does anyone know: when did mathematics and philosophy split, btw?
Angela _________
This is a very complicated and difficult question. In a sense there is no split and never will be, because mathematics is a form of philosophy. In other words you can express a philosophical idea in mathematical terms. And, at its most abstract, mathematics requires philosophical judgments as to the nature of mathematics. For example, what is mathematics, what is number, what is a point, what is any mathematically defined entity? It really doesn't do to just repeat some definition or axiom, because the formation of these is also in question.
But historically, what happened as far as I can tell is that mathematics became so intensely difficult and comprehensive in scope that it became almost impossible to master enough of either philosophy or mathematics to put the two together--that is for a single mind. I would guess that point was closely approached some time around the turn of the 19th Century. And, contemporary with these developments was Kant's division of knowledge, which was a sort of formal announcement of the divorce, but with joint custody for the children.
The late 18th and early 19th C. was the period when there was an major shift in mathematics to reform the calculus methods of Newton and Leibniz. Before this era, calculus was practiced and taught as a complicated and messy form of geometry. Roughly, the idea of a limit for example was described as the decreasing geometric area between an arc and a straight line--a tangent to a curve. Under the reform movements of the late 18th C the idea of a limit changed over to arithmetic forms by exploring and developing infinite series as sums and ratios.
But right away the use of infinite series caused a philosophical stir. There is no such thing as infinity--that is to say, you can't write an infinite collection down. How can you sum an infinite series, if you never get to the end of it to do the sum? If some finite calculation is supposed to reach a limit on an infinite collection, how does that work, exactly? And, then there are are many questions about the nature of numbers, just below the surface of all these questions.
Under all of that is the question of the relationship between the ideal and the concrete. The practice of mathematics is concrete--you write it down. But this symbolic practice hardly squares with the idealized realm represented. Mathematics always lurks just at the boundary between these realms, so there are entire schools of thought that fall on one side or the other.
On the thread on quarks, there was a discussion on what does "up" mean. Here we go, lurking in the boundary region. The sci-math types will give you a definition and consider the issue closed.
Up is spin orientation. But this sort of definition is already many, many layers into a symbolism that boggles the mind at every turn. Elementary particles spin and this idea explains their electro-magnetic properties. But what is spinning? Well, electric charge spins so it can follow the rules of motion in a magnetic field (which are cock-eyed). But if it is a spinning particle how come it defracts like a wave? Enter the particle-wave duality. Blah, blah, blah. See, it just gets worse and worse. At some point, you are babbling in Jabberwocky. I used to drive my son crazy by claiming that electrons don't exist (he was a chemistry major).
For a history of the development of calculus, see Carl Boyer, _The History of the Calculus_, Dover, NY, 1949. This is a really good book because it covers the detailed sequence of ideas and mathematicians who set the stage for the current textbook courses in calculus. But it doesn't complete the story. It leaves off exactly were the real controversies and issues of this century begin with Cantor. Cantor started off by analyzing trigonometric series. His and Dedekind work started to unearth all the philosophical implications that were hidden in the elaborate techniques just developed to make calculus such a useful application in physics and engineering.
And, it's deja vu all over again.
Charles, just asked were Archimedes and Euclid philosophers? And the answer is yes.
Take a look at the very closing sections to Euclid's books. These sections contain a long development of how to construct the Platonic solids (rendering the ideal concrete!). Formally, these were the most difficult constructions and required a lot of the hardware from the rest of the books to accomplish. So they represented the philosophical pinnacle of the collected work. These constructions depend heavily on dividing the circle which immediately brings up all sorts of philosophical issues over the nature of numbers.
Archimedes came very close to inventing the analytical concepts (concrete, geometric methods) to deal with infinity and limits. It was his work on conic sections (areas, volumes) that laid out where Kepler, Cavalieri, Fermat, Leibniz and Newton began. The technique is a geometric version of the philosophical method of exhaustion--arguments ad infinitum. Boyer lays this connection out in the chapter, 'Conceptions in Antiquity'.
I should be clear and say that Boyer considers the separation between mathematics and philosophy to have occurred in antiquity under the Greeks. He considers Euclid and Archimedes to be mathematicians in the modern sense of experts in a technical field.
But I don't go along with that interpretation. There was a separation between schools so that Athens had a more comprehensive philosophical bend while Alexandria had a more specialist and technical bend. In my eccentric version of history, I think of Athens as the snotty know-it-alls with no hands-on work--all talk and politics. Where as, in Alexandria it was all about technicans--less talk and more work, willing to get down and make something out of ideas. Maybe it was just that in Alexandria, political discourse was a dangerous occupation since the place was littered with fanatics from all over the ancient world.
Chuck Grimes