0 if the numbern of hairs onx’s chin< L,

and, for the in between cases, i.e., L≤ the numbern of hairs onx’s chin≤ M,

I (B(x)) = (n− L)/(M − L).

The true problem with this solution is that, if I happen to see someonewith (no specified number of) hairs on his chin, I am not going to countthe number n of hairs and, above all, I will definitely not utter a statementof the form “That he has a beard, I am willing to accept for about 0.8733”.In a sense, the fuzzy logic solution does way too much.

Talking quite generally, vagueness has something to do with being un-decided, with open cases, doubtful cases, with borderline cases and so on.Thus any solution that proposes truth values defined everywhere, whetherit is the classical set {0,1} or the continuum interval [0,1] has a seriousdefect. Therefore, I will use a solution that is, of course, not withoutits own problems, but that avoids, to some extent, the difficulties of thesolutions discussed above. I am referring to thesupervaluationalinterpret-ation of vague predicates. To be absolutely clear, I see supervaluations asa useful first-order approximation to a “decent” solution to the problemof vagueness. To put it differently: as it is my task to show that alternativemathematics is possible and, if a first-order approximation suffices to showthis, then probably more sophisticated approaches will do even better and,therefore, I can and will restrict myself to this approximation.

Let me briefly summarize the general idea of a supervaluationalapproach. Given a standard first-order languageL,

(i) Start with a notion of a “classical” modelM, i.e., a model where thevaluation functionv is such thatv(A) = 0 or 1, for all wffsA in thelanguageL,

(ii) Define a setM of such modelsM,(iii) Then define a supervaluationV , such that:


V (A) = 1 if for all valuationsv in modelsM ∈ M , v(A) = 1

V (A) = 0 if for all valuationsv in modelsM ∈ M , v(A) = 0

V (A) = u in all other cases, whereu stands for undecided orundetermined.

As is well-known, one of the major advantages of this approach is the factthat classical truths remain true. This will imply that the vague theories,if they contain the classical predicates, are in fact extensions of the clas-sical theory. As is equally well-known, there is a problem with validity,since there are two possible notions: local and global validity. Thus, givenpremissesA1, A2, . . . ,An andB, B is a locally valid consequence iff inany modelMi , if v(A1) = v(A2) = · · · = v(An) = 1 thenv(B) = 1. Bis globally valid consequence iff ifV (A1) = V (A2) = · · · = V (An) = 1thenV (B) = 1. As most of the theorems below involve single statements,I will not address this issue here.

The most serious drawback of the supervaluational approach is, ofcourse, that it is assumed that the vague predicate can be made precisein a number of ways, thus, it is, in principle at least, eliminable. Does itthen make sense to speak of alternative mathematics, if classical modelsare present in the background? First, the point I want to make is not thatalternative mathematics should be in some sense “incommensurable” withexisting mathematics, rather that it can be sufficiently different. Second,I could of course have presented the alternative in vague terms straightaway, but I doubt that many readers would accept that a statement such as“Small numbers have few prime numbers” is a theorem of vague numbertheory, unless some interpretation is given. Third, all these considerationsdo not exclude the possibility that historically vague mathematics couldhave developed relying on predicates such as “small”, “few”, and so on, asprimitives.

3. THE GENERAL STRATEGY

If, with all the necessary precautions outlined above, a supervaluationalinterpretation is used for vague predicates, then the following strategy canbe formulated for introducing vagueness into an existing mathematicaltheory.

(i) Consider any classical first-order mathematical theory MT. In thelanguageLMT we will have constants, variables, predicates, and logicalconnectors. A modelM will be a triple〈D, I, v〉, whereD is the domain of


the model,I the interpretation function that maps constants and variablesonD and predicatesP of rankr on a subset ofDr , ther-cartesian productof D. Finally v is the valuation function that maps sentencesA onto {0,1}according to some set of semantical rules.

(ii) In particular, the interpretation of a predicateP (of rank r) willbe I+(P ) ⊆ Dr . Call I+(P ) the positiveextension andDr\I+(P ) =I−(P ), the negativeextension. Thus to a predicateP in the languagewill correspond the full interpretationI (P ) = 〈I+(P ), I−(P )〉 such thatI+(P ) ∪ I−(P ) = Dr andI+(P ) ∩ I−(P ) = ∅.

(iii) Now consider a setM of modelsM1, M2, . . . ,Mn such that, if,in Mi the predicateP has an interpretationIi(P ) = 〈Ii+(P ), Ii−(P )〉and such that, ifi < j , thenIi+(P ) ⊆ Ij+(P ) (and, hence,Ij−(P ) ⊆Ii−(P ). Thus we obtain a “nested” set of models.

(iv) Finally, apply supervaluation to the setM to obtain truths, false-hoods and undetermined cases.

For simplicity, I will assume that the classical theory MT has a pre-ferred interpretation, call it the standard interpretation. Thus for the naturalnumbers, I am referring, classically speaking, to the standard model.

Note that this strategy can be applied in two ways (as I will do in whatfollows). Either one can introduce new predicates – such as “being a smallnumber” – or one can introduce for an existing classical sharp predicate avague “companion”, e.g., the predicate “being almost a multiple ofn” canbe associated to the predicate “being a multiple ofn”.

4. NUMBER THEORY: THE VAGUE WAY

Throughout this paragraph, I will assume that a classical model for numbertheory is a tripleM = 〈D, I, v〉, such that

(i) D is the standard set of natural numbers, indicated by bold letters,i.e.,D = {0, 1, 2, . . . },

(ii) I is the interpretation function that maps 0 to0, and that mapsn to n,(iii) v is a classical valuation function, i.e.,

v(∼A) = 1 iff v(A) = 0,

v(A&B) = 1 iff v(A) = v(B) = 1,

v((∃x)A(x)) = 1 iff there is an interpretation functionI ′that differs fromI at most in the value ofI (x), and suchthatv(A(x)) = 1 under that interpretation.


Example 1: Adding new predicates. The predicates “small” orS(n) and“large” orL(n) can be added to the language. Both are rank 1. As to theirinterpretation, what one has to do is to specify the setM of models. Wecan select two numbersS1 andSn, S1 < Sn, such that, for any numberSi ,such thatS1 ≤ Si ≤ Sn, there is corresponding modelMi, where:

Ii(S) = 〈Ii+(S), Ii−(S)〉= 〈{0,1, . . . , Si}, {Si+1, Si+2, . . .}〉.

It is easy to see that the condition: ifi < j , thenIi+(S) ⊆ Ij+(S), issatisfied.

The same procedure can be carried out forL(n). Again, we can selecttwo numbersL1 andLm,L1 > Lm, such that, for any numberLi, such thatL1 ≥ Li ≥ Lm, there is corresponding modelMi , where:

Ii(L) = 〈Ii+(L), Ii−(L)〉= 〈{Li+1Li+2, . . .}, {0,1, . . . , Li}〉.

It is easy to see that the condition: ifi < j , thenIi+(L) ⊆ Ij+(L), issatisfied in this case as well.

The first thing to deal with is how to combine the set of models for“small” with the set of models for “large”. Many possibilities can beexplored: each classical model for “small” can be combined with everyclassical model for “large”, or, more restrictedly, for every “small”-model,there is one “large”-model and vice-versa. In other words, in terms of thedescription above,n = m. I will continue with this simplified approach.

Obviously, without any further specifications, the following theorem istrivial to prove:

THEOREM 1. (∀n)(S(n)∨∼S(n)) & (∀n)(L(n)∨∼L(n)). (Any numberis either small (large) or not small (large)).

In order to get more interesting results, we have to say something aboutthe connection betweenSi andLi. Suppose that for everyMi , Si < Li (orequivalently,Sn < Ln). Then it is easy to prove the following:

THEOREM 2. (∀n)(L(n) ⊃ ∼S(n)) & (∀n)(S(n) ⊃ ∼L(n)). (If anumbern is large (small), then it is not small (large)).

THEOREM 3. (∃n)(∼S(n) & ∼L(n)) or, equivalently,∼(∀n)(S(n) ∨L(n)). (There is a number n that is neither small nor large, or, equivalently,it is not the case that every number is either small or large).


As indicated, all of this is rather trivial. What follows is a less trivial ex-ample: it concerns the introduction of a finite counterpart of the classicalnotion of feasibility. Let us further refine the numbersSi andLi. Supposethat the connection betweenSn andLn is such that there exists a numberk,such thatLn = Skn, for 2< k < S1. Call an arithmetical operationf (n,m)feasible iff if n andm are small, thenn+m is not large, possibly small.

THEOREM 4.(∀n)(∀m)((S(n) & S(m)) ⊃ ∼L((n + m)). (Addition isfeasible).

Proof: In any modelMi , Si < Sn < Ln = Skn < Li. If S(n) andS(m), thenn, m ≤ Si and hencen +m ≤ 2. si , hencen +m < Li, i.e.,∼ L(n+m). AsMi is arbitrary, it holds for all. QED.

THEOREM 5.(∀n)(∀m)((S(n) & S(m)) ⊃ ∼L(n · m)). (Multiplicationis feasible).

Proof: Just note that inMi, n ·m ≤ S2i < S

ki < Skn < Li. QED.

If furthermore we now add the condition thatSS11 ≥ L1, then one can prove

that

THEOREM 6.(∃n)(∃m)(S(n) & S(m) & L(nm)). (There are small num-bers such that the exponentiation is large, i.e., exponentiation is notfeasible).

Proof: In a modelMi , taken andm such thatn, m = S1 < Si, thennm = SS1

1 , hencenm ≥ L1 andL1 > Li, hencenm is large. QED

Although these theorems have not really impressive or deep results, they doshow that it is possible to reason coherently with vague predicates and toprove theorems about them. Furthermore, it is quite interesting to see thatrelations and connectionsbetweenvague predicates can be made precise.In terms of the above models, it is only because some relation betweenlower and upper limits is specified, that particular theorems can be shownto hold.

Example 2: Vague counterparts for sharp predicates. Take the classicalpredicate “is a prime number” or Pr(n). The vague counterpart is “to havea few prime numbers” orFPr(n). For the interpretation ofFPr(n) definethe following:

− If the decomposition ofn = 5pmii , thenNPr(n) =6mi is the numberof primes composingn,


− Select two numbersk1 andkn, such that to every modelMi a numberki is associated,k1 ≤ ki ≤ kn.

The interpretationIi(FPr) = 〈Ii+(FPr), Ii−(FPr)〉 can now be definedas

Ii+(FPr) = {n|NPr(n) ≤ ki}.If we now suppose that for every numbern ≤ Si, NPr(n) < ki , then it isstraightforward to prove the following theorem:

THEOREM 7. (∀n)(S(n) ⊃ FPr(n)). (Small numbers have few primenumbers).

Proof: In anyMi, if S(n), thenn ≤ Si and henceNPr(n) < ki . QED

In the theorem where there are infinitely many prime numbers, there nowexists a straightforward vague counterpart:

THEOREM 8. (∃n)(L(n) & FPr(n)). (There exists large numbers thathave few prime numbers).

I want to emphasize that this theorem presents a new result comparedwith the proof of the infinity of the primes. For it could well be the casethat in the realm of the large numbers, we have either prime numbers ornumbers with many prime numbers (i.e., more thankn upon interpretation).

Perhaps one might have the impression that all these theorems are rathertrivial and this is the best one can expect from vague predicates. To counterthis claim, here is a nice non-trivial theorem.

THEOREM 9. There exists arbitrarily long sequences of numbersn1, n2,. . . ,nk such that for everyni: L(ni) & ∼FPr(ni)).

This theorem is a generalisation of the classical theorem that betweenn!+2andn! + n, there are no prime numbers since all the numbersn! + i arethe product ofi andn!/i + 1. However it does not follow that every oneof the numbersn!/i + 1 has many prime numbers. In 1967, Jarden provedthe theorem that says in classical terms what it says above in vague terms:For everyN ands, there existN successive terms with at leasts distinctprime factors.7


5. INFINITY: THE VAGUE WAY

It is obvious that a more sophisticated theory can be constructed by com-bining predicates such as “small”, “large”, “few” and “many”. Through asuitable choice of interpretation, it becomes possible to make claims suchas: The sum of a few small numbers is not large. This can be consideredas an extension of Theorem 4. Depending on what one considers to belarge, a stronger theorem can result: The sum of many small numbers isnot large. I will not pursue these ideas here, but I will instead turn to themost intriguing problem of them all: infinity.

Even a rough glance at the history of mathematics shows that the notionof infinity had a difficult start. If it was referred to at all, it was often un-derstood as an unstructured something, not as an entity possessing specificproperties. In short, infinity had some features of vagueness. In terms ofthe approach outlined above, one can characterize the infinite through thenotion of “large” on the one hand and a notion of “rough equality” on theother. I start from the setM of modelsMi with predicates “large”,L(n),and “small”,S(n), as defined above. To each of these models correspondsa predicate of rough equality,n ≈ m. The interpretationIi(≈) = 〈Ii+(≈),Ii−(≈)〉 is such that

Ii+(≈) = {〈i, i〉 | i < Li} ∪ {〈i, j〉 | i, j ≥ Li}

Ii−(≈) = D ×D\Ii+(≈).

In other words, belowLi, rough equality corresponds with classical equal-ity, aboveLi everything is equal to everything else. I will list a number oftheorems without proofs as they are all quite elementary:

THEOREM 10.(∀n)(∀m)((n = m) ⊃ (n ≈ m)). (Equality implies roughequality).

THEOREM 11.(∀n)(n ≈ n). (Reflexivity of rough equality).

THEOREM 12.(∀n)(∀m)(n ≈ m) ⊃ (m ≈ n)). (Symmetry of roughequality).

THEOREM 13.(∀n)(∀m)(∀k)(((n ≈ m) & (m ≈ k)) ⊃ (n ≈ k)).(Transitivity of rough equality).

THEOREM 14.(∃k)(∀n)(L(k)& (S(n) ⊃ (k+n ≈ k))). (There is a largenumberk such that for all small numbersn, k + n is roughly equal ton).


THEOREM 15.(∃k)(L(k)& (k+k ≈ k). (There is a large number roughlyequal to twice itself).

Of special interest are theorems that show that “infinitely” largenumbers have no special properties, such as:

THEOREM 16.(∃k)(∃n)(L(k) & ((k ≈ 2.n) & (k ≈ 2.n + 1))). (Largenumbers are roughly both even and odd).

In other words, in terms of rough equality, the infinitely large is both evenand odd. Along the same lines, it will be roughly equal both to a prime andto a composite number. Generally, it will have roughly all properties andhence, so to speak, none. Note that this quite simple interpretation leads toa form of strict finitism, if all numbers beyondLi are interpreted as a singlenumber, or, better still, if a single name has been selected for all those(roughly (!) speaking) undifferentiated numbers. Note, furthermore, that itfollows that, if a number can be differentiated, then it is finite (equivalentin this framework to either being small or not large):

THEOREM 17.(∀n)((∀m)((n ≈ m) ⊃ (n = m)) ⊃ Fin(n)). (If, for anumbern, rough equality implies equality, then the number is finite).

Thus, a positive characterization of “finite” (in the sense ofS(n) ∨∼L(n)) is possible in vague terms. If one insists that classical equality can-not be used, then another formulation could be this(∀n)(∼(n ≈ n+ 1) ⊃Fin(n)). Here too, many sophistications are possible that I will not explorefurther in this paper. Let me instead return in a concluding paragraph tothe history of mathematics itself.

6. BACK TO HISTORY

It is one thing to claim that mathematics could have retained vague pre-dicates and properties in its language and could have worked and reasonedwith them reaching the same rigour as has been achieved in mathematicsas we know it now. It is quite another thing to claim that this type ofvague mathematics could have developed in different ways. Let me giveone example of such a difference.

Everyone is familiar with the famous analysis of Imre Lakatos (see his(1976)) of Euler’s theorem concerning polyhedra, namely that

V (number of vertices)−E (number of edges) +F (number offaces) = 2.


Counterexamples were found early on, soon after Euler’s proof (or“proof”). One technique that was applied was the so-called “monster-barring”. The counterexample is ignored or rejected for being bizarre,far-fetched, and so on. Lakatos suggests that this method should be avoidedin all cases. But suppose a situation where there are only a few (!) counter-examples. On the one hand, if the theorem says “For all so-and-sox,A(x)”then it is possible of course to include the exceptions, to obtain “For allso-and-sox, if different from this and that, thenA(x)”. What cannot beexpressed is whether there are few exceptions or not. However, in vagueterms, apart from the quantifier “For allx, . . . ”, one can also use the quan-tifier “For mostx, . . . ” or “For nearly allx, . . . ”. In this sense, we get adouble picture of mathematics: the “rough” approach and the “detailed”approach. From this perspective, it seems “natural” to continue on the onehand with the standard cases, and, on the other, make a separate study ofthe exceptions. This represents a quite different development from whatwe see today, where mathematicians continue to look for a single, simpletheory that will include all cases. Often the price to pay is an increasein the level of abstractness, making it more difficult to relate the generaltheorem, covering all cases, to a special, specific case. Perhaps, after all, itis possible that the rough picture is the true(r) picture.

NOTES

1 As an example, the theorem that there are an infinite number of prime numbers, can betranslated as:∼(∃n)(∀m)(m ≥ n ⊃ Comp(m)), i.e., there is no natural numbern such thatall larger numbers are composite.2 Actually, it is the work of Newton da Costa and his colleagues on the one hand, and thework of Richard Sylvan, Graham Priest and, above all, Chris Mortensen (see his (1995)) onthe other, that inspired me to think about alternative mathematics and to write this paper.3 I must emphasize that I am definitely not the first one to pose this question. In fact, thispaper was inspired by chapter 6, “Can There Be an Alternative Mathematics?” of Bloor’s(1991, 107–130).4 An English translation is to be found in Newman (1956), Vol. I, 420–429.5 I must add here that I was also deeply inspired by the work of Brian Rotman, see his(1987, 1993, 1997) for a survey, and the work of Keith Devlin, especially his (1997), wherehe draws attention to the idea of a “soft” mathematics (pp. 282–290).6 This is, of course, a very crude treatment of this position. See Haack (1996), for someonewho believes that vagueness should be eliminated: “So it seems most economicalnotto modify logic to cope with vagueness, but rather to regard classical logic as an ideal-isation of which arguments in ordinary discourse fall short, but to which they can beapproximated” (p. 125)7 The theorem is referred to in Ribenboim (1989, 191). The original paper was publishedin theFibonacci Quarterlyin 1967.


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Rotman, Brian: 1987,Signifying Nothing. The Semiotics of Zero, Stanford University Press,Stanford.

Rotman, Brian: 1993,Ad Infinitum. The Ghost in Turing’s Machine. Taking God Out ofMathematics and Putting the Body Back In, Stanford University Press, Stanford.

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Centrum voor Logica en WetenschapsfilosofieVrije Universiteit BrusselUniversity of GhentBelgiumE-mail: [email protected]

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Alternative Mathematics: The Vague Way