Realizing Their Separate Identities By a Mathematical Reflection into English Syntax
by Robert H. Schmidt
In the notes that accompany a translation of an early mathematical writing, it is the custom to restate at least the most interesting propositions and their arguments in terms of our modern algebraic notation. Indeed, the literal translation is sometimes omitted altogether in favor of an algebraic paraphrase. In the case of the geometrical writings of the ancient Greeks, this practice is often justified by the supposition of a Greek geometrical algebra.
The present translation (Apollonius' On Cutting Off a Ratio) avoids any use of algebraic candidate for such treatment. After all, modern algebra got its start in the late Renaissance as an attempt to restore the ancient method of analysis and synthesis.
It is not that we have any scruple about using the operational symbolism of algebra per se where the Greeks used their own natural language. It may in fact be the case that what is essential about analytical reasoning is reproduced just as well by formalization in a calculus as by translation into a spoken language. There may be something to the argument that a formal language is simply a cleaner, clearer way of talking about mathematical things, one that does not distract us with irrelevancies.
Nor do we refrain from algebraic translation because of the still unsettled problems of intentionality in Greek mathematics - I refer to the question of whether the objects under consideration, say in the second book of Euclid, are rectangles and hence intrinsically figural and geometrical, or products of "pure" or general magnitudes and hence subject to algebra - for I don't see how algebraic calculation obliges us to make a choice here.
No, not for either of the above reasons, but because the mode of reasoning itself which is inseparable from algebraic argument does not reflect the way of thinking that is characteristic of the Greek method or analysis and synthesis. That is, the Greek art is virtually untranslatable by algebraic concepts. This is a more fundamental difficulty than either of the above. It is the purpose of this editorial to contrast these two types of reasoning.
It is an unexpected corollary to this that the supposition of a Greek geometrical algebra (at least one dealing with pure magnitudes) is probably erroneus, even in the fifth book of the Elements. I will also point out how the central analytical concept may provide the basis for a true formalization of the Greek art.
The Approach
The most conspicuous feature in the surviving examples of Greek analysis is the constant repetition of the word dedomenon. As has been explained at the end of the Apollonius translation, this word is a participial form of the common Greek verb meaning 'to give'. It describes how some geometrical figure ought to be regarded in an analysis. And as a participle of a verb that can take both a direct and indirect object, it implies a fairly involved syntax. One might say that the very essence of the Greek method of analysis and synthesis is distilled into this single word.
Therefore, to begin my exegesis of Greek analysis, I will present schematically a geometrical situation in which the Greek concept of dedomenon emerges in all its grammatical complexity. I will also elaborate various satellite concepts that are naturally suggested by the central concept of dedomenon so understood. These will correspond to the natural interpretations of other terms that are important in Greek analysis-such as the words 'analysis' and 'synthesis' themselves - and this agreement should give an increasing amount of weight to my original interpretation of the word 'dedomenon'.
At that point I will go into some historical matters that hint at a difference between algebraic reasoning and analytical reasoning in the style of the Greeks. Then I will establish this more rigorously by contrasting the concept of dedomenon with the algebraic concept of the known, which is often treated as the analogue to the Greek concept. I will heighten this contrast somewhat with some remarks about the character of the "hypothetical" beginning appropriate to each method. And, by way of conclusion, I will point out how the ultimate intention behind each way of thinking is quite different.
A Geometrical Context for the Greek Concept Dedomenon
Again recalling the explication of the word 'dedomenon' at the end of the translation, my immediate task is to present a mathematically meaningful geometrical situation that can be grammatically schematized as follows: a subject has something else given to it so irrevocably that it becomes a permanent possession of that subject under such conditions that it is possible to assign the responsibility for this action. That is, I must show the geometrical significance of the verb 'to give' as it is influenced by its occurrence in the perfect tense, and I must point to both a direct and indirect object of this giving. Furthermore there must be an identifiable agent of the giving.
Say that the Geometer comes upon a straight line that has certain describable geometrical relations to other points, lines, angles, figures. Perhaps it falls across two parallel straight lines while making some angle with them. Or perhaps it is drawn from a certain point on a straight line to make some angle with this line. The Geometer need not know the antecedent history of this line that resulted in its being "just-so" - that is, which geometrical construction resulted in its having just the length it does and in its being in just such a place - but only that it has the specified relations to the other elements in the figure.
Now, it may be the case that the line in question is related to the other geometrical figures in such a way that fixing some of their accidents of size, position and shape in various ways results in the fixity of one of the accidents of the drawn line. Witness the first example, wherein all the lines so drawn must be equal, as long as the two parallel lines are put back in the same places and the transversal always cuts them at the same angle; these same conditions, however, do not constrain these lines to the same place. Or the second example, wherein, as long as the line is returned to its place, and the point fixed in place on it, and the line is erected at the same angle, the lines so drawn are allowed the freedom to assume different lengths, although they are all constrained to be spatially oriented in the same way.
How should the Geometer conceptualize the fixity of accident that the other geometrical figures would impose on the drawn lines? As a determination of that line? Too vague a term. As assigning the line to a certain place or magnitude? Closer. As forcing the line to take up again the same place being held in readiness by the other figures, or to become equal to the same magnitude? In other words, by making such provision that it keeps the magnitude or place that it already possesses? This is sufficiently clear for our purposes. We wll call such a line 'recipient in magnitude' or 'recipient in position' to emphasize that it has been so provided for. (Cf. common Greek usage eu didonai tini.) For the way the other geometrical entities can be responsible for this fixity, compare the Aristotelean material cause to tinon onton anagke tout' einai, which might be rendered into English as "being necessarily so from certain things being so." Thus the other geometrical elements can be regarded as agents in the sense of material causes.
Related Concepts
A whole host of related concepts come up when we consider ways of ascertaining whether such a line is recipient. Basically, we try to show that the line in question has an additional identity, that it possesses implicit relationships to the other figures in the question which are already known to fix it in position or in magnitude. For example, by performing a simple construction that introduces a line corresponding to the line in the second example, the line in the first example above can assume the identity of a line which is equal to a line that is known to be recipient in magnitude. In other words, we show that the magnitude of the line is more properly an accident that pertains to this second identity of the line. That is, we show that this particular accident is not immediately dependent on the conditions under which the line was drawn, but rather on simpler conditions implicit in them.
In some other example we might show that the place of some line was more properly the place of some other line that it was coincident with.
There are two movements to be distinquished here. One is the movement by which ever more basic identities are uncovered for some figure. This may be called anagoge, a "leading-upward" or "leading-backward," often called "reduction" It is the movement that establishes the predicate of positional or extensional recipience for the figure in question.
The second is the movement by which we have been able to ascertain the standard of magnitude (or place) for the line to be the magnitude (or place) of some other figure. In a sense we have detached the magnitude (or place) from the line originally in question. That is, we have analyzed it. This is the movement that actually locates the magnitude or place held in readiness by the fixed figures, which is to serve as a guide or standard for the replacing or reextending of the figure in question.
So here analysis does not mean "taking apart figures". Nor does it have anything to do with "deductive connections" on the purely logical side. Nor is it "solution backwards", supposing that this is all that Pappus meant by his etymology. Analysis may have the character of such a backwards movement, or anagoge, as a result of its own proper function of detaching the place, magnitude or shape of a figure from that figure itself, but that is not why it is called analysis. (If we go over in our minds what happens in a ordinary geometrical demonstration, we should have a fair insight into the reason why Aristotle called his logical writings The Analytics.)
Ultimately this anagoge must come to an end. We uncover the fundamental identity of the figure in question. We cannot detach the place or magnitude of this fundamental figure by further construction. But by this point we should be able to see that this figure is fixed with respect to one of its accidents. How? By imagining that it is introduced by means of an ordinary Euclidean construction that we perform on the places, magnitudes, etc., represented by the other figures in the question. (In the case of the second example above, it would be the Euclidean problem of drawing a line at a given point on a given line which makes an angle equal to a given angle.) The figure so drawn will match the one in question. We will have been able to provide one the same as it out of our own resources. This is a sufficient guarantee that this line would end up having the same magnitude or passing over the same place if it were introduced again in accordance with the same conditions that originally put it there-that it is recipient. I am seriously tempted to call this procedure epagoge, or "leading-onto" or "leading-forward."
The only task that now remains for ascertaining that the line originally in question is recipient is to make sure that the place or magnitude detached from it, which was seen to be the place or magnitude of another figure, is in fact the place or magnitude of the figure originally in question. This can be done with a twofold argument. First of all the Euclidean construction which would serve to match the same line as the line in its basic identity is carried out. Then it is demonstrated that this line not only has the properties of the line in its basic identity, but also the properties and relationships of the line originally in question. But this is not enough, because it could be that there were other lines having these same relationships, and so we would not be sure that we had found the place or magnitude of the line we are interested in. This possibility may be eliminated by showing that no line other than the one we have introduced possesses these relationships. If this can be shown, then we know we have restored the place or magnitude to the line in question. This procedure is fittingly called synthesis. Let me only point out here that the second stage of this argument is more obviously needed in the investigation of loci in the Greek manner.
Thus, the Geometer must be concerned not only with figures (taking this term widely enough to include lines and angles) and the properties that belong to figures due to their kinds, but also with the effects that figures have on one another due to their accidents of size, shape, and position. The propositions of this latter part of geometry will assert that if certain things (or their constituent parts) are recipient in position, magnitude; or shape, something (or its constituent part) standing in describable relations to these also is recipient in some respect.
The kind of proposition in which we need to provide a place, magnitude or shape for preexisting figure was called a porism. One way of distinguishing a porism from a theorem and a problem is to contrast the different intentions that the Greek geometer had when he was proving something about a geometrical figure. That is; he sought not only to verify the properties of figures or their relations to one another through a analysis & synthesis to investigate it if we turn it into a porism.
This is done by means of a hypothesis. Before specifying those very given figures with which the problem must be constructed, draw an analogous diagram. As appropriate, suppose only that these analgous figures occupy some places that we can provide out of our own resources that is, give - or are equal to magnitudes we can give, etc. Thus these figures are all recipient in position or in magnitude. Draw a line to correspond to the line, say, which in the problem must cut off a ratio the same as a given one. Assume only that the ratio that this line in fact cuts off is the same as any given ratio whatsoever. Show that the line is recipient in position, as in any porism. Perform the synthesis in the ordinary manner.
If we can perform this synthesis, and restore the detached place to the line in question for the analysis, the same procedure would work if we used preassigned given figures, in the manner of an ordinary problem. Only we would not be restoring a place to a line, but merely preparing a determinate setting for the appropriate line. Basically we just need to use the determinate problem embedded in the porism itself. That is, the analysis & synthesis have discovered the rule that can now be applied to any determinate givens whatever.
In the editorial commentary to the first locus in the translation itself, I have suggested that diorismos, the determination of the limits and conditions of possibility in some problem, may have used the analytical procedure to detach the limit (horos) of the ratios, and synthesis to ascertain that it was a true limit. It is actually this particular application of analysis & synthesis that furnishes the most direct link to Aristotle's Analytics.
A Historical Conjecture
It may happen that the locus to which the point or figure in question is confined in a porism does not coincide with it, and is not yet present. For example, when a certain point must be bound to a straight line. Then the word 'porism' has a particularly conspicuous application-namely, the line itself must be introduced, instead of just being uncovered. In fact, all the above concepts find their most natural expression in the geometrical situation of loci. We conjecture that they were first introduced in that context, and that the method of analysis was only later extended to the solution of determinate problems.
In support of this, let me here only mention the curious matter of ephectic loci (ephektikoi topoi), in which the place of a point is a point, a line a line, etc. Isn't this in fact what happens in the analysis of a determinate problem such as the ones in On Cutting Off A Ratio, in which the place of the line drawn cutting off a ratio is shown to be the place occupied by another line which cuts some line so that a certain application, say, is possible. Euclid's definition of 'recipient in position', the predicate used at the conclusion of those analyses, actually uses the word epechei.
I have tried to show that it is most natural to think of problematic analysis as an adaptation of an art originally developed to investigate loci. As far as I know there is no historical evidence to refute this. It does account for the somewhat artificial but sophisticated character of the class of ephectic loci.
Perhaps this is the meaning of the legend that Plato invented analysis - namely, that he saw how to adapt it to the investigation of determinate problems.
This is exactly opposite to the historical development in algebra. In a geometrical context, algebra was originally employed most successfully on determinate problems. It was only with difficulty and finesse that it was extended to loci.
Is this a sign of an intrinsic difference in the two arts?
The Known and Its Related Concepts
There is a great deal of confusion over the various algebraic pairs 'known & unknown', 'determinate & indeterminate', 'constant & variable', partially due to the fact that they were introduced at different stages in the development of algebra. It will be an extremely difficult matter to sort out the consistencies, overlays, and exclusions among these concepts. (Compare the difficulty Frege has with the concepts of 'variable' and 'constant' in his paper What Is A Function?)
Historically, at least, the central algebraic concept is 'the known'. This is the concept that held sway during the Renaissance restructuring of algebra. Behind this concept is the concept of 'the given'. This may be seen most clearly in Regiomontanus' work On Triangles. And 'the given' is "that whose measure we can find and give (and hence know) in terms of other quantities that are measured" - something we can provide ourselves, if not in itself, then at least with respect to its position or size, etc.
At this point we could similarly carry out a reduction of the other concepts of algebra, such as reduction, hypothesis, etc., to this concept, but this has already been done sufficiently for our present purposes by Descartes (in the Regulae) and others.
Now the concept of 'the known' as just discussed corresponds closely enough to the attributive Greek participle dothen as I have explicated it in the points following the translation. And that corresponds to the direct object of an act of giving of which we are the agent. But the Greek art of analysis was centered around the dedomenon, which was the indirect object of an act of giving for which the other geometrical figures were responsible. And all the basic concepts of the analytic art were reduced to this concept, while all the basic concepts of algebra could be reduced to the knozon. Thus, by reflecting these two arts into the grammatical structure of the English language in this way, we can conclude that analysis and algebra are to one another as the indirect and direct objects of fundamentally different acts of giving.
Further Comparisons
In view of the above, there is no reason that the concepts in analysis should have a one-to-one correspondence with those in algebra, except for the fact that algebra was reconceptualized in the late Renaissance with one eye on analysis. As a consequence of this, there are concepts in algebra & in analysis that have the same name but radically different meanings. A good example is the "hypothetical" starting points of algebra and analysis.
When we draw a line in algebra and suppose that it solves a problem, we are assuming that it is the very line that satisfies the particular conditions given ahead of time. But there is no way of even knowing whether there is such a line at this point. The line is a fiction. We begin from a situation that may well be impossible, and only redize the impossibility if we encounter a contradiction.
When we draw the same line in analysis we do not assume that it fulfills certain specific preassigned conditions. We only assume, say, that it cuts off a ratio the same as some ratio we could provide ourselves. There are clearly an infinity of such ratios and there is nothing fictional in assuming that this line cuts off one of them. The lines and figures in the analysis are just what they are. They are only the analogues of the lines and figures with which the actual construction of the problem will have to be carried out. So we begin from a situation that is possible, and only then explore the frontiers of possibility.
It would be fair to say that the arguments in analysis are theoretical, while those in algebra are hypothetical. I am reminded of Aristotle's remark that hypothetical syllogisms cannot be analysed.
These different kinds of hypothesis reflect the different intentions behind the two arts. What algebra seeks is the quantity which is supposed to fulfill certain conditions - it seeks a fact. In the course of this investigation it also turns up the relationships that are sufficient to prove that the quantity so found does in fact fulfill the conditions of the problem. Analysis works the other way around. It seeks to show that one of the accidents that the figure actually possesses has a certain fixity - that is, it seeks the cause for this fixity. In the course of its investigation it discovers a rule by which analogous figures could be found in a corresponding problem.
Conclusion
Thus the whole effort of algebraizing analysis is mistranslation in the most fundamental sense.
Two Corollaries
'Recipient' is a proper predicate of a figure. What is provided for the figure is the fixity of one of its accidents of position, magnitude or shape. The first twenty-four propositions of Euclid's Recipients, commonly (but misleadingly) called the Data, prove propositions about the recipience of magnitudes and their ratios. They use straight lines just like the propositions in the fifth book of Euclid. It is usual to think that the objects of such propositions are pure quantities, which the lines symbolize, and not particular quantities, which the lines would illustrate and be the most convenient examples of. But 'recipient' is not the proper predicate of a pure quantity. So it seems that the intentionality of the figures in these propositions is not different than that in any other Greek geometrical proposition. And there is no reason to think that the lines in these propositions of Recipients signify their objects in any different way than the propositions in the fifth book of Euclid.
Regarding the possibility of the formalization of Greek analysis: In my editorial to Bonasoni's Algebra Geometrica, I tried to distinguish two different kinds of signification for mathematical symbols. In ordinary signification the SIGN directs our attention to some other object. We see right through it, as it were. But the kind of symbol I have called a COUNTERPART attracts attention to itself. Manipulations of the counterparts effectively drag along their objects. This kind of symbol is most appropriate for formaliza tion.
Now, the objects in a Greek analysis are the analogues of the objects in the corresponding problem. The objects in the problem are the givens, those in the analysis the recipients or the given-to' s. The direction of giving in the analysis is toward the objects in the analysis, and would favor using its objects as counterparts to the objects in the problem. The difficulty is that the givens and the recipients pertain to different acts of giving. Both the given in the problem and the given-to in the analysis would have to be reconceptualized somewhat before their relationship could be that of object to counterpart.