Part II The Origins of
II.6 The Development of the Idea of Proof
Leo Corry
1 Introduction and Preliminary Considerations
In many respects the development of the idea of proof is coextensive with the development of mathematics as a whole. Looking back into the past, one might at first consider mathematics to be a body of scien- tific knowledge that deals with the properties of num- bers, magnitudes, and figures, obtaining its justifica- tions from proofs rather than, say, from experiments or inductive inferences. Such a characterization, how- ever, is not without problems. For one thing, it imme- diately leaves out important chapters in the history
of civilization that are more naturally associated with mathematics than with any other intellectual activity.
For example, the Mesopotamian and Egyptian cultures developed elaborate bodies of knowledge that would most naturally be described as belonging to arithmetic or geometry, even though nothing is found in them that comes close to the idea of proof as it was later prac- ticed in mathematics at large. To the extent that any justification is given, say, in the thousands of math- ematical procedures found on clay tablets written in cuneiform script, it is inductive or based on experi- ence. The tablets repetitively show—without additional explanation or attempts at general justifications—a given procedure to be followed whenever one is pursu- ing a certain type of result. Later on, in the context of Chinese, Japanese, Mayan, or Hindu cultures, one again finds important developments in fields naturally asso- ciated with mathematics. The extent to which these cul- tures pursued the idea of mathematical proof—a ques- tion that is debated among historians to this day—
was undoubtedly not as great as it was in Greek tra- dition, and it certainly did not take the specific forms we typically associate with the latter. Should one nev- ertheless say that these are instances of mathematical knowledge, even though they are not justified on the basis of some kind of general, deductive proof? If so, then we cannot characterize mathematics as a body of knowledge that is backed up by proofs, as suggested above. However, this litmus test certainly provides a useful criterion—one that we do not want to give up too easily—for distinguishing mathematics from other intellectual endeavors.
Without totally ignoring these important questions, the present account focuses on a story that started, at some point in the past, usually taken to be before or around the fifth centuryb.c.e. in Greece, with the realization that there was a distinctive body of claims, mainly associated with numbers and with diagrams, whose truth could be and needed to be vindicated in a very special way—namely, by means of a general, deductive argument, or “proof.” Exactly when and how this story began is unclear. Equally unclear are the direct historical sources of such a unique idea. Since the emphasis on the use of logic and reason in constructing an argument was well-entrenched in other spheres of public life in ancient Greece—such as politics, rhetoric, and law—much earlier than the fifth centuryb.c.e., it is possible that it is in those domains that the origins of mathematical proof are to be found.
The early stages of this story raise additional ques- tions, both historical and methodological. For instance, Thales of Miletus, the first mathematician known by name (though he was also a philosopher and scientist), is reported to haveprovedseveral geometric theorems, such as, for instance, that the opposite angles between two intersecting straight lines are equal, or that if two vertices of a triangle are the endpoints of the diam- eter of a circle and the third is any other point on the circle then the triangle must be right angled. Even if we were to accept such reports at face value, sev- eral questions would immediately arise: in what sense can it be asserted that Thales “proved” these results?
More specifically, what were Thales’s initial assump- tions and what inference methods did he take to be valid? We know very little about this. However, we do know that, as a result of a complex historical process, a certain corpus of knowledge eventually developed that comprised known results, techniques employed, and problems (both solved and yet requiring solution).
This corpus gradually also incorporated the regulatory idea of proof: that is, the idea that some kind of gen- eral argument, rather than an example (or even many examples), was the necessary justification to be sought in all cases. As part of this development, the idea of proof came to be associated withstrictly deductiveargu- ments, as opposed to, say, dialogic (meaning “nego- tiated”) or “probabilistically inferred” truth. It is an interesting and difficult historical question to estab- lish why this was the case, and one that we will not address here.
euclid’s [VI.2] Elements was compiled some time around the year 300b.c.e.It stands out as the most suc- cessful and comprehensive attempt of its kind to orga- nize the basic concepts, results, proofs, and techniques required by anyone wanting to master this increasingly complex body of knowledge. Still, it is important to stress that it was not the only such attempt within the Hellenic world. This endeavor was not just a matter of compilation, codification, and canonization, such as one can find in any other evolving field of learning at any point in time. Instead, the assertions it contained were of two different kinds, and the distinction was vitally important. On the one hand there were basic assumptions, oraxioms, and on the other there were theorems, which were typically more elaborate state- ments, together with accounts of how they followed from the axioms—that is, proofs. The way that proof was conceived and realized in theElementsbecame the paradigm for centuries to come.
This article outlines the evolution of the idea of deductive proof as initially shaped in the framework of Euclidean-style mathematics and as subsequently practiced in the mainstream mathematical culture of ancient Greece, the Islamic world, Renaissance Europe, early modern European science, and then in the nine- teenth century and at the turn of the twentieth. The main focus will be on geometry: other fields like arith- metic and algebra will be treated mainly in relation to it. This choice is amply justified by the subject mat- ter itself. Indeed, much as mathematics stands out among the sciences for the unique way in which it relies on proof, so Euclidean-style geometry stood out—at least until well into the seventeenth century—among closely related disciplines such as arithmetic, algebra, and trigonometry.
Results in these other disciplines, or indeed the disciplines as a whole, were often regarded as fully legitimate only when they had been provided with a geometric (or geometric-like) foundation. However, important developments in nineteenth-century math- ematics, mainly in connection with the rise of non- euclidean geometries [II.2 §§6–10] and with prob- lems in thefoundations of analysis[II.5], eventually led to a fundamental change of orientation, where arith- metic (and eventuallyset theory[IV.22]) became the bastion of certainty and clarity from which other math- ematical disciplines, geometry included, drew their legitimacy and their clarity. (See the crisis in the foundations of mathematics [II.7] for a detailed account of this development.) And yet, even before this fundamental change, Euclidean-style proof was not the only way in which mathematical proof was con- ceived, explored, and practiced. By focusing mainly on geometry, the present account will necessarily leave out important developments that eventually became the mainstream of legitimate mathematical knowledge. To mention just one important example in this regard, a fundamental question that will not be pursued here is how the principle of mathematical induction originated and developed, became accepted as a legitimate infer- ence rule of universal validity, and was finally codified as one of the basic axioms of arithmetic in the late nine- teenth century. Moreover, the evolution of the notion of proof involves many other dimensions that will not be treated here, such as the development of the inter- nal organization of mathematics into subdisciplines, as well as the changing interrelations between math- ematics and its neighboring disciplines. At a different level, it is related to how mathematics itself evolved as
a socially institutionalized enterprise: we shall not dis- cuss interesting questions about how proofs are pro- duced, made public, disseminated, criticized, and often rewritten and improved.
2 Greek Mathematics
Euclid’sElements is the paradigmatic work of Greek mathematics, partly for what it has to say about the basic concepts, tools, results, and problems of syn- thetic geometry and arithmetic, but also for how it regards the role of a mathematical proof and the form that such a proof takes. All proofs appearing in theEle- ments have six parts and are accompanied by a dia- gram. I illustrate this with the example of proposi- tion I.37. Euclid’s text is quoted here in the classical translation of Sir Thomas Heath, and the meaning of some terms differs from current usage. Thus, two tri- angles are said to be “in the same parallels” if they have the same height and both their bases are contained in a single line, and any two figures are said to be “equal”
if their areas are equal. For the sake of explanation, names of the parts of the proof have been added: these do not appear in the original. The proof is illustrated in figure 1.
Protasis (enunciation). Triangles which are on the same base and in the same parallels are equal to one another.
Ekthesis (setting out). Let ABC, DBC be triangles on the same base BC and in the same parallels AD, BC.
Diorismos (definition of goal). I say that the triangle ABC is equal to the triangle DBC.
Kataskeue (construction). Let AD be produced in both directions to E, F; through B let BE be drawn parallel to CA, and through C let CF be drawn parallel to BD.
Apodeixis (proof). Then each of the figures EBCA, DBCF is a parallelogram; and they are equal, for they are on the same base BC and in the same parallels BC, EF. Moreover the triangle ABC is half of the par- allelogram EBCA, for the diameter AB bisects it; and the triangle DBC is half of the parallelogram DBCF, for the diameter DC bisects it. Therefore the triangle ABC is equal to the triangle DBC.
Sumperasma (conclusion). Therefore triangles which are on the same base and in the same parallels are equal to one another.
This is an example of a proposition that states a prop- erty of geometric figures. TheElements also includes propositions that express a task to be carried out. An
E
B C
A D F
Figure 1Proposition I.37 of Euclid’sElements.
A B
G D
H K
L F
E
C
Figure 2 Proposition IX.35 of Euclid’sElements.
example is proposition I.1: “On a given finite straight line to construct an equilateral triangle.” The same six parts of the proof and the diagram invariably appear in propositions of this kind as well. This formal struc- ture is also followed in all propositions appearing in the threearithmetic books of theElements and, most importantly, all of them are always accompanied by a diagram. Thus, for instance, consider proposition IX.35, which in its original version reads as follows:
If as many numbers as we please be in continued pro- portion, and there be subtracted from the second and the last numbers equal to the first, then, as the excess of the second is to the first, so will the excess of the last be to all those before it.
This cumbersome formulation may prove incompre- hensible on first reading. In more modern terms, an equivalent to this theorem would state that, given a geometric progressiona1, a2, . . . , an+1, we have
(an+1−a1):(a1+a2+ · · · +an)=(a2−a1):a1. This translation, however, fails to convey the spirit of the original, in which no formal symbolic manipulation is, or can be, made. More importantly, a modern alge- braic proof fails to convey the ubiquity of diagrams in Greek mathematical proofs, even where they are not needed for a truly geometric construction. Indeed, the accompanying diagram for proposition IX.35 is shown
as figure 2 and the first few lines of the proof are as follows:
Let there be as many numbers as we please in contin- ued proportion A, BC, D, EF, beginning from A as least and let there be subtracted from BC and EF the num- bers BG, FH, each equal to A; I say that, as GC is to A, so is EH to A, BC, D. For let FK be made equal to BC and FL equal to D.. . .
This proposition and its proof provide good exam- ples of the capabilities, as well as the limitations, of ancient Greek practices of notation, and especially of how they managed without a truly symbolic language.
In particular, they demonstrate that proofs were never conceived by the Greeks, even ideally, as purely logical constructs, but rather as specific kinds of arguments that one applied to a diagram. The diagram was not just a visual aid to the argumentation. Rather, through the ekthesis part of the proof, it embodied the idea referred to by the general character and formulation of the proposition.
Together with the centrality of diagrams, the six- part structure is also typical of most of Greek math- ematics. The constructions and diagrams that typi- cally appeared in Greek mathematical proofs were not of an arbitrary kind, but what we identify today as straightedge-and-compass constructions. The reason- ing in theapodeixispart could be either a direct deduc- tion or an argument by contradiction, but the result was always known in advance and the proof was a means to justify it. In addition, Greek geometric thinking, and in particular Euclid-style geometric proofs, strictly adhered to a principle of homogeneity. That is, magni- tudes were only compared with, added to, or subtracted from magnitudes of like kind—numbers, lengths, areas, or volumes. (Seenumbers[II.1 §2] for more about this.) Of particular interest are those Greek proofs con- cerned with lengths of curves, as well as with areas or volumes enclosed by curvilinear shapes. Greek mathe- maticians lacked a flexible notation capable of express- ing the gradual approximation of curves by polygons and an eventual passage to the infinite. Instead, they devised a special kind of proof that involved what can retrospectively be seen as an implicit passage to the limit, but which did so in the framework of a purely geo- metric proof and thus unmistakably followed the six- part proof-scheme described above. This implicit pas- sage to the infinite was based on the application of a continuity principle, later associated witharchimedes [VI.3]. In Euclid’s formulation, for instance, the princi-
A
D R
Q C P B
O E
K F
L G
M
H S
Figure 3Proposition XII.2 of Euclid’sElements.
ple states that, given two unequal magnitudes of the same kind,A,B(be they two lengths, two areas, or two volumes), withAgreater thanB, and if we subtract from Aa magnitude which is greater than A/2, and from the remainder we subtract a magnitude that is greater than its half, and if this process is iterated a sufficient number of times, then we will eventually remain with a magnitude that is smaller thanB. Euclid used this prin- ciple to prove, for instance, that the ratio of the areas of two circles equals the ratio of the squares of their diameters (XII.2). The method used, later known as the exhaustion method, was based on adouble contradic- tionthat became standard for many centuries to come.
This double contradiction is illustrated in figure 3, the accompanying diagram to the proposition.
If the ratio of the square on BD to the square on FH is not the same as the ratio of circle ABCD to circle EFGH, then it must be the same as the ratio of circle ABCD to an area S either larger or smaller than cir- cle EFGH. The curvilinear figures are approximated by polygons, since the continuity principle allows the dif- ference between the inscribed polygon and the circle to be as close as desired (e.g., closer than the differ- ence betweenSand EFGH). The “double contradiction”
is reached if one assumes thatS is either smaller or larger than EFGH.
Forms of proof and constructions other than those mentioned so far are occasionally found in Greek math- ematical texts. These include diagrams based on what is assumed to be the synchronized motion of two lines (e.g., the trisectrix, or Archimedes’ spiral), mechanical devices of many sorts, or reasoning based on ideal- ized mechanical considerations. However, the Euclid- ean type of proof described above remained a model to be followed wherever possible. There is a famous Archimedes palimpsest that provides evidence of how less canonical methods, drawing on mechanical consid- erations (albeit of a highly idealized kind), were used to
deduce results about areas and volumes. However, even this bears testimony to the primacy of the ideal model:
there is a letter from Archimedes to Eratosthenes in which he displays the ingenuity of his mechanical meth- ods but at the same time is at pains to stress their heuristic character.
3 Islamic and Renaissance Mathematics Just as Euclid is now considered to represent an entire mainstream tradition of Greek mathematics, so al- khw¯arizm¯ı [VI.5] is regarded as a representative of Islamic mathematics. There are two main traits of his work that are relevant to the present account and that became increasingly central to the development of mathematics, starting with his works in the late eighth century and continuing until the works ofcar- dano[VI.7] in sixteenth-century Italy. These traits are a pervasive “algebraization” of mathematical thinking, and a continued reliance on Euclidean-style geometric proof as the main way of legitimizing the validity of mathematical knowledge in general and of algebraic reasoning in mathematics in particular.
The prime example of this combination is found in al-Khw¯arizm¯ı’s seminal text al-Kit¯ab al-mukhtas.ar f¯ı h.is¯ab al-jabr wa’l-muq¯abala(“The compendious book on calculation by completion and balancing”), where he discusses the solutions of problems in which the unknown length appears in combination with numbers and squares (the side of which is an unknown). Since he only envisages the possibility of positive “coefficients”
and positive rational solutions, al-Khw¯arizm¯ı needs to consider six different situations each of which requires a different recipe for finding the unknown: the full- grown idea of a general quadratic equation and an algo- rithm to solve it in all cases does not appear in Islamic mathematical texts. For instance, the problem “squares and roots equal to numbers” (e.g.,x2+10x =39, in modern notation) and the problem “roots and numbers equal to squares” (e.g., 3x+4 = x2) are considered to be completely different ones, as are their solutions, and accordingly al-Khw¯arizm¯ı treats them separately.
In all cases, however, al-Khw¯arizm¯ıproves the validity of the method described by translating it into geomet- ric terms and then relying on Euclid-like geometric the- orems built around a specific diagram. It is noteworthy, however, that the problems refer to specific numeri- cal quantities associated with the magnitudes involved, and these measured magnitudes refer to the accompa- nying diagrams as well. In this way, al-Khw¯arizm¯ıinter- estingly departs from the Euclidean style of proof. Still,
f
e
d c
b a
Figure 4 Al-Khw¯arizm¯ı’s geometric justification of the formula for a quadratic equation.
the Greek principle of homogeneity is essentially pre- served, as the three quantities usually involved in the problem are all of the same kind, namely, areas.
Consider, for instance, the equation x2+10x = 39, which corresponds to the following problem of al-Khw¯arizm¯ı.
What is the square which combined with ten of its roots will give a sum total of 39?
The recipe prescribes the following steps.
Take one-half of the roots[5]and multiply them by itself[25]. Add this amount to 39 and obtain 64. Take the square root of this, which is eight, subtract from it half the roots, leaving three. The number three there- fore represents one root of this square, which itself, of course, is nine.
Thejustificationis provided by figure 4.
Here ab represents the said square, which for us is x2, and the rectangles c, d, e, f represent an area of
10
4xeach, so that all of them together equal 10x, as in the problem. Thus, the small squares in the cor- ners represent an area of 6.25 each, and we can “com- plete” the large square, being equal to 64, and whose side is therefore 8, thus yielding the solution 3 for the unknown.
Abu Kamil Shuja, just one generation after al-Khw¯ar- izm¯ı, added force to this approach when he solved additional problems while specifically relying on theo- rems taken from theElements, including the accompa- nying diagrams, in order to justify his method of solu- tion. The primacy of the Euclidean-type proof, which was already accepted in geometry and arithmetic, thus also became associated with the algebraic methods that eventually turned into the main topic of inter- est in Renaissance mathematics. Cardano’s 1545Ars