Greek Mathematical Thought and the Origin of Algebra

Rating:
Summary: On 'arithmos' and 'general magnitude'
Review: It's hard to say something about this wonderful book without sounding pompous. Generally, I try to avoid terms like 'classic' and 'essential', but they keep coming to mind.

The original was written in the mid 1930s. As Klein writes in this version's preface, "This study was originally written and published in Germany during rather turbulent times."

The late Jacob Klein spent his post war years teaching Platonic philosophy at St. John's College. There, he was known as something of a lovable elitist. Professors tell a story about Klein being partial to the number 12. He claimed that there were an exclusive 12 philosophers, 7 Greek and 5 German. The word got out and he soon received a letter from 4,000 American philosophers begging to differ with his opinion.

While many might call this book 'philosophy of math,' I doubt Dr. Klein would agree. The book is without much in the way of serious math. It is more concerned with the symbols of math and how they are used. Quoting from the first paragraph of the introduction:

"Creation of a formal mathematical language was of decisive significance for the constitution of modern mathematical physics. If the mathematical presentation is regarded as a mere device, preferred only because the insights of natural science can be expressed by "symbols" in the simplest and most exact manner possible, the meaning of the symbolism as well as of the special methods of the physical disciplines in general will be misunderstood. True, in the seventeenth and eighteenth century it was still possible to' express and communicate discoveries concerning the "natural" relations of objects in non mathematical terms, yet even then -or, rather, particularly then - it was precisely the mathematical form, the mos geometricus, which secured their dependability and trustworthiness. After three centuries of intensive development, it has finally become impossible to separate the content of mathematical physics from its form. The fact that elementary presentations of physical science which are to a certain degree nonmathematical and appear quite free of presuppositions in their derivations of fundamental concepts (having recourse, throughout, to immediate "Intuition") are still in vogue should not deceive us about the fact that it is impossible, and has always been impossible, to grasp the meaning of what we nowadays call physics independently of its mathematical form. Thence arise the insurmountable difficulties in which discussions of modern physical theories become entangled as soon as physicist or nonphysicists attempt to disregard the mathematical apparatus and to present the results of scientific research in popular form. The intimate connection of the formal mathematical language with the content of mathematical physics stems from the special kind of conceptualization which is a concomitant of modern science and which was of fundamental importance in its formation."

While this iconoclastic promise is a bit difficult to extract from the somewhat professional philosophic prose, there is a wonderful essay in "Biographies of Scientific Objects," edited by Lorraine Daston that serves as an excellent commentary. The essay called "Mathematical Entities in Scientific Discourse" credits Klein with a new perspective from which to interpret the transition of ancient and medieval traditions to the new mathematical physics of the seventeenth century. His was the seemingly narrow-but only deceptively so-perspective of the ancient concept of "arithmos", compared to the concept of number in its modern, symbolic sense. In Klein's own words, the underlying thematics of the book never loses sight of the "general transformation, closely connected with the symbolic understanding of number, of the scientific consciousness of later centuries."

Although the Greek conceptualization of mathematical objects was indeed based upon the notion of arithmos, this notion should not be thought of as a concept of "general magnitude." It never means anything other than "a definite number of definite objects," or an "assemblage of things counted". Likewise, geometric figures and curves, commensurable and incommensurable magnitudes, ratios, have their own special ontology which directs mathematical inquiry and its methods.

In contradistinction to Greek parlance, "general magnitude," according to Klein, is clearly a modern concept. Proving this case is the project of both books.

I think you will find reading this material an interesting journey.

Rating:
Summary: On 'arithmos' and 'general magnitude'
Review: It's hard to say something about this wonderful book without sounding pompous. Generally, I try to avoid terms like 'classic' and 'essential', but they keep coming to mind.

The original was written in the mid 1930s. As Klein writes in this version's preface, "This study was originally written and published in Germany during rather turbulent times."

The late Jacob Klein spent his post war years teaching Platonic philosophy at St. John's College. There, he was known as something of a lovable elitist. Professors tell a story about Klein being partial to the number 12. He claimed that there were an exclusive 12 philosophers, 7 Greek and 5 German. The word got out and he soon received a letter from 4,000 American philosophers begging to differ with his opinion.

While many might call this book 'philosophy of math,' I doubt Dr. Klein would agree. The book is without much in the way of serious math. It is more concerned with the symbols of math and how they are used. Quoting from the first paragraph of the introduction:

"Creation of a formal mathematical language was of decisive significance for the constitution of modern mathematical physics. If the mathematical presentation is regarded as a mere device, preferred only because the insights of natural science can be expressed by "symbols" in the simplest and most exact manner possible, the meaning of the symbolism as well as of the special methods of the physical disciplines in general will be misunderstood. True, in the seventeenth and eighteenth century it was still possible to' express and communicate discoveries concerning the "natural" relations of objects in non mathematical terms, yet even then -or, rather, particularly then - it was precisely the mathematical form, the mos geometricus, which secured their dependability and trustworthiness. After three centuries of intensive development, it has finally become impossible to separate the content of mathematical physics from its form. The fact that elementary presentations of physical science which are to a certain degree nonmathematical and appear quite free of presuppositions in their derivations of fundamental concepts (having recourse, throughout, to immediate "Intuition") are still in vogue should not deceive us about the fact that it is impossible, and has always been impossible, to grasp the meaning of what we nowadays call physics independently of its mathematical form. Thence arise the insurmountable difficulties in which discussions of modern physical theories become entangled as soon as physicist or nonphysicists attempt to disregard the mathematical apparatus and to present the results of scientific research in popular form. The intimate connection of the formal mathematical language with the content of mathematical physics stems from the special kind of conceptualization which is a concomitant of modern science and which was of fundamental importance in its formation."

While this iconoclastic promise is a bit difficult to extract from the somewhat professional philosophic prose, there is a wonderful essay in "Biographies of Scientific Objects," edited by Lorraine Daston that serves as an excellent commentary. The essay called "Mathematical Entities in Scientific Discourse" credits Klein with a new perspective from which to interpret the transition of ancient and medieval traditions to the new mathematical physics of the seventeenth century. His was the seemingly narrow-but only deceptively so-perspective of the ancient concept of "arithmos", compared to the concept of number in its modern, symbolic sense. In Klein's own words, the underlying thematics of the book never loses sight of the "general transformation, closely connected with the symbolic understanding of number, of the scientific consciousness of later centuries."

Although the Greek conceptualization of mathematical objects was indeed based upon the notion of arithmos, this notion should not be thought of as a concept of "general magnitude." It never means anything other than "a definite number of definite objects," or an "assemblage of things counted". Likewise, geometric figures and curves, commensurable and incommensurable magnitudes, ratios, have their own special ontology which directs mathematical inquiry and its methods.

In contradistinction to Greek parlance, "general magnitude," according to Klein, is clearly a modern concept. Proving this case is the project of both books.

I think you will find reading this material an interesting journey.

Rating:
Summary: Klein's work is a masterpiece of philosophical exegesis.
Review: Klein's work examines the generally unsuspected foundations of modern algebraic mathematics. He charts the development of a new kind of intentionality which lies at the heart of modern mathematical practice, with an explicit affirmation that this mode of intentionality is exemplary for all of modern thought. Beginning from the classical foundations of mathematics, he follows the subject carefully through every turn of ideation until he has completed his thesis. On the basis of this thorough-going evaluation and exegesis of mathematical thought, he identifies Francois Viete as the true founder of this modern symbolic intentionality. But he does not rest with this, proceeding to show how Descartes, Stevin, and Wallis each draw out of this foundation conclusions which are familiar to the modern thinker. This reader knows of no other work of this kind that has so deeply penetrated the foundations of what we call modernity.