Between philosophy and mathematics: examples of interactions in classical Islam.In this paper the author raises the question of the existence and the extension of philosophy of mathematics in classical Islam. He considers three types of interactions between mathematics and theoretical philosophy. The first is that used by alKindi who uses the means and methods of mathematics to reconstruct his philosophical system. The second type emerges when the mathematician alTusi tries to solve a philosophical question: the emanation emanation, in philosophy emanation (ĕmənā`shən) [Lat.,=flowing from], cosmological concept that explains the creation of the world by a series of radiations, or emanations, originating in the godhead. of the multiplicity. The third type is comes from the mathematician alSijzi, who philosophically solves a mathematical problem Mathematical problem may mean two slightly different things, both closely related to mathematical games:
Keywords: Philosophy and mathematics in Islamic tradition; links between philosophy and science; history of Islamic science
********** Historians of Islamic philosophy Islamic philosophy (الفلسفة الإسلامية) is a branch of Islamic studies, and is a longstanding attempt to create harmony between philosophy (reason) and the religious teachings of Islam have a keen interest in what is often readily called falsafa. According to according to prep. 1. As stated or indicated by; on the authority of: according to historians. 2. In keeping with: according to instructions. 3. their understanding, it deals with the doctrines of Being and the Soul, as developed by the authors of the Islamic culture, without any consideration for other fields of knowledge and independent of all determination, if it were not for the link these doctrines may have with religion. Thus, Islamic philosophers are perceived to be evolving within the Aristotelian tradition of neo Platonism, and to be no more than heirs of late Antiquity Late Antiquity is a rough periodization (c. AD 300  600) used by historians and other scholars to describe the interval between Classical Antiquity and the Middle Ages in both mainland Europe and the Mediterranean world: generally between the decline of the western Roman Empire , albeit with an Islamic 'touch'. This kind of historical understanding appears to assure us that from the ninth century onwards, the passage of Aristotle, Plotinus, and Proclus, among others, to the Islamic philosophers had taken place safely, without the slightest trouble. But this historical account is advanced not without some serious consequences: not only does it paint for us a very pale and an impoverished picture of philosophical activity, it also transforms the historian into an archaeologist whose skills he happens to lack. Indeed, it is not at all rare that historians set themselves the task of rummaging through the terrain of Islamic philosophy in search of vestiges of Greek works whose originals are lost but are thought to be preserved in the Arabic translations, or that, for lack of better resources, they are at least satisfied with the traces of writings of the philosophers of Antiquity, often studied with the competence and talent characteristic of the historians of Greek philosophy on modern philosophy, as well as modern science. Clear unbroken lines of influence lead from ancient Greek and Hellenistic philosophers, to medieval Muslim philosophers and scientists, to the European Renaissance and Enlightenment, to the secular sciences of the modern day. . It is in this manner that the history of Islamic philosophy is transformed into archaeology, so to speak. It is true that aside from the Greek heritage some historians have recently turned their attention to doctrines developed in other disciplines, such as philosophy of law, masterly developed by the jurists The following lists are of prominent jurists, including judges, listed in alphabetical order by jurisdiction. See also list of lawyers. Antiquity
Science and mathematics, however, are far from having received the same favourable attention given to law, kalam, linguistics, or Sufism. Furthermore, the examining of the relationships, which we regard as essential, between the sciences and philosophy, and notably between mathematics and philosophy, is completely disregarded. This serious lacuna lacuna /la·cu·na/ (lahku´nah) pl. lacu´nae [L.] 1. a small pit or hollow cavity. 2. a defect or gap, as in the field of vision (scotoma). is not the sole responsibility of the historians of philosophy; it is that of the historians of science as well. It is true that the examination of these relationships requires various competencies and an indepth knowledge of these areas: in addition to a linguistic knowledge which goes beyond that called for in geometry (there one might do with elementary syntax and a poor lexicon), one is also asked to have a grasp of the history of philosophy itself. If we realize these requirements are not often met, and on top of that the conception of the relationships between science and philosophy is inherited from an oftambient positivism positivism (pŏ`zĭtĭvĭzəm), philosophical doctrine that denies any validity to speculation or metaphysics. Sometimes associated with empiricism, positivism maintains that metaphysical questions are unanswerable and that the only , we may come to better appreciate the profound indifference exhibited by the historians of science visavis the kind of examination we are calling for. This is despite the wellknown fact that the links which science and philosophy enjoy are but an integral part of the history of sciences. As a matter of fact, the situation is quite paradoxical: while scientific and mathematical research of the most advanced standards had been developed and worked out in Arabic in the urban centres of the Islamicate for a period of seven centuries, is it at all conceivable that philosophers, who were often themselves mathematicians, doctors, and so forth, would have remained recluse in their philosophical activity, totally oblivious to the mutations taking place under their eyes, and completely blind to the successive scientific results that were then being achieved? Moreover, when faced with such an outburst of new disciplines and also success (an astronomy critical of Ptolemaic models, optics reformed and renewed, an algebra created, an algebraic geometry algebraic geometry, branch of geometry, based on analytic geometry, that is concerned with geometric objects (loci) defined by algebraic relations among their coordinates (see Cartesian coordinates). invented, a Diophantine analysis Di·o·phan·tine analysis n. A collection of methods for determining integral solutions of certain algebraic equations. [After Diophantus, thirdcentury a.d. transformed, a theory of the parallels discussed, projective pro·jec·tive adj. 1. Extending outward; projecting. 2. Relating to or made by projection. 3. Mathematics Designating a property of a geometric figure that does not vary when the figure undergoes projection. methods developed, and so on) can it be imagined that philosophers remained unperturbed by these developments, as to deduce that they were strictly confined to the relatively narrow frame of the Aristotelian tradition of neoPlatonism? The seemingly impoverished philosophy in the classical period of Islam, which emerges as a result of such accounts, is due to historians rather than history. Actually, to examine the relationships between philosophy and science, or between philosophy and mathematicswhich is our only concern here as they appear in the works of the 'pure' philosophers, amounts to only a third of the quest desired here. There is a further need to address in this instance, and that is to bring in the views of both the mathematicianphilosophers and the 'pure' mathematicians. But why have we from the beginning taken the view that we should limit our consideration in this instance to mathematics? This idea deserves an explanation, particularly as this approach is by no means exclusive to Islamic philosophy. No other scientific disciplines have contributed to the genesis of theoretical philosophy as mathematics have, and none other than mathematics have established links with philosophy which are not only numerous but also ancient. Indeed, since antiquity, mathematics had never ceased to present to the philosophers' reflection central themes of vital importance: it is owing to owing to prep. Because of; on account of: I couldn't attend, owing to illness. owing to prep → debido a, por causa de mathematics that philosophers acquired their methods of exposition, their argumentation procedure, and were provided even with instruments which they appropriated for their analysis. Mathematics are themselves presented to the philosopher as an object of study, when the latter devotes himself to the clarification of mathematical knowledge, studying its object, methods, whilst inquiring on the characters of its apodictic ap·o·dic·tic adj. Necessarily or demonstrably true; incontrovertible. [Latin apod . Throughout the history of philosophy, questions about this mathematical knowledge (its genesis, its power of extension, the nature of certitude cer·ti·tude n. 1. The state of being certain; complete assurance; confidence. 2. Sureness of occurrence or result; inevitability. 3. it attains, its place in the classification of knowledge) have been asked relentlessly. In that regard, philosophers of Islam during the classic period are no exception to the rule: alKindi, alFarabi, Avicenna, Ibn Bajja Ibn Bajja: see Avempace. , Maimonidesto name only a feware no exception to the rule. Links other than these have also been established between mathematics and theoretical philosophy, albeit in a more subtle and discrete fashion. Whether the aim is forging a method, even logic (as in the meeting of Aristotle and Euclid in relation to the axiomatic method axiomatic method In logic, the procedure by which an entire science or system of theorems is deduced in accordance with specified rules by logical deduction from certain basic propositions (axioms), which in turn are constructed from a few terms taken as primitive. ), or the appeal of alTusi's combinatory analysis in order to solve the philosophical problem of the emanation from the one, their collaboration is again not rare. There is, however, in all those possible forms of relationship, one that is of a particular significance, and one which is instigated by the mathematician and not the philosopher: those doctrines mathematicians worked out in order to justify their own practice. The most opportune conditions for such theoretical constructions are gathered at the time when the mathematician spearheading the research of his period is hindered by an unsurmountable difficulty: namely, when the available mathematical techniques are deemed inadequate in the face of the emerging and spreading new objects. One could think of the diverse variants from the theory of parallels, particularly from the time of Thabit ibn Qurra, to the sort of analysis situs analysis situs: see topology. conceived by Ibn alHaytham Ibn alHaytham (ĭb`ən älhīthäm`) or Alhazen (ălhəzĕn`), 965–c.1040, Arab mathematician. , for the doctrines of the invisibles in the seventeenth century. The relationships between theoretical philosophy and mathematics are essentially found in three types of intellectual oeuvre: those of the philosophers; those of the philosophersmathematicians like alKindi, Nasir alDin Nasir alDin, Nasir adDin , Nasiruddin (meaning "Protector of the Faith") and several other transliterations may refer to one of the following.
We have endeavoured on many occasions to exhibit some of the themes of this philosophy of mathematics simply by digging here and there, our aim being to unearth samples demonstrating the richness of this domain rather than its systematic examination, which is not our purpose here. We do recognize that such a project deserves quite a large book. Yet, what seems most suitable to us is to move away from the pure relation of the views, which philosophers may have expressed on mathematics and their importance; our way is mostly interested in the themes that were debated, in the intimate relationships that unite mathematics to philosophy, and their role in the propping up or scaffolding of doctrines or systems. We are, in other words Adv. 1. in other words  otherwise stated; "in other words, we are broke" put differently , seeking the organising role of mathematics. We will first highlight how the philosophermathematicians proceed in their search for mathematical solutions to philosophical problems, a fruitful approach which generates new doctrines and even new disciplines. As we look at the mathematicians we will bring into relief their attempts at solving philosophically mathematical problems. It will become clear that such an intellectual enterprise is indeed necessary and of farreaching consequences. To clarify the aforementioned typology typology /ty·pol·o·gy/ (tipol´ahje) the study of types; the science of classifying, as bacteria according to type. typology the study of types; the science of classifying, as bacteria according to type. , I will now mention some examples in brief. I. AlKindi as PhilosopherMathematician An appreciation of the relationships between philosophy and mathematics is essential for the reconstitution of alKindi's system. Is it not this very dependence which drives the philosopher to write a book entitled Philosophy Can Only be Acquired Through the Mastery of Mathematics and an epistle epistle (ĭpĭs`əl), in the Bible, a letter of the New Testament. The Pauline Epistles (ascribed to St. Paul) are Romans, First and Second Corinthians, Galatians, Ephesians, Philippians, Colossians, First and Second Thessalonians, First and by the title On the Quantity of Aristotle's Works, in which he presents mathematics as a foundational course prior to the teaching of philosophy? In that epistle he even goes so far as to call the student of philosophy, warning him that he is in fact before the following alternative: either to start with the study of mathematics prior to embarking on the works of Aristotle, as grouped and ordered by alKindi, and then the student could hope to become a real philosopheror else he could skip the study of mathematics altogether, and be only an imitator, so long as his memory does not fail him in this task. It is clear that for alKindi mathematics form the very base of the philosophical course. If we were to delve into their role in the philosophy of alKindiwhich is not what we are doing herewe could then grasp more rigorously the specificity of his oeuvre. Historians tend to look at alKindi's philosophical edifice under two clearly distinct lights. According to the first interpretation, alKindi comes across as a Muslim who represents the Aristotelian tradition of neoPlatonismthat is, a philosopher of a doublylate antiquity. The second sees in him a continuator con·tin·u·a·tor n. One that continues, especially a person who carries on the work of another. of speculative theology theology as founded upon, or influenced by, speculation or metaphysical philosophy. See also: Theology (kalam)that is, a theologian who had to switch languages in order to speak of Greek philosophy. But if we were to grant mathematics the true role they play in the elaboration of philosophy, then alKindi's fundamental options would be brought to relief: one of them stems from his Islamic convictions, as they are explicated and formulated in the tradition of the speculative theology, particularly that of alTawhid, which holds that revelation brings us the truth to which reason can attain; the other falls back and refers to Euclid's Elements as a model and as method: the rational can also be attained by way of the truths inherent to reason, which need to comply with the criteria of the geometric proof, and are therefore independent of revelation. These truths of reason, which here serve as primitive notions and postulates, are, during the period of alKindi, brought about by the Aristotelian Tradition of neoPlatonism. It is these truths that are then chosen to replace the truths offered by revelation to speculative theology, so long as they can satisfy the exigencies of geometric thinking and allow for an exposition that is axiomatic ax·i·o·mat·ic also ax·i·o·mat·i·cal adj. Of, relating to, or resembling an axiom; selfevident: "It's axiomatic in politics that voters won't throw out a presidential incumbent unless they think his challenger will in all appearances. It is in this way that 'the mathematical examination' (alfahs alriyadi) has become the instrument of metaphysics. Epistles EPISTLES, civil law. The name given to a species of rescript. Epistles were the answers given by the prince, when magistrates submitted to him a question of law. Vicle Rescripts. in theoretical philosophy such as First Philosophy as well as On the Finitude fin·i·tude n. The quality or condition of being finite. Noun 1. finitude  the quality of being finite boundedness, finiteness of the Universe are in fact a case in point. In relation to the latter, for instance, alKindi proceeds in an orderly manner in order to demonstrate the inconsistency of the concept of the infinite body. There he begins by defining the primitive terms: magnitude and homogenous homogenous  homogeneous magnitudes. He then introduces what he calls an "absolute/assertoric proposition" (qadiyya haqq), or as he explains elsewhere, "first premises, true and intelligible through immediate inference" (almuqaddimat aluwal alhaqqiyya almacqula bila tawassut), or, again, "first premises, evident, true and immediately intelligible" (almuqaddimat alula alwadiha alhaqqiyya almacqula bila tawassut); that is to say tautological tau·tol·o·gy n. pl. tau·tol·o·gies 1. a. Needless repetition of the same sense in different words; redundancy. b. An instance of such repetition. 2. propositions. These are formulated in terms of primitive notions, relations of order thereof, operations of reunions and separation thereof, and predications: finite and infinite. It is to do with propositions such as of homogenous magnitudes, the magnitudes of which are not greater than the others are equal; or, if to one of the equal homogenous magnitudes is added a magnitude homogenous to it, the magnitudes would be unequal. Finally, alKindi proceeds by demonstration, through reductio ad absurdum [Latin, Reduction to absurdity.] In logic, a method employed to disprove an argument by illustrating how it leads to an absurd consequence. , using a hypothesis: the part of an infinite magnitude is necessarily finite. II. Nasir alDin alTusi as MathematicianPhilosopher (or A Mathematical Resolution of a Philosophical Question) In this example, we are called to reflect on other relationships between mathematics and philosophy in classical Islam: the ties that are established between the two in the instance when the philosopher borrows an instrument from mathematics with the aim of solving a logicometaphysical question. Nevertheless, the situation which is of particular interest to us here has a specific trait: this borrowing returns dividends to the mathematical domain which provided the instrument, enhancing its progress and advancement. The exchange between combinatorics combinatorics (kŏm'bənətôr`ĭks) or combinatorial analysis (kŏm'bĭnətôr`ēəl) and metaphysics is an excellent illustration of this double movement: did not Ibn Sina Ibn Sina: see Avicenna. give, on the basis of his ontological and cosmogonical cos·mog·o·ny n. pl. cos·mog·o·nies 1. The astrophysical study of the origin and evolution of the universe. 2. A specific theory or model of the origin and evolution of the universe. conceptions, a formulation of the doctrine of the emanation from the One? And did not alTusi, when endeavouring to derive multiplicity from the One, manage to see that the doctrine as developed by Ibn Sina could still be dotted with a combinatorial armature armature, in art: see sculpture. Armature That part of an electric rotating machine which includes the main currentcarrying winding. , which was borrowed then from the algebraists? For alTusi's move to work, however, the rules of combination of the algebraists had to be interpreted in a combinatorial manner. And it is this very combinatorial interpretation that finally marked the birth of this discipline called combinative analysis, much exploited after alTusi by mathematicians like alFarisi and Ibn alBannab, among others. On the basis of this contribution, alHalabi, a philosopher of a later period, will attempt to organise the elements of the new discipline, assigning to it a name as a way of demarcating it and underscoring its autonomy. It is imperative for us to, nonetheless, distinguish this movement from the kind of progression like that of Raymond Lulle. Lulle has combined notions following mechanical rules, the results of which appeared later to be arrangements and combinations. But Lulle has borrowed nothing from mathematics and his approach does not have the slightest consideration for mathematics. This is unlike alTusi whose approach comes very close to that of Leibniz, despite the differences that may exist between the two projects: the first, as we have already stated, intends to solve mathematically the issue of the emanation of the multiplicity from the One, which in the end brought him to galvanise Verb 1. galvanise  to stimulate to action ; "..startled him awake"; "galvanized into action" galvanize, startle ball over, blow out of the water, floor, shock, take aback  surprise greatly; knock someone's socks off; "I was floored when I heard that I was the Avicenian doctrine of creation with a combinatorial armature; the second wanted in fact to build an ars inveniendi on the combinatorial. The emanation of the intelligences, of the celestial bodies as well as the other worldsthat of nature and that of corporeal Possessing a physical nature; having an objective, tangible existence; being capable of perception by touch and sight. Under Common Law, corporeal hereditaments are physical objects encompassed in land, including the land itself and any tangible object on it, that can be thingsfrom the One, is one of the central doctrines of Avicenna's metaphysics. This doctrine raises a question which is ontological and noetic no·et·ic adj. Of, relating to, originating in, or apprehended by the intellect. [Greek no at the same time: how is it that from one being, unique and simple, there could emanate a multiplicity, which is also a complexity that comprises in the end both the matter of things and the form of the bodies and the human souls? This ontological and noetic duality raises an obstacle, as both a logical and metaphysical difficulty that must be disentangled. From this we understand, at least in part, why Ibn Sina time and again had returned to this doctrine, and implicitly to this question. A study of the historical evolution of Ibn Sina's thought on this issue, in light of his different writings, would reveal to us how he managed to amend an initial formulation with such a difficulty in mind. Let us confine our field to his alShifa' and alIsharat wa alTanbihat. There Ibn Sina exposes the principles of this doctrine as well as the rules of the emanation of multiples from a simple unity. His explication ex·pli·cate tr.v. ex·pli·cat·ed, ex·pli·cat·ing, ex·pli·cates To make clear the meaning of; explain. See Synonyms at explain. [Latin explic seems articulate and orderly, but is short of constituting a rigorous proof: in fact, he does not provide there the syntactical rules apt to espouse the semantic of the emanation, while it is precisely in this that the difficulty of the question related to the derivation of the multiplicity from the One resides. Indeed, this issue of derivation had been perceived as a problem for a long time. The mathematician, philosopher, and commentator of Ibn Sina, Nasir alDin alTusi (12011273), not only grasped the difficulty, but he also wanted to come up with the syntactical rules which were then lacking. In his commentary on alIsharat wa alTanbihat, alTusi introduces the language and the procedures of combination to follow the emanation until the third order of beings. He then stops the application of these procedures to conclude: "if we were to exceed these three orders [the first three], there may exist an uncountable uncountable  countable multiplicity (la yuhsa cadaduha) in one single order, and to the infinite". AlTusi's intention is thus clear, just as the procedure he applied for the first three orders leaves us with no doubt: the proof and the means that Ibn Sina lacked must be put forward. At this stage, however, alTusi is still far from the aim. It is one thing to proceed by combinations for a number of objects, and it is another thing altogether to construct a language with its syntax. Here, this language would be that of combinations. And it is precisely for the introduction of this language that alTusi strives in a separate essay, whose title leaves no ambiguity in the air: On the Demonstration Concerning the Mode of the Emanation of Things in Infinite [Numbers] from the Beginning of the First and Unique Principle. In this instance, alTusi proceeds generally through the combinatorial analysis. AlTusi's text and the results it contains will not disappear with their author; they are found again in a later treatise which was completely devoted to the combinative analysis. Thus, not only does alTusi's solution mark out a style of research in philosophy, it also represents an interesting contribution to the very history of mathematics. III. AlSijzi as Mathematician (or A Philosophical Solution to a Mathematical Problem) In proposition fourteen of his book On the Conics Con´ics n. 1. That branch of geometry which treats of the cone and the curves which arise from its sections. 2. Conic sections. , Apollonius proposes to demonstrate that the asymptotes and the hyperbole come closer to one another indefinitely without actually ever meeting. This proposition obviously calls for that formidable notion of the infinite. Firstly, the infinite is presented as an object of knowledge, since what is at issue are mathematical beings whose existence entail infinite processes. This is but a trait peculiar to any asymptomatic behaviour. The idea of the infinite, however, also comes to the fore as a means for knowledge called for by the infinite mathematical construction, such as the infinite construction of the rest of the distances between the curve and its asymptote asymptote In mathematics, a line or curve that acts as the limit of another line or curve. For example, a descending curve that approaches but does not reach the horizontal axis is said to be asymptotic to that axis, which is the asymptote of the curve. : one needs to ascertain that it is always possible to reiterate the same construction. Now, it is not hard to understand that this notion of the infinite, as professed by Apollonius, must have bothered both mathematicians and philosophers. For if the former could not have remained indifferent to an obvious difficulty in the demonstration, mainly due to the use of a notion which was never clearly drawn, the latter must have been attuned at·tune tr.v. at·tuned, at·tun·ing, at·tunes 1. To bring into a harmonious or responsive relationship: an industry that is not attuned to market demands. 2. to a new problem which happens to be emerging at that very conjuncture con·junc·ture n. 1. A combination, as of events or circumstances: "the power that lies in the conjuncture of faith and fatherland" Conor Cruise O'Brien. 2. , and whose traces had continued to persist even during the eighteenth century viz., the gap between our ability to conceive of Verb 1. conceive of  form a mental image of something that is not present or that is not the case; "Can you conceive of him as the president?" envisage, ideate, imagine a property and our capacity to actually rigorously establish it. Can we establish a mathematical property which we can not distinctly conceive? We may need to take a few steps back in order to localise v. t. 1. Same as localize. Verb 1. localise  identify the location or place of; "We localized the source of the infection" localize, place the commencement of this philosophical interrogation interrogation In criminal law, process of formally and systematically questioning a suspect in order to elicit incriminating responses. The process is largely outside the governance of law, though in the U.S. . In the eyes of alSijzi, this proposition covers up a problematic which he tries to elicit in the rather inflected in·flect v. in·flect·ed, in·flect·ing, in·flects v.tr. 1. To alter (the voice) in tone or pitch; modulate. 2. Grammar To alter (a word) by inflection. 3. language of Islamic Aristotelian philosophy. In line with the Aristotelian Islamic philosophers, alSijzi seems in fact to admit that mathematical knowledge, like any other knowledge, may be characterised by the twosome "conception/judgment" (tasawwur/tasdiq); whereas in mathematics this twosome is confined to that of "conception/demonstration", in that here judgment is considered but a demonstrative LEGACY, DEMONSTRATIVE. A demonstrative legacy is a bequest of a certain sum of money; intended for the legatee at all events, with a fund particularly referred to for its payment; so that if the estate be not the testator's property at his death, the legacy will not fail: but be payable syllogism syllogism, a mode of argument that forms the core of the body of Western logical thought. Aristotle defined syllogistic logic, and his formulations were thought to be the final word in logic; they underwent only minor revisions in the subsequent 2,200 years. . Again in line with the Aristotelians, alSijzi only recognizes that conception which is 'essential' and revealed by way of a rational intuition or expressed in a definition. In his case, as in others indeed, we may paraphrase that famous text of the Second Analytics: conception "shows what a thing is, whereas demonstration shows whether a thing is or is not attributed to a given other". Following this terminology, Apollonius' proposition raises the problem of those affirmations which are demonstrable while they remain inconceivable or hardly conceivable at the very least. Having said that, we know that to establish the proposition of Apollonius rigorously one needed to make use of concepts and techniques which al Sijzi as a mathematician had not yet possessed: these are the concepts and the means for the analysis. But, in this case, it is worth noting that philosophical elucidation allows the mathematician to actually elbow in and make a dent till the plotting of future mathematical pathways is laid out. And if the mathematical difficulty calls for a philosophical thematic, philosophical explication is in turn presented as a means for the reflection of the mathematician. It is these two tasks together which completely characterise alSijzi's approach. In the first instance, he is driven to make a comparison between conception and demonstration with the aim of establishing a typology of mathematical propositions, which will then permit him to close in on the exact type of Apollonius proposition. Following the Aristotelian philosophers, he begins by recognising two extreme types, the confrontation of which shows clearly that there can not be a conception of all that lends itself to demonstration: this is precisely the case with the proposition advanced by Apollonius. We could, on the other hand, seize the essence of the object of a proposition, and conceive of it without having recourse to a demonstration. Between these two extreme types are the others, the intermediary ones; and it is then that alSijzi brings to relief the classification of the mathematical propositions. 1. Propositions which are directly conceivable on the basis of philosophical principles. 2. Propositions which are conceivable prior to attempting their demonstration. 3. Propositions which are conceivable only once an idea of their demonstration is formed. 4. Propositions which are conceivable only once they are demonstrated. 5. Propositions which are hardly conceivable, even after they had been demonstrated. Thus, alSijzi has provided us with one of the earliest classifications of mathematical propositions from the twosome "conception/ demonstration". IV. The Philosophical Problem of the Unity of Mathematics: the Mathematicians' Examination Let us recall, although it goes without saying, that the heirs of Hellenic mathematics had been accumulating results and methods through active research for more than two centuries, and were thus led to conceive of disciplines unknown to the Greeks: algebra, the entire Diophantine analysis, and the algebraic 1. (language) ALGEBRAIC  An early system on MIT's Whirlwind. [CACM 2(5):16 (May 1959)]. 2. (theory) algebraic  In domain theory, a complete partial order is algebraic if every element is the least upper bound of some chain of compact elements. theory of the cubic equations, to name only a few. Again, with their access to works in astronomy, optics, and statistics, mathematicians were also led to 'renew' Hellenic geometry, introducing new chapters to it. Among the renewed disciplines, one could list infinitesimal in·fin·i·tes·i·mal adj. 1. Immeasurably or incalculably minute. 2. Mathematics Capable of having values approaching zero as a limit. n. 1. geometry, spheric geometry, and so on. As for the new chapters, they deal with geometry of position and form, particularly on the study of geometrical transformations. The language of the quadrivium quad·riv·i·um n. pl. quad·riv·i·a The higher division of the seven liberal arts in the Middle Ages, composed of geometry, astronomy, arithmetic, and music. had proven inept to contain such diversity, and things had already got a bit jammed with that of the theory of proportions. Consequently, a search began for another mode of demonstration that could be algebraic and undertake projections both through proportions and foldingbacks. It still remained that the global landscapethe development of an increasing diversity on the one hand and the persistence of a wooden language on the otherwas begging, as it were, for both a logical examination and a philosophical elucidation. Philosophers of the calibre of alFarabi appear to have anticipated some of the difficulties engendered by this situation. The latter had for instance conceived a new ontology ontology: see metaphysics. ontology Theory of being as such. It was originally called “first philosophy” by Aristotle. In the 18th century Christian Wolff contrasted ontology, or general metaphysics, with special metaphysical theories of the mathematical object and an architecture other than that of the quadrivium, for the purpose of the composition of an encyclopaedia of mathematics and of other forms of knowledge in general. However, for reasons that were at once theoretical as well as practical, it behoved the mathematicians alone to face to these difficulties, and indeed, soon thereafter they came up against them, especially in their compositions on the analysis and the synthesis. This encyclopaedic Adj. 1. encyclopaedic  broad in scope or content; "encyclopedic knowledge" encyclopedic comprehensive  including all or everything; "comprehensive coverage"; "a comprehensive history of the revolution"; "a comprehensive survey"; "a comprehensive education" aspect of the analysis and the synthesis has since recalled a vibrant problem, but one obscured in this context: to render an account of the new disciplines, and restore the unity of mathematics. By the end of the ninth and beginning of the tenth century, the term 'mathematic' and the term 'geometric' pertained to a host of dispersed disciplines, which from then onwards could be contained by the increasingly narrow frame of the quadrivium. As a matter of fact, it is no longer possible to gather all of these disciplines under one denomination, like that of the 'theory of the magnitudes', for instance. How is the unity of mathematics thought out in these conditions? This is a question at once necessary and difficult: there had been no means at the time, and for a long time still, to attain this unity. Algebra was still far from being the discipline of the algebraic structures In universal algebra, a branch of pure mathematics, an algebraic structure consists of a set closed under one or more operations, satisfying a number of axioms, including none. Abstract algebra is primarily the study of algebraic structures and their properties. , and was not formalised Adj. 1. formalised  concerned with or characterized by rigorous adherence to recognized forms (especially in religion or art); "highly formalized plays like `Waiting for Godot'" formalistic, formalized at all. It could only do some partial unifications, such as the geometry of conics and the theory of equations. As the algebra of structures was yet to be created, mathematicians had no other choice but to find another way: the idea was to find a discipline that was logically prior to all of the other mathematical disciplines, but which did not predate them historically, and was necessarily posterior to all of them, so that it was able to actually provide them with the unifying principles. In the meantime Adv. 1. in the meantime  during the intervening time; "meanwhile I will not think about the problem"; "meantime he was attentive to his other interests"; "in the meantime the police were notified" meantime, meanwhile , no determination of the nature of this discipline or its methods and objects was a priori a priori In epistemology, knowledge that is independent of all particular experiences, as opposed to a posteriori (or empirical) knowledge, which derives from experience. called for. The analysis and the synthesis had manifestly played the role of this unifying discipline. Ibn Sinan (909946) does not preoccupy pre·oc·cu·py tr.v. pre·oc·cu·pied, pre·oc·cu·py·ing, pre·oc·cu·pies 1. To occupy completely the mind or attention of; engross. See Synonyms at monopolize. 2. himself with the entire field of mathematics, but only with geometry, although the unification he did is brought to bear on the gamut of the analysis and synthesis procedures, and the reasonings deployed, independently of the realm of geometry in which they apply. The discipline which justifies the method, viz. the analysis and synthesis as a discipline, is a sort of programmatic logic, in that it allows one to associate an ars inveniendi to an ars demonstrandi. Ibn Sinan's contribution is of a particular interest: it is the first substantial essay, to our knowledge, which deals with this kind of philosophical logic Philosophical logic is the study of the more specifically philosophical aspects of logic. The term contrasts with mathematical logic, and since the development of mathematical logic in the late nineteenth century, it has come to include most of those topics traditionally of mathematics. The author has thus brought the fundamental problem of the unity of geometry to this logicophilosophical discipline of the analysis and the synthesis, inaugurating in this way an entire tradition that can be traced throughout the tenth century all the way to the algebraist al Samawbal in the twelfth century. Indeed, it is also following Ibn Sinan and against him that Ibn alHaytham would develop his project. With Ibn Sinan we cannot say that we have reached the middle of the great century when the mathematical activity is at its zenith. The differentiation between the disciplines was, nonetheless, following its course; geometry of projections had received a strong jolt at the hands of mathematicians like alQuhi and Ibn Sahl For the poet, see Ibn Sahl of Sevilla. Ibn Sahl (Abu Sa`d al`Ala' ibn Sahl) (c. 9401000) was an Arabian mathematician and optics engineer associated with the court of Baghdad. ; geometrical transformations had become an object of reflection and application for the mathematicians; a chapter on geometrical constructions with the help of the conics had taken shape and developed. For geometrical demonstrations we now increasingly have recourse to a foldingback, to punctual punc·tu·al adj. 1. Acting or arriving exactly at the time appointed; prompt. 2. Paid or accomplished at or by the appointed time. 3. Precise; exact. 4. transformations, and to the asymptotic properties of conical curves to demonstrate their points of intersection. Simply put, two types of exigencies had then emerged: demonstrative structures have to be conceived for the new objects as well as providing for their plan of existence. It was recognized the accomplishment of these two intimately linked approaches also requires that the methods employed would have to be founded on the basis of a discipline. This would have to also be general enough as to be able to offer the means of existence to the new geometrical objects, without being reduced to a pure logic; but additionally, it would have to precede logically all the other mathematical disciplines in order to provide foundations to the diverse demonstrative structures. It is this monumental task that Ibn alHaytham has tackled, no doubt by choice, but by necessity as well. We owe to him the undertaking of innovative research of the highest degree not only in all the branches of geometry, but also in arithmetic and the Euclidean theory of numbers, and these are precisely the domains that would occupy him the most. We have indicated above four situations in which philosophermathematicians, mathematicianphilosophers, and 'pure' mathematicians delved into the topic of the philosophy of mathematics, and brought forth as much evidence as is possible for the blossoming of this field from the ninth century onwards. To forget these contributions not only impoverishes the history of philosophy, but it goes as far as to truncate To cut off leading or trailing digits or characters from an item of data without regard to the accuracy of the remaining characters. Truncation occurs when data are converted into a new record with smaller field lengths than the original. the history of mathematics as well. Roshdi Rashed is Professor of History and Philosophy of Science The history and philosophy of science (HPS) is an academic discipline that encompasses the philosophy of science and the history of science. Although many scholars in the field are trained primarily as either historians or as philosophers, there are degreegranting departments of , Centre National de la Recherche Scientifique The Centre national de la recherche scientifique ("National Scientific Research Centre", CNRS) is the largest governmental research organization in France. It involves 26,000 permanent staff (researchers, engineers, and administrative staff) and a further 4,000 temporary , Paris, France; 8 allee du Val de Bievre, 92340 Bourg bourg n. 1. A market town. 2. A medieval village, especially one situated near a castle. [French, from Old French, from Late Latin burgus, fortress, la Reine, Paris, France. Email: rashed@paris7.jussieu.fr. Translated from the original French by Redha Ameur 

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