Critical Analysis of Egyptian, Mesopotamia and Greek MathematicsGraduate Study/Course: M.A.Math.Ed. History of MathematicsMasterand: Johnmark L.Gorgonio, CE, ME IProfessor: Merlyn M. Sanchez, Ph.D. ____________________________________________________________________________Egyptian MathematicsOur first knowledge of mankinds use of mathematics beyond mere counting comes from the Egyptians and Babylonians. Both civilizations developed mathematics that was similar in some ways but different in others. The mathematics of Egypt, at least what is known from the papyri, can essentially be called applied arithmetic. It was practical information communicated via example on how to solve specific problems. This point, that mathematics was communicated by example, rather than by principle, is significant and is different than todays mathematics that is communicated essentially by principle with examples to illustrate principles. The reasons for this are unknown but could be due partly to the fact that symbolism, the medium of principles, did not exist in these early times. Indeed, much of mathematics for many centuries was communicated in this way. It may be much easier to explain to a young student an algorithm to solve a problem and for them to learn to solve like problems, than to explain the abstract concept first and basing examples upon this concept.

1. Basic facts about ancient Egypt.Egyptian hieroglyphics are in great abundance throughout Egypt. They were essentially indecipherable until 1799 when in Alexandria the trilingual Rosetta Stone was discovered. The Rosetta stone, an irregularly shaped tablet of black basalt measuring about 3 feet 9 inches by 2 feet 4 inches, was found near the town of Rosetta (Rashid) just a few miles northwest of Alexandria. Written in the two languages (Greek and Egyptian but three writing systems (hieroglyphics, its cursive form demotic script, and Greek, it provided the key toward the deciphering of hieroglyphic writing. The inscriptions on it were the benefactions conferred by Ptolemy V Epiphanes (205 - 180 BCE) were written by the priests of Memphis. The translation was primarily due to Thomas Young (1773 - 1829) and Jean Francois Champollion (1790-1832), who, very early in his life was inspired to Egyptology by the French mathematician Jean Baptiste Joseph Fourier (1768 - 1830). Champollion completed the work begun by Young and correctly deciphered the complete stone. An Egyptologist of the first rank, he was the first to recognize the signs could be alphabetic, syllabic, or determinative (i.e. standing for complete ideas) He also established the original language of the Rosetta stone was Greek, and that the hieroglyphic text was a translation from the Greek. An unusual aspect of hieroglyphics is that they can be read from left to right, or right to left, or vertically (top to bottom). It is the orientation of the glyphs that gives the clue; the direction of people and animals face toward the beginning of the line. For the Egyptians writing was an esthetic experience, and they viewed their writing signs as Gods words.. This could explain the unnecessary complexity, in face of the fact that obviously simplifications would certainly have occurred if writing were designed for all citizens. Rosetta Stone The demotic script was for more general use, the hieroglyphics continued to be used for priestly and formal applications. The Egyptians established an annual calendar of 12 months of 30 days each plus five feast days. Religion was a central feature of Egyptian society. There was a preoccupation with death. Many of Egypts greatest monuments were tombs constructed at great expense, and which required detailed logistical calculations and at least basic geometry. Construction projects on a massive scale were routinely carried out. The logistics of construction require all sorts of mathematics. You will see several mensuration (measurement) problems, simple algebra problems, and the methods for computation. Our sources of Egyptian mathematics are scarce. Indeed, much of our knowledge of ancient Egyptian mathematics comes not from the hieroglyphics (carved sacred letters or sacred letters) inscribed on the hundreds of temples but from two papyri containing collections of mathematical problems with their solutions. The Rhind Mathematical Papyrus named for A.H.Rhind (1833- 1863) who purchased it at Luxor in 1858. Origin: 1650 BCE but it was written very much earlier. It is 18 feet long and 13 inches wide. It is also called the Ahmes Papyrus after the scribe that last copied it. The Moscow Mathematical Papyrus purchased by V. S. Golenishchev (d. 1947). Origin: 1700 BC. It is 15 ft long and 3 inches wide. Two sections of this chapter offer highlights from these papyri. Papyrus, the writing material of ancient times, takes its name from the plant from which it is made. Long cultivated in the Nile delta region in Egypt, the Cyperus papyrus was grown for its stalk, whose inner pith was cut into thin strips and laid at right angles on top of each other. When pasted and pressed together, the result was smooth, thin, cream-colored papery sheets, normally about five to six inches wide. To write on it brushes or styli, reeds with crushed tips, were dipped into ink or colored liquid.A remarkable number of papyri, some dating from 2,500 BCE, have been found, protected from decomposition by the dry heat of the region though they often lay unprotected in desert sands or burial tombs.2. Counting and Arithmetic - basicsThe Egyptian counting system was decimal. Though non positional, it could deal with numbers of great scale. Yet, there is no apparent way to construct numbers arbitrarily large. (Compare that with modern systems, which is positional, which by its nature allows and economy for expressing huge numbers.) The number system was decimal with special symbols for 1, 10, 100, 1,000, 10,000, 100,000, and 1,000,000. Addition was accomplished by grouping and regrouping. Multiplication and division were essentially based on binary multiples. Fractions were ubiquitous but only unit fractions, with two exceptions, were allowed. All other fractions were required to be written as a sum of unit fractions. Geometry was limited to areas, volumes, and similarity. Curiously, though, volume measures for the fractional portions of the hekat a volume measuring about 4.8 liters, were symbolically expressed differently from others. Simple algebraic equations were solvable, even systems of equations in two dimensions could be solved.

It seems certain that the Egyptians understood general rules for handling fractions.3. The Ahmes PapyrusThe Ahmes was written in hieratic, and probably originated from the Middle Kingdom: 2000-1800 BC. It claims to be a thorough study of all things, insight into all that exists, knowledge of all obscure secrets. In fact, it is somewhat less. It is a collection of exercises, substantially rhetorical in form, designed primarily for students of mathematics. Included are exercises in fractions notation arithmetic algebra geometry mensurationThe practical mathematical tools for construction? To illustrate the level and scope of Egyptian mathematics of this period, we select several of the problems and their solutions as found in the two papyri. For example, beer and bread problems are common in the Ahmes.

Geometry and Mensuration Most geometry is related to mensuration. The Ahmes contains problems for the areas of isosceles triangles (correct) isosceles trapezoids (correct) quadrilaterals (incorrect) frustum (correct) circle (incorrect) curvilinear areasIn one problem the area for the quadrilateral was given by

which of course is wrong in general but correct for rectangles. Yet the .Rope stretchers. of ancient Egypt, that is the land surveyors, often had to deal with irregular quadrilaterals when measuring areas of land. This formula is quite accurate if the quadrilateral in question is nearly a rectangle.

On Rigor. There is in Egyptian mathematics a search for relationships, but the Egyptians had only a vague distinction between the exact and the approximate. Formulas were not evident. Only solutions to specific problems were given, from which the student was left to generalize to other circumstances. Yet, as we shall see, several of the great Greek mathematicians, Pythagoras , Thales, and Eudoxus to name three, studied in Egypt. There must have been more there than student exercises to learn!

The are numerous myths about the presumed geometric relationship among the dimensions of the Great Pyramid.

Heres one:[perimeter of base]= [circumference of a circle of radius=height] Such a formula would yield an effective = 3(1/7), not = 3(1/6).

4 The Moscow Papyrus

The Moscow papyrus contains only about 25, mostly practical, examples. The author is unknown. It was purchased by V. S. Golenishchev (d. 1947) and sold to the Moscow Museum of Fine Art. Origin: 1700 BC. It is 15 feet long and about 3 inches wide.

Heres the picture that is found in the Moscow Papyrus

Heres the modern version of the picture and a perspective drawing

5. Implication of Egyptian MathematicsIn the few bullet items below we give a summary of known Egyptian mathematical achievements. Records of conquests of pharohs and other facts of Egyptian life are in abundance throughout Egypt, but of her mathematics only traces have been found. These fragments, from a civilization that lasted a millennium longer than the entire Christian era, that undertook constructions projects on a seen not seen again6 until this century, and that created abundance from a desert, allow only the following conclusions. Egyptian mathematics remained remarkably uniform through out time. It was built around addition. Little theoretical contributions were evident. Only the slightest of abstraction is evident. Yet exact versions of difficult to find formulas were available. It was substantially practical. The texts were for students. No principles. are evident, neither are there laws, theorems, axioms postulates or demonstrations; the problems of the papyri are examples from which the student would generalize to the actual problem at hand. The papyri were probably not written for self study. No doubt there was a teacher present to assist the student learning the examples and then giving exercises for the student to solve. There seems to be no clear differentiation between the concepts of exactness and approximate. Elementary congruencies were used only for mensuration. Yet, there must have been much more to Egyptian mathematics. We know that Thales, Pythagoras and others visited Egypt to study. If there were only applied arithmetic methods as we haveseen in the papyri, the trip would have had little value. But where are the records of achievement? Very likely, the mathematics extant was absorbed into the body of Greek mathematics . in an age where new and better works completely displaced the old, and in this case the old works written in hieroglyphics. Additionally, the Alexandrian library, one place where ancient Egyptian mathematical works may have been preserved, was destroyed by about 400 CE.________________________________

Babylonian Mathematics

Basic FactsThe Babylonian civilization has its roots dating to 4000BCE with the Sumerians in Mesopotamia. Yet little is known about the Sumerians. Sumer was first settled between 4500 and 4000 BC by a non-Semitic people who did not speak the Sumerian language. These people now are called Ubaidians, for the village Al-Ubaid, where their remains were first uncovered. Even less is known about their mathematics. Of the little that is known, the Sumerians of the Mesopotamian valley builthomes and temples and decorated them with artistic pottery and mosaics in geometric patterns. The Ubaidians were the first civilizing force in the region. They drained marshes for agriculture, developed trade and established industries including weaving, leatherwork, metalwork, masonry, and pottery. The people called Sumerians, whose language prevailed in the territory, probably came from around Anatolia, probably arriving in Sumer about 3300 BC. For a brief chronological outline of Mesopotamia. The early Sumerians did have writing for numbers as shown below. Owing to the scarcity of resources, the Sumerians adapted the ubiquitous clay in the region developing a writing that required the use of a stylus to carve into a soft clay tablet. It predated the cuneiform (wedge) pattern of writing that the Sumerians had developed during the fourth millennium. It probably antedates the Egyptian hieroglyphic may have been the earliest form of written communication. The Babylonians, and other cultures including the Assyrians, and Hittites,inherited Sumerian law and literature and importantly their style of writing.Here we focus on the later period of the Mesopotamian civilization which engulfed the Sumerian civilization. The Mesopotamian civilizations are often called Babylonian, though this is not correct. Actually, Babylon3 was not the first great city, though the whole civilization is called Babylonian. Babylon, even during its existence, was not always the center of Mesopotamian culture. The region, at least that between the two rivers, the Tigris and the Euphrates, is also called Chaldea.

Babylonian NumbersIn mathematics, the Babylonians (Sumerians) were somewhat more advanced than the Egyptians. Their mathematical notation was positional but sexagesimal. They used no zero. More general fractions, though not all fractions, were admitted. They could extract square roots. They could solve linear systems. They worked with Pythagorean triples. They solved cubic equations with the help of tables. They studied circular measurement. Their geometry was sometimes incorrect.For enumeration the Babylonians used symbols for 1, 10, 60, 600, 3,600, 36,000, and 216,000, similar to the earlier period. Below are four of the symbols. They did arithmetic in base 60, sexagesimal.

Babylonian AlgebraIn Greek mathematics there is a clear distinction between the geometric and algebraic. Overwhelmingly, the Greeks assumed a geometric position wherever possible. Only in the later work of Diophantus do we see algebraic methods of significance. On the other hand, the Babylonians assumed just as definitely, an algebraic viewpoint. They allowed operations that were forbidden in Greek mathematics and even later until the 16th century of our own era. For example, they would freely multiplyareas and lengths, demonstrating that the units were of less importance. Their methods of designating unknowns, however, does invoke units. First, mathematical expression was strictly rhetorical, symbolism would not come for another two millenia with Diophantus, and then not significantlyuntil Vieta in the 16th century. For example, the designation of the unknown was length. The designation of the square of the unknown was area. In solving linear systems of two dimensions, the unknowns were length and breadth, and length, breadth, and width for three dimensions.Square Roots. Recall the approximation of 2. How did they get it? There are two possibilities: (1) Applying the method of the mean. (2)Applying the approximation

Solving Quadratics. The Babylonian method for solving quadratics is essentially based on completing the square. The method(s) are not as clean. as the modern quadratic formula, because the Babylonians allowed only positive solutions. Thus equations always were set in a form for which there was a positive solution. Negative solutions (indeed negative numbers) would not be allowed until the 16th century CE.

The rhetorical method of writing a problem does not require variables. As such problems have a rather intuitive feel. Anyone could understand the problem, but without the proper tools, the solution would be impossibly difficult. No doubt this rendered a sense of the mystic to the mathematician.

Solving Cubics. The Babylonians even managed to solve cubic equations, though again only those having positive solutions. However, the form of the equation was restricted tightly.

Solving linear systems. The solution of linear systems were solved in a particularly clever way, reducing a problem of two variables to one variable in a sort of elimination process, vaguely reminiscent of Gaussian elimination

Pythagorean TriplesAs we have seen there is solid evidence that the ancient Chinese were aware of the Pythagorean theorem, even though they may not have had anything near to a proof. The Babylonians, too, had such an awareness. Indeed, the evidence here is very much stronger, for an entire tablet of Pythagorean triples has been discovered. The events surrounding them reads much like a modern detective story, with the sleuth being archaeologist Otto Neugebauer. We begin in about 1945 with the Plimpton322 tablet, which is now the Babylonian collection at Yale University, and dates from about 1700 BCE. It appears to have the left section broken away. Indeed, the presence of glue on the broken edge indicates that it was broken after excavation. What the tablet contains is fifteen rows of numbers, numbered from 1 to 15.

Babylonian Geometry

Circular Measurement. We find that the Babylonians used = 3 for practical computation. But, in 1936 at Susa (captured by Alexander the Great in 331 BCE), a number of tablets with significant geometric results were unearthed. One tablet compares the areas and the squares of the sides of the regular polygons of three to seven sides. For example, there is the approximation

Implication of Babylonian Mathematics

That Babylonian mathematics may seem to be further advanced than that of Egypt may be due to the evidence available. So, even though we see the development as being more general and somewhat broader in scope, there remain many similarities. For example, problems contain only specific cases. There seem to be no general formulations. The lack of notation is clearly detrimental in the handling of algebraic problems.

There is an absence of clear cut distinctions between exact and approximate results.

Geometric considerations play a very secondary role in Babylonian algebra, even though geometric terminology may be used. Areas and lengths are freely added, something that would not be possible in Greek mathematics. Overall, the role of geometry is diminished in comparison with algebraic and numerical methods. Questions about solvability or insolvability are absent. The concept of .proof. is unclear and uncertain. Overall, there is no sense of abstraction. In sum, Babylonian mathematics, like that of the Egyptians, is mostly utilitarian . but apparently more advanced.

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The Greek Mathematics

Though the Greeks certainly borrowed from other civilizations, they built a culture and civilization on their own which is The most impressive of all civilizations The most influential in Western culture The most decisive in founding mathematics as we know it.

The impact of Greece is typified by the hyperbole of Sir Henry Main: Except the blind forces of nature, nothing moves in this world which is not Greek in its origin. Including the adoption of Egyptian and other earlier cultures by the Greeks, we find their patrimony in allphases of modern life. Handicrafts, mining techniques, engineering, trade, governmental regulation of commerce and more have all come down to use from Rome and from Rome through Greece. Especially, our democracies and dictatorships go back to Greek exemplars, as well do our schools and universities, our sports, our games. And there is more. Our literature and literary genres, our alphabet, our music, oursculpture, and most particularly our mathematics all exist as facets of the Greek heritage. The detailed study of Greek mathematics reveals much about modern mathematics, if not the modern directions, then the logic and methods.

The Sources of Greek MathematicsIn actual fact, our direct knowledge of Greek mathematics is less reliable than that of the older Egyptian and Babylonian mathematics, because none of the original manuscripts are extant.

There are two sources:

Byzantine Greek codices (manuscript books) written 500-1500 years after the Greek works were composed. Arabic translations of Greek works and Latin translations of the Arabic versions. (Were there changes to the originals?)

Moreover, we do not know even if these works were made from the originals. For example, Heron made a number of changes in Euclids Elements, adding new cases, providing different proofs and converses. Likewise for Theon of Alexandria (400 A. D.). The Greeks wrote histories of Mathematics:

Eudemus (4th century BCE), a member of Aristotles school wrote histories3 of arithmetic, geometry and astronomy (lost), Theophrastus (c. 372 - c. 287 BCE) wrote a history of physics (lost). Pappus (late 3rdcentury CE) wrote the Mathematical Collection, an account of classical mathematics from Euclid to Ptolemy (extant). Pappus wrote Treasury of Analysis, a collection of the Greek works themselves (lost). Proclus (410-485 CE) wrote the Commentary, treating Book I of Euclid and contains quotations due to Eudemus (extant). various fragments of others.

Major Schools of Greek MathematicsThe Classical Greek mathematics can be neatly divided in to several schools, which represent a philosophy and a style of mathematics. Culminating with The Elements of Euclid, each contributed in a real way important facets to that monumental work. In some cases the influence was much broader.

The Ionian SchoolThe Ionian School was founded by Thales (c. 643 - c. 546 BCE). Students included Anaximander (c. 610 - c. 547 BCE) and Anaximenes (c. 550- c. 480BCE), actually a student of Anaximander. He regarded air as the origin and used the term air as god. Thales is the first of those to write on physics physiologia, which was on the principles of being and developing in things. His work was enthusiastically advanced by his student Anaximander. Exploring the origins of the universe, Axaximander wrote that the first principle was a vast Indefinite-Infinite (apeiron), a boundless mass possessing no specific qualities. By inherent forces, it gradually developed into the universe.In his system, the animate and eternal but impersonal Infinite is the only God, and is unvarying and everlasting. Thales is sometimes credited with having given the first deductive proofs. He is credited with five basic theorems in plane geometry, one being that the every triangle inscribed in a semicircle is a right triangle. Another result, that the diameter bisects a circle appears in The Elements as a definition. Therefore, it is doubtful that proofs provided by Thalesmatch the rigor of logic based on the principles set out by Aristotle and climaxed in The Elements. Thales is also credited with a number of remarkable achievements, from astronomy to mensuration to businessacumen, that will be taken up another chapter. The importance of the Ionian School for philosophy and the philosophy of science is without dispute.

The Pythagorean SchoolThe Pythagorean School was foundedby Pythagoras in about 455 BCEA brief list of Pythagorean contributions includes:

1. Philosophy.2. The study of proportion3. The study of plane and solid geometry.4. Number theory.5. The theory of proof.6. The discovery of incommensurables.

For another example, Hippocrates of Chios (late 5th century BCE), computedthe quadrature of certain lunes. This, by the way, is the first correct proof of the area of a curvilinear figure, next to the circle, though the issue is technical. He also was able to duplicate the cube by finding two mean proportionals.

The Eleatic SchoolThe Eleatic School from the southern Italian city of Elea was founded by Xenophanes of Colophon, but its chief tenets appear first in Parmenides, the second leader of the school. Melissus was the third and last leader of the school. Zeno of Elea (c. 495 - c. 430 BCE), son of Teleutagoras and pupil and friend of Parmenides, no doubt strongly influenced the school. Called by Aristotle the inventor of dialectic, he is universally known for his four paradoxes. These, while perplexing generations of thinkers, contributed substantially to the development of logical and mathematical rigor. They were regarded as insoluble until the development of precise concepts of continuity and infinity.It remains controversial that Zeno was arguing against the Pythagoreans who believed in a plurality composed of numbers that were thought of as extended units. The fact is that the logical problems which his paradoxes raise about a mathematical continuum are serious, fundamental, and were inadequately solved by Aristotle.

Zeno made use of three premises:

1. Any unit has magnitude2. That it is infinitely divisible3. That it is indivisible.

Yet he incorporated arguments for each. In his hands, he had a very powerful complex argument in the form of a dilemma, one horn of which supposed indivisibility, the other infinite divisibility, both leading to a contradiction of the original hypothesis who brought to the fore the contradictions between the discrete and the continuous, the decomposable and indecomposable.

Zenos ParadoxesZeno constructed his paradoxes to illustrate that current notions of motionare unclear, that whether one viewed time or space as continuous or discrete, there are contradictions. Paradoxes such as these arose because mankind was attempting to rationally understand the notions of infinity for the first time. The confusion centers around what happens when the logic of the finite (discrete) is used to treat the infinite (infinitesimal) and conversely, when the infinite is perceived within the discrete logical framework. They areDichotomy To get to a fixed point one must cover the halfway mark, and then the halfway mark of what remains, etc.Achilles Essentially the same for a moving point.Arrow An object in flight occupies a space equal to itself but that which occupies a space equal to itself is not in motion.Stade Suppose there is a smallest instant of time. Then time must be further divisible!

The Sophist SchoolThe Sophist School (e480 BCE) was centered in Athens, just after the final defeat of the Persians.7 There were many sophists and for many years, say until 380 BCE, they were the only source of higher education in the more advanced Greek cities. Of course such services were provided for money. Their influence waned as the philosophic schools, such as Platos academy, grew in prestige. Chief among the sophists were most important were Protagoras, Gorgias, Antiphon, Prodicus, and Thrasymachus. In some regards Socrates must be considered among them, or at least a special category of one among them. Plato emphasized, however, that Socrates never accepted money for knowledge. Greece because Athens was a democracy, young men needed instruction in politics. Sophists provided that instruction, teaching men how to speak and what arguments to use in public debate. A Sophistic education became popular among older families and the upwardly mobile withoutfamilies. Among the instruction given were ways to argue against traditionalvalues, which Plato thought unfair and unjustified. However, he learned that even to defend traditional values, one must use a reasoned argument, not appeals to tradition and unreflecting faith.

The Trisectrix. Here is how to construct the trisectrix. A rotating arm begins at the vertical position and rotates clockwise as a constant rate to the 3 oclock position. A horizontal bar falls from the top (12 oclock position) to the x-axis at a constant rate, in the same time. The locus of points where the horizontal bar intersects the rotating arm traces the trisectrix. (In the figure below, you may assume that radian measure isused with a (quarter) circle of radius /2. Thus the time axis ranges in [0, /2 ].)

The Platonic SchoolThe Platonic School, the most famous of all was founded by Plato (427- 347BCE) in 387 BCE in Athens as an institute for the pursuit of philosophical and scientific teaching and research. Plato, though not a mathematician, encouraged research in mathematics. Pythagorean forerunners of the school, Theodorus of Cyrene and Archytas10 of Tarentum, through their teachings, produced a strong Pythagorean influence in the entire Platonic school. Little is known of Platos personality and little can be inferred from his writing. Said Aristotle, certainly his most able and famous student, Plato is a man whom it is blasphemy in the base even to praise. This meant that even those of base standing in society should not mention his name, so noble was he. Much of the most significant mathematical work of the 4th century was accomplished by colleagues or pupils of Plato. Members of the school included Menaechmus and his brotherDinostratus and Theaetetus(c. 415- 369 BCE) According to Proclus, Menaechmus was one of those who made the whole of geometry more perfect.

The School of EudoxusThe School of Eudoxus founded by Eudoxus (c. 408 BCE), the most famous of all the classical Greek mathematicians and second only toArchimedes. Eudoxus developed the theory of proportion, partly to account for and study the incommensurables (irrationals). He produced many theorems in plane geometry and furthered the logical organization of proof. He also introduced the notion of magnitude. He gave the first rigorous proof on the quadrature of the circle. (Proposition. The areas of two circles are as the squares of their diameters.)

The School of AristotleThe School of Aristotle, called the Lyceum, founded by Aristotle (384- 322 BCE) followed the Platonic school. It had a garden, a lecture room, and an altar to the Muses. Of his character more is known than for others we have considered. He seems to have been wealthy with holdings from Stagira. Therefore, he had the leisure to study. He apparently used sums of money to purchase books. So many books did he read that Plato referred to him as the reader, indicating a bit of contempt or perhaps rivalry. While still a member of Platos academy, his early writings works were dialogues were concerned with thoughts of the next world and the worthlessness of this one, themes familiar to him from Platos writing (e.g. Phaedo). Anecdotes about him show him as a kindly and affectionate. They show hardly a trace of the self importance that some scholars claim to detect in his works. His will has survived and exhibits the same kindly traits; he references a happy family life and takes solicitous care of his children, as well as his servants. His apparent joy of life is reflected in the literary On Philosophy, which was completed in about 348. Afterwards, he devoted his energies to research, teaching, and writing of technical treatises.

Aristotle set the philosophy of physics, mathematics, and reality on a foundations that would carry it to modern times. He viewed the sciences as being of three types theoretical (math physics, logic and metaphysics), productive (the arts), and the practical (ethics, politics).

He contributed little to mathematics however,

...his views on the nature of mathematics and its relations to the physical world were highly influential. Whereas Plato believed that there was an independent, eternally existing world of ideas which constituted the reality of the universe and that mathematical concepts were part of this world, Aristotle favored concrete matter or substance

Aristotle regards the notion of definition as a significant aspect of argument. He required that a definition may not reference prior objects. The following definition,

A point is that which has no part,

which is the first definition from the first book of Euclids Elements, would be unacceptable. Aristotle also treats the basic principles of mathematics, distinguishing between axioms and postulates. Axioms include the laws of logic, the law of contradiction, etc. The postulates need not be self-evident, but their truth must be sustained by the results derived from them.

Euclid uses this distinction. Aristotle explored the relation of the point to the line again the problem of the indecomposable and decomposable.

Aristotle makes the distinction between potential infinity and actual infinity. He states only the former actually exists, in all regards.

Aristotle is credited with the invention of logic, through the syllogism. He cites two laws studied by every student.

1. The law of contradiction. (A statement may not be T and F)

2. The law of the excluded middle. (A statement must be T or F, there is no other alternative.)

His logic remained unchallenged until the 19th century. Even Aristotle regarded logic as an independent subject that should precede scienceand mathematics.______________END______________