Iamblichus: On the General Science of Mathematics

John Dillon, J.O. Urmson, John Dillon, J.O. Urmson

• Éditeur: Bloomsbury Academic
• Année: 2020
• ISBN:
  9781350127678
• Édition: 1
• Sujet: Mathematics
• Sous-sujet: History & Philosophy of Mathematics

On the General Science of Mathematics is the third of four surviving works out of ten by Iamblichus ( c. 245 CE–early 320s) on the Pythagoreans. He thought the Pythagoreans had treated mathematics as essential for drawing the human soul upwards to higher realms described by Plato, and downwards to understand the physical cosmos, the products of arts and crafts and the order required for an ethical life. His Pythagorean treatises use edited quotation to re-tell the history of philosophy, presenting Plato and Aristotle as passing on the ideas invented by Pythagoras and his early followers. Although his quotations tend to come instead from Plato and later Pythagoreanising Platonists, this re-interpretation had a huge impact on the Neoplatonist commentators in Athens. Iamblichus' cleverness, if not to the same extent his re-interpretation, was appreciated by the commentators in Alexandria.

Iamblichus

On the General Science of Mathematics

Translation

Iamblichus of Chalcis in Coele Syria on the General Science of Mathematics

Chapters of the Third Discourse¹

1. What is the aim as a whole of the present book and what are the particular targets under that aim, divided into what topics, and whence do they derive the primary grounds of their special study, and from what sort of essential being?²

2. What is the whole theory behind mathematics in general, and what the nature of mathematical science, whence it derives, and from what sources its definition is to be acquired; also how great is its compass, and into how many general kinds it extends.

3. What the principles of the whole of mathematics are and how they differ from other principles which are of other subject-matters; also how such sorts of principle provide the ground common to the whole of mathematics.

4. What the special principles of each branch of mathematics are and what their special character, their differences from each other and from all the other principles of all existent things.

5. What common features underlie all the mathematical sciences with which students of mathematics are engaged, and how it is possible to study them universally.

6. What is the best use of the study of mathematics, and to what goal one should refer the best sort of study of it.

7. What the special object of knowledge is which is the basis of each branch of mathematics, and how it is possible by division to make a general distinction of them, so as to know both the one and the manifold in mathematics, its character, and how it should be defined.

8. What is the general criterion for the whole of mathematics and how it is discovered from the division of the line which the Pythagoreans hand down.

9. About those that propose a definition of the essence of mathematics, of whom the first viewpoint is presented as being that of those who relate it to the soul; several causes of such a conception are stated, and starting points for the whole study of mathematics.

10. How the essence (ousia) of the soul is composed out of all the mathematical sciences, and according to what distinction the mixture of them within it may be determined, and whether the soul encompasses within itself the whole level of being of mathematics, or if some other principle of mathematics is also to be considered.

11. What is the function (ergon) of the study of mathematics, and how it arises, and that it is called mathematics conformably to these.

12. What are the capacities (dynameis) of mathematical science, the orders within them, the number of differences by which they are divided, and in how many ways they are conceived.

13. What elements (stoikheia) and kinds (gene) of mathematical science there are, and how the same ones are present as elements and how as kinds; by what these are different from those in the other sciences and kinds of things, both the intelligible and such as are in the realm of generation.

14. Concerning the similarities and dissimilarities in mathematics, what they are and how far they extend; how they are present in the subject-matter of mathematics, and in what respect they differ from similarly named kinds, both of intelligible and of sensible objects.

15. How the whole of mathematical science extends, both itself, its kinds, its elements, and its principles, into the whole of philosophy and all the parts of philosophy; how it is involved with them and by what sort of contribution.

16. How many goods it provides to the arts and crafts, universally to them as a whole and to their different kinds, as it does to the theoretical, creative, and practical crafts; and succinct instruction about them.

17. What the order of presentation in mathematics is, and if it has a natural order also for learning; and if each ordering is appropriate to each, and the two to each other.

18. The special methods of the Pythagorean presentation of the mathematical sciences, how and in what matters they made use of them, and for whom; also that they always took due account both of the subject-matter and of learners.

19. The Pythagorean division of mathematical science as a whole into genera and the most important species, which creates a common study of them.

20. What is the method of definition in mathematics and how it comes about; the kind of help it contributes to science; that mathematics has a goal and what sort of goal it has.

21. Who were the originators of Pythagorean mathematics, and what according to him are the salient points of such a science; how, according to him, one should organize mathematics. A general survey.

22. The distinctive practice of mathematical science according to Pythagoras, and how many uses of it he envisaged for the soul and for mankind; also how they practised it throughout the whole of their own life.

23. That the Pythagoreans extended mathematics not blindly, but with a view to the life of necessary utility; a record of the reasons for this, with many examples.

24. What was the group practice in the Pythagoreans’ occupation with mathematical pursuits and what was their training and preparation in sciences?

25. Which of the Pythagoreans were Mathematicals, and in what respect did they differ from the Acousmatics? What was their function and of what kind the form of their arguments and demonstrations?

26. Objections to mathematics as being worthless; replies thereto, and comparisons of these arguments in detail.

27. What the really educated person should ask for from the mathematician; also how one should judge his study and by what standards determine its correctness.

28. When the problem or the method of demonstration is mathematical or belongs to another science; a scientific distinction.

29. Concerning the mathematical syllogisms, and the mathematical divisions and definitions; how mathematical science should use them, whether it gets its principles in its own way or from dialectic.

30. That mathematics makes great contributions to philosophy in general and to all its branches, supporting it in every respect; and especially that of the Pythagoreans, which is much superior to other mathematics.

31. That the Pythagoreans used the same mathematics for many different purposes; that they conducted many studies that clarified the same problem, and for what reasons they did that.

32. How sometimes we tackle also questions about perceptible things mathematically, and how often this occurs; also how in mathematics many things lead on to others and why this is so.

33. What is common to the whole of mathematical science and specific to it with regard to the differences observed in its many species, and how one should divide it in accordance with scientific division from one into two and then more species.

34. Whence mathematical science got its name and what its characteristic is; also to what one should pay attention in judging the species of mathematics.

35. Summary of the general account of all mathematics, a justification of the order of the chapters, and also a reminder at the same time about the right way to divide up the whole conspectus of the discipline.

Chapter 1³

What is the aim as a whole of the present book and what are the particular targets⁴ under that aim, divided into what topics, and whence do they derive the primary grounds of their special study, and from what sort of essential being?⁵

9,4 The object of the present investigation is to indicate the general study of mathematics, what it is as a whole, and what single ground and fundamental subject-matter it has.⁶ Following on this first topic, let us investigate whether there are perhaps two principles of it,⁷ and, after this dichotomy, we shall 10 attempt to work out whether there is some determinate number of kinds of mathematics by means of a scientific division.⁸ Then we shall also investigate the common species of all mathematics from some common point of focus,⁹ though without yet dealing with individual theorems.¹⁰ In each of the above we shall indicate¹¹ the subject-matter with which each kind and species of 15 mathematics is concerned and what each of them contributes to the totality, and we shall not omit a discussion of their relation to each other, what their kinship is and whence it comes, by what principles they are bound together, 20 and upon what more basic principles of the study they depend; also how one may discover them; also, of what use the subject is¹² and to how many good things it leads; also, that it is both choiceworthy in itself and also through the 10,1 sciences arising from it; also, that it converts¹³ the reasoning faculty¹⁴ to the whole of philosophy and to the whole science of things with true being that are intelligible. That is the sum of the things that lie before us to deal with in this 5 book. Let us start by taking up the first of the topics mentioned above.

Let the following postulates be assumed in common for the totality of the objects of mathematics: that their mode of existence is incorporeal and self-subsistent, 10 holding an intermediate rank between indivisible beings and those that are divided about bodies,¹⁵ both as forms and as reason-principles,¹⁶ being assigned the middle status between the simple and the divisible, being purer than the latter, more variegated than the former; that they make use of composition¹⁷ and division, but oversee the synthesized and the divided 15 without coming to be and eternally; they are inferior to intelligible beings but prior to natural ones; they are in beauty, in order, and in exactitude superior to things visible but inferior to the intelligible, and similarly they have an intermediate symmetry and compatibility; they have the power to transport¹⁸ 20 and lead over to the indivisible forms, since they are akin to them, and they lead away from corporeal concerns those who have become accustomed to them, converting them to the divine beings, as if on a ladder leading up to the heights.

11,1 But one must view the gift of incorporeal essence not as coming to them in a secondary way from only a single kind of being, but from all kinds whatsoever that inhere in true being and Intellect. For from all these there descends to the intermediate natures of the mathematical objects the mediating role between the causal principles and t...

Table des matières

• Cover
• Half-Title Page
• Series
• Title Page
• Contents
• Preface
• Conventions
• Abbreviations
• Introduction
• Iamblichus’ Projects and Genres
• Textual Emendations
• Translation
• Notes
• Bibliography
• English–Greek Glossary
• Greek–English Index
• Index of Passages
• Subject and Name Index
• Copyright