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The Moscow Puzzles : Mathematical Re Speed Mathematics Simplified Dover Scie An Adventurer's Guide to Number Theory Click on the cover image above to read some pages of this book! This original anthology assembles eleven accessible essays by a giant of modern mathematics. Hermann Weyl made lasting contributions to number theory as well as theoretical physics, and he was associated with Princeton's Institute for Advanced Study, the University of Gottingen, and ETH Zurich.

Spanning the ss, these articles offer insights into logic and relativity theory in addition to reflections on the work of Weyl's mentor, David Hilbert, and his friend Emmy Noether. Country of Publication: US Dimensions cm : Help Centre. My Wishlist Sign In Join. Be the first to write a review. Add to Wishlist. 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. 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.

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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. 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.

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.

The standard English translation is given first, and, where appropirate, other more idiomatic translations. Greek is in parentheses. Aristotle Aristotle, General Topics: logic. Aristotle and Mathematics First published Fri Mar 26, Introduction 2.

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Demonstration and Mathematics 5. What Mathematical Sciences Study: 4 Puzzles 7. Aristotle's Treatment of Mathematical Objects 7. Universal Mathematics 9. Place and Continuity of Magnitudes Unit monas and Number arithmos Mathematics and Hypothetical Necessity The Infinite apeiron Introduction The late fifth and fourth centuries B. The Structure of a Mathematical Science: First Principles 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's list here includes the most general principles such as non-contradiction and excluded middle, and principles more specific to mathematicals, e. Aristotle divides posits thesis into two types, definitions and hypotheses: A hypothesis hupothesis asserts one part of a contradiction, e. For more information, see the following supplementary document: Aristotle and First Principles in Greek Mathematics 3. Mathematical example: point is on every line i.

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An alternative stronger? In this case, A and B are called, in modern discussions, commensurate universals. Demonstration and Mathematics 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.

The Relation Between Different Sciences: Autonomy and Subalternation A science is defined by the genus or kind it studies and by a group of specifiable properties which belong to that kind. 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.

What Mathematical Sciences Study: Puzzles Aristotle's principal concern in discussing ontological issues in mathematics is to avoid various versions of platonism. 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. 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.

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. This is the problem of non-revisionism sometimes also called naturalism. So Aristotle says Met. 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. 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. 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.

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. 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. 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.

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. Matter is the genus, e. De anima i. The parts of a mathematical object which do not occur in the definition of the object, e.

Universal Mathematics Ancillary to his discussions of being qua being and theology Metaphysics vi. 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. 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.

C is one point in number. C is two points in its being or formula logos. For more information, see the following supplementary document: Place and continuity of Magnitudes Their conception involves: A number is constructed out of some countable entity, unit monas. Numbers are more like concatenations of units and are not sets.


To draw a contrast with modern treatments of numbers, a Greek pair or a two is neither a subset of a triple, nor a member of a triple. It is a part of three. If I say that ten cows are hungry, then I am not saying that a set is hungry. So too, these ten units are a part of these twenty units: 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.

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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 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. 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. 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. The notion of incomparable numbers lacks the basic conception of numbers as concatenations of units. 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. 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. Mathematics and Hypothetical Necessity It is often supposed, for Aristotle, mathematical explanation plays no role in the study of nature, especially in biology.

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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. The Infinite apeiron 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.

For more information, see the following supplementary document: Aristotle on the Infinite 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.

For more information, see the following supplementary document: Aristotle and Greek Mathematics Glossary The standard English translation is given first, and, where appropirate, other more idiomatic translations. Bibliography Collections of essays and journals referred to below with abbreviations Barnes : Barnes, Jonathan, Schofield, Malcolm, and Sorabji, Richard, eds.

London: Duckworth. Berti : Berti, Enrico, ed.. Padova: Editrice Antenore. Graeser : Graeser, A. Akten des X. Symposium Aristotelicum. Bern: Haupt. Aristotelis loca mathematica ex universis ipsius operibus collecta et explicata. Bologna: Sumptibus Hieronymi Tamburini. Heath, Thomas L. Mathematics in Aristotle. Oxford: Oxford University Press, reprint.

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New York: Garland Press, Heiberg, I. Leipzig: Teubner. Works on philosophical issues in Aristotle on mathematics or use of mathematics in discussing philosophical discussions Annas, Julia, , Aristotle's Metaphysics Books M and N English translation and commentary. Oxford Clarendon Aristotle, ed. Oxford: Oxford University Press. Annas, Julia, , "Aristotle, Number and Time. Chicago: Chicago University Press.

Barnes, Jonathan, , Aristotle's Posterior Analytics. Barnes, Jonathan, , "Aristotle's Arithmetic. Barnes, Jonathan, , "Aristotle's Theory of Demonstration.