A Brief Guide to the Great Equations Read Online Free PDF

for another reason. Mocking it as a ‘mousetrap’ proof that lures readers in and then ‘springs’ a trap on them, Schopenhauer thought it logically true but overtly complicated, did not like the fact that not all its steps were intuitive (he much preferred proofs that appealed to intuition), and maintained that Euclid’s proof is a classical illustration of a misleading demonstration. Indeed, he saw Euclid’s proof of the Pythagorean theorem as emblematic of all that was
wrong
with the philosophy of his day, for it emphasized the triumph of sheer logic over insight and educated intuition. Hegel’s philosophical system, for Schopenhauer, was in effect no more than one huge conceptual mousetrap. 20
    Third, the Pythagorean theorem makes accessible the visceral thrill of discovery. Whenever we prove it, we can hardly be said to be ‘learning’ anything, for we learned the hypotenuse rule as schoolchildren. But as the proof proceeds – as we set the problem in a bigger context, and as the little pieces begin to snap together with an awesome inevitability – we seem to be taken out of the here and now to someplace else, a realm of truths far more ancient than we, a place we can reach with a little bit of effort no matter where we are. In that place, this particular right-angled triangle is nothing special; all are the same and we do not have to start the proof all over again to be certain of it. Something lies behind this particular triangle, of which it is but an instance. The experience is comforting, even thrilling, and you do not forget it. The proof arrives as the answer to a puzzle in a language that you did not have beforehand, a language that arrives in that instant yet which, paradoxically, you senseyou already possessed. Without that moment of insight, the Pythagorean theorem remains a rule handed down authoritatively, rather than a proof gained insightfully.
The Pythagorean Theorem in Plato’s Meno
    All three components of the magic of the Pythagorean theorem are evident in the earliest known, most celebrated, and most complexly described story of a journey to the Pythagorean theorem. That occurs in Plato’s dialogue
Meno
, written about 385 BC , or somewhat more than a century after Pythagoras and almost a century before Euclid’s
Elements
. It is the first extended illustration of the mathematical knowledge of ancient Greece that exists. In the
Meno
, Socrates coaxes a slave boy, ignorant of mathematics, to prove a particular instance of the theorem, one involving an isosceles right-angled triangle.
    The principal participants are Socrates and Meno, a handsome youth from Thessaly. Meno is impatient, balks at difficult ideas, and likes impressive-sounding answers – a teacher’s nightmare. He’s been pestering Socrates about how it is possible to learn virtue. Socrates finds it difficult to get Meno’s mind going; his name, appropriately, means ‘stand fast’ or ‘stay put.’ The word ‘education’ means literally ‘to lead out.’ Socrates cannot lead Meno much of anywhere.
    At one point, Meno throws up his hands and asks Socrates – in a famous query known as Meno’s paradox – how it is possible to learn anything at all. If you don’t know what you are looking for, you won’t be able to recognize it when you come across it – while if you do know you don’t bother to go looking for it. Meno is implying that it is fruitless even to try.
    The paradox arises, as philosophers say today, from the mistaken assumption that knowledge comes in disconnected bits and pieces. In reality, we humans notice that something is unknown thanks to the whole matrix of things that we know already. We can extend this matrix – and fill in and flesh out gaps and thin areas – byapplying what we know to find what we don’t, bringing everything else to bear on it, inevitably uncovering new holes and weaknesses in the process. Acquiring knowledge is not like putting things someone else gives us in a mental