I present a systematic interpretation of the foundational purpose of constructions in ancient Greek geometry. I argue that Greek geometers were committed to an operationalist foundational program, according to which all of mathematics—including its entire ontology and epistemology—is based entirely on concrete physical constructions. On this reading, key foundational aspects of Greek geometry are analogous to core tenets of 20th-century operationalist/positivist/constructivist/intuitionist philosophy of science and mathematics. Operationalism provides coherent answers to a range of traditional (...) philosophical problems regarding classical mathematics, such as the epistemic warrant and generality of diagrammatic reasoning, superposition, and the relation between constructivism and proof by contradiction. Alleged logical flaws in Euclid can be interpreted as sound operationalist reasoning. Operationalism also provides a compelling philosophical motivation for the otherwise inexplicable Greek obsession with cube duplication, angle trisection, and circle quadrature. Operationalism makes coherent sense of numerous specific choices made in this tradition, and suggests new interpretations of several solutions to these problems. In particular, I argue that: Archytas’s cube duplication was originally a single-motion machine; Diocles’s cissoid was originally traced by a linkage device; Greek conic section theory was thoroughly constructive, based on the conic compass; in a few cases, string-based constructions of conic sections were used instead; pointwise constructions of curves were rejected in foundational contexts by Greek mathematicians, with good reason. Operationalism enables us to view the classical geometrical tradition as a more unified and philosophically aware enterprise than has hitherto been recognised. (shrink)

Some philosophers hold that mathematics depends on idealising assumptions. While these thinkers typically emphasise the role of idealisation in set theory, Edmund Husserl argues that idealisation is constitutive of the early Greek geometry that is codified by Euclid. This paper takes up Husserl's idea by investigating three major developments of Greek geometry: Thalean analogical idealisation, Hippocratean dynamic idealisation, and Archimedean mechanical idealisation. I argue that these idealisations are not, as Husserl held, primarily a matter of ‘smoothing (...) out’ sensory reality to produce ideal, ‘perfect’ figures. Rather, Greek geometry depends on assuming some falsehoods – equidistance from a source of illumination, a perfectly unwavering hand, or a machine that weights abstract objects – in order to make complex problems tractable. Although these idealisations were rarely discussed explicitly in antiquity, they can be systematically reconstructed from our extant sources. (shrink)

We present an approach in which ancient Greek mathematical proofs by Hippocrates of Chios and Euclid are addressed as a form of (guided) intentional reasoning. Schematically, in a proof, we start with a sentence that works as a premise; this sentence is followed by another, the conclusion of what we might take to be an inferential step. That goes on until the last conclusion is reached. Guided by the text, we go through small inferential steps; in each one, (...) we go through an autonomous reasoning process linking the premise to the conclusion. The reasoning process is accompanied by a metareasoning process. Metareasoning gives rise to a feeling-knowing of correctness. In each step/cycle of the proof, we have a feeling-knowing of correctness. Overall, we reach a feeling of correctness for the whole proof. We suggest that this approach allows us to address the issues of how a proof functions, for us, as an enabler to ascertain the correctness of its argument and how we ascertain this correctness. (shrink)

in this paper we return to Marshall Clagett’s view about the existence of an ancient Greek geometry of motion. It can be read in two ways. As a basic presentation of ancient Greek geometry of motion, followed by some aspects of its further development in landmark works by Galileo and Newton. Conversely, it can be read as a basic presentation of aspects of Galileo’s and Newton’s mathematics that can be considered as developments of a (...) geometry of motion that was first conceived by ancient Greek mathematicians. (shrink)

The purpose of this work is to address the relation existing between ancient Greek practical geometry and ancient Greek pure geometry. In the first part of the work, we will consider practical and pure geometry and how pure geometry can be seen, in some respects, as arising from an idealization of practical geometry. From an analysis of relevant extant texts, we will make explicit the idealizations at play in pure geometry (...) in relation to practical geometry, some of which are basically explicit in definitions, like that of segments in Euclid’s Elements. Then, we will address how in pure geometry we, so to speak, “refer back” to practical geometry. This occurs in two ways. One, in the propositions of pure geometry. The other, when applying pure geometry. In this case, geometrical objects can represent practical figures like, e.g., a practical segment. (shrink)

Peg Rawes examines a "minor tradition" of aesthetic geometries in ontological philosophy. Developed through Kant’s aesthetic subject she explores a trajectory of geometric thinking and geometric figurations--reflective subjects, folds, passages, plenums, envelopes and horizons--in ancient Greek, post-Cartesian and twentieth-century Continental philosophies, through which productive understandings of space and embodies subjectivities are constructed. Six chapters, explore the construction of these aesthetic geometric methods and figures in a series of "geometric" texts by Kant, Plato, Proclus, Spinoza, Leibniz, Bergson, Husserl and (...) Deleuze. In each text, geometry is expressed as a uniquely embodies aesthetic activity because each respective geometric method and figure is imbued with aesthetic sensibility and geometric sense (rather than as disembodies scientific methods). An ontology of aesthetic geometric methods and figures is therefore traced from Kant’s Critical writings, back to Plato and Proclus Greek philosophy, Spinoza and Leibniz’s post-Cartesian philosophies, and forwards to Bergson’s "duration" and Husserl’s "horizons" towards Deleuze’s philosophy of sense. (shrink)

Some ancient Greek coins from the island state of Aegina depict peculiar geometric designs. Hitherto they have been interpreted as anticipations of some Euclidean propositions. But this paper proposes geometrical constructions which establish connections to pre-Euclidean treatments of incommensurability. The earlier Aeginetan coin design from about 500 bc onwards appears as an attempt not only to deal with incommensurability but also to conceal it. It might be related to Plato’s dialogue Timaeus. The newer design from 404 bc onwards (...) reveals incommensurability, namely in the context of ‘doubling the square’. It thereby covers the same topic but a different geometry as passages in Plato’s dialogue Meno (385 bc). This coin design incorporates important elements of ancient Greek geometrical analysis of the fifth century bc like the gnomon, Hippocrates’ squaring of the lunule (ca. 430 bc), and a geometrical version of monetary equivalence. Through this venue, the design’s conceptual lineage might be traced as far back as Heraclitus’ cosmology of about 500 bc. (shrink)

As we will show in the present work, the historical development of pure geometry did not arise as a direct “transition” from practical geometry into pure geometry, at least as these are usually understood. We can discern four phases related to this evolution. Initially, we have practical geometry as applied in ancient Greece and other ancient civilizations. This surveyors’ practical geometry was somewhat transformed in “didactic” contexts when applied to problem-solving. This not-so-practical (...) class='Hi'>geometry is the direct antecedent of the first form of pure geometry, that of Hippocrates of Chios. Only afterward did pure geometry, as we usually understand it, emerge. The form of pure geometry that we find in Euclid’s Elements. (shrink)

This article is primarily concerned with Aristotle’s theory of the syllogistic, and the investigation of the hypothesis that logical symbolism and methodology were in these early stages of a geometrical nature; with the gradual algebraization that occurred historically being one of the main reasons that some of the earlier passages on logic may often appear enigmatic. The article begins with a brief introduction that underlines the importance of geometric thought in ancient Greek science, and continues with a short (...) exposition of Aristotle’s views and methods in regard to logic. We then offer an interpretation of syllogisms, as well as of the main proof methods that are utilized by Aristotle, in terms of diagrams that can be seen as analogous to those of Euclidean geometry; where the various proofs proceed by appropriately manipulating an initially drawn diagram, in a way parallel to constructions that occur in Euclid’s Elements. In this way, logic is presented as following, to a large degree, methodological tenets established by the practice of geometry. Finally, we present a diagrammatic decision procedure that allows one to directly read off the validity of syllogisms from a corresponding diagram, in a way such that no further manipulation of it is necessitated. (shrink)

Greek ancient and early modern geometry necessarily uses diagrams. Among other things, these enter geometrical analysis. The paper distinguishes two sorts of geometrical analysis and shows that in one of them, dubbed “intra-confgurational” analysis, some diagrams necessarily enter as outcomes of a purely material gesture, namely not as result of a codifed constructive procedure, but as result of a free-hand drawing.

This chapter contains section titled: Discussion.

James Gow's A Short History of Greek Mathematics provided the first full account of the subject available in English, and it today remains a clear and thorough guide to early arithmetic and geometry. Beginning with the origins of the numerical system and proceeding through the theorems of Pythagoras, Euclid, Archimedes and many others, the Short History offers in-depth analysis and useful translations of individual texts as well as a broad historical overview of the development of mathematics. Parts I (...) and II concern Greek arithmetic, including the origin of alphabetic numerals and the nomenclature for operations; Part III constitutes a complete history of Greek geometry, from its earliest precursors in Egypt and Babylon through to the innovations of the Ionic, Sophistic, and Academic schools and their followers. Particular attention is given to Pythagorus, Euclid, Archimedes, and Ptolemy, but a host of lesser-known thinkers receive deserved attention as well. (shrink)

ArgumentThe Fourth Postulate of Euclid’s Elements states that all right angles are equal. This principle has always been considered problematic in the deductive economy of the treatise, and even the ancient interpreters were confused about its mathematical role and its epistemological status. The present essay reconsiders the ancient testimonies on the Fourth Postulate, showing that there is no certain evidence for its authenticity, nor for its spuriousness. The paper also considers modern mathematical interpretations of this postulate, pointing out (...) various anachronisms. It further discusses the validity of the ancient proof by superposition of the Fourth Postulate. Finally, the article proposes an interpretation of the history of the concept of angle in Greek geometry between Euclid and Apollonius, and puts forward a conjecture on the interpolation of the Fourth Postulate in the Hellenistic age. The essay contributes to a general reassessment of the axiomatic foundations of ancient mathematics. (shrink)

This paper aims to show that in ancient Greek architecture, it is possible to find a genesis of the geometric modeling of visual perception present in propositions of Euclid's Optics, considering mathematical knowledge as a human wisdom expression. Let us start by emphasizing that mathematical thinking is not exclusively rooted in mathematical disciplines, but also includes the broad spectrum of human activities, including activities that come from everyday life. Based on this, we present a socio-cultural characterization of human (...) experience as the source and sustainer of mathematical knowledge. Thereafter, on the basis of a content and contextual analysis of _Euclid's Optics_, we explain the use and development of geometry in the study of various optical effects of visual perception in the architectural art embodied in the Acropolis of Athens; and we focus our analysis on one of the most important of these effects, the Acropolis of Athens. We focus our analysis on one of the most emblematic architectural works of ancient Greek culture: the Parthenon of Athens. (shrink)

In lieu of an abstract, here is a brief excerpt of the content: Unmasking the Maxim: An Ancient Genre And Why It Matters Now W. ROBERT CONNOR We live surrounded by maxims, often without even noticing them. They are easily dismissed as platitudes, banalities or harmless clichés, but even in an age of big data and number crunching we put them to work almost every day. A Silicon Valley whiz kid says, Move Fast and Break Things. Investors try to (...) Buy Low and Sell High. Investigative reporters Follow the Money. Others Follow their Bliss. The lavish host thinks, The More the Merrier, while the Modernist architect is sure that Less is More. Captains of Industry whisper to themselves, Me First; captains of sinking ships shout, Women and Children First. Be Prepared was the Boy Scouts’ marching song, while the Proud Boys Stand Down and Stand By, awaiting their marching orders. Maxims are perhaps the smallest members in a large family of speech acts, which among the Greeks included proverbs, oracles, riddles, blessings, curses and lamentations. Not all of these continue in use, but maxims have found companions—mottos, mantras, advertising tag lines and jingles, slogans, political rallying cries, hashtags, three- or four-letter acronyms and 280 character tweets from the goddess Twitter. Familiarity, however, can breed contempt, or at least inattentiveness to the full range of their influence. Sometimes what sounds at first like empty verbiage turns into action. Bruce Lee adapted an old Taoist saying when he said, Be Water, and thereby provided what was for a while a surprisingly effective strategy for protesters in Hong Kong. arion 28.3 winter 2021 6 unmasking the maxim Maxims empower, for good or ill. In the hands of bigots or ideologues, they can turn deadly. Sic Semper Tyrannis shouted John Wilkes Booth after shooting Abraham Lincoln. A manufacturer of assault rifles urged potential customers to Earn Your Man Card. How better to earn it than to obey 8chan, a web site favored by white nationalists, with its own maxim, Embrace Infamy. A devotee of the site did just that, perpetrating a mass shooting in El Paso, Texas. Maxims, then, should not be lightly dismissed. Since the ancient Greeks so relished maxims, they should surely be interrogated to help us better understand the power of these often underestimated speech acts. Before turning to the Greeks, however, the English word maxim needs a closer look. english “maxim” english borrowed maxim from a Latin phrase and boiled it down for everyday use. In Latin writings about logic, the expression maxima propositio, biggest proposition, denoted a statement that did not need to be proved, but could provide the basis for proof of other, lesser propositions. This phraseology goes back at least to Boethius in the sixth century of our era. It’s the equivalent of the generalizations that serve as the major premises in syllogistic logic. If this makes maxims sound like the axioms of geometry, that is historically right. The starting points of plane geometry are propositions that deserve to be accepted; even without proof that they are worthy of belief. That’s the meaning of the term axiōma. No one needs to prove that things equal to the same thing are equal to each other. Think about it; after a while it seems self-evident. It is worthy of assent. We might equally well call such an axiom a maxim; indeed, the earliest (1426) attested use of maxim in English is described in the Oxford English Dictionary as “an axiom; a self-evident proposition assumed as a premise in mathematical, or dialectical reasoning.” W. Robert Connor 7 This usage is now obsolete, but Thomas Jefferson understood the idea behind it, since he believed that there were such truths, waiting to be put to work in building a society where all people were equal and possess certain inalienable rights. He did not feel he had to prove these propositions; they were in his view self-evident. In using these ideas in his draft of the Declaration of Independence, he was, in effect, transferring to politics what Euclid had done in geometry. It was a swift, strategic stroke on his part, for, since self-evident truths require no argument or explanation, they focus discussion not on... (shrink)

This radical, profoundly scholarly book explores the purposes and nature of proof in a range of historical settings. It overturns the view that the first mathematical proofs were in Greek geometry and rested on the logical insights of Aristotle by showing how much of that view is an artefact of nineteenth-century historical scholarship. It documents the existence of proofs in ancient mathematical writings about numbers and shows that practitioners of mathematics in Mesopotamian, Chinese and Indian cultures knew (...) how to prove the correctness of algorithms, which are much more prominent outside the limited range of surviving classical Greek texts that historians have taken as the paradigm of ancient mathematics. It opens the way to providing the first comprehensive, textually based history of proof. (shrink)

Descartes’ Rules for the direction of the mind presents us with a theory of knowledge in which imagination, considered as an “aid” for the intellect, plays a key role. This function of schematization, which strongly resembles key features of Proclus’ philosophy of mathematics, is in full accordance with Descartes’ mathematical practice in later works such as La Géométrie from 1637. Although due to its reliance on a form of geometric intuition, it may sound obsolete, I would like to show that (...) this has strong echoes in contemporary philosophy of mathematics, in particular in the trend of the so called “philosophy of mathematical practice”. Indeed Ken Manders’ study on the Euclidean practice, along with Reviel Netz’s historical studies on ancient Greek Geometry, indicate that mathematical imagination can play a central role in mathematical knowledge as bearing specific forms of inference. Moreover, this role can be formalized into sound logical systems. One question of general epistemology is thus to understand this mysterious role of the imagination in reasoning and to assess its relevance for other mathematical practices. Drawing from Edwin Hutchins’ study of “material anchors” in human reasoning, I would like to show that Descartes’ epistemology of mathematics may prove to be a helpful resource in the analysis of mathematical knowledge. (shrink)

In this article, I tackle a heretofore unnoticed difficulty with the application of Pyrrhonian scepticism to science. Sceptics can suspend belief regarding a dogmatic proposition only by setting up opposing arguments for and against that proposition. Since Sextus provides arguments exclusively against particular geometrical definitions in Adversus Mathematicos III, commentators have argued that Sextus’ method is not scepticism, but negative dogmatism. However, commentators have overlooked the fact that arguments in favour of particular geometrical definitions were absent in ancient (...) class='Hi'>geometry, and hence unavailable to Sextus. While this might explain why they are also absent from Sextus’ text, I survey and evaluate various strategies to supply arguments in support of particular geometrical definitions. (shrink)

The discussions of conatus – force, tendency, effort, and striving – in early modern metaphysics have roots in Aristotle’s understanding of life as an internal experience of living force. This paper examines the ways that Spinoza’s conatus is consonant with Aristotle on effort. By tracking effort from his psychology and ethics to aesthetics, I show there is a conatus at the heart of the activity of the ψυχή that involves an intensification of power in a way which anticipates many of (...) the central insights of early modern and 20th century European philosophy. The first section outlines how Aristotle’s developmental conception of the soul as geometrically ordered lays the foundation for his understanding of effort. The developmental series of powers of the soul are analogous to the series of shapes in mathematics. The second section links the striving of the soul to the gradual acquisition of virtues as a directed activity unifying multiplicity. The third examines the paradigm of self-awareness that Aristotelian effort involves. In the final section I show how ancient Greek theories of music were founded on the experience of striving. The “nature” of music is defined by Aristoxenus, and Theophrastus, in relation to the passion and intentionality of the soul. The geometrical order, as a synthesis of elements in geometry, music, or ethics, is a generative process in which past elements are retained and reintegrated in later stages of development. It requires effort to think geometrically, and the progress of knowledge itself is an integral aspect of all effort. Effort is the lived and self-aware cause which, moving step by step in an orderly and deliberate way, grows and advances upon itself. For both Spinoza and Aristotle, effort is the immanent intelligence which accomplishes what is in the purview of its understanding. Thus, will, in this conception of effort, is not something we already possess innately, but emerges gradually by an effort aimed at improvement. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Kant's Transcendental Deduction by Alison LaywineKatherine DunlopAlison Laywine. Kant's Transcendental Deduction. Oxford: Oxford University Press, 2020. Pp. iv + 318. Hardback, $80.00.Alison Laywine's contribution to the rich literature on Kant's "Transcendental Deduction of the Categories" stands out for the novelty of its approach and conclusions. Laywine's declared "strategy" is "to compare and contrast" the Deduction with the Duisburg Nachlaß, an important set of manuscript jottings from the 1770s (...) (10). But her approach is also deeply informed by Kant's writings on metaphysics from the 1750s and 1760s; moreover, she gives attention to ancient Greek geometry and its importance for Kant's thought.I believe Laywine's most important interpretative claim is that the Transcendental Deduction's "final step" addresses "the question of how nature is possible" (210). Here, 'nature' is understood in what Kant calls the "formal sense," as "the totality of rules under which appearances must stand if they are to be thought as joined in one experience" (Prolegomena, §36; cf. B 165). This constitutes a novel answer to the problem posed in Dieter Henrich's landmark 1969 paper "The Proof Structure of the Transcendental Deduction" (Review of Metaphysics 22 [1969]: 640–59): how to understand §§21–26 of the (B-edition) Deduction as proving more than the thesis already stated in §20 (that the manifold in an intuition necessarily stands under the categories).Crucial support for Laywine's interpretation comes from §26 of the Deduction, which claims the possibility of cognizing objects "a priori through categories... as far as the laws of their combination are concerned, thus the possibility of as it were prescribing the law to nature and even making [nature] possible," is now "to be explained" (B 159). Kant does not use the term 'world' in this passage (as he does in Prolegomena §36). But Laywine contends that here the word 'nature' "has unmistakable, deliberate, cosmological connotations," and in fact "means 'world' in the sense of Kant's early cosmology": roughly, a whole unified by means of laws (12–13).The "cosmological" language of §26 does not recur in the following, officially concluding, section of the Deduction. So, we might ask whether Kant's explanation of how the understanding prescribes laws to nature is integral to the Deduction; Henry Allison, for instance, describes this explanation as an "appendix" (Kant's Transcendental Deduction: An Analytical-Historical Commentary [Oxford: Oxford University Press, 2015], 9). The question of whether it belongs integrally has bite because, as Laywine makes clear in her "Conclusion," the universal laws at issue are just the Analogies of Experience. Thus, the passage could be read as the "promissory note with a forward-looking reference to the System of Principles" that she finds missing from the Deduction (289). Laywine meets such worries by arguing that "the reappropriation [from the pre-Critical period] of the general cosmology is actually [End Page 162] doing... a lot of hard work" in the Deduction (13). This sustained argument comes to a head in chapter 4, which contends that each of the two steps into which Laywine divides the first half of the (B-edition) Deduction, treated in chapters 2 and 3 (respectively), relies on cosmological presuppositions.Chapter 2 analyzes the conception of knowledge as relation to an object that is asserted (in §17) to rely on the synthetic unity of apperception. Laywine traces this conception to the Duisburg Nachlaß's account of "exposition," which relates concepts a priori to appearances, in something like the way that geometrical construction relates them to pure intuition. (Laywine further connects the Latin term expositio to the Greek ekthesis, which expositio translates in the context of geometrical proof. She thereby gives the notion of ekthesis broader importance than does Jaakko Hintikka, who noted its relevance to Kant's philosophy of mathematics in "Kant on the Mathematical Method" [Monist 51 (1967): 352–75]. Laywine claims that Borelli's edition of Euclid could have been Kant's source for the term, but I doubt it was known to Kant and his contemporaries, since it is not among the editions cited by Christian Wolff; Commandino seems likelier to me. Chapter 2 is concerned to articulate the notion of... (shrink)

The cognitive basis of geometry is still poorly understood, even the ‘simpler’ issue of what kind of representation of geometric objects we have. In this work, we set forward a tentative model of the neural representation of geometric objects for the case of the pure geometry of Euclid. To arrive at a coherent model, we found it necessary to consider earlier forms of geometry. We start by developing models of the neural representation of the geometric figures of (...) ancient Greek practical geometry. Then, we propose a related model for the earliest form of pure geometry – that of Hippocrates of Chios. Finally, we develop the model of the neural representation of the geometric objects of Euclidean geometry. The models are based on the hub-and-spoke theory. In our view, the existence of specific models opens the possibility of addressing the relationship between geometric figures and geometric objects, in a novel way, in terms of their neural representation. (shrink)

We discuss critically some recent theses about geometric cognition, namely claims of universality made by Dehaene et al., and the idea of a “natural geometry” employed by Spelke et al. We offer arguments for the need to distinguish visuo-spatial cognition from basic geometric knowledge, furthermore we claim that the latter cannot be identified with Euclidean geometry. The main aim of the paper is to advance toward a characterization of basic, practical geometry – which in our view requires (...) a combination of experiments on visuo-spatial cognition with studies in cognitive archaeology and comparative history. Examples from these fields are given, with special emphasis on the comparison of ancient Chinese and ancient Greek geometric ideas and procedures. (shrink)

Major shifts in the field of model theory in the twentieth century have seen the development of new tools, methods, and motivations for mathematicians and philosophers. In this book, John T. Baldwin places the revolution in its historical context from the ancient Greeks to the last century, argues for local rather than global foundations for mathematics, and provides philosophical viewpoints on the importance of modern model theory for both understanding and undertaking mathematical practice. The volume also addresses the impact (...) of model theory on contemporary algebraic geometry, number theory, combinatorics, and differential equations. This comprehensive and detailed book will interest logicians and mathematicians as well as those working on the history and philosophy of mathematics. (shrink)

This book explores and articulates the concepts of the continuous and the infinitesimal from two points of view: the philosophical and the mathematical. The first section covers the history of these ideas in philosophy. Chapter one, entitled ‘The continuous and the discrete in Ancient Greece, the Orient and the European Middle Ages,’ reviews the work of Plato, Aristotle, Epicurus, and other Ancient Greeks; the elements of early Chinese, Indian and Islamic thought; and early Europeans including Henry of Harclay, (...) Nicholas of Autrecourt, Duns Scotus, William of Ockham, Thomas Bradwardine and Nicolas Oreme. The second chapter of the book covers European thinkers of the sixteenth and seventeenth centuries: Galileo, Newton, Leibniz, Descartes, Arnauld, Fermat, and more. Chapter three, 'The age of continuity,’ discusses eighteenth century mathematicians including Euler and Carnot, and philosophers, among them Hume, Kant and Hegel. Examining the nineteenth and early twentieth centuries, the fourth chapter describes the reduction of the continuous to the discrete, citing the contributions of Bolzano, Cauchy and Reimann. Part one of the book concludes with a chapter on divergent conceptions of the continuum, with the work of nineteenth and early twentieth century philosophers and mathematicians, including Veronese, Poincaré, Brouwer, and Weyl. Part two of this book covers contemporary mathematics, discussing topology and manifolds, categories, and functors, Grothendieck topologies, sheaves, and elementary topoi. Among the theories presented in detail are non-standard analysis, constructive and intuitionist analysis, and smooth infinitesimal analysis/synthetic differential geometry. No other book so thoroughly covers the history and development of the concepts of the continuous and the infinitesimal. (shrink)

In this paper, we focus on the development of geometric cognition. We argue that to understand how geometric cognition has been constituted, one must appreciate not only individual cognitive factors, such as phylogenetically ancient and ontogenetically early core cognitive systems, but also the social history of the spread and use of cognitive artifacts. In particular, we show that the development of Greek mathematics, enshrined in Euclid’s Elements, was driven by the use of two tightly intertwined cognitive artifacts: the (...) use of lettered diagrams; and the creation of linguistic formulae. Together, these artifacts formed the professional language of geometry. In this respect, the case of Greek geometry clearly shows that explanations of geometric reasoning have to go beyond the confines of methodological individualism to account for how the distributed practice of artifact use has stabilized over time. This practice, as we suggest, has also contributed heavily to the understanding of what mathematical proof is; classically, it has been assumed that proofs are not merely deductively correct but also remain invariant over various individuals sharing the same cognitive practice. Cognitive artifacts in Greek geometry constrained the repertoire of admissible inferential operations, which made these proofs inter-subjectively testable and compelling. By focusing on the cognitive operations on artifacts, we also stress that mental mechanisms that contribute to these operations are still poorly understood, in contrast to those mechanisms which drive symbolic logical inference. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Logos and Alogon: Thinkable and Unthinkable in Mathematics, from the Pythagoreans to the Moderns by Arkady PlotnitskyNoam CohenPLOTNITSKY, Arkady. Logos and Alogon: Thinkable and Unthinkable in Mathematics, from the Pythagoreans to the Moderns. Cham: Springer, 2023. xvi + 294 pp. Cloth, $109.99The limits of thought in its relations to reality have defined Western philosophical inquiry from its very beginnings. The shocking discovery of the incommensurables in Greek (...) mathematics not only reintroduced these limits with greater force but also instigated two different forms of coping with the unthinkable in Western philosophy, mathematics, and science. As Arkady Plotnitsky sets forth, in response to the incommensurability crisis in the ancient Greek intellectual world, there appeared two different attitudes toward the alogon, that is, the unthinkable or irrational. Plato and his followers sought to overcome the unthinkable, of which the incommensurables in mathematics are only an example, through philosophy. In contrast, the Pythagoreans continued to hold on to mathematics as the primary means of knowing the world, but put a larger emphasis on geometry. Plotnitsky [End Page 359] claims that by not putting aside the primacy of mathematics, the Pythagorean shift from arithmetic to geometry established in Western thought a certain tolerance toward the unthinkable within the realm of the thinkable, “a possibility of the alogon in the logos.” Until modern times this remained no more than a possibility. However, in modern mathematics and mathematical physics Plotnitsky identifies a certain fulfillment of this potential, a “radical Pythagorean mathematics.” In his comprehensive and thought-provoking study, Plotnitsky recounts the development of such radical modern Pythagoreanism, not only to tell its history but also to suggest that the unthinkable is a condition of possibility for thought, and that a proper recognition of this role it plays is crucial for the advancement of mathematical thinking and thinking in general.Plotnitsky begins by portraying in broad strokes the two defining characteristics of radical Pythagorean mathematics: the presence of the unthinkable within mathematical thought itself, and the interplay of algebra and geometry. Relying on an extensive knowledge of the history of modern mathematics and physics, in the first two chapters of the book Plotnitsky gradually demonstrates the key role that the alogon plays in some theories of modern mathematical thought, such as Gödel’s incompleteness theorems and quantum mechanics. He also presents and clarifies his own conception of the history of Western thought as a creative process of concept-forming. By creating rather than discovering concepts, great mathematicians break new ground for thinking. The book as a whole argues in length for these two main theses.In chapter 3, the author addresses the thought of Bernhard Riemann to make the case that Riemann’s mathematical thought puts an emphasis on creative conceptual thinking and is not defined merely by calculations or logic. Such thinking is problem-posing rather than axiomatic. Riemann’s central innovation in this respect is the concept of manifold (Mannigfaltigkeit), which enabled him to think anew space and geometry in a radical non-Euclidean fashion, implying an infinite number of possible geometries. Plotnitsky argues—borrowing Deleuze and Guattari’s terminology—that this opened up a “plane of immanence,” which is to say that Riemann created a concept-generating movement of thought. However, we cannot grasp the generation of concepts in modern mathematics without giving proper attention to the interplay of algebra and geometry that defines it. Chapter 4 demonstrates this interweaving of the two fields by recounting the history of the idea of the curve in modernity, from the analytic geometry and calculus of Fermat and Descartes to its modification in the concept of a Riemann surface, up to its more recent developments in the algebraic geometry of Weil and Grothendieck. It gives a historical account of the algebraization of the curve that at the same time brings into relief the irreducible geometrical aspect, continuity, of all modern mathematical conceptualizations of the curve.The next two chapters provide further case studies. Chapter 5 expands and solidifies the argument of chapter 2. It discusses three examples of mathematical practice, each displayed by a pair of important [End Page 360] mathematicians: Abel and Galois, Lobachevsky and Riemann, Weil and Grothendieck. Each case... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:“Even the Papuan is a Man and not a Beast”: Husserl on Universalism and the Relativity of CulturesDermot Moran (bio)“[A]nd in this broad sense even the Papuan is a man and not a beast.” ([U]nd in diesem weiten Sinne ist auch der Papua Mensch und nicht Tier, Husserl, Crisis, 290/Hua. VI.337–38)1“Reason is the specific characteristic of man, as a being living in personal activities and habitualities.” (Vernunft ist das (...) Spezifische des Menschen, als in personalen Aktivitäten und Habitualitäten lebenden Wesens, Husserl, Crisis, 338/Hua. VI.272)1. Particular Historicities and Universal ReasonIn this paper2 i shall explore—and suggest a way of resolving—the evident tensions to be found in Husserl’s Crisis of European Sciences (and associated texts) [End Page 463] between his commitment to the universality of reason as the goal or telos of European humanity (founded on the ancient Greek “breakthrough” to philosophy and science) and his recognition of the empirical plurality and “relativity” (Relativität) of individual peoples and nations (e.g. Indian, Chinese, Papuan, Bantu) locked into their own particular “socialities” (Sozialitäten), communal worlds, and historical trajectories (what Husserl broadly calls “historicities,” Geschichtlichkeiten, Historizitäten). 3 I shall examine the complex relations and tensions between Husserl’s conception of universality, whereby the same reason functions in every human as animal rationale, “no matter how primitive he is” (“Origin of Geometry,” Crisis, 378/ Hua. VI.385), and his concept of the self-enclosed particularity of individual peoples with their own cultural forms. Indeed, Husserl often emphasizes that the most prominent feature of cultural plurality is precisely its relativity: “relativity belongs to the normal course of life.”4 I shall evaluate Husserl’s response, which defends the project of realizing the ideal of a critical universal rationality, by situating his discussion in terms of the cultural conflict of the time with the rising National Socialist commitment to racial particularism, and by showing Husserl’s commitment to the inherent universality of the one shared life-world.In his research manuscripts of the 1920s and 1930s Husserl frequently discusses the complex relationships that exist between different cultures and traditions; different cultures have their specific historicities (Crisis, 274/Hua. VI.320), dialects, norms, ways of life, and so on. This is simply a matter of fact. There are, furthermore, some well-known and controversial passages in Husserl’s “Vienna Lecture” of May 1935 and also in his Crisis texts (both in the main 1936 published text, Part One [§6] and in the then unpublished Crisis Part Three A §365) where Husserl speaks of the universality inherent in European philosophical culture of the logos and contrasts it with various other communal forms, which are, in his view, merely “empirical-anthropological” types, enclosed in their particular historicities and relativities. Indeed, in this context, Husserl regularly invokes the idea of the “relativity of everything historical” (die Relativität alles Historischen, Crisis, 373/Hua. VI.382). In addition, there are a number of related texts, collected in the Crisis supplementary volume (Hua. XXIX),6 in the Intersubjectivity volumes (especially Hua. XV),7 as well as in the recent volume on the life-world [End Page 464] (Hua. XXXIX),8 which discuss the empirical differences between peoples and also the layers and strata of social groups, nations, and even the idea of larger international collectivities or “supernations” (Übernationen), such as Europe, or the League of Nations. Furthermore, there are—as Husserl indicates in his 1935 letter to the prominent French anthropologist Lucien Lévy-Bruhl—even cultures that are completely “self-enclosed” (abgeschlossene) unities, cut off, some of which know no history. According to Husserl, into this classical world of closed cultures, the ancient Greeks bring a new form of universality, one that leads them, through the grasp of idealization to infinite tasks, to break through the finite horizons of their environing world (Umwelt), which presents itself to them as a “near-world” (Nahwelt, Crisis, 324/Hua. VI.303; Hua. XXVII.228), and to arrive at the highly refined concept of the “true world” or the “scientific world [which] is a purposeful structure (Zweckgebilde) extending to infinity” (Crisis, 382... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:A Humanist History of Mathematics?Regiomontanus's Padua Oration in ContextJames Steven ByrneIn the spring of 1464, the German astronomer, astrologer, and mathematician Johannes Müller (1436–76), known as Regiomontanus (a Latinization of the name of his hometown, Königsberg in Franconia), offered a course of lectures on the Arabic astronomer al-Farghani at the University of Padua. The only one of these to survive is his inaugural oration on the history and utility (...) of the mathematical arts.1 Regiomontanus tells his audience that the purpose of the oration is torelate first the origin of our arts, and among which nations they first began to be cultivated, in what way they were at last translated from various foreign tongues into Latin, which of our ancestors were famed in these disciplines, and to whom in our lifetimes recognition should be granted.2To this end, he offers a history of the quadrivial arts (arithmetic, geometry, music, and astronomy) and other important mathematical disciplines from [End Page 41] antiquity to his own time, praises their utility, and exhorts his audience to revive the languishing study of mathematics at Padua. Astrology, a discipline with which Regiomontanus himself was closely associated, is singled out for particular praise.Traditionally, Regiomontanus's Padua oration has been seen through the lens of its rather obvious humanism. In particular, the Padua oration has come to be understood as the rhetorical embodiment of the fifteenth-and-sixteenth-century revival of ancient (Greek) mathematics. Just as this revival is inextricable from the rise of humanism, so Regiomontanus has come to be seen as an exemplar of humanist mathematics.3 It is the aim of this paper to examine the Padua oration in the context both of contemporary humanist rhetoric and of Regiomontanus's own intellectual background in order to argue that, while the oration is stylistically consistent with humanist norms, the vision of mathematics presented in it is also deeply grounded in the university mathematical curriculum and in Regiomontanus's own reading of mathematical texts.Regiomontanus was educated primarily at the University of Vienna (he was also briefly at the University of Leipzig), where he enrolled in 1450, completed his baccalaureate in 1452 and became a master in 1457.4 He remained at Vienna until 1461, when the death of his friend and teacher Georg Peurbach prompted him to travel to Italy with his patron, Cardinal Bessarion.5 In Regiomontanus's day Vienna was probably the most important of the German universities, rivaled only by Prague, whose prestige had declined after it was stripped of its theology faculty in the aftermath of the Hussite Wars.6 The curriculum at the University of Vienna was modeled [End Page 42] after that of Paris, and Parisian scholars played a major role both in its founding in 1365 and in its re-establishment (this time with a theology faculty) in 1384.7Most importantly for the purposes of this paper, the Viennese mathematical curriculum of Regiomontanus's day included all of the traditional authorities taught at Paris in the fourteenth century. For arithmetic and algebra, various "algorisms" (prose or poetry instructions for carrying out arithmetical operations that often also included a small amount of number theory) were the basic texts, supplemented in the fourteenth century by the Quadripartitum numerorum of Jean de Murs. Jordanus de Nemorare's De numeris datis and al-Khwarizmi's Algebra were common sources for those engaged in more advanced studies (i.e., they were not normally the subject of ordinary lectures, but were readily available to interested students, and would perhaps have been the subject of occasional extraordinary lectures). Euclid's Elements, supplemented by the commentaries of Pappus and Campanus, was the central text for geometry, and a number of medieval texts on practical and speculative geometry were in circulation as well. For astronomy, the Sphere of Sacrobosco and the Theorica planetarum were the most commonly used teaching texts, sometimes supplemented by al-Farghani's Elements of Astronomy (the subject of Regiomontanus's Padua lectures). Advanced students could read numerous more specific treatises by Arabic and Latin authors. For optics, the Perspectiva communis of John Peckham was the most common basic text, with texts... (shrink)

This paper is a contribution to our understanding of the constructive nature of Greek geometry. By studying the role of constructive processes in Theodoius’s Spherics, we uncover a difference in the function of constructions and problems in the deductive framework of Greek mathematics. In particular, we show that geometric problems originated in the practical issues involved in actually making diagrams, whereas constructions are abstractions of these processes that are used to introduce objects not given at the outset, (...) so that their properties can be used in the argument. We conclude by discussing, more generally, ancient Greek interests in the practical methods of producing diagrams. (shrink)

John Corcoran. 1979 Review of Hintikka and Remes. The Method of Analysis (Reidel, 1974). Mathematical Reviews 58 3202 #21388. -/- The “method of analysis” is a technique used by ancient Greek mathematicians (and perhaps by Descartes, Newton, and others) in connection with discovery of proofs of difficult theorems and in connection with discovery of constructions of elusive geometric figures. Although this method was originally applied in geometry, its later application to number played an important role in the (...) early development of algebra [Jacob Klein, English translation, Greek mathematical thought and the origin of algebra, especially pp. 154–157, M.I.T. Press, Cambridge, Mass., 1968]. -/- It is universally agreed that the method of analysis begins by “assuming the thing sought after” (e.g., in geometry, the truth of the proposition to be proved or the existence of the geometric figure to be constructed). Aside from this, little else can be taken for granted. There is disagreement concerning the “direction of analysis”, i.e. whether one is to seek implications of the assumption or whether one is to seek implicants of it. There is also disagreement concerning what is to be “anatomized” (analyzed), i.e., whether one analyzes mathematical objects (figures), mathematical propositions (the axioms, known theorems, and analytic assumption) or an imagined proof (of the analytic assumption from axioms and known theorems). (shrink)

For more than a century, there has been some discussion about whether medieval Arabic al-jabr has its roots in Indian or Greek mathematics. Since the 1930s, the possibility of Babylonian ultimate roots has entered the debate. This article presents a new approach to the problem, pointing to a set of quasi-algebraic riddles that appear to have circulated among Near Eastern practical geometers since c. 2000 BCE, and which inspired first the so-called “algebra” of the Old Babylonian scribal school and (...) later the geometry of Elements II. The riddles also turn up in ancient Greek practical geometry and Jaina mathematics. Eventually they reached European mathematics via the Islamic world. However, no evidence supports a derivation of medieval Indian algebra or the original core of al-jabr from the riddles. (shrink)

The derivative is a basic concept of differential calculus. However, if we calculate the derivative as change in distance over change in time, the result at any instant is 0/0, which seems meaningless. Hence, Newton and Leibniz used the limit to determine the derivative. Their method is valid in practice, but it is not easy to intuitively accept. Thus, this article describes the novel method of differential calculus based on the double contradiction, which is easier to accept intuitively. Next, the (...) geometrical meaning of the double contradiction is considered as follows. A tangent at a point on a convex curve is iterated. Then, the slope of the tangent at the point is sandwiched by two kinds of lines. The first kind of line crosses the curve at the original point and a point to the right of it. The second kind of line crosses the curve at the original point and a point to the left of it. Then, the double contradiction can be applied, and the slope of the tangent is determined as a single value. Finally, the meaning of this method for the foundation of mathematics is considered. We reflect on Dehaene’s notion that the foundation of mathematics is based on the intuitions, which evolve independently. Hence, there may be gaps between intuitions. In fact, the Ancient Greeks identified inconsistency between arithmetic and geometry. However, Eudoxus developed the theory of proportion, which is equivalent to the Dedekind Cut. This allows the iteration of an irrational number by rational numbers as precisely as desired. Simultaneously, we can define the irrational number by the double contradiction, although its existence is not guaranteed. Further, an area of a curved figure is iterated and defined by rectilinear figures using the double contradiction. (shrink)