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Ancient Mathematics

The history of mathematics.

Mathematics, also known as the Queen of Sciences, permeates every area of our lives. Whether you are filling in your accounts, building a cabinet, or watching the stars, you are using mathematical principles laid down through the ages, and it is a discipline that underpins life as we know it.

This article is a part of the guide:

• History of the Scientific Method
• Aristotle's Psychology
• Who Invented the Scientific Method?

Browse Full Outline

• 1 History of the Scientific Method
• 2 Who Invented the Scientific Method?
• 3.1 Medicine
• 3.2 Physics
• 3.3 Mathematics
• 3.4 Chinese Alchemy
• 3.5 Chinese Astronomy
• 3.6 Mayan Astronomy
• 3.7 Indian Astronomy
• 3.8 Egyptian Astronomy
• 3.9 Egyptian Mathematics
• 4.1 Mesopotamian Astronomy
• 4.2 Neolithic Astronomy
• 4.3 Babylonian Mathematics
• 5.1 Alchemy and the Philosopher’s Stone
• 5.2 Aristotle’s Zoology
• 5.3 Aristotle's Psychology
• 5.4 Greek Astronomy
• 5.5.1.1 War Machines
• 5.5.2 Euclid
• 5.5.3 Pythagoras
• 5.5.4 Thales and the Deductive Method
• 5.6.1 Heron's Inventions
• 6 Building Roman Roads
• 7.1 Alchemy
• 7.2 Astronomy
• 7.3 Scholars and Biology
• 7.4 Medicine
• 7.5 Ophthalmology
• 7.6 Psychology
• 8.1 Psychology in the Middle Ages
• 8.2 Part II
• 8.3 St. Augustine
• 8.4 Collective Psychology
• 8.5 Thomas Aquinas
• 8.6 Mental Hospitals

Alongside the development of language, mathematics has shaped human civilization and has given us the mental tools to expand our knowledge in other areas.

The History of Mathematics - Applied and Pure Mathematics

In today's world, mathematics has two broad divisions:

Applied mathematics, which gives us the tools we need to shape the world around us. From the simple arithmetic of counting your change at the store, to the complex functions and equations used to design jet turbines, this field is the practical, hands on side of math.

Pure math is the esoteric part of the discipline, where mathematicians seek proofs and develop theorems. I studied pure mathematics (not very successfully) at school and it is almost like a different language; professional mathematicians seem to see the world in a different way, their elegant theorems and mathematical functions giving them a different insight onto the world.

The Development of Mathematics - The Egyptians and the Babylonians

Of course, this division into two broad fields is a little crude and arbitrary, with statistics and probability, topography, geometry, and calculus all standalone subjects in their own right. They use their own language and methods, as different from each other as biology is from physics, or psychology from engineering. However, this division into two disciplines hails back to the formation of the subject thousands of years ago.

Applied math developed because of necessity, as a tool to watch the stars and develop calendars, or build architectural marvels. The Egyptians devised a mathematical system designed to meet their needs, based around the need for accurate surveying. Their methods were functional and approximate, using brute force and trial and error to find solutions. As their great monuments attest, this worked for them and, for example, they did not need to know the value of Pi down to 40 decimal places, only an approximation that did the job.

By contrast, the Babylonians, with their skill in astronomy and the need to devise ever more accurate calendars, began to look at the theoretical side of mathematics, studying relationships between numbers and patterns. Like the Egyptians, they passed much of their knowledge on to the Greeks, with great mathematicians such as Thales and Pythagoras learning from these great cultures.

The Greeks and the Romans - From Applied to Pure and Back Again

The Greeks were the first mathematicians to concentrate upon pure mathematics, believing that all mathematical knowledge could be derived from deduction and reasoning . They used geometry to lay down certain axioms and built theories upon those, refining the idea of seeking proof through deduction alone, without empirical measurements .

Of course, the idea that the Greek mathematicians focused upon pure, theoretical mathematics does not mean that they did not contribute to applied math. Greek mathematicians and inventors created many instruments for watching the stars or surveying the land, all built upon mathematical principles. However, their insistence upon a deductive method is what defined their work, and the Greek mathematicians laid down elaborate rules that their modern counterparts still use.

By contrast, the Romans once again took math into the realm of the tangible, trying to seek a practical use for the discipline. The were amongst the greatest engineers that the world has ever seen and their use of techniques based upon applied math for surveying, building bridges, tunnels and temples.

With them, the pendulum sung back towards engineering and pragmatism, and theoretical math became less important until the time of the Islamic Golden Age.

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Martyn Shuttleworth (May 16, 2010). Ancient Mathematics. Retrieved May 26, 2024 from Explorable.com: https://explorable.com/ancient-mathematics

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Egyptian Mathematics

Babylonian Mathematics

Greek Geometry

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Writing the History of Mathematics: Interpretations of the Mathematics of the Past and Its Relation to the Mathematics of Today

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• First Online: 15 May 2019
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• Johanna Pejlare 2 &
• Kajsa Bråting 3  

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In the present chapter, interpretations of the mathematics of the past are problematized, based on examples such as archeological artifacts, as well as written sources from the ancient Egyptian, Babylonian, and Greek civilizations. The distinction between history and heritage is considered in relation to Euler’s function concept, Cauchy’s sum theorem, and the Unguru debate. Also, the distinction between the historical past and the practical past , as well as the distinction between the historical and the nonhistorical relations to the past, are made concrete based on Torricelli’s result on an infinitely long solid from the seventeenth century. Two complementary but different ways of analyzing the mathematics of the past are the synchronic and diachronic perspectives, which may be useful, for instance, regarding the history of school mathematics. Furthermore, recapitulation, or the belief that students’ conceptual development in mathematics is paralleled to the historical epistemology of mathematics, is problematized emphasizing the important role of culture.

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Pejlare, J., Bråting, K. (2019). Writing the History of Mathematics: Interpretations of the Mathematics of the Past and Its Relation to the Mathematics of Today. In: Sriraman, B. (eds) Handbook of the Mathematics of the Arts and Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-70658-0_63-1

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Aristotle and Mathematics

Aristotle uses mathematics and mathematical sciences in three important ways in his treatises. Contemporary mathematics serves as a model for his philosophy of science and provides some important techniques, e.g., as used in his logic. Throughout the corpus, he constructs mathematical arguments for various theses, especially in the physical writings, but also in the biology and ethics. Finally, Aristotle's philosophy of mathematics provides an important alternative to platonism. In this regard, there has been a revival of interest in recent years because of its affinity to physicalism and fictionalisms based on physicalism. However, his philosophy of mathematics may better be understood as a philosophy of exact or mathematical sciences.

This article will explore the influence of mathematical sciences on Aristotle's metaphysics and philosophy of science and will illustrate his use of mathematics.

1. Introduction

2. the structure of a mathematical science: first principles, 3. three notions in demonstration: ‘of every’, ‘per se’, ‘universal’, 4. demonstration and mathematics, 5. the relation between different sciences: autonomy and subalternation.

• 6. What Mathematical Sciences Study: 4 Puzzles

7.1 Objects From Abstraction or ‘Removal’ ( ta ex aphaireseôs )

7.2 precision ( akribeia ), 7.3 as separated ( hôs kekhôrismenon ), 7.4 x qua ( hêi ) y, 7.5 intelligible matter ( noêtikê hylê ), 8. universal mathematics.

• 9. Place and Continuity of Magnitudes

10.1 Background

10.2 measure ( metron ), 11. mathematics and hypothetical necessity, 12. the infinite ( apeiron ).

• 13. Aristotle and Evidence for the History of Mathematics

Other Internet Resources

Related entries.

The late fifth and fourth centuries B.C.E. saw many important developments in Greek mathematics, including the organization of basic treatises or elements and developments in conceptions of proof, number theory, proportion theory, sophisticated uses of constructions (including spherical spirals and conic sections), and the application of geometry and arithmetic in the formation of other sciences, especially astronomy, mechanics, optics, and harmonics. The authors of such treatises also began the process of creating effective methods of conceiving and presenting technical work, including the use of letters to identify parts of diagrams, the use of abstract quantities marked by letters in proofs instead of actual numerical values, and the use of proofs. We cannot know whether Aristotle influenced the authors of technical treatises or merely reflects current trends.

In this context, Plato's Academy was fertile ground for controversy concerning how we are to know mathematics (the sorts of principles, the nature of proofs, etc.) and what the objects known must be if the science is to be true and not vacuous. Aristotle's treatments of mathematics reflect this diversity. Nonetheless, Aristotle's reputation as a mathematician and philosopher of mathematical sciences has often waxed and waned.

In fact, Aristotle's treatises display some of the technically most difficult mathematics to be found in any philosopher before the Greco-Roman Age. His technical failures involve conceptually difficult areas involving infinite lines and non-homogenous magnitudes.

Commentators on Aristotle from the 2nd century on tended to interpret Aristotle's mathematical objects as mental objects, which made Aristotle more compatible with neo-Platonism. Later the mechanistic movement in the late Renaissance treated Aristotle as divorcing mathematics from physical sciences in order to drive a deeper wedge between their views and his. Because of this, it has been very easy to discount Aristotle as subscribing to a version of psychologism in mathematics. These tendencies contribute to the common view that Aristotle's views mathematics are marginal to his thought. More recently, however, some sympathetic readers have seen Aristotle as expressing a fictionalist version of physicalism, the view that the objects of mathematics are fictional entities grounded in physical objects. To the extent that this view is regarded as a plausible view about mathematics, Aristotle has regained his position.

There are two important senses in which Aristotle never presents a philosophy of mathematics. Aristotle considers geometry and arithmetic, the two sciences which we might say constitute ancient mathematics, as merely the two most important mathematical sciences. His explanations of mathematics always include optics, mathematical astronomy, harmonics, etc. Secondly, Aristotle, so far as we know, never devoted a treatise to philosophy of mathematics. Even Metaphysics xiii and xiv, the two books devoted primarily to discussions of the nature of mathematical objects, are really concerned with diffusing Platonist positions that there are immutable and eternal substances over and beyond sensible substances and Pythagorean positions that identify numbers with sensible substances.

Aristotle's discussions on the best format for a deductive science in the Posterior Analytics reflect the practice of contemporary mathematics as taught and practiced in Plato's Academy, discussions there about the nature of mathematical sciences, and Aristotle's own discoveries in logic. Aristotle has two separate concerns. One evolves from his argument that there must be first, unprovable principles for any science, in order to avoid both circularity and infinite regresses. The other evolves from his view that demonstrations must be explanatory. (See subsections A, B, and C of §6, Demonstrations and Demonstrative Sciences, of the entry Aristotle's logic .)

Aristotle distinguishes ( Posterior Analytics i.2) Two sorts of starting points for demonstration, axioms and posits .

An axiom ( axiôma ) is a statement worthy of acceptance and is needed prior to learning anything. Aristotle's list here includes the most general principles such as non-contradiction and excluded middle, and principles more specific to mathematicals, e.g., when equals taken from equals the remainders are equal. It is not clear why Aristotle thinks one needs to learn mathematical axioms to learn anything, unless he means that one needs to learn them to learn anything in a mathematical subject or that axioms are so basic that they should form the first part of one ‘s learning.

Aristotle divides posits ( thesis ) into two types, definitions and hypotheses:

A hypothesis ( hupothesis ) asserts one part of a contradiction, e.g., that something is or is not. A definition ( horismos ) does not assert either part of a contradiction (or perhaps is without the assertion of existence or non-existence).

Since a definition does not assert or deny, Aristotle probably intends us to understand definitions as stipulations or as defining expressions which are equivalent in some way to the defined term. The definition of unit as ‘indivisible in quantity’ will not presuppose that units do or do not exist. Hence, the syllogistic premise, ‘A unit is indivisibile in quantity,’ if taken as presupposing the existence of units, will not be a definition in this sense. Later, of course, Aristotle will allow for many other kinds of definitions.

There are many views as to what Aristotle's hypotheses are: (i) existence claims, (ii) any true assumption within a science, and (iii) the stipulation of objects at the beginning of a typical proof in Greek mathematics. Examples might be, ‘Let A be a unit,’ (where the object is stipulated to be a unit) or, more characteristically of Greek mathematics, ‘Let there be a line AB ’ (where a line is stipulated to exist, namely AB ). In fact, all these interpretations may have a modicum of what Aristotle means. In that case, Aristotle implies that any assumption within a science that asserts or denies something is a hypothesis. However, he singles out existence claims. How do existence claims work in Aristotle's conceptions of science? From Physics iv, we have claims such as ‘There is place,’ and ‘There is no void.’ However, the examples that Aristotle uses in the Posterior Analytics are claims such as that the genus exists, or specifically that there are units, or that there are points and lines. Aristotle also points out that sometimes the hypothesis of the genus is omitted as too obvious. Only by comparing these general claims with their use in Aristotelian mathematics can we get a sense of what Aristotle means. Aristotle intends us to understand that prior to the demonstrations in a scientific treatise, the treatise should state starting propositions. These include general claims broader than the science, definitions which are stated as stipulations and not as assertions, and a claim that the basic entities ‘exist’. What counts as an acceptable existence claim is relative to the actual science. The opening of a proof, ‘Let there be a line AB ,’ is an application of the basic hypothesis of the science. Since Aristotle regards such proofs through particular lines as general proofs, the opening claim is actually to be understood as standing for the general claim that there are lines. This is how the hypothesis is used as a premise. The stipulation, ‘Let there be a triangle ABC ,’ would not be a hypothesis on this interpretation, since he holds that the existence of triangles is to be proved, so that this instantiates a derived proposition.

A science consists of a genus ( genos ), what the science is about, and a collection of attributes, what the science says about the genus. The genus or kind is both defined and hypothesized to exist. From his examples (points and lines for geometry), it would seem that the genus is to be understood loosely as the fundamental entities in the science. The attributes are defined but are not hypothesized as existing. One must prove that the attributes belong to various members of the genus. For example, one must prove that triangles exist, e.g., that some [constructible] figures are triangles.

If we take very seriously the common view that Aristotle claims that every immediate premise of a demonstration expresses something about an existing entity, then one may well wonder how the principles of demonstration, axioms and posits, can be premises of demonstrations. Existence claims and stipulations do not express something about an existing entity. Since Aristotle calls the axioms, ‘those from which (demonstration arises),’ some have suggested that the axioms alone form the premises for a science and that a proof in any science arises by placing genus terms and their definitions in the axioms and then substituting terms like ‘triangle’ for their definitions when they arise in proofs. However, besides pointing to the inadequacy of the axioms for this job, it may be objected that Aristotle also calls the principles of demonstration immediate statements protaseis ), i.e., axioms and posits. Another possibility is that he regards even stipulations and existence claims as premises, as well as other hypotheses, but treats the axioms as somehow more fundamentally the source of proofs. In this case, he has a looser conception of what counts as a premise than many readers would expect. In any case, if his proof theory is to work at all, he must allow many more immediate premises than one would find in the introduction to a standard text of ancient Greek mathematics.

For more information, see the following supplementary document:

Aristotle and First Principles in Greek Mathematics

In the Posterior Analytics i.4, Aristotle also develops three notions crucial to his theory of scientific claims: ‘of every’, ‘per se’ ( kath’ hauto ) or ‘in virtue of itself’ (in four ways) and ‘universally’ ( katholou ). Although his exposition of these notions is tailored to his proof theory, the notions are designed also to characterize the basic features of any scientific claim, where the principal examples come mostly from mathematics. (See §6 Demonstrations and Demonstrative Sciences of the entry on Aristotle's logic .)

A holds true ‘of every’ B iff A holds of B in every case always. Note that this is a stronger condition than is meant in the Prior Analytics by ‘ A belongs to all B ’. Mathematical example: point is on every line (i.e., every line has points on it). A is per se 1 with respect to B iff ‘ A ’ is in the account which gives the essence of B . Note that Aristotle does not say that A belongs to all B (e.g., ‘hair’ occurs in the definition of bald, but ‘having hair’ does not belong to a bald person), yet it is presupposed by the use Aristotle makes of it. Aristotle allows that there are immediate statements of the form, A belongs to no B . Mathematical examples: ‘line’ is in the definition of triangle, ‘point’ is in the definition of line. A is per se 2 with respect to B iff ‘B’ is in the account which gives the essence of A and A belongs to B . Mathematical examples: straight and circular-arc belong to line, odd and even to number. Some commentators have held that it is the disjunction which belongs per se 2 (e.g., straight or circular-arc belongs per se 2 to all lines); others that the examples are that each predicate belongs per se 2 to the subject (e.g., straight belongs per se 2 to (some) line). However, Aristotle should know that not all lines are straight or circular. A is per se 3 iff ‘A’ indicates ‘a this’ ( tode ti ), i.e., ‘A’ refers to just what A is. At Post. An. i.22, Aristotle identifies the per se 3 with substance, the rock bottom of a syllogistic chain. However, one might well ask whether there must be an analogous notion within a science. If so, A would be per se 3 if A is a basic entity in a given science, an instance of the kind studied by the science. If so, the per se 3 items in arithmetic would be units. A is per se 4 with respect to B iff A belongs to B on account of A . Either no mathematical example is given or the examples are (depending on how we read the text): straight or curved belongs to line and odd or even belongs to number, but these may be cases of per se 2 . The non-mathematical example is: in getting its throat cut it dies in virtue of the throat-cutting. A belongs to B universally iff A belongs to all B and A belongs to B per se (in virtue of B ) and qua itself (qua B ). Here the notion of ‘per se’ seems to be slightly different from those previously mentioned (it has been suggested that the sense is per se 4 ), but, in any case, is said to be equivalent to ‘qua itself’. Perhaps we need a fifth notion of per se. B has/is A per se 5 (i.e., in virtue of B ) iff A belongs to B qua B , i.e., there is no higher genus or kind C of B such that A belongs to C and so to B in virtue of belonging to C . Again, Aristotle does not mark out per se 5 as a separate notion, so that the notion may be subsumed under per se 4 . Note that unlike per se 1 and per se 2 , per se 5 is in virtue of the subject of the predication.

The idea here seems to be that:

A belongs to B universally iff A belongs to all B and A belongs per se 5 to B .

An alternative (stronger?) interpretation is that:

A belongs to B universally iff A belongs per se to all B and B belongs to all A . In this case, A and B are called, in modern discussions, commensurate universals . (cf. An. Post , B 16-17)

Aristotle describes the property that a triangle has angles equal to two right angles as being per se 5 (= per se 4 ) and universal, but also the property of ‘having internal angles equal to two right angles’ as per se accidens ( kath’ hauto sembebêkôs ) of triangle. It is commonly thought that these are somehow essential accidents . Since these follow from per se properties by necessity, it seems strange to call them accidents at all. Sometimes, however, it is more appropriate to think of accidents as concomitants, the result of different demonstrative chains. Alternatively, Aristotle frequently uses the same word to indicate consequences. In that case, they should be called per se consequences .

It should be noted that the proof theory of Aristotle requires that all predicates in demonstrations be either per se 1 or per se 2 . What is neither per se 1 nor per se 2 is accidental. Hence, per se 4 (or per se 5 if it is a separate notion), and per se accidens should be reducible to these notions in any case.

Because of the formal success of his logical theory, Aristotle also considers most mathematical proofs as having the form of a universal affirmative syllogism , namely Barbara. (See the section on The Syllogistic in the entry on Aristotle's logic .) This means that most mathematical theorems are one thing A said of another C and that every mathematical demonstration has a middle term B which explains the connection between A and C . Aristotle provides several examples of such triads of terms in mathematics, e.g., two right angles-angles about a point-triangle, or right angle-half two right angles-angle in a semicircle. It has long been noted by commentators that mathematical proofs work with a particular case through universal instantiation ( ekthesis ) and then universalize to the general claim, and that not all propositions have the form: A is said of B , e.g., Elements 1 1, “To construct an equilateral triangle on a given line.” A more modern objection is that the formal theory of the syllogism as presented in Prior Analytics 1 1, 3-7 is woefully inadequate to express a theory involving conditionals and many-many relations, as is the case with all ancient mathematics. Nonetheless, Aristotle does think that most mathematical proofs actually do have this form. Those that wouldn't would certainly be negative propositions and possibly existential propositions. (We simply do not know enough about how Aristotle conceived of the logical form of existential propositions.) From a careful reading of the rest of the Prior Analytics , it becomes clear that Aristotle has a flexible notion of “one thing said of another” and that he regards standard mathematical proofs as really being in a universal form, which we express for purposes of comprehension as particular.

A science is defined by the genus or kind it studies and by a group of specifiable properties which belong to that kind. Secondly, the properties studied within a science are defined in terms of the genus of the science (per se 2 ). Hence, it follows that it will commonly be impossible to prove one thing using a different science. For one would have to prove that a property within one genus applies to a completely different genus. Hence, every science is autonomous . Aristotle makes this claim, however, in the context of his rejection of Plato's view that sciences are subordinate to knowledge of the Good. What he actually claims is much more modest. If one genus comes under another genus, it will be possible, in some cases incumbent, to prove that a property belongs to a genus by using a theorem from another science. In such a case the one science is said to be under ( subalternate with) the other science.

Here are the sciences along with their relations which Aristotle mentions in the Analytics:

Geometry Stereometry (solid geometry) Arithmetic | | | Optics (mathematical) Astronomy (mathematical) Mechanics Harmonics (mathematical) | | | Concerning the Rainbow Nautical Astronomy, Phenomena, Empirical Acoustical Science

Aristotle treats the science at the lowest level, descriptions of the rainbow, astronomical phenomena, and acoustical harmonics, as descriptive, providing the fact that something is the case, but not the explanation, which is provided by the higher science. It is very easy to speculate how Aristotle would fill in the relations in the table; e.g., would he put stereometry below geometry, as Plato does in Rep. vii? Similarly, the explanatory relation between mathematical optics and geometry is not the same as the relation of optics to empirical optics. This example of the rainbow seems to refer to the argument in Meteorology iii.5, where the observed fact that rainbows are never more than a semicircle (true in flat lands) is explained by a proof in optics that is thoroughly geometrical in character. Once the basic set-up and principle of reflection is provided, the rest is geometrical.

A different situation obtains when one science is not under another science, but some of the properties come from the other science. Aristotle's example is the fact that that round wounds heal more slowly [than slashes]. The medical property depends on the area of the wound and its perimeter.

Aristotle's point about autonomy is that a theorem in arithmetic (even less a theorem in harmonics) cannot be used to prove something in geometry. Here, arithmetic is probably understood as the number theory found in Euclid, Elements vii-ix, and not mere calculation of numbers, which, of course, is used in geometry. This allows the anti-Platonic point that theorems about the beautiful and theorems in mathematics have nothing to do with one another, even if some theorems are beautiful.

Elsewhere (esp. Physics ii.2 and Metaphysics xiii.3), Aristotle provides different accounts of the relations between mathematical sciences.

6. What Mathematical Sciences Study: Puzzles

Aristotle's principal concern in discussing ontological issues in mathematics is to avoid various versions of platonism. Aristotle shares with Plato the view that there are objects of understanding, that these must be universal and not particular and that they have to satisfy certain “Parmenidean” conditions, such as being unchanging and eternal. However, Aristotle rejects the view of Plato that objects of understanding are separate from particulars. This is a general problem in Aristotle's metaphysics. However, in the case of mathematical objects, there are three important difficulties. First, if physical objects are the objects of mathematical understanding and satisfy the standard definitions of line, circle, etc., then it is arguable that they manifestly fail in two ways (cf. Met . iii 2 997b25-8a19):

• The physical straight lines we draw are not straight; a physical tangent line does not really touch a circle at a point. In other words, physical objects fail to have the mathematical properties we study. This is the problem of precision .
• Physical mathematical objects lack properties which we require of objects of understanding. They are not separate or independent of matter. Hence, they are not eternal or unchanging. This is the problem of separability .

Although these two problems are distinct, Aristotle may hold that this failure is at least partly responsible for the failure of mathematical objects to have the mathematical properties we study. Platonic Forms fail in a third way.

• Suppose that there is a Form for each kind of triangle. There still would be only one Form for each kind. A mathematical theorem about diagonals of rectangles might mention two equal and similar triangles which are, nonetheless, distinct. Mathematical sciences require many objects of the same sort. This is the problem of plurality (cf. Met . iii 2 and Met . iii.1-2).

A fourth problem is not explicitly stated by Aristotle, but is clearly a presupposition of his discussion.

• An account of mathematics should not impinge on mathematical practice so as to make it incoherent or impossible. If mathematicians talk about triangles, numbers, etc., the account of mathematical objects should at least explain the discourse. This is the problem of non-revisionism (sometimes also called naturalism ). So Aristotle says (Met. xiii.3 1077b31-33) of his own account of mathematics that “it is unqualifiedly true to say of the mathematicals that thay exist and are such as they <the mathematicians say>” (cf. Phys . iii.7 207b27-34 for an application of the principle).

To solve the problems of separation and precision, contemporary philosophers such as Speusippus and possibly Plato posited a universe of mathematical entities which are perfect instances of mathematical properties, adequately multiple for any theorem we wish to prove, and separate from the physical or perceptible world. Aristotle calls them mathematicals or intermediates , because they are intermediate between the Forms and physical objects, in as much as they are perfect, eternal, and unchanging like the Forms, but multiple like physical objects (cf., for example, Met . i.6 987b14-18, iii 2, xiii.1-2). This solution is the ancestor of many versions of platonism in mathematics.

Aristotle's rejection of intermediates involves showing that their advocates are committed to an unwieldy multiplicity of mathematical universes, at least one corresponding to each mathematical science, whether kinematics, astronomy, or geometry. However, he also sets out to show that such ontologies are not merely pleonastic, but also that an alternative account can be given free of all the difficulties mentioned. In other words, Aristotle's strategy is best seen as diffusing some versions of platonism.

Aristotle also rejects a compromise as merely compounding this difficulty, the view that either Forms or intermediates are immanent in things (separate but coextensive), since these different worlds will now have to exist bundled together.

7. Aristotle's Treatment of Mathematical Objects

The account here of Aristotle on the status of mathematical objects will center on five concepts, which Aristotle uses in his discussions: ‘abstraction’ or ‘taking away’ or ‘ removal ’ or ‘subtraction’ ( aphairesis ), ‘ precision ’ ( akribeia ), ‘ as separated ’ ( hôs kekhôrismenon ), ‘ qua ’ or ‘in the respect that’ ( hêi ), and ‘ intelligible matter ’ ( noêtikê hylê ). Principal sources are the Posterior Analytics , De Anima iii.6-8, Metaphysics iii.2, vi.1, vii.10-11, ix.9, x.1-2, xi.2-3, 7, xiii.1-3, Physics ii.2.

Aristotle occasionally refers to mathematical objects as things by, in, from, or through removal (in different works Aristotle uses different expressions: ta aphairesei , ta en aphairesei , ta ex aphaireseôs, ta di’ aphaireseôs ). It is also clear that this usage relates to logical discussions in the Topics of definitions where one can speak of adding a term or deleting a term from an expression and seeing what one gets as a result. Our principal task is to explain what this logical/psychological removal is and how it solves the four puzzles. Aristotle starts with the class of perceptible or physical magnitudes . The examination of these is a part of physics (cf. Physics iii.4). The ontological status of these does not concern him, but we may suppose that they consitute the category of quantities: the bodies, surfaces, edges, corners, places, and times, sounds, etc. of physical substances ( Categories 6).

In the Analytics, where the notion of matter is absent, Aristotle begins with a particular geometrical perceptible figure. What is removed is its particularity and all that comes with this, including its being perceptible. What is left then is a universal of some sort. Aristotle also does not seem to think in this work that there is any conflict between the plurality problem and thinking of all terms in a mathematical deduction as universals. However, since he allows that a term can be a very complex expression, it can designate a rich complex for which there would probably be no corresponding Form in a Platonic theory.

Elsewhere, Aristotle usually seems to mean that the attributes not a part of the science are removed. What is left may be particular, a quasi-fictional entity. It is the status of this entity which leads to much controversy. Is it a representation in the soul or is it the perceptible object treated in a special way? Ancient and medieval readers tended to take the former approach in their interpretation of Aristotle, that the object left is a stripped down representation with only the required properties. (Cf. Mueller (1990) and neo-Platonic foundations for these interpretations of mathematics, such as Proclus in his commentary on Euclid.)

Most modern readers, perhaps influenced by the critiques of Berkeley and Hume against the first position and certainly less committed to neo-Platonism, take the second approach. The objects studied by mathematical sciences are perceptible objects treated in a special way, as a perceived representation, whether as a diagram in the sand or an image in the imagination. Furthermore, perhaps as a response to Frege's devastating critique of psychologism and Husserl's first attempt at a psychological account of arithmetic, some have suggested that Aristotle has no need for a special faculty of abstracting. Rather the mind is able to consider the perceived object without some of its properties, such as being perceived, being made of sand, marble, bronze, etc. However, this is analogous to the logical manipulation of definitions, by considering terms with or without certain additions. Hence, Aristotle will sometimes call the material object, the mathematical object by adding on. As a convenience, the mind conceives of this as if the object were just that. On this view, abstraction is no more and no less psychological than inference.

Conceptually, we might think of the process as the mind rearranging the ontological structure of the object. As a substantial artifact, what-it-is, the sand box has certain properties essentially. The figure drawn may be incidental to what-it-is, i.e., an accident. In treating the object as the figure drawn, being made of sand is incidental to it. Hence, ‘things by removal’ may be one way of explaining perceptible magnitudes qua lengths. This is the concept which does most of the work for Aristotle.

In his discussions of precision, Aristotle states that those sciences which have more properties removed are more precise. Arithemetic, about units, is more precise than geometry, since a point is a unit having position. A science of kinematics (geometry of moving magnitudes) where all motion is uniform motions is more precise than a science that includes non-uniform motions in addition, and a science of non-moving magnitudes (geometry) is more precise than one with moving magnitudes. However, one might infer that ‘precision’ here means nothing more than ‘clarity’ (or perhaps ‘refinement’, with all its ambiguity). Does this concept of ‘precision’ provide a framework for solving the problem of precision?

Aristotle solves the separability problem with a kind of fictionalism. The language and practice of mathematicians is legitimate because we are able to conceive of perceptible magnitudes in ways that they are not. The only basic realities for Aristotle remain substances, however we are to conceive them. A primary characteristic of substances is that they are separate. Yet we are able to speak of a triangle, a finite surface, merely the limit of a body, and hence not separate, as if it were separate ( hôs kekhôrismenon ). It is a subject in our science (in our discourse in the science). The mental and logical mechanism by which we accomplish this is the core of Aristotle's strategy in diffusing platonisms.

The word Aristotle uses is commonly translated with the English word ‘ qua ’ which itself translates the Latin relative pronoun ‘ qua ’, but with one important grammatical difference. The English adverb is normally followed by a noun phrase.

As a relative adverbial pronoun in the dative case, the Greek word captures all possible meanings of the dative, including, ‘where’, ‘in the manner that’, ‘by-means-of-the-fact-that’, or ‘in-the-respect-that’. Some have suggested translating it with the word ‘because’, although it is arguable that the English word at best intersects with the appropriate Greek meaning (perhaps ‘just or precisely because’ works better. Hence, ‘ X qua Y ’ should be understood as elliptic for:

‘ X in the respect that X is Y ’

‘ X by means of the fact that X is Y ’ (or ‘ X precisely because X is Y ’).

Here ‘ X ’ is normally a noun phrase referring to an entity in any ordinary way, e.g., arithmetic studies Leopold qua unit, this man qua unit, this musician qua unit.

In the context of a scientific claim, ‘ X by-means-of-the-fact-that or in-the-respect-that X is Y is F ’ maintains that ‘ Y is F ’ is a theorem, where Y is the most universal or appropriate subject for F , and that X is F in virtue of the fact that X is Y . For example, Figure ABC qua triangle has internal angles equal to two right angles, but qua right triangle has sides AB 2 + BC 2 = AB 2 .

In the case where we examine or study an object X qua Y or X in-the-respect-that X is Y , we study the consequences that follow from something's being an Y . In other words, Y determines the logical space of what we study. If X is a bronze triangle (a perceptible magnitude), to study X qua bronze will be to examine bronze and the properties that accrue to something that is bronze. To study X qua triangle is to study the properties that accrue to a triangle. Unless it follows from something's being a triangle that it must be bronze, the property of being bronze will not appear in one's examination.

Note that there is no necessity that ‘ qua ’ operators be of the form ‘ qua Y ’, where Y is a noun phrase. For example, Aristotle says ( De anima iii.4.429b25-6) that two things affect and are affected “ qua something in common belongs to both.” Similarly, as evidence that ‘ qua ’ does not in these contexts always mean ‘because’ (usually, the context is too ambiguous to precisely decide whether it means ‘because’ or ‘in the respect that’), consider Nicomachean Ethics i.3.1102b8-9, “Sleep is an inactivity of the soul qua it is called good or bad,” but certainly not because it is.

With only one or two possible exceptions, it seems that whenever Aristotle speaks of F ( X ) qua G ( X ), G ( X ) must be true. We can study a perceptible triangle qua triangle because it is a triangle. For convenience, we can call this principle qua -realism .

The Account of Mathematical Objects with ‘ Qua ’ . We begin with perceptible magnitudes. These are volumes, surfaces, edges, and corners. They change in position and size. They are made of some material and are the quantities of substances and their interactions. The volumes, surfaces, and edges have shape. Times and corners do not. Different sciences treat different perceptible magnitudes qua different things.

Moreover, since there are many perceptible magnitudes, there will be enough, qua line, to prove any theorem that involves lines. The plurality problem is trivially solved.

The separability problem is solved because if we examine X qua Y , we will talk about Y as if it is a separate entity, as a subject, and will pay no attention to the way in which we captured Y through a description ‘ X ’, in the sense that only the residue of the qua -filter are studied. The science will speak of Y . This too will not interfere with mathematical practice and so will not violate non-revisionism.

In Metaphysics vi.1, Aristotle argues that physics concerns things which have change, but are substances, that at least some of the things that mathematics is about do not change and are eternal but are not substances (exceptions would probably include stars and spheres in mathematical astronomy and bodies in mathematical kinematics), while first philosophy or theology is about things which are substances but do not change and are eternal. We can now characterize the way in which mathematical objects are eternal and lack change. Namely, generation and change are not among the predicates studied by geometry or arithmetic. Hence, it is correct to say that qua lines, perceptible lines lack generation, destruction, and change (with appropriate provisos for kinematics and mathematical astronomy).

Whether the precision problem is also solved and how it is solved is more controversial. On the ancient and medieval interpretation, the problem of precision is solved by allowing mental representations to be as precise as one chooses. The contemporary interpretation of considering Aristotle's mathematical objects as physical object treated in a special way has a more difficult task. There are five ways in which Aristotle may attempt to solve the precision problem.

• Many scholars today seem to hold to a view that for Aristotle if one can speak of X qua Y , then X must be Y precisely. This means that any theorem about triangles will only hold of the rare perfect triangles, wherever they may be (a thesis once suggested by Descartes).
• To increase the number of instances of exact triangles in the ontology, some scholars turn to Met. xiii.3, where Aristotle notes that being is said in two ways, the one in actuality and the other materially, he may be pointing to the fact that mathematical entities exist in continua as potentialities. Hence, a perfect line exists potentially in the sand, even if the one I have drawn is not. (Some have also seen support for this in Met. ix.9.) Hence, although there may be no actual triangles right now, at least there are an infinity of potential ones. The difficulty is that the argument is not about precision. It concerns an objection to Aristotle that man qua man is indivisible, but geometry studies man qua divisible. Since man is not divisible, the principle of qua -realism, that if one can study X qua Y then X is Y , is violated. Aristotle says that man is in actuality indivisible (you cannot slice a man in two and still have man or men), but is materially divisible. It is enough that X is Y materially or in actuality to study X qua Y . Nonetheless, the solution to the puzzle could point to an Aristotelian solution to the problem of precision.
• Alternatively, it is arguable that Aristotle allows that ‘ X is Y ’ may be true only imprecisely. For example, I may study a triangle in a diagram ABC qua triangle, but ABC is only a triangle imprecisely. Many Hellenistic treatises involving applied sciences set up convenient but false premises for the purposes of mathematical manipulation, including, notably, Aristotle's own account of the rainbow (Meteorology iii.5). Hence, an appeal to potentialities to get more exact triangles will do nothing to eliminate these apparent violations of qua -realism.
• In providing his hierarchy of precision in sciences, Aristotle may think that from filtering out more properties one gets greater precision. One finds more precise straight lines in geometry than in kinematics. Besides the obscurity of the position, it is not clear that he intends any such thing (see Section 7.2 above).
• One possibility is that Aristotle thinks that if a description has more properties removed from consideration, the entity studied is more precise in that there will be instances materially or actually that exactly exhibit satisfy qua -realism. For example, there are precise instances of units or of corners or points so that arithmetic and geometry are precise, while astronomy might not be so precise, since the planets are imprecisely points, but are studied qua points.

Our difficulty is that while Aristotle raises the problem of precision, he does not explicitly explain his solution to it.

Perceptible magnitudes have perceptible matter. A bronze sphere is a perceptible magnitude. For solving the plurality problem, Aristotle needs to have many triangles with the same form. Since perceptible matter is not part of the object considered (in abstraction or removal), he needs to have a notion of matter which is the matter of the object: bronze sphere MINUS bronze (perceptible matter). Since this object must be a composite individual to distinguish it from other individuals with the same form, it will have matter. He calls such matter intelligible or mathematical matter. Aristotle has at least four different conceptions of intelligible matter in the middle books of the Metaphysics , Physics iv, and De anima i:

• The form of a magnitude is its limit ( Metaphysics v.17); hence, the matter is what is between the limits of the magnitude, its extension ( Physics iv 2).
• Matter is the genus, e.g., in the sense that magnitude (and not perceptible-magnitude) is the kind for triangles (e.g., Metaphysics v.28, viii.6).
• The ‘non-perceptible’ matter of a perceptible magnitude, which is in the perceptible matter ( Metaphysics vii.10, cf. De anima i.1).
• The parts of a mathematical object which do not occur in the definition of the object, e.g., acute angle is not in the definition of right angle, but is a part of it and so is a non-perceptible material part of the angle ( Metaphysics vii.10, 11).

(1) and (3) are compatible; (2) may be a separate notion having more to do with the unity of definition and seems incompatible with (4); Aristotle treats (3) and (4) as the same notion. Since Aristotle's concern in discussing (4) is with the nature of the parts of definitions and not with questions of extended matter, it is unclear whether the non-definitional parts are potential extended parts or merely forms of extended parts, although the former seems more plausible.

Ancillary to his discussions of being qua being and theology ( Metaphysics vi.1, xi.7), Aristotle suggests an analogy with mathematics. If the analogy is that there is a super-science of mathematics coverying all continuous magnitudes and discrete quantities, such as numbers, then we should expect that Greek mathematicians conceived of a general mathematical subject as a precursor of algebra, Descartes' mathesis universalis (universal learning/mathematics), and mathematical logic.

Aristotle reports ( Posterior Analytics . i.5, cf. Metaphysics xiii.2) that whereas mathematicians proved theorems such as a : b = c : d => a : c = b : d (alternando) separately for number, lines, planes, and solids, now there is one single general or universal proof for all (see Section 3). The discovery of universal proofs is usually associated with Eudoxus' theory of proportion. For Aristotle this creates a problem since a science concerns a genus or kind, but also there seems to be no kind comprising number and magnitude. Some scholars have proposed that a universal science of ‘posology’ (a science of quantity) takes the whole category of quantity as its subject.

Aristotle seems more reticent, describing the proofs as concerning lines, etc., qua having such and such increment ( An. Post. ii.17). He seems to identify such a super-mathematics ( Metaphysics vi.1, xi.7), but seems to imply that it does not take a determinate kind as its subject. Another possibility is that the common science has theorems which apply by analogy to the different mathematical kinds.

Elsewhere ( Metaphysics xi.4), where Aristotle builds an analogy with the science of being qua being, he seems to suggest that universal proofs of quantities (here too including numbers) concern continuous quantity (unlike the similar passage in Metaphysics iv.3). If so and if this is by Aristotle, it would correspond to the general theory of proportions as it comes down to us. One may well wonder if scholars have been led astray by a hyperbole about universal proofs.

Ironically, extant Greek mathematics shows no traces of an Aristotelian universal mathematics. The theory of ratio for magnitudes in Euclid, Elements v is completely separate from the treatment of ratio for number in Elements vii and parts of viii, none of which appeals to v, even though almost all of the proofs of v could apply straightforwardly to numbers. For example, Euclid provides separate definitions of proportion (v def. 5, and vii def. 20). Compare the rule above (alternando),which is proved at v.16, while the rule follows trivially for numbers from the commutivity of multiplication and vii.19: ad = bc ⇔ a : b = c : d .

9. Place and continuity of Magnitudes

In Plato's Academy, some philosophers suggested that lines are composed of indivisible magnitude, whether a finite number (a line of indivisible lines) or a infinite number (a line of infinite points). Aristotle builds a theory of continuity and infinite divisibility of geometrical objects. Aristotle denies both conceptions. Yet, he needs to give an account of continuous magnitudes that is also free from paradoxes that these theories attempted to avoid. The elements of his account may be found principally in Physics iv.1-5 and v.1 and vi. Aristotle's account pertains to perceptible magnitudes. However, it is clear that he understands this to apply to magnitudes in mathematics as well.

Aristotle has many objections to thinking of a line as composed of actual points (likewise, a plane of lines, etc.), including:

• No point in a line is adjacent to another point.
• If a line is composed of actual points, than to move a distance an object would have to complete an infinite number of tasks (as suggested by Zeno's arguments against motion)

To say that a line is comprised of an infinity of potential points is no more than to say that a line may be divided (with a line-cutter, with the mind, etc.) anywhere on it, that any potential point may be brought to actuality. The continuity of a line consists in the fact that any actualized point within the line will hold together the line segments on each side. Otherwise, it makes no sense to speak of a potential point actually holding two potential lines together.

Suppose I have a line AB and cut it at C . The lines AC and CB are distinct. Is C one point or two?

C is one point in number. C is two points in its being or formula ( logos ).

This merely means that we can treat it once or twice or as many times as we choose. Note that Aristotle says the same thing about a continuous proportion. in a : b = b : c , b is one magnitude in being, but is used as two.

Place and continuity of Magnitudes

10. Unit ( monas ) and Number ( arithmos )

Greek mathematicians tend to conceive of number ( arithmos ) as a plurality of units. Perhaps a better translation, without our deeply entrenched notions, would be ‘count’. Their conception involves:

• A number is constructed out of some countable entity, unit (monas).
• One (a unit) typically is not a number (but Aristotle is ambivalent on this), since a number is a plurality of units.
• At least in theoretical discussions of numbers, a fractional part is not a number.
• In other words, numbers are members of the series: 2, 3, …, with 1 conceived as the ‘beginning’ ( archê ) of number or as the least number.
• In early Greek mathematics (5th century), numbers were represented by arrangements of pebbles. Later (at least by the 3rd century BCE) they were represented by evenly divided lines.

For Aristotle and his contemporaries there are several fundamental problems in understanding number and arithmetic:

• The precision problem of mathematicals is similar in the case of geometrical entities and units (see Section 6). Consider, for example, Plato's discussion of incompatible features of a finger as presenting one or two things to sight. Aristotle deals with the problem in his discussion of measure (see Section 10.1).
• The separability problem is the same as for geometry (see Section 6).
• The plurality problem of mathematicals (Section 6) is similar in the case of geometrical entities and units, with some differences. To count ‘perceptible’ units, or rather units from abstraction (cf. Section 7.1), one needs some principle of individuating units, what one is counting, whether cows or categories of predication. Aristotle says that one can always find an appropriate classification (we may assume that some classifications would be fairly convoluted, but that this is at best an aesthetic and not a logical problem). For units, one will use the same principle that allows one to individuate triangles. This is why Aristotle can describe a point as unit-having-position. Arithmetic involves the study of entities qua indivisible.
• The unity problem of numbers: This problem bedevils philosophy of mathematics from Plato to Husserl. What makes a collection of units a unity which we identify as a number? It cannot be physical juxtaposition of units. Is it merely mental stipulation?

Aristotle does not seem bothered by:

• The overlap problem : What guarantees that when I add this 3 and this 5 that the correct result is not 5, 6, or 7, namely that some units in this 3 are not also in this 5.

Aristotle presents three Academic solutions to these problems. Units are comparable if they can be counted together (such as the ten cows in the field). Units are not comparable, if it is conceptually impossible to count them together (a less intuitive notion).

• Incomparable Units : Form numbers are conceived as ordinals, with units conceived as being well ordered. What makes this number 3 is not that it is a concatenation of three units, but that its unit is the third unit in this series of units. Hence, it is simply false that there is a unity of the first three units forming a number three. What makes an ordinary concatenation, e.g., a herd of cows, ten cows is that they can be counted according to the series of Form-numbers. The notion of incomparable numbers lacks the basic conception of numbers as concatenations of units.
• Comparable/Incomparable Units : Form numbers are, of course, special. Each is a complete unity of units. For example, the Form of 3 is a unity of three units. Since it is a unity, it cannot be an accident that these three units form this unity. They are comparable with each other in the sense that together they comprise Three Itself and perhaps cannot be conceived separately. Hence, they cannot be parts of any other Form number. We cannot take 2 units from the Three Itself and add them to 4 units in the Six itself, to get a Form-number of the Seven itself. Myles Burnyeat once suggested an analogy with a sequence of playing cards of one suit, say diamonds. [ 1 ] Each card from ace (unit) to ten contains the appropriate diamonds (from 1 to 10) on each card, unified by their being on their particular card. Yet we don't count up two diamonds from the deuce and two from the trey, but treat each card as a complete unity.
• Comparable Units : These are intermediate or mathematical numbers (see Section 6). There is a unlimited number of units (enough to do arithmetic), which are arranged and so forth. Comparable numbers solve the plurality problem, but not the unity problem.

Aristotle reports that some Academics opted for a version of Incomparable or Comparable/Incomparable Units to solve the unity problem and introduced comparable units as the objects of mathematical theorems, e.g., given some comparable units, they are even if they can be divided in half, into two concatenations corresponding to (participating in) the same Form-number.

10.2.1 Background

Greeks used an Egyptian system of fractions. With the exception of 2/3, all fractions are proper parts which modern readers will see as unit fractions: 1/n. For example, 2/5 is 1/3 1/15 (that is, the sum of 1/3 and 1/15). Additionally, Greeks used systems of measure, as we do, with units of measure being divided up into more refined units of measure. 1 foot is 16 finger-widths (inches). Hence, one can always eliminate fractions by going, as we do, to a more refined measure (1, 1/2, 1/4 feet or 1 foot 12 inches). This feature of measure may be reflected in Plato's observation ( Rep . vii) that in arithmetic, one can always eliminate parts of units.

10.2.2 Aristotle's views

Since Aristotle (esp. Metaphysics x.1-2) treats measurement under his discussions of units (and hence number), it turns out that the precision problem becomes a problem of sorting out precise units of measure. Hence, in the case discrete quantities, such as cows, the unit is very precise, one cow. In the case of continuous quantities, the most precise unit of time is the time it takes for the fixed stars (the fastest things in the universe) to move the smallest perceptible distance. But a point or indivisible with position removed or a cow qua unit, i.e., a mathematical unit, is precise.

Aristotle's treatment of time ( Physics iv.10-14) includes some observations about numbers which come closest to being an account of number. Aristotle defines time as the number or count of change and then proceeds to distinguish two senses of number, what is counted (e.g., ten cows as measured by the cow-unit, or ten feet as measured by the foot-unit) and that by which we count. Time is number in the first sense, not as so-many changes, but as so-much change as measured by a unit of change. Aristotle clarifies the distinction between what is counted (or is countable) and that by which we count. These five black cats (number as what is counted) are different from these five brown cats, but their number (that by which we count) is the same.

But what are the numbers by which we count? Aristotle says nothing, but we may speculate that the five by which we count is the single formal explanation of what makes the five black cats five and what makes the five brown cats five. Hence, Aristotle probably subscribes to an Aristotelian version of the distinction between intermediate and Form-numbers.

Aristotle's discussion of time also gives us some insight into the unity problem. What gives the five black cats unity is just that they can be treated as a unity. From this it follows for Aristotle that there can be no number without mind. Nothing is countable unless there exists a counter.

It is often supposed, for Aristotle, mathematical explanation plays no role in the study of nature, especially in biology. This conception is, most of all, a product of anti-scholastics of the late Renaissance, who sought to draw the greatest chasm between their own mechanism and scholasticism. Mathematics plays a vital role in both. The principal way in which mathematics enters into biological explanation is through hypothetical necessity:

If X is to have feature Y (which is good for X ), then it is a feature of its matter that Z be the case.

Z may be a constraint determined by a mathematical fact. For example, animals by nature do not have an odd number of feet. For if one had an odd number of feet, it would walk awkardly or the feet would have to be of different lengths ( De incessuanimalium 9). To see this, imagine an isosceles triangle with a altitude drawn.

Aristotle famously rejects the infinite in mathematics and in physics, with some notable exceptions. He defines it thus:

The infinite is that for which it is always possible to take something outside.

Implicit in this notion is an unending series of magnitudes, which will be achieved either by dividing a magnitude (the infinite by division) or by adding a magnitude to it (the infinite by addition). This is why he conceives of the infinite as pertaining to material explanation, as it is indeterminate and involves potential cutting or joining (cf. Section 7.5).

Aristotle argues that in the case of magnitudes, an infinitely large magnitude and an infinitely small magnitude cannot exist. In fact, he thinks that universe is finite in size. He also agrees with Anaxagoras, that given any magnitude, it is possible to take a smaller. Hence, he allows that there are infinite magnitudes in a different sense. Since it is always possible to divide a magnitude, the series of division is unending and so is infinite. This is a potential, but never actual infinite. For each division potentially exists. Similarly, since it is always possible to add to a finite magnitude that is smaller than the whole universe continually smaller magnitudes, there is a potential infinite in addition. That series too need never end. For example, if I add to some magnitude a foot board, and then less than a half a foot, and then less than a fourth, and so forth, the total amount added will never exceed two feet. Aristotle claims that the mathematician never needs any other notion of the infinite.

However, since Aristotle believes that the universe has no beginning and is eternal, it follows that in the past there have been an infinite number of days. Hence, his rejection of the actual infinite in the case of magnitude does not seem to extend to the concept of time.

Aristotle on the Infinite

13. Aristotle and the Evidence for the History of Mathematics

As philosophers usually do, Aristotle cites simple or familiar examples from contemporary mathematics, although we should keep in mind that even basic geometry such as we find in Euclid's Elements would have been advanced studies. The average education in mathematics would have been basic arithmetical operations (possibly called logistikê ) and metrological geometry (given certain dimensions of a figure, to find other dimensions), such as were also taught in Egypt. Aristotle does allude to this sort of mathematics on occasion, but most of his examples come from the sort of mathematics which we have come to associate with Greece, the constructing of figures from given figures and rules, and the proving that figures have certain properties, and the ‘discovery’ of numbers with certain properties or proving that certain classes of numbers have certain properties. If we attend carefully to his examples, we can even see an emerging picture of elementary geometry as taught in the Academy. In the supplement are provided twenty-five of his favorite propositions (the list is not exhaustive).

Aristotle also makes some mathematical claims that are genuinely problematic. Was he ignorant of contemporary work? Why does he ignore some of the great problems of his time? Is there any reason why Aristotle should be expected, for example to refer to conic sections? Nonetheless, Aristotle does engage in some original and difficult mathematics. Certainly, in this Aristotle was more an active mathematician than his mentor, Plato.

Aristotle and Greek Mathematics

The standard English translation is given first, and, where appropirate, other more idiomatic translations. Greek is in parentheses.

• a this ( tode ti )
• abstraction, by removal, by subtraction, by taking away ( aphairesis )
• axiom ( axiôma )
• common notion ( koinê ennoia )
• definition ( horismos ), ( horos )
• hypothesis ( hupothesis )
• infinite ( apeiron )
• intelligible matter ( noêtikê hulê )
• intermediates ( ta metaxu ); also mathematicals ( ta mathêmatika )
• of everything ( kata pantos )
• per se, in virtue of itself ( kath hauto )
• per se accidens, per se accidents or per se consequences ( kath’ hauto sembebêkôs )
• posits ( thesis )
• postulate ( aitêma )
• qua, in respect that, because of the fact that — also sometime translated as ‘in so far as’ ( hêi )
• subalternate ( hupo )
• unit ( monas )
• universal ( katholou )

Collections of essays and journals referred to below with abbreviations

Collections of mathematical passages in aristotle with extensive discussions.

• Blancanus, Josephus (Guiseppi Biancani). 1615. Aristotelis loca mathematica ex universis ipsius operibus collecta et explicata . Bologna: Sumptibus Hieronymi Tamburini.
• Heath, Thomas L. 1949. Mathematics in Aristotle . Oxford: Oxford University Press, (reprint. New York: Garland Press, 1980).
• Heiberg, I.L. 1904. "Mathematisches zu Aristoteles," in Abhandlungen zur Geschichte der Mathematischen Wissenschaften . Vol. 18: 1-49. Leipzig: Teubner.

Works on philosophical issues in Aristotle on mathematics or use of mathematics in discussing philosophical discussions

• Annas, Julia, 1976, Aristotle's Metaphysics Books M and N (English translation and commentary). Oxford Clarendon Aristotle, ed. J.L. Ackrill. Oxford: Oxford University Press. 2nd ed., 1988.
• Annas, Julia, 1975, "Aristotle, Number and Time." Philosophical Quarterly 25: 97-113.
• Annas, Julia, 1987, "Die Gegenstände der Mathematik bei Aristotles." In Graeser (1987), pp. 131-47.
• Apostle, Hippocrates George, 1952, Aristotle's Philosophy of Mathematics . Chicago: Chicago University Press.
• Barnes, Jonathan, 1975, Aristotle's Posterior Analytics . Oxford: Oxford University Press. 2nd ed., 1993.
• Barnes, Jonathan, 1985, "Aristotle's Arithmetic." Revue de Philosophie Ancienne 3: 97-133.
• Barnes, Jonathan, 1969, "Aristotle's Theory of Demonstration." Phronesis 14: 123-52. Revised in Articles. Vol. 1, ed. Barnes et al., 65-87.
• Barnes, Jonathan, 1981, "Proof and the Syllogism." In Berti (1981), 17-59.
• Cleary, John, 1985, "On the Terminology of ‘Abstraction’ in Aristotle." Phronesis 300: 13-45.
• Cleary, John, 1989, "Commentary on Halper's ‘Some Problems in Aristotle's Mathematical Ontology.’" P roceedings of the Boston Area Colloquium in Ancient Philosophy 5: 277-290.
• Cleary, John J. 1995, Aristotle & Mathematics: Aporetic Method in Cosmology & Metaphysics . Philosophia Antiqua 67. Leiden: Brill.
• Gaukroger, Stephen, 1980, "Aristotle on Intelligible Matter." Phronesis 25: 187-197.
• Gomez-Lobo, A, 1977, "Aristotle's Hypotheses and Euclid's Elements." Review of Metaphysics 30: 430-439.
• Görland, Albert, 1899, Aristoteles und die Mathematik . Marburg: N.G. Elwert'sche Verlagsbuchhandlung (reprint: Frankfurt/Main: Minerva, 1985).
• Halper, Edward, 1989, "Some Problems in Aristotle's Mathematical Ontology." Proceedings of the Boston Area Colloquium in Ancient Philosophy 5: 247-276.
• Hintikka, Jaakko, 1972, "On the Ingredients of an Aristotelian Science." Nous 6: 55-69
• Hintikka, Jaakko, 1980, "Aristotelian Induction." Revue Internationale de Philosophie 34: 422-440.
• Hintikka, Jaakko, 1973, Time and Necessity: Studies in Aristotle's Theory of Modality . Oxford: Oxford University Press.
• Hussey, Edward, 1983, Aristotle's Physics III & IV . Trans. and commentary. Clarendon Aristotle Series. Oxford: Oxford University Press.
• Hussey, Edward, 1991, "Aristotle on Mathematical Objects," in I. Mueller (ed.), Peri tôn Mathêmatôn . Apeiron 24 nr. 4 (Dec.): 105-134.
• Jones, Joe, 1983, "Intelligible Matter and Geometry in Aristotle." Apeiron 17: 94-102.
• Kouremenos, Theokritos, 1995. Aristotle on Mathematical Infinity . Stuttgart: Franz Steiner.
• Lear, Jonathan, 1982 "Aristotle's Philosophy of Mathematics." Philosophical Review 91: 161-92.
• Lear, Jonathan, 1979, "Aristotelian Infinity." Proceedings of the Aristotelian Society 80: 187-210.
• Lear, Jonathan, 1988, Aristotle: The Desire to Understand , ch. 6 §2. Cambridge: Cambridge University Press.
• Lee, H.D.P., 1935, "Geometrical Method and Aristotle's Account of First Principles." Classical Quarterly 29: 113-123.
• Mendell, Henry, 1998, "Making Sense of Aristotelian Demonstration". Oxford Studies in Ancient Philosophy , 16, 160-225.
• Mendell, Henry, 1987, "Topoi on Topos: the Development of Aristotle's Theory of Place." Phronesis 32: 206-231.
• Milhaud, Gaston, 1903, "Aristote et les mathematiques." Archiv für Geschichte der Philosophie 16: 367-92.
• Modrak, Deborah, 1989, "Aristotle on the Difference between Mathematics and Physics and First Philosophy." Apeiron 22: 121-139.
• Mueller, Ian, 1978, Review of Julia Annas, Aristotle's Metaphysics Books M and N (1st. ed., Oxford: Oxford University Press, 1976). Philosophical Review 87: 479-485
• Mueller, Ian, 1970, "Aristotle on Geometrical Objects." Archiv für die Gesch. der Philosophie 52: 156-171 (reprint. in Barnes, Articles , vol. 3)
• Mueller, Ian, 1990, "Aristotle's Doctrine of Abstraction in the commentators." In Richard Sorabji (ed.), Aristotle Transformed: the Ancient Commentators and their Influennce , Ithaca: Cornell University Press, 463-479.
• McKirahan, R.D., 1992, Principles and Proofs: Aristotle's Theory of Demonstrative Science . Princeton: Princeton University Press.
• Mignucci, M, 1987, "Aristotle's Arithmetic." In Graeser, 175-211.
• Philippe, M.D., 1948, "Aphairesis, Prosthesis, khôrizein dans la philosophie d'Aristote." Revue Thomiste 48: 461-79.
• Sorabji, Richard, 1973, "Aristotle, Mathematics, and Color." Classical Quarterly 22: 293-308.
• Tiles, J.E., 1983, "Why the Triangle has 2 Right Angles Kath’ Hauto." Phronesis 28: 1-17.
• White, Michael J., 1992. The Continuous and the Discrete . Oxford: Oxford University Press.

Works focussing on mathematical issues either in Aristotle or importantly related to Aristotle

• Becker, Oskar, 1933, "Eudoxos-Studien I. Eine voreudoxische Proportionenlehre und ihre Spuren bei Aristoteles und Euklid." QS 2: 311-333.
• Becker, Oskar, 1933, "Eudoxos-Studien II. Warum haben die Griechen die Existenz der vierten Proportionale angenommen?" QS 2: 369-387.
• Becker, Oskar, 1936, "Eudoxos-Studien III. Spuren eines Stetigkeitsaxioms in der Art des Dedekind'schen zur Zeit des Eudoxos." QS 3: 236-244.
• Becker, Oskar, 1936, "Eudoxos-Studien IV. Das Prinzip des ausgeschlossenen Dritten in der griechischen Mathematik." QS 3: 370-388.
• Euclid, 1926, The Elements of Euclid . 3 vols. 2nd. ed. Eng. trans. with comm. T.L. Heath. Cambridge: Cambridge U. P., (reprint: New York, Dover, 1956).
• Euclide, 1994, 1998, 1998 Les Éléments . 4 volumes (3 completed) French trans. with commentary by Bernard Vitrac, Paris: Presses Universitaires de France.
• Einarson, Benedict, 1936, "On Certain Mathematical Terms in Aristotle's Logic." American Journal of Philology 57: 33-44 (Part I), 151-172 (Part II).
• Fowler, D.H., 1987, The Mathematics of Plato's Academy . Oxford: Oxford University Press, (2nd ed., 1999).
• Heath, T.L., 1921, A History of Greek Mathematics . Oxford: Oxford University Press, (reprint: New York, Dover, 1981)
• Knorr, Wilbur R., 1978, "Archimedes and the Pre-Euclidean Proportion Theory." Archives Internationales d'Histoire des Sciences 28: 183-244.
• Knorr, Wilbur R., 1983, "Construction as Existence Proof in Ancient Geometry", Ancient Philosophy 3, 125-49.
• Knorr, Wilbur R., 1975, The Evolution of the Euclidean Elements . Synthese Historical Library 15. Dordrecht: Reidel.
• Knorr, Wilbur R., 1986, The Ancient Tradition of Geometric Problems . Boston: Birkhäuser, (reprint, New York: Dover).
• Mendell, Henry R., 1984, "Two Geometrical Examples from Aristotle's Metaphysics." Classical Quarterly 34: 359-72.
• Mendell, Henry R., 2001, "The Trouble with Eudoxus". In Pat Suppes, Julius Moravcsik, and Henry Mendell (eds.), Ancient and Medieval Traditions in the Exact Sciences: Essays in Memory of Wilbur Knorr , Stanford: CSLI (distr. University of Chicago Press), 59-138.
• Mueller, Ian, 1969, "Euclid's Elements and the Axiomatic Method." British Journal for the Philosophy of Science 20: 289-309.
• Mueller, Ian, 1974, "Greek Mathematics and Greek Logic." In John Corcoran (ed.), Ancient Logic and its Modern Interpretations (Dordrecht, Reidel), 35-70.
• Mueller, Ian, 1981, Philosophy of Mathematics and Deductive Structure in Euclid's Elements . Cambridge: MIT Press.
• Netz, Reviel, 1999, The Shaping of Deduction in Greek Mathematics: a Study in Cognitive History . Cambridge: Cambridge University Press.
• Proclus Diadochus, 1992, A Commentary on the First Book of Euclid's Elements . Trans. with intro. and notes by Glenn R. Morrow. Forward by Ian Mueller. Princeton: Princeton University Press.

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The Concept of Deduction in Ancient Greek and Egyptian Mathematics Essay

Unlike Egyptian mathematics, Greek mathematics refers to the texts typically written during and when ideas stem from the Archaic through the Roman and Hellenistic eras. The work of the famous and great Ancient Greek mathematicians has permeated every aspect and part of life, especially from sending rockets into space to accounting, architecture, and even the DIY field. In addition, Greek deduction is not only regarded as one of the most pivotal and incredible legacies that came into this world but is also considered the primary foundation of modern society. Remarkably, the concept of deduction in Greek has played a vital role in setting ancient Greek mathematics apart from famous ancient Egyptian mathematicians.

The Greeks copied many methods and techniques from the Babylonians and the Egyptians. Nonetheless, the Greeks were among the first to make math a realm of theory through reasoning and the deduction concept rather than measurement (Fraser & Schroter, 2019). In contrast, Egyptian mathematics was used for practical purposes. As an illustration, mathematicians from Egypt were uninterested and did not bother finding the exact pi figure. However, an approximate figure was adequate as long as their goals and uses were achieved.

Nonetheless, Egypt’s mathematical concepts and principles were based and built around the addition. The Greeks and Egyptians had and used different mathematical principles. The latter focused on practical arithmetic concepts, while their counterparts concentrated mainly on mathematical concepts, regulations, and ideas in their Greek deduction (Herrmann, 2023). However, most historians believe that the Egyptians did not perceive numbers as abstract quantities, but they primarily thought of a collection of eight items, especially when this number was mentioned.

Therefore, the mathematics of Egypt invented tricky methods to overcome this shortcoming because most of their numbers were inappropriate for multiplication, as illustrated in the Rhind papyrus. The ancient mathematicians also utilized a wide range of mathematical concepts and principles. These included the numeral system, which played a vital role in helping the counting process and solving written mathematical problems that often entail fractions and multiplications (Herrmann, 2023). The Great Pyramid of Egypt was also constructed using the mathematical Golden Ratio proportions, as evidenced by its properly adjusted height, base length, and angles. However, most of the evidence in ancient Egyptian mathematics is limited to the fact that there is a low amount of sources written on the Rhind papyrus.

Nevertheless, the Babylonians went the extra mile by looking at the relationship and the connection behind the numbers, but most of their work was in an empirical form. Nevertheless, the Greeks completely changed this perception by thoroughly exploring the underlying rules and relationships governing the functions and numbers (Christianidis & Megremi, 2019). The Greeks believed that because the universe is perfect, they had to develop and use the Greek deduction techniques to establish and identify mathematical facts without the challenges of inaccuracy and impurity from empirical measurements.

The above motive was crucial in the advancements they later made in algebra, calculus, geometry, and, most importantly, the mathematical form of reasoning, which is the primary foundation of logical arguments (Christianidis & Megremi, 2019). Having been influenced by famous ancient Egyptian mathematicians, Greek mathematics made several breakthroughs in Pythagoras’ theory of the right-angled triangles. It also focused on the abstract, bringing clarity and, primarily, precision to the age-old mathematical problems and concepts.

In conclusion, ancient Egyptian arithmetic greatly influenced the deductions in ancient Greek mathematics. It is the process of solving specific problems and the organization of the subjects. The Greeks invented and borrowed many techniques from the Babylonians and Egyptians. In addition, it is clear that ancient Egyptian mathematics also significantly affected the development of a modern system of weights used in the scaling of products.

Christianidis, J., & Megremi, A. (2019). Tracing the early history of algebra: Testimonies on Diophantus in the Greek-speaking world (4th–7th century CE) . Historia Mathematica, 47(2), 16-38.

Fraser, C., & Schroter, A. (2019). Past, present and anachronism in the historiography of mathematics . CMS Notes, 51(3), 16-17.

Herrmann, D. (2023). Ancient mathematics: History of mathematics in Ancient Greece and Hellenism . Springer

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IvyPanda. (2022, December 18). The Concept of Deduction in Ancient Greek and Egyptian Mathematics. https://ivypanda.com/essays/the-concept-of-deduction-in-ancient-greek-and-egyptian-mathematics/

"The Concept of Deduction in Ancient Greek and Egyptian Mathematics." IvyPanda , 18 Dec. 2022, ivypanda.com/essays/the-concept-of-deduction-in-ancient-greek-and-egyptian-mathematics/.

IvyPanda . (2022) 'The Concept of Deduction in Ancient Greek and Egyptian Mathematics'. 18 December.

IvyPanda . 2022. "The Concept of Deduction in Ancient Greek and Egyptian Mathematics." December 18, 2022. https://ivypanda.com/essays/the-concept-of-deduction-in-ancient-greek-and-egyptian-mathematics/.

1. IvyPanda . "The Concept of Deduction in Ancient Greek and Egyptian Mathematics." December 18, 2022. https://ivypanda.com/essays/the-concept-of-deduction-in-ancient-greek-and-egyptian-mathematics/.

Bibliography

IvyPanda . "The Concept of Deduction in Ancient Greek and Egyptian Mathematics." December 18, 2022. https://ivypanda.com/essays/the-concept-of-deduction-in-ancient-greek-and-egyptian-mathematics/.

• Science & Math
• Sociology & Philosophy
• Law & Politics
• Ancient Mathematics: Egyptians, Babylonians, Greeks

Ancient knowledge of the sciences was often wrong and wholly unsatisfactory by modern standards.  However, not all of the knowledge of the more learned peoples of the past was false.  In fact, without people like Euclid or Plato, we may not have been as advanced in this age as we are.  Mathematics is an adventure in ideas.  Within the history of mathematics, one finds the ideas and lives of some of the most brilliant people in the history of mankind’s populace upon Earth.

First, man created a number system of base 10.  Certainly, it is not just a coincidence that man just so happens to have ten fingers or ten toes, for when our primitive ancestors first discovered the need to count they definitely would have used their fingers to help them along just like a child today.  When primitive man learned to count up to ten he somehow differentiated himself from other animals. 

As an object of higher thinking, man invented ten number-sounds.  The needs and possessions of primitive man were not many.  When the need to count over ten aroused, he simply combined the number-sounds related with his fingers.  So, if he wished to define one more than ten, he simply said one-ten.  Thus our word eleven is simply a modern form of the Teutonic “ein-lifon” (”one over”). 

Since those first sounds were created, man has only added five new basic number-sounds to the ten primary ones.  They are “hundred,”  “thousand,”  “million,” “billion” (a thousand million in America, a million millions in England), “trillion” (a million millions in America, a million-million millions in England).  Because primitive man invented the same number of number-sounds as he had fingers, our number system is a decimal one, or a scale based on ten, consisting of limitless repetitions of the first ten number sounds.

Undoubtedly, if nature had given man thirteen fingers instead of ten, our number system would be much changed.  For instance, with a base thirteen number system we would call fifteen, two-thirteen’s. While some intelligent and well-schooled scholars might argue whether or not base ten is the most adequate number system, base ten is the irreversible favorite among all the nations. Of course, primitive man most certainly did not realize the concept of the number system he had just created.  Man simply used the number-sounds loosely as adjectives.  So an amount of ten fish was ten fish, whereas ten is an adjective describing the noun fish.

Soon the need to keep tally on one’s counting raised. The simple solution was to make a vertical mark.  Thus, on many caves, we see a number of marks that the resident used to keep track of his possessions such a fish or knives.  This way of record keeping is still taught today in our schools under the name of tally marks.

The earliest continuous record of mathematical activity is from the second millennium BC  When one of the few wonders of the world were created mathematics was necessary.  Even the earliest Egyptian pyramid proved that the makers had a fundamental knowledge of geometry and surveying skills.  The approximate time period was 2900 BC

The first proof of mathematical activity in written form came about one thousand years later.  The best known sources of ancient Egyptian mathematics in the written format are the Rhind Papyrus and the Moscow Papyrus.  The sources provide undeniable proof that the later Egyptians had intermediate knowledge of the following mathematical problems:  applications to surveying, salary distribution, calculation of area of simple geometric figures’ surfaces and volumes, simple solutions for first and second degree equations.

Egyptians used a base ten number system most likely because of biological reasons (ten fingers as explained above).  They used the Natural Numbers (1,2,3,4,5,6, etc.) also known as the counting numbers.  The word digit, which is Latin for finger, is also another name for numbers which explains the influence of fingers upon numbers once again.

The Egyptians produced a more complex system than the tally system for recording amounts.  Hieroglyphs stood for groups of tens, hundreds, and thousands.  The higher powers of ten made it much easier for the Egyptians to calculate into numbers as large as one million.  Our number system which is both decimal and positional (52 is not the same value as 25) differed from the Egyptian which was additive, but not positional.

The Egyptians also knew more of pi than its mere existence.  They found pi to equal C/D or 4(8/9)ª  whereas a equals 2.  The method for ancient peoples arriving at this numerical equation was fairly easy. They simply counted how many times a string that fit the circumference of the circle fitted into the diameter, thus the rough approximation of 3.

The biblical value of pi can be found in the Old Testament (I Kings vii.23 and 2 Chronicles iv.2)in the following verse:

“Also, he made a molten sea of ten cubits from

brim to brim, round in compass, and five cubits

the height thereof; and a line of thirty cubits did

compass it round about.”

The molten sea, as we are told is round, and measures thirty cubits round about (in circumference) and ten cubits from brim to brim (in diameter).  Thus the biblical value for pi is 30/10 = 3.

Now we travel to ancient Mesopotamia, home of the early Babylonians.  Unlike the Egyptians, the Babylonians developed a flexible technique for dealing with fractions.  The Babylonians also succeeded in developing a more sophisticated base ten arithmetic that were positional and they also stored mathematical records on clay tablets.

Despite all this, the greatest and most remarkable feature of Babylonian Mathematics was their complex usage of a sexagesimal place-valued system in addition to a decimal system much like our own modern one.  The Babylonians counted in both groups of ten and sixty.  Because of the flexibility of a sexagismal system with fractions, the Babylonians were strong in both algebra and number theory.  The remaining clay tablets from the Babylonian records show solutions to first, second, and third-degree equations.

Also, the calculations of compound interest, squares and square roots were apparent in the tablets. The sexagismal system of the Babylonians is still commonly in usage today.  Our system for telling time revolves around a sexagesimal system.  The same system for telling time that is used today was also used by the Babylonians.  Also, we use base sixty with circles (360 degrees to a circle).

Usage of the sexagesimal system was principally for economic reasons.  Being, the main units of weight and money were mina,(60 shekels) and talent (60 mina).  This sexagesimal arithmetic was used in commerce and in astronomy.

The Babylonians used many of the more common cases of the Pythagorean Theorem for right triangles.  They also used accurate formulas for solving the areas, volumes and other measurements of the easier geometric shapes as well as trapezoids.  The Babylonian value for pi was a very rounded off three.  Because of this crude approximation of pi, the Babylonians achieved only rough estimates of the areas of circles and other spherical, geometric objects.

The real birth of modern math was in the era of Greece and Rome.  Not only did the philosophers ask the question “how” of previous cultures, but they also asked the modern question of “why.”  The goal of this new thinking was to discover and understand the reason for mans’ existence in the universe and also to find his place.  The philosophers of Greece used mathematical formulas to prove propositions of mathematical properties.  Some of who, like Aristotle, engaged in the theoretical study of logic and the analysis of correct reasoning.  Up until this point in time, no previous culture had dealt with the negated abstract side of mathematics, or with the concept of mathematical proof.

The Greeks were interested not only in the application of mathematics but also in its philosophical significance, which was especially appreciated by Plato (429-348 BC).  Plato was of the richer class of gentlemen of leisure.  He, like others of his class, looked down upon the work of slaves and craftsworker.  He sought relief, for the tiresome worries of life, in the study of philosophy and personal ethics. 

Within the walls of Plato’s academy at least three great mathematicians were taught, Theaetetus, known for the theory of irrational, Eodoxus, the theory of proportions, and also Archytas (I couldn’t find what made him great, but three books mentioned him so I will too).  Indeed the motto of Plato’s academy “Let no one ignorant of geometry enter within these walls” was fitting for the scene of the great minds who gathered here.

Another great mathematician of the Greeks was Pythagoras who provided one of the first mathematical proofs and discovered incommensurable magnitudes, or irrational numbers.  The Pythagorean theorem relates the sides of a right triangle with their corresponding squares. The discovery of irrational magnitudes had another consequence for the Greeks:  since the length of diagonals of squares could not be expressed by rational numbers in the form of  A over B, the Greek number system was inadequate for describing them.

As you might have realized, without the great minds of the past our mathematical experiences would be quite different from the way they are today.  Yet as some famous (or maybe infamous) person must have once said “From down here the only way is up,”  so you might say that from now, 1996, the future of mathematics can only improve for the better.

Bibliography

Ball, W. W. Rouse. A Short Account of The History of Mathematics. Dover Publications Inc.

Mineloa, N.Y. 1985

Beckmann, Petr. A History of Pi. St. Martin’s Press. New York, N.Y. 1971

De Camp, L.S. The Ancient Engineers. Double Day. Garden City, N.J. 1963

Hooper, Alfred. Makers of Mathematics.  Random House. New York, N.Y. 1948

Morley, S.G. The Ancient Maya. Stanford University Press. 1947.

Newman, J.R. The World of Mathematics. Simon and Schuster. New York, N.Y. 1969.

Smith, David E. History of Mathematics. Dover Publications Inc. Mineola, N.Y. 1991.

Struik, Dirk J. A Concise History of Mathematics. Dover Publications Inc. Mineola, N.Y. 1987

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First formulated by Albert Einstein in 1915, the theory of general relativity remains our best and most accurate understanding of how gravity works on medium to large scales. 

Yet, zoom out even farther to view enormous groups of gravitationally bound galaxies interacting, and some inconsistencies appear to emerge. This suggests that gravity, which is theorized to be a constant across all times and scales, could actually become slightly weaker at cosmic distances. 

In a study published March 20 in the Journal of Cosmology and Astroparticle Physics , researchers described this discrepancy as a "cosmic glitch," and they say their proposed fix for it could help us understand some of the universe's most enduring mysteries.

"[It's] like making a puzzle on the surface of a sphere, then laying the pieces on a flat table and trying to fit them together," study co-author Niayesh Afshordi , a professor of astrophysics at the University of Waterloo in Ontario, told Live Science. "At some point, the pieces on the table will not quite fit each other, because you are using the wrong framework.

Related: James Webb telescope confirms there is something seriously wrong with our understanding of the universe  

"The glitch is the smoking gun for a fundamental violation of Einstein's equivalence principle (or Lorentz symmetry), which could point to radically different pictures for quantum gravity, the Big Bang , or black holes," Afshordi added.

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Cosmos for concern 

Einstein's theory of general relativity is remarkably good at describing the universe above quantum scales, and it has even predicted other aspects of our cosmos, including black holes , the gravitational lensing of light, gravitational waves, and the Big Bang.

Yet some discrepancies between theory and reality remain. First, attempts to scale down general relativity to describe how gravity operates on quantum scales transform its usually robust equations into incomprehensible nonsense. 

Second, completing our current model of the universe required the introduction of two mysterious additions, known as dark matter and dark energy . Believed to make up most of the contents of the universe, these entities have never been directly detected and fail to explain why our cosmos is expanding at different speeds depending on where we look . 

In response to these problems, the authors of the new paper came up with a simple suggestion: a tweak to Einstein's theory at different distance scales.

"The modification is very simple: We assume the universal constant of gravitation is different on cosmological scales, compared to smaller (like solar system or galactic) scales," Afshordi said. "We call this a cosmic glitch."

Afshordi said this tweak makes changes to patterns found in the cosmic microwave background — the leftover radiation produced 380,000 years after the Big Bang — and in the universe's structure and expansion. These adjustments are subtle, but the implication that the laws of gravity change over distance scales could be profound.

"We find evidence for the glitch: cosmic gravity is about 1% weaker than galactic/solar-system gravity," he added. 

— Mysterious 'unparticles' may be pushing the universe apart, new theoretical study suggests

— 'It could be profound': How astronomer Wendy Freedman is trying to fix the universe

— James Webb telescope discovers oldest black hole in the universe  

The researchers said the glitch's existence could be confirmed by next-generation galaxy surveys, including those performed with the European Space Agency 's Euclid space telescope , the Dark Energy Spectroscopic Instrument and the Simons Observatory . They say that these instruments should make measurements of the glitch four times more precise than is currently possible and, therefore, confirm or rule out their theory.

However, some scientists say a simple modification of Einstein's relativity might not be enough. In fact, it's possible that the discrepancies revealed by astronomical observations are hints that our understanding of the universe needs a complete rewrite.

"It's not that surprising that this new model is a slightly better fit to the data, but maybe that is telling us something," said Scott Dodelson , a professor of physics and the chair of the physics department at Carnegie Mellon University, who was not involved in the study.

"If so, it means we understand even less than we thought we did," he told Live Science. "My hunch is that instead of adding more new stuff, we need a new paradigm. But no one has come up with anything that makes any sense yet."

Ben Turner is a U.K. based staff writer at Live Science. He covers physics and astronomy, among other topics like tech and climate change. He graduated from University College London with a degree in particle physics before training as a journalist. When he's not writing, Ben enjoys reading literature, playing the guitar and embarrassing himself with chess.

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An ancient manuscript up for sale gives a glimpse into the history of early Christianity

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An important piece of early Christian history, the Crosby-Schøyen Codex, is up for auction at Christie’s in London. This codex is a mid-fourth century book from Egypt containing a combination of biblical and other early Christian texts.

The Crosby-Schøyen Codex was discovered alongside more than 20 other codices near Dishna, Egypt, in 1952. These manuscripts are collectively known as “the Dishna Papers” or “the Bodmer Papyri,” after the Swiss collector Martin Bodmer.

Though often overshadowed by other 20th century discoveries, this trove of ancient manuscripts represents one of the most significant finds for understanding the history of early Christianity. As an expert on early Christian reading practices , I consider the Dishna Papers an invaluable witness to the formation of the Christian Bible. This ancient library shows how, before the consolidation of the Bible, early Christians read canonical and non-canonical scriptures – as well as pagan classics – side by side.

An overshadowed discovery

The middle decades of the 20th century were exciting years for scholars of early Christianity.

In 1945, a collection of 13 ancient codices was discovered near Nag Hammadi, Egypt. These contained dozens of otherwise unknown works, mostly associated with minority and marginalized forms of early Christianity. With titles like “The Gospel of Thomas” and “The Secret Revelation of John,” this cache of non-canonical scriptures captured the public’s imagination and inspired a bestseller .

The very next year, Bedouin shepherds discovered ancient Hebrew scrolls hidden in a cave at Qumran on the northwestern shore of the Dead Sea.

The “ Dead Sea Scrolls ” found in this and a dozen subsequently discovered caves constituted a massive library of Jewish texts, including biblical works and hitherto unknown texts with remarkable parallels to the writings of the New Testament. This find was celebrated in news stories , documentaries and other publications as among the greatest discoveries of the 20th century.

At the very same time, the Dishna Papers were discovered, smuggled out of Egypt and sold to European collectors with considerably less fanfare. No headline hailed the discovery of the Dishna Papers. Instead, pieces of this collection were sold to the highest bidders, scattering the ancient library across the globe .

The Dishna Papers

Though less exotic than Nag Hammadi or Qumran, the contents of the Crosby-Schøyen Codex and the 20-some additional codices discovered near Dishna have proved every bit as important for our understanding of early Christianity.

Two manuscripts of the canonical gospels, Luke and John, belonging to this ancient library predate almost every other surviving copy of these gospels. Scholars used these new manuscripts to revise the text of the New Testament .

For instance, the vast majority of manuscripts of the Gospel of John describe Jesus as “the only-begotten Son” (1:18). But the early manuscripts discovered at Dishna read “the only-begotten God.” Here and elsewhere, English translations of the Bible were changed to reflect the contents of the Dishna Papers.

But the library discovered near Dishna did not consist entirely of texts that ended up in the Christian Bible. Scriptures that were not included in the Christian canon, like Paul’s “ Third Letter to the Corinthians ” and “ The Shepherd of Hermas ,” were also found among the Dishna Papers.

One codex from Dishna contains the “ Acts of Paul ,” an extra-Biblical account of Paul’s travels and martyrdom. Another contains the “ Infancy Gospel of James ,” a non-canonical story about the life of Mary, Jesus’ mother. The discoveries at Dishna provide evidence that these writings, though unfamiliar to modern readers of the Bible, spent centuries on the periphery of Christian scripture.

The Dishna Papers included a few additional literary texts. One codex in this mostly Christian library contains several comedies by the Hellenistic playwright Menander. Another codex binds together a chapter of Thucydides’ “History of the Peloponnesian War” with a Greek version of the biblical Book of Daniel.

Evidently, the owner of this Christian library had no aversion to the arts and sciences of pre-Christian Hellenism. In this library, pagan classics and Christian scripture stood side by side.

But whose library was this?

The Crosby-Schøyen Codex, which is now up for sale, actually supplies several important clues to the origin of the Dishna Papers with which it was found.

Thanks to recent radiocarbon dating of this codex and the contents of a closely related manuscript , the Crosby-Schøyen Codex can be dated with some measure of confidence to the middle of the fourth century – roughly 325 to 350 C.E.

The Crosby-Schøyen Codex itself contains five texts in Sahidic Coptic, a dialect of the ancient Egyptian language. Three texts are Biblical: Jonah, Second Maccabees 5:27-7:41, and 1 Peter. The rest of the codex contains part of a well-known Easter homily and a brief otherwise unknown exhortation.

These texts, argue scholars Albert Pietersma and Susan Comstock, may have been collected into a single codex for use as an Easter lectionary . A lectionary is a collection of readings used in Christian worship services. Such lectionaries were used in Pachomian monasteries, like the one located only a few miles west of Dishna.

This monastery was established in the mid-330s by Pachomius, the reputed founder of communal monasticism . His Pachomian Rule , by which the monks would have ordered their communal life, makes frequent reference to the public and private use of books. Pachomius’ monasteries even taught illiterate monks to read.

It seems likely that this eclectic library of canonical and non-canonical scriptures, early Christian writings and pagan classics belonged to these book-loving monks in central Egypt. One of the Pachomian rules allowed monks to borrow books from the monastic library for up to one week.

Today, for a few million dollars , one such book can be yours forever. On June 11, 2024, the Crosby-Schøyen Codex will go to the highest bidder.

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