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Ibn Al-Haytham's Revision of the Euclidean Foundations Of Research Collection Journal Article Ibn al-Haytham’s Revision of the Euclidean Foundations of Mathematics Author(s): Ighbariah, Ahmad; Wagner, Roy Publication Date: 2018-03 Permanent Link: https://doi.org/10.3929/ethz-b-000315867 Originally published in: HOPOS: The Journal of the International Society for the History of Philosophy of Science 8(1), http:// doi.org/10.1086/695957 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library Copyright The International Society for the History of Philosophy of Science 2017. Preprint (not copyedited or formatted). Please use DOI when citing or quoting. DOI: 10.1086/695957 Abstract: This article studies Ibn al-Haytham’s treatment of the common notions from Euclid’s Elements (usually referred to today as the axioms). We argue that Ibn al-Haytham initiated a new approach with regard to these foundational statements, rejecting their qualification as innate, self-evident or primary. We suggest that Ibn al-Haytham’s engagement with experimental science, especially optics, led him to revise the framing of Euclidean common notions in a way that would fit his experimental approach. In his great work on optics, al-ManÁÛir (henceforth: Optics), Ibn al-Haytham does not restrict himself to explaining the sensual processing of visual phenomena (involving light, the eye of the viewer, the position of the observed object with respect to the eye, etc.). He also studies the process of producing cognitive concepts and judgments at the intellectual level. In this system, the common notions are not exceptional primary statements because they undergo the same epistemological process of validation as other statements: they are a product of sensual observations processed by intellectual faculties. Ibn al-Haytham’s approach is echoed in his second commentary of Euclid’s Elements, but contrasts with statements in his first, earlier commentary. In this article we will survey Ibn al- Haytham’s philosophy of mathematics and philosophy of mind as expressed in the two commentaries of the Elements and in the Optics. We will situate this philosophy in its scholarly context, and analyze the motivation that led to its revision. In broad terms, Ibn al-Haytham’s philosophy is in line with that of many of his important contemporaries and predecessors: it is true to a basic form of Aristotelian “empiricism”, but depends also on innate intellectual capacities that carry a Neo-Platonic flavor. However, his approach is uniquely tailored for his own special philosophical project, which involves his 1 This content downloaded from 205.208.116.024 on November 17, 2017 02:32:08 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). Copyright The International Society for the History of Philosophy of Science 2017. Preprint (not copyedited or formatted). Please use DOI when citing or quoting. DOI: 10.1086/695957 engagement with experimentation.1 Ibn al-Haytham’s cognitive architecture provides a detailed description of how concepts and judgments are formed from sense impressions by means of the faculties of imagination (takhayyul) and distinction (tamyÐz), connecting experiential, experimental and mathematical and knowledge in ways that are crucial for the unity of mathematized natural science. The impact of the Latin translation of his Optics on early modern European science is still debated, but it definitely did not go unnoticed; Ibn al-Haytham’s approach continued to resonate well beyond his life span and original cultural context (Omar 1979; Smith 2001, vol. 1, introduction, part 6; Elghonimi 2015; Lindberg 1976, 86). Ibn al-Haytham, the Elements, and the two commentaries Al-Íasan ibn al-Íasan ibn al-Haytham, known in the Latin west as Alhazen or Alhacen, was born around 965 in the city of Basra in today’s southern Iraq. He then moved to Baghdad, extended his education and became famous as a great scientist specializing in a variety of sciences (Ibn ÞAbÐ ÞUÒaybiÝa 1995, vol. 2, 90).2 He was later invited by the Fatimid caliph of Egypt to assist in a project aiming to regulate the flow of the Nile (Sabra 1972, vol. 6, 189).3 He died in Cairo after 1040. Among his interests, we can specify his pioneering studies in optics, astronomy, mathematics, philosophy, physics and other intellectual areas. Philosophically, he was an avid reader of Aristotle, as he testifies in his own autobiography,4 and wrote some epitomes on Aristotle’s works (Ibn ÞAbÐ ÞUÒaybiÝa 1995, vol. 2, 94–5). Ibn al-Haytham is considered to be one of the great mathematicians of Islam, both because of his contributions to the field, and because of his copious writings that came to around twenty-five treatises covering different branches of mathematics.5 2 This content downloaded from 205.208.116.024 on November 17, 2017 02:32:08 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). Copyright The International Society for the History of Philosophy of Science 2017. Preprint (not copyedited or formatted). Please use DOI when citing or quoting. DOI: 10.1086/695957 One of the books that Ibn al-Haytham chose to engage was Euclid’s Elements, which served as a canonical reference and an ideal of scientific presentation ever since its appearance.6 Ibn al-Haytham composed two commentaries on the Elements, where in line with his contemporary scholarly tradition, he interpreted the text (as he and his predecessors understood it), challenged it with criticism, and attempted to resolve these critical charges (Al-QifÔÐ 2005, 56; Ibn al-NadÐm 1996, 129). The first commentary, SharÎ muÒÁdarÁt kitÁb UqlÐdis, has a modern Arabic edition, and was partly translated as Commentary on the Premises of Euclid’s Elements.7 The second commentary is called Íall shukÙk kitÁb UqlÐdis fÐ al-ÞuÒÙl wa sharÎ maÝÁnÐhi (On the Resolution of Doubts8 in Euclid’s Elements and Interpretation of its Special Meanings), and is preserved as a manuscript with a modern facsimile edition.9 When browsing through both commentaries we find many statements indicating that the second came to fill the gaps in the first, such as “this treatise shall become, together with our Commentary on the Premises of Euclid’s Elements, a complete interpretation for the entire treatise” (Ibn al-Haytham 1985, 4; see also Rashed 2002, 8). However, a comparison of Ibn al- Haytham discussion of the premises (muqaddimÁt) in the two texts (definitions, postulates and – most importantly for us – common notions, referred to as ÝulÙm Þuwal, lit. first knowledge), reveals essential differences between the two commentaries. Indeed, SharÎ muÒÁdarÁt is linked to the philosophical tradition which treated the common notions as self-evident primary propositions that cannot be doubted (ÝulÙm Þuwal yashhad bihÁ al-fahm, wa lÁ yaÝriÃu fÐhÁ al-shakk) (Ibn al-Haytham 2005, 140). However, in the subsequent Íall shukÙk, we notice a new development in Ibn al-Haytham’s thinking, as he undermines the foundational status of these common notions. Indeed, he claims that the knowledge expressed by the common notions (or any such statements) is not innate, unprovable 3 This content downloaded from 205.208.116.024 on November 17, 2017 02:32:08 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). Copyright The International Society for the History of Philosophy of Science 2017. Preprint (not copyedited or formatted). Please use DOI when citing or quoting. DOI: 10.1086/695957 or self-evident, but has acquired its foundational status as a result of the frequent use that people made of it. This paper will attempt to analyze and explain this transition. Our analysis will suggest that the new ideas of Íall shukÙk were developed after or in proximity to the methods of the Optics. Note that the development of ideas does not necessarily coincide with the writing of texts, but we should bring evidence at least against the possibility that Íall shukÙk was completed before the Optics was embarked upon. This evidence can be found in Ibn ÞAbÐ ÞUÒaybiÝa’s list of Ibn al-Haytham’s works.10 Rosenfeld and Ihsanoºlu date Íall shukÙk and the Optics to later than and earlier than 1038 respectively, because only the latter appears in Ibn al-Haytham’s list of works up to that year (Rosenfeld and Ihsanoºlu 2003, 132). This claim is strengthened by the fact that while the Optics is cited in several of works of Ibn al-Haytham’s other works, this is never the case for Íall shukÙk. However, Íall shukÙk may include variations of earlier treatises, some of which are mentioned in Ibn ÞAbÐ ÞUÒaybiÝa’s third list (in places 40, 55 and 56 – the second concerning the first book of the Elements, and is therefore specifically relevant for us). But Sabra showed, based on internal citations, that the order in Ibn ÞAbÐ ÞUÒaybiÝa’s third list is approximately chronological. In that list, the Optics appears as number 3, and is cited by treatises in places 36, 38, 49 and later. This means that even the possible drafts of Íall shukÙk are likely to postdate the Optics, and that the most likely chronological sequence of redaction is SharÎ muÒÁdarÁt, the Optics, and Íall shukÙk. We will start by providing some background concerning the premises of the Elements in the Greek and Arabic traditions, and then turn briefly to Arabic theories of mind. Then we will briefly review Ibn al-Haytham’s SharÎ muÒÁdarÁt, move on to a deeper analysis of the second commentary, and draw our conclusions.
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