Jaakko Hintikka ANALYZING (AND SYNTHESIZING) ANALYSIS The history of mathematics might not seem a promising field for conspiracy theories. Yet such a theory was rampant in the seventeenth century. No less a thinker than Descartes believed that the geometers of antiquity employed a sort of analysis which they went on to apply to the solution of every problem, though they begrudged revealing it to posterity (Regulae, Adam & Tannery vol.1, p. 373) Equally surprisingly, Descartes s paranoid belief was shared by several contemporary mathematicians, among them Isaac Barrow, John Wallis and Edmund Halley. (Huxley 1959, pp. 354-355.) In the light of our fuller knowledge of history it is easy to smile at Descartes. It has even been argued by Netz that analysis was in fact for ancient Greek geometers a method of presenting their results (see Netz 2000). But in a deeper sense Descartes perceived something interesting in the historical record. We are looking in vain in the writings of Greek mathematicians for a full explanation of what this famous method was. And I will argue for an answer to the question why this lacuna is there: Not because Greek geometers wanted to hide this method, but because they did not fully understand it. It is instructive to note the ambivalent attitude of the most rigorous mathematician of the period, Isaac Newton, to the method of analysis. He used it himself in his own mathematical work and in the expositions of that work. Yet when the mathematical push came to physical and cosmological shove, he formulated his Principia entirely in C:\Hintikka.ANALYZING (AND SYNTHESIZING) ANALYSIS.0406.doc.7/30/2008
synthetic terms. In his mathematical heart of hearts he clearly thought that modern mathematicians indulge far too much in speculations about analysis. He was not only critical of its uses by Descartes and others, but suspicious of the method itself. It will in fact turn out that, if I am right, ancient Greek mathematicians were in their practice more keenly attuned to what is involved in the method of analysis conceptually and logically than their recent interpreters, even though they did not have a framework to describe the method in general logical terms. (This lack is not surprising, if I am right in arguing that certain crucial elements of this framework were recognized only a few years ago.) By saying this, I am placing an onus on myself of giving a better analysis of this famous method of analysis. Before tackling this task, a number of preliminary remarks are in order. First, in a birds-eye historical perspective, the method of analysis (or perhaps of analysis and synthesis) has two aspects, if not two meanings. Sometimes analysis seems to mean assuming the desired result, be it a proof or a construction, and reasoning from it backwards until a bridge to already known results is established. I will call this the directional sense of analysis. It is what Pappus seems to be describing in the most extensive surviving ancient account of analysis. Now analysis is a method of taking that which is sought as though it were > / admitted and passing from it through its concomitants ( α κολουθων ) in order to something which is admitted as a result of synthesis; for in analysis we suppose that which is sought to be already done, and we inquire what it is from this comes about, and again what the antecedent cause of the latter, and so on until, by retracing our steps, we light upon something already known or ranking as a first principle; and such a method we call analysis, as being a reverse solution. But in synthesis, proceeding in the opposite way, we suppose to be already true that which was last reached in the analysis, and arranging in their natural order as 2
consequents what were formerly antecedents and linking them one with another, we finally arrive at the construction of what was sought; and this we call synthesis. Now analysis is of two kinds, one, whose object is to seek the truth, being called theoretical, and the other, whose object is to find something set for finding, being called problematical. In the problematical kind we suppose that which is set as already known, \ (γνωσθε ν), and then pass through its concomitants in order, as though they were true, up to something admitted; then, if what is admitted be possible and can be done, (ποριστον) that is, if it be what the mathematicians call given, what was originally set will also be possible, and the proof will again be the reverse of the analysis (Hultsch 634, 3-636.30, Thomas 1941, vol. 2, pp.596-599.) For earlier interpretations of this passage, see Tannery (1903), Cornford (1932), Gulley (1958), Mahoney (1968-69) and Hintikka and Remes (1974). On the other hand, analysis seems to mean (or at least emphasize) something else, viz. a study of the interrelations of different geometrical objects in certain figures, that is, in certain geometrical configurations. (Cf. Hintikka and Remes 1974.) This sense might be called analysis as an analysis of configurations. This is the sense of analytic in analytic geometry, which came about when interdependencies of different geometrical objects in a given configuration began to be expressed algebraically. How these two notions of analysis are related to each other (if they are) is examined below. This interrelation is one of the most interesting features of the saga of the analytic method. Another matter that can be disposed of is the question of the direction of analysis. Interpretationally, this is largely a pseudo-problem. In theoretical analysis, the steps are mediated by relations of logical consequence. Pappus makes it clear that in them we inquire into what the conclusion comes from. Hence, in them the direction is unequivocal; from the desired theorem backwards what is already known. In problematical analysis, the steps are mediated by dependence relations between 3
geometrical objects, for instance relations expressed by algebraic equations. Such equations can usually be solved for either term. In that sense, the direction usually does not matter. It is also relevant to note that understanding the ancient method of analysis is not an isolated problem in the history of mathematics and in the history of thought even more generally. In the history of Greek mathematics, related problems concern the understanding of such studies as pertained to what were known as data or known as porismoi. Euclid wrote an entire book on each of these topics, although only the Data has survived. It is far from obvious even what their subject matter is, let alone what they were needed for. They have no obvious counterparts in modern mathematical practice. Why not? In later periods, the history of the method of analysis is connected with the development of algebra and with its uses in analytic geometry. Some scholars have even seen in the idea of an analytical experiment the methodological Leitmotif of early modern science. (See Becker 1959, especially pp. 20-25.) Another preliminary observation is that the role of the method of analysis in mathematical practice makes little difference to the framework that should be used in analyzing it. Whether it was a method of proof, a heuristic method of finding proofs and constructions, or a method of exposition, its precise argumentative structure must be identified. For instance, in so far as heuristic techniques can be rationally discussed, they can be construed as strategic methods. But if so, they can in principle be examined, formulated and theorized about in as explicit fully formal terms as rules of proof. (Needless to say, it may and should be asked which in principle formalizable strategies 4
are heuristically better or worse than others.) Hence analytic argumentation will be examined here in the same terms as deductive argumentation. For similar reasons, it does not make much difference if an interpreter tries to evoke the famous holy cow called mathematical practice. If such practice is not haphazard, it must be governed by some tacit rules which must be discussed on a par with explicitly codified ones. The fact that the structural analysis of the ancient Greek method is largely independent of its pragmatic uses has not always been recognized. For instance, some scholars think that if we interpret analysis as a search for premises, it becomes a nondeductive, intuitive procedure (Rehder 1982, p. 356). This is a gross misunderstanding. Precisely the same rules as govern steps of deduction ipso facto govern the search for premises. We merely have to apply the same rules in the opposite direction. For another example, Netz thinks that the direction makes a difference whether analysis is considered as a method of reasoning or a method of exposition. Sometimes this mistake is merely an instance of the common prejudice that strategic rules of deduction or of other kinds of reasoning can only be heuristic rules of thumb and not as precise in principle as definitory rules of deduction. With these methodological precepts in mind, one can suggest that there is an excellent explicit model of analytic reasoning in mathematics already in existence. It has in fact been available for more than half a century. It is the technique of argumentation in our usual basic logic, first-order logic, known as the tableau method of E. W. Beth. In his very first presentation of this method (Beth 1955, p. 319) the historically perceptive Beth noted that his 5
approach realizes to a considerable extent the conception of a purely analytical method, which has played such an important role in the history of logic and philosophy. Interpreters have not paid enough attention to Beth s suggestion. But what precisely is Beth s approach and what is the extent to which it realizes the old Greek idea of analytical method in geometry? Let s deal with the first question first. The tableau method is a procedure for looking for a proof that a certain first-order proposition G is a logical consequence of another one, say F. As an example, let F be (1.1) ( x) ( y) Nxy which you can think of as saying someone in envied by everybody. Likewise, let G be (1.2) ( z) ( u) Nzu in words, everyone envies somebody. The tableau could then look as follows (1) ( x) ( y) Nyx (2) ( z) ( u) Nzu (3) ( y) Nyα from (1) (4) ( u) Nβu from (2) (5) Nβα from (3) (6) Nβα from (4) What happens here is almost self-explanatory, and so is its connection with the ideas of analysis and synthesis. The tableau construction is a kind of combination of synthesis and analysis. On the left side we take the premise F and see what the world must be like if it is true. On the right side we take the conclusion and see what the world 6
might be like if it were to make the conclusion G true. For instance, if (1) is true, there must exist some individual or other, call it α, to make it true. Hence the step to (3). (Of course, we need not know who α is otherwise.) By symmetry, if (4) is true for any arbitrary individual β of whom we need not know anything else, then (2) is true. (This necessitates considering some instantiating terms like β in (2) playing the role of a variable bound to a universal quantifier.) Thus on the right side we reason backwards from the desired conclusion to the conditions that would make it true. The steps (5) and (6) are merely applications of general truths to particular instances. Once we have the same proposition on both sides, we have built a bridge and run up the right side in order to reach the conclusion G = (2) from the premise F = (1). The right side thus exemplifies analytic reasoning in the directional sense while the left side can be thought of as synthetic reasoning in the directional sense. Similar examples are easily found in geometrical reasoning. They would be more complicated than this one, however. A comparison between an explicitly logical argument and an ordinary mathematical one is facilitated by the fractional division of a geometrical theorem in Euclid (or elsewhere in the classical tradition). This division is explained e.g. in Heath 1926, vol.1, pp. 129-131. Thus in this traditional terminology, steps (3) and (4) would be cases of ekthesis while (5) and 6) would be parts of apodeixis. This geometrical comparison illustrates an integral aspect of Beth s rational reconstruction of the analytic method viz. the fact that both sides of a tableau can be thought of as exemplifying analytic reasoning in the analysis-of-configurations sense. On both sides, we are reasoning about certain given or postulated individuals. In the sample tableau, we considered the configuration formed by the individual α who was assumed to 7
be envied by everyone and by the postulated individual β who was hoped to envy someone or else. The introduction of such representative individuals into a geometrical argument was a routine part of ancient mathematicians practice. This was what the ekthesis part of the argument establishing a geometrical proposition mentioned above. In more complicated arguments, the reasoning might have to branch into two or more disjunctive lines of thought. Furthermore, individuals other than those instantiating the premise and the conclusion might have to be introduced. (As an example, you might for instance construct a tableau for the inference from someone is envied by all unmarried persons to every unmarried person envies somebody.) Such further introductions are common in geometrical arguments. They were called by the ancients auxiliary constructions. Moreover, in steps (5) and (6) we are in an obvious sense analyzing the configuration formed by α and β. We are asking what envying relations there might obtain between them. In a geometrical case, the corresponding relations might be expressible by algebraic equations. There are plenty of perfectly valid rules of logical inference that do not lead themselves to such an interpretation for instance, adding a disjunction of the form (S v ~S) as an extra premise (for instance to the left side of a tableau) preserves truth and hence constitutes a valid inference. Such additions might even help an argument significantly. However, they cannot be interpreted as dealing with some particular configuration of individuals so far postulated or introduced in the course of the argument in question. The naturalness of this interpretation of the method of analysis can be seen in several different ways. The idea of analysis as an analysis of configurations requires that 8
each step of an analytic argument can be interpreted as saying something about some given and/or constructed figure, sometimes pertaining to the interrelations of the ingredients and sometimes adding a new geometrical object to it. This is of course what Greek geometers routinely did, not only in what are technically known as their analyses. Even though this practice has prompted little comment, the possibility of doing so consistently is not at all obvious from a logical point of view. The possibility of viewing each step of an argument as analytically as pertaining to an already given configuration of geometrical objects is virtually tantamount to the requirement that the logically explicit argument satisfies what is called the subformula property. What this principle requires is that each new sentence propounded in the course of an argument is a subformula of an earlier one, including substitution-instances of open subformulas. It is not difficult to see how the subformula requirement makes it possible to consider a geometrical (or any other logical) argument as pertaining to a given configuration. Of the two main rules of inference needed, universal instantiation with respect to names already present in the argument means simply applying an assumed or already established general truth to an object in the given configuration. Existential instantiation means enriching the configuration in question by introducing a new arbitrary sample object into one s line of reasoning. Now the subformula principle is the most important logical feature characterizing the tableau method. The possibility of transforming any first-order proof into a form in which the subformula principle is satisfied is the first and foremost result of modern proof theory. Its first version is known as Gentzen s first Hauptsatz. (See Szabo 1969.) Since in the usual Gentzen type formulations of first-order logic the rules that do not 9
satisfy the subformula principle are condemned into what is known as the cut rule, this fundamental result is also known as the cut elimination theorem. Greek geometers analytical practice thus relied tacitly on the possibility of cut elimination. This vantage point offers also an insight into the relation of the two ingredients of the method of analysis. It is not hard to see that we can maintain the subformula principle in all proofs from premises only if some of the argumentation proceeds from the hopedfor consequences backwards toward the premises. For instance, suppose that the right side of a tableau is in effect a deductive argument proceeding from bottom up, if so, that deductive argument could not satisfy the subformula principle. For instance, in our sample tableau a step from (6) to (4) would be valid, but it would not satisfy the subformula principle. In this sense, the analysis-of-configurations sense practically implies the directional sense. This insight in turn enhances the interest and value of the tableau reconstruction of the method of analysis. These remarks deserve elaboration. There is a remarkable connection between the entire tableau method and the Greek geometrical practice. This practice involved considering geometrical arguments, synthetic as well as analytic, by reference to certain geometrical configurations illustrated by figures. Now it is far from obvious and indeed false that any proof in elementary geometry can be so considered. For instance, application of modus ponens cannot be so considered in any natural way. Cognitive psychologists apparently considered it initially as an interesting novelty that human deductive reasoning sometimes uses figure-like mental models ; see Johnson-Laird (1983). 10
In order to be interpretable as dealing with a specific figure, deductive reasoning must satisfy what is known as the subformula principle. The name of this principle describes it. What it says is that each step in explicit formal reasoning must be a subformula (or a substitution-instance of one) of a previous formula. Only if this principle is satisfied can each step of reasoning be construed either as a statement about previously considered objects or else as an introduction of a new individual bearing a certain definite relation to the earlier ones. Without the subformula property, a deductive argument in geometry cannot always be interpreted as being about a certain figure. And this is how Greek geometers looked upon their proofs. Hence they were in effect assuming the subformula property. Now the subformula property can be considered as the characteristic mark of the tableau method. This method naturally results as soon as we assume the two-column format (premises in the left or true column and the conclusion in the right or false column) and then require that the subformula property holds in each column. What this means for the interpretation of the method of analysis is that the analysis-of-figures sense of analysis virtually leads us to some variant of the tableau method. But if so, the analysis-of-figures sense also leads to the directional sense of analysis. For from logical theory we know that in a tableau proof we typically have to consider on each side more individuals mentioned in the initial premise or in the conclusion indeed, unpredictably, many individuals. As a consequence, there will usually have to be a large number of steps also on the right-hand side, which manifests the directional sense of analysis. Hence the interpretation of the method of analysis and synthesis as a tableau procedure shows that there is an actual connection between the two 11
apparently unrelated senses of analysis. As a synthesizing slogan, it can be said that analysis in the directional sense is needed if we are to implement the analysis-of-figures sense. Since the tableau approach and the method of analysis and synthesis are closely related, these considerations strongly suggested that the method of analysis and synthesis is deeply rooted in the Greek practice of thinking of geometric arguments as arguments about some one figure or kind of figure. Moreover, the idea of arguing about figures has an obvious connection with the heuristic usefulness of analysis. Geometrical invention does not mean that one geometrical statement somehow suggests another statement. Such an invention typically amounts to an insight into the interrelations of geometrical configurations. Indeed, such configurations need not even be geometric in a narrow sense, but can be discrete figures as in the pebble configurations of the Pythagoreans. (See Mäenpää 2006.) Among other things, this rational reconstruction shows the subtlety of the relation between analysis and synthesis. What was pointed out above is that the deduction that inverts the analytic reasoning as it were on the right side of a tableau cannot itself satisfy the subformula principle, that is, be analytic. We might express that by saying that analyticity in the analysis-of-figures sense presupposes analyticity in the directional sense. But the converse also holds. One cannot conduct the entire argument analytically in the directional sense. If by analysis one means looking for premises from which the conclusion can be derived deductively, this search cannot satisfy the subformula principle, that is, be analytic in the analysis-of-figures sense. For on the left hand side we cannot proceed upwards without violating the subformula principle. What follows is that 12
if a geometrical argument from a premise to a conclusion is to be interpretable as an argument about certain figures (i.e. certain kinds of geometrical configuration), it must contain both an analytic and a synthetic component. And obviously Greek mathematicians wanted their arguments to be about figures in this sense. Hence analysis and synthesis are structurally not just movements in opposite directions. A visualizable geometrical argument normally has to have both an analytic and a synthetic component. Analysis was not practiced by Greek mathematicians merely as a heuristic technique, much less (pace Netz) as an expository device. It was a consequence of their practice of viewing geometrical arguments as arguments about geometrical figures. This consequence is not unavoidable. It can be avoided by using indirect proofs. This would be like switching from the tableau method of deductive proof to the wellknown tree method which was discovered at the same time as the tableau method. Interpretationally, this switch would mean considering a table construction as an attempt to describe one can even say, construct an isomorphic picture of the world as it would have to be if the premise(s) were true but the conclusion false. This interpretation would enable us to view the entire argument as being a kind of picture construction and hence as dealing with figures. Thus, such an interpretation seems to have been foreign to the Greeks. It seems that we can see traces of the complementary relation of analysis and synthesis in the texts. It may for instance be significant that Pappus never says that analysis should always be carried back to the first principles. Rather, he says that analysis ends when the analyst has reached things already known or having the status of a first principle or reached something admitted. It is unnatural to take these phrases to refer 13
to earlier theorems. Rather, a phrase like what is admitted is relative to a stage of an ongoing argument and cannot naturally refer to its potential ultimate premises. This is ever clearer in the case of a problem-solving analysis. The tableau reconstruction also throws some light on the problem concerning the direction of a step of analysis as compared with the direction of a deductive inference. According to Pappus, in analysis we hypothesize what is sought as being and as true and then proceed through the things that follow [or accompany, akolouthein] in order. Is such a procedure a series of steps in deductive inferences or their increases? In the strictest logical sense possible, the tableau interpretation yields the interesting answer: None of the above. For even on the left side of a tableau, the successive lines (which may be disjunctive, the disjuncts being left hand sides of a subtableau) are not logical consequences of the premise. Only their existential closures are such consequences. In our sample analysis above, (3) is not a deductive consequence of (1). Rather, it illustrates it through the arbitrary object α. But what this means is that the existential closure of (3) is (1). In intuitive terms, even in a synthetic argument, we do not just draw deductive inferences. Rather, we examine what the consequences of those inferences amount to in the case of a sample configuration of individuals, some of which may be what has been called arbitrary individuals. Similar comments pertain to what is going on on the right hand side of a tableau. (More will be said on this matter below.) In view of such subtle distinctions, it perhaps is not surprising if Pappus did not manage to give us an unequivocal explanation of the relationship of steps of analysis to deductive inference relations. It is very dubious whether a philological exegesis of 14
Pappus s words promises any real illumination. (It may none the less be relevant to note that the key word akolouthein elsewhere in the Greek philosophical texts sometimes means accompanying ( going together ) rather than consequence. See Hintikka 1962.) An approach to the method of analysis and synthesis by reference to tableau proofs thus throws important light on the method. Nevertheless, the most interesting and constructive feature of the tableau reconstruction is that it fails or more specifically the ways in which it fails in the form in which it has just been presented. The principal shortcomings of the tableau reconstruction can be given labels. These labels are problem and construction. It is no accident that these two notions have been central in recent discussions about Greek mathematics in general. To take the notion of construction first, where should we place it in the tableau scheme? Constructions were taken in Greek geometrical reasoning to introduce new objects into the argument. In the tableau technique of logical proof, such new individuals are introduced on the left side by existential instantiation and on the right side by a mirror image rule of universal instantiation. But such instantiations are not constructions. What are introduced are not objects constructed out of the familiar ones, but John Doe-like arbitrary individuals. In a left-hand existential instantiation the argument has reached the tentative conclusion that there exist individuals of a certain kind. The arguer then in effect says (to himself and/or to others), Let us consider one case in point and call it α. A judge in a court of law might likewise say of an unknown perpetrator, Call him John Doe. In fact in the early modern period, Wallis (1683, p. 66, quoted by Klein 1968, p. 321) claimed that the use of symbols for unknowns in algebra was inspired by such a legal usage. This comment, whether or not it is historically accurate, is suggestive in that 15
it shows that an early modern algebraist thought of the introductions of new individuals into an argument, not just as constructions in a geometrical sense, but more generally as instantiations in the sense of modern logic. In any case, the use of such arbitrary individuals presupposes that we do not know or assume anything about them except which existential proposition they instantiate, for instance know how they are related to objects already known or postulated, for instance how they are constructed out of the known ones. Hence in this sense constructions yield too much information. They also yield too little information, in that geometrical objects can be known to exist that cannot be introduced by any obvious construction. Moreover, as Wilbur Knorr among others has argued, Greek mathematicians were familiar with examples of such nonconstructive existence. Moreover, on the right side of a tableau new individuals instantiate universally quantified variables. The intuitive meaning of such a variable in terms of a figure is to stand for an arbitrary sample object, as if another Jane Doe or Richard Roe. It makes no prima facie sense to think of the new objects to be actually constructed, for the construction would have to start from merely notional and possibly impossible entities. Hence the tableau interpretation cannot explain or even accommodate the crucial role of constructions in analysis or, for that matter, in other kinds of mathematical reasoning. Nor does it have anything to say about what problems were and how they differed from theorems, in spite of the fact that it is generally agreed that the main use of analysis was in problem-solving. Zeuthen (1896) already suggested that Greek geometrical problems were in effect existence theorems. Recently, his view has been trenchantly criticized by Knorr (1986, especially pp. 74-80, 360) and others. However, a 16
convincing positive answer to the question as to what problems were is not found in the literature. The reason for this absence of a recorded answer is perhaps that the answer is too obvious. Problems are questions, and their logic is the logic of questions and answers. And since questions are requests of information, the logic of questions and answers is a part of epistemic logic. The logic of ancient Greek mathematicians is not the usual firstorder logic. It is epistemic logic (or equivalent). The question whether the tableau interpretation can be extended so as to cover problems is a special case of the wider question whether the logical techniques of epistemic logic can yield a logic of problem solving. The main difficulty is obvious. In order to practice the tableau method, an analyst must know the desired conclusion to be established. It is what is put on the top of the right side. But in problem solving this conclusion is precisely what it is sought for. This is in effect the logical gist of Meno s paradox. This seemingly makes the tableau procedure completely useless in discussing the analytical method in its directional sense, for the method was supposed to consist in analyzing the desired conclusion. In the case of a problem, the configuration instantiating the conclusion should include the hoped-for construction. But how can one possibly analyze an unknown configuration? The systematic logical problem we are facing here has recently been solved. The tableau method can be used as a framework of problem-solving, more explicitly as a framework for looking for an answer to a question in the same sense as it can be used as a framework for looking for a proof of a logical consequence relation. This can be done by 17
introducing the epistemic element into the reasoning. In mathematical practice, this element is often tacit. It can be made explicit by adding to the usual first-order logic an it is known that operator K. In the tableau method it can always remain sentenceinitial. Furthermore, a distinction must be made between known and (possibly) unknown individuals. I will spare you the technical details and indicate only their manifestation. It turns out that quantifiers ranging over known individuals are those that are independent of the K operator. This is indicated by writing them as ( x/k) or ( y/k). Such quantifiers can be instantiated only by known individuals. That an individual b is known is expressed by K ( y / K)(b = y). That a function f(x) is known is likewise expressed by K ( x)( y / K)(f(x) = y), which is equivalent with K ( g / K)( x)(f(x) = g(x)). Given a question, we can now always form its desideratum, which expresses the epistemic state which the questioner aims at. For instance, if the question is Who murdered Roger Ackroyd?, the desideratum is it is known who murdered Roger Ackroyd ; in symbols, K ( y / K)(y murdered R.A.). If the question is, How does the area of a square depend on the length of its sides? the desideratum is of the form K ( x)( y / K)(x is the length of a side of a square y is the area of the same square). 18
Now the tableau method can be applied to problem-solving simply using the desideratum of the question to be answered as the desired conclusion. (Naturally the premises must be thought of as being preceded by the K operator.) This procedure bears an obvious similarity to what Pappus described as hypothesizing what is sought as being and as true and more than mere similarity with what Pappus refers to as supposing that which is set as already known. The desideratum is that crucial hypothesis. The possibility of being so is the genuine secret of the method of analysis. Beth was on the right track in seeing a connection between the idea of a purely analytic method and tableau proofs. However, the right interpretation can be reached only by means of the concept of desideratum and by relying to that extent on epistemic logic. In practice, problem-solving tableau arguments do not look very different from familiar first-order tableau arguments, especially when the sentence-initial K s are left tacit. The main difference is that we now have to keep track of which individuals are known and which quantifiers range over known individuals. The following is an example of such a tableau argument. The question in question is of the form, Given an individual x, what individual y is related to it as in D(x,y)? For instance, the question might be, How is distance covered by a freely falling body related to time? The argument shows that a functional dependence y = g(x) is an answer to this question, if g(x) is a known function. 19
(1) K ( y)d(x,g(x)) (assumption) (3) K ( x)( y/k)d(x,y) (desideratum) (2) K ( x)( y/k)(g(x) = y) (assumption) (4) K ( y/k)d(α,y) from (3) (5) K D(α,g(α)) from (1) (10) K D(α,β) from (4), (8) (6) K ( y/k)(g(α) = y) from (2) (7) K (g(α) = β) from (6) (8) K ( y / K)(β = y) from (6), (7) (9) K D(α,β) from (5), (7) Here the desideratum says that it is known what individual y is related to any given x by the relation D(x,y). Again, similar geometrical arguments are easy to find. In Euclidspeak, step (4) of our argument is an ekthesis and step (7) a kataskeue. Assumption (2) says that the fraction g(x) is given. This guarantees that g(α) in (6) is given. Moreover, (8) says that β is given, which enables it to be substituted for the variable y in (4) so as to obtain (10), even though y is slashed in (4). This small sample argument illustrates the nature of the epistemic tableau method. If the epistemic operators K are omitted, the tableau looks very much like an ordinary nonepistemic tableau. The only exception are the slashes indicating which 20
quantifiers range over known individuals. There is no similar syntactical indicator for names and/or dummy names like α and β. If you look at the tableau, you can see that α is introduced to instantiate an unslashed quantifier. This α is therefore not assumed to be known. Hence it cannot be substituted for a variable bound to a slashed quantifier. It can be substituted for the x in (3) or for the y in (1). However, it could not be substituted for the y in (4). In contrast, β is introduced as an instantiation of the slashed variable y in (6). That it refers to an individual hypothesized to be known is expressed in (8). Hence it can be substituted for the slashed variable y in (4). Modest though this little example is, it illustrates an important feature of the conceptual situation. The premise (1) says that it is known that, for each x, g(x) satisfies the condition D(x,g(x)). Here g(x) is what used to be called function-in-extension, which is a mere set-theoretical entity, the class of ordered pairs <x,g(x)>. From the argument it is seen that this assumption (1) does not alone entail (3), that is, entail that it is known how g(x) depends on x. Formally, since g(α) is not assumed to be known, it cannot be substituted for the slashed variable y in (4). The desideratum follows only with the help of the assumption (2), that is, only because g(x) is assumed to be a known function. Hence what separates an epistemic tableau from an ordinary one is essentially that some of the instantiations will have to be with respect to known individuals. These instantiations cannot be with respect to arbitrary John Doe individuals. Wallis may have been right about the parentage of algebraic instantialization, but a similar idea does not explain the introduction of new individuals in a Greek analysis. 21
Hence, clearly constructions were not the only way of proving the existence of a geometrical object in Greek mathematics. What matters is not so much that the object in question can be produced as of whether it is determined (known, given ). It is now my main thesis that the epistemic tableau method is logically speaking the analytic method of Greek geometers. What is my evidence for saying so? The main characteristic of the epistemic tableau method is the need of keeping track of known individuals in arguments. Now not only can we find a methodology for doing so in Greek mathematics. Realizing the logical function of this methodology solves one of the main open problems concerning the interpretation of Greek mathematics in general. For this methodology is precisely the business of that important part of the Greek mathematical corpus that dealt with what were known as data. Euclid s so-called book is only the best known example of this preoccupation with the given, as the usual literal translation. But the intended meaning is obviously what for the purposes of some particular argument is assumed to be known. Later, Arabic mathematicians in fact spoke in so many words of the known instead of the given, and assumed that this is what their Greek predecessors meant. (See Berggren and van Brummelen 2000.) Whatever qualifications this assimilation needs will be commented on below. The need of tracking down individuals in an epistemic tableau argument necessitates the formulation of explicit rules of what is given in a geometrical argument where something else is given (known). This is the purpose of the entire data literature. The rules concerning what is given are logically speaking rules authorizing instantiations. And such rules are needed as soon as we have to deal with an epistemic element in mathematical reasoning, as we do in explicit problem-solving. But 22
the interpretation of the method of analysis as the epistemic tableau technique also solves another outstanding general problem concerning Greek geometrical theory. We can first of all see how Zeuthen went wrong. The true conclusion in a problematic analysis is not an existence theorem, but the desideratum of the problem in question. But this does not mean that existential propositions do not play a role in solving geometrical problems analytically. Problems are naturally construed as questions. But before one can legitimately ask a question, one has to establish the presupposition, as we know from the logic of questions and answers. And the presupposition of a question is the proposition obtained from its desideratum by omitting all slashes. In the case of general wh-questions, which is what we are dealing with here, such presuppositions are existential propositions. Such existential propositions hence play an important role in analytic problem-solving as prerequisites of the enterprise. More specifically, they guarantee the existence of a solution. Hence it must be expected, if I am right, that Greek geometers should have paid special attention to these existential propositions that served as presuppositions of their problems. This expectation is fulfilled. It is fulfilled by what were known as porismoi. Like data, porismoi attracted a great deal of attention on the part of Greek geometers. For instance, not only did Euclid write a book called literally Data (See Ito, 1980, Simson 1806), he also wrote a book on the porismoi, which unfortunately has been almost completely lost. Diophantus, too, is known to have written on the subject. In spite of this apparent importance of porismoi, their nature and their role in geometrical argumentation has not been previously understood adequately. This may have been the situation already in antiquity. Pappus says that 23
many geometers understand them (sc. porisms) only in a partial way and are ignorant of the essential textures of their contents. The epistemic tableau procedure now helps us to understand what is implied. First, what is a porismos supposed to be like? Pappus tells us that the aim of a porism is not a demonstration as in a theorem, nor a construction, as in a problem, but the producing of the thing proposed. But what is a production that is not a construction? The natural interpretation is that what is meant is the finding of the object sought. And to show that something can be found is tantamount to showing that it exists. Hence porisms are naturally taken to be existence theorems, in other words, the existential statements that are the presuppositions of problems when they are considered as questions. This role of porisms as establishing the presuppositions of the questions that problems are is somewhat obscured by the conventions of exposition. When a question is explicitly asked, its presupposition is normally presented as an earlier result that serves to guarantee the existence of an answer. In contrast, in Greek mathematical practice, porisms were not necessarily proved independently of the analytic arguments by means of which problems were solved. A successful analysis proves the existence of the solution, but an analysis may also end up showing the impossibility of a solution, that is, the falsity of the presupposition. It is in this role that the very word ποριστóν occurs in Pappus s statement (see above). Proclus also explains the notion of porism by using the notion of finding in so many words, saying that porism is the name given to things that are sought but need some finding and are neither pure bringing into existence nor simple theoretical argument. 24
Indeed, as Heath (1926, vol. 1, p. 13) puts it, the usual form of a porism was to prove that it is possible to find a point with such and such a property In several languages, can be found or some equivalent locution is in fact used to express existence. A medieval Arab mathematician is even more explicit. Speaking of what is knowable he writes: The essence of the knowable notions does not require that one perceives them or that they are actually produced. Rather, if the proof of the possibility of the notion has been provided, the notion is sound, whether one has actually produced it or not (Hogendijk, p. 96, quoted by Berggren and Van Brummelen, 2000, p. 26). In order to understand what is involved, it can be noted that in a solution of a problem by the epistemic tableau method what is established is the existence of a known individual, whereas in the corresponding existence theorem it is only shown that there exists ( can be found ) a sought-for individual. Mahoney (1968-69, p. 344) notwithstanding, the content and meaning of any one porism lends itself to unambiguous description in purely mathematical language. What is puzzling is the role of the porismoi in the overall process of problem-solving. Yet this role was in practice understood better by the Greeks than by their latter-day commentators. These insights into the nature of the Greek method of analysis help to confirm the interpretation of this method as being in effect an epistemic tableau technique. But what about the idea of construction? In a loose sense all introductions of new objects (including John Doe-like arbitrary objects ) into an argument can sometimes be called auxiliary construction in the sense of Euclidian kataskeue. But in a more specific sense, 25
the construction of a from b and c is what serves to make sure that when b and c are given, a is also given. Hence constructions in this specific sense, while not being needed in the proof of a theorem, are vital in a solution of a problem in the same sense as considerations of what is given. Thus everything seems to be explained neatly by the interpretation of the method of analysis in terms of epistemic tableaux. However, at a closer look there lurks a scary skeleton in the closet of Greek analysis. Once again the tableau framework serves well in spelling out the situation. The notion of construction fits well in what happens on the left side of a problem-solving tableau. However, the same does not hold of the right side. There new individuals, including known ones, are introduced by universal instantiation. But what universally quantified variables represent intuitively speaking are hypothesized individuals, not in any natural sense given or known ones. Such merely notional individuals cannot serve as inputs into a construction in any clear sense. What is supposed to happen on the right side is a coexistence and cooperation of the two aspects of the analytic method, the analysis-of-figures sense and the directional (backwardsmoving) sense. It now seems that this coexistence is impossible. This conundrum is reflected in the confusions and paradoxes that actually beset discussions about data in antiquity. Typically, the source of puzzles in them is in effect the idea that in the right side the looked-for solution is already known. But this is not a difficulty for an interpreter like myself. It was a problem for the Greek geometers themselves. They were looking at the method of analysis in a wrong way. Theirs is not a coherent model-theoretical interpretation of the tableau method. The correct one is that an a tableau analysis the mathematician is trying to describe (and in a sense construct) a 26
model(scenario, situation or possible world ) in which the premise is true but the conclusion false. If the attempt is frustrated, the consequence is valid. In the epistemic case, an analyst is likewise trying to describe a situation in which the premise is known but the conclusion is not. The individuals hypothesized on the right side are therefore no the individuals that would make the conclusion known, but the ones that would make it unknown, in the jargon of modern mathematics, they are the unknowns, not the data, of the problem in question. By definition, they cannot be thought of as being known. This conundrum manifests itself also in the natural model-theoretical interpretation of the tableau method. On this interpretation a search for a tableau proof is viewed as an experimental attempt to construct a counter model for the consequence relation, that is to say, to construct a model in which the premise is true but the conclusion false. If such an attempted construction leads to an overt contradiction in all directions, the conclusion is seen to be valid and the tableau which shows this serves as the proof of the conclusion. But in such a proof, steps on the right side cannot be understood as part of a construction of an actual figure. They have to be viewed as steps in an attempted construction of a model (figure, configuration) that is in fact impossible. Are such signs real constructions? In what sense? The same problematic is in evidence already in the sample analyses presented above. In the first one, the term β does not refer to any particular object in the kind of figure that can be used to illustrate a geometrical argument. As a consequence, it cannot serve as a starting point of a construction in any natural sense. As was noted above, it operates formally like a universally quantified variable, not like a name. 27
Somewhat similar things can be said of the term α of the second (epistemic) sample analysis. It, too, can only be thought of as standing for an arbitrary object. It cannot serve as the basis of a construction, as is witnessed by the fact that it cannot be substituted for the variable of a slashed quantifier. To formulate the same problem in other words, it is perfectly possible to consider conclusions from a merely postulated theorem. But one cannot carry out constructions starting from a merely postulated geometrical object. A merely postulated entity apparently cannot be given. Insofar as the Greeks emphasized actual constructions as the way in which new objects are introduced into a geometrical arguments, they could not fully understand what happens in the backwards-moving part of the procedure of analysis and synthesis. In sum, the Greeks could not accommodate what happens on the right hand side with the idea that geometrical arguments and geometrical analyses should be thought of as dealing with certain figures (configurations). I believe that this failure of Greek mathematicians to understand fully the logic of the analytic method is what made them reluctant to explain it. Thus Descartes and his contemporaries had in fact something to be paranoid about, even if they misconstrued Greek mathematicians motivation. The method of analysis is understood and used correctly only when the unknowns are treated on a par with known entities. In formal terms, this means allowing universal instantiation on the right side to operate just like existential instantiation on the left side. This presupposes that the links between individuals, for instance between geometrical objects, that make them given or known in the sense manifesting itself in their being 28