Is God a Mathematician_ - Mario Livio [28]
Figure 12
The drama surrounding the palimpsest is only fitting for a document that gives us an unprecedented glimpse of the great geometer’s method.
The Method
When you read any book of Greek geometry, you cannot help but be impressed with the economy of style and the precision with which the theorems were stated and proved more than two millennia ago. What those books don’t normally do, however, is give you clear hints as to how those theorems were conceived in the first place. Archimedes’ exceptional document The Method partially fills in this intriguing gap—it reveals how Archimedes himself became convinced of the truth of certain theorems before he knew how to prove them. Here is part of what he wrote to the mathematician Eratosthenes of Cyrene (ca. 276–194 BC) in the introduction:
Figure 13
I will send you the proofs of these theorems in this book. Since, as I said, I know that you are diligent, an excellent teacher of philosophy, and greatly interested in any mathematical investigations that may come your way, I thought it might be appropriate to write down and set forth for you in this same book a certain special method, by means of which you will be enabled to recognize certain mathematical questions with the aid of mechanics [emphasis added]. I am convinced that this is no less useful for finding the proofs of these same theorems. For some things, which first became clear to me by the mechanical method, were afterwards proved geometrically, because their investigation by the said method does not furnish an actual demonstration. For it is easier to supply the proof when we have previously acquired, by the method, some knowledge of the questions than it is to find it without any previous knowledge.
Archimedes touches here on one of the most important points in scientific and mathematical research—it is often more difficult to discover what the important questions or theorems are than it is to find solutions to known questions or proofs to known theorems. So how did Archimedes discover some new theorems? Using his masterful understanding of mechanics, equilibrium, and the principles of the lever, he weighed in his mind solids or figures whose volumes or areas he was attempting to find against ones he already knew. After determining in this way the answer to the unknown area or volume, he found it much easier to prove geometrically the correctness of that answer. Consequently The Method starts with a number of statements on centers of gravity and only then proceeds to the geometrical propositions and their proofs.
Archimedes’ method is extraordinary in two respects. First, he has essentially introduced the concept of a thought experiment into rigorous research. The nineteenth century physicist Hans Christian Ørsted first dubbed this tool—an imaginary experiment conducted in lieu of a real one—Gedankenexperiment (in German: “an experiment conducted in the thought”). In physics, where this concept has been extremely fruitful, thought experiments are used either to provide insights prior to performing actual experiments or in cases where the real experiments cannot be carried out. Second, and more important, Archimedes freed mathematics from the somewhat artificial chains that Euclid and Plato had put on it. To these two individuals, there was one way, and one way only, to do mathematics. You had to start from the axioms and proceed by an inexorable sequence of logical steps, using well-prescribed tools. The free-spirited Archimedes, on the other hand, simply utilized every type of ammunition he could think of to formulate new problems and to solve them. He did not hesitate to explore and exploit the connections between the abstract mathematical objects (the Platonic forms) and physical reality (actual solids or flat objects) to advance his mathematics.
A final illustration that further solidifies Archimedes’ status as a magician is his anticipation of integral and differential calculus—a branch of mathematics formally developed by Newton (and independently by the German mathematician Leibniz)