Author: Stephen Hawking
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Then, someone (whose name we don’t know) hit upon a deep insight. Suppose that the square root of 2 can be expressed as the ratio of two whole numbers and that these two whole numbers share no common divisors except 1, their common unit. Call these whole numbers p and q with the property that the square P erected on a side of length p is exactly twice the area of a square Q erected on a side of length q. Now if P contains twice the number of units than Q, then P must contain an even number of units!
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Eventually, the argument reaches its climax. It began by supposing that the sides p and q had no divisor in common except 1 and ended up with a contradiction: that they shared the divisor 2!
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During the height of the Newtonian era, philosophers, such as Immanuel Kant, never doubted the truth of Euclid’s parallel postulate. They merely inquired about the nature of its truth. Was the parallel postulate necessarily true of the cosmos or only contingently true? Of course, since the advent of the Einsteinian revolution, we know that the parallel postulate isn’t true at all about the cosmos. The Einsteinian space-time cosmos that we inhabit is curved. Euclidean geometry and Newtonian physics are only approximations.
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This should come as no surprise to us moderns who have the benefit of 2,500 years of hindsight and know that the Pythagorean theorem is false in non-Euclidean geometries.
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If the proof of the irrationality of the square root gave us the first crisis in mathematics, it also gave us the first example of the form of argument known since ancient times as reductio ad absurdum, reduction to absurdity.
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Proving the infinitude of prime numbers is stunningly simple. Suppose there is a largest prime number P. Multiply together all of the prime numbers up to an including P. Now add 1. The result is not divisible by P, nor is it divisible by any of the prime numbers less than P, because P and all of the prime numbers less than it evenly divides into their product before 1 is added to it. Supposing that there is a largest prime number leads to a contradiction. Reductio ad absurdum!
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What is a perfect number? A perfect number is the sum of its integer divisors greater than or equal to 1 but less than itself, what are called its proper divisors.
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If (2n − 1) is a prime number then 2n−1 x (2n − 1) is a perfect number and that even perfect numbers must have this form.
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Whereas the Pythagoreans had tried to arithmetize geometry and failed, when Eudoxus tried to geometrize arithmetic he succeeded!
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And when the lines containing the angle are straight, the angle is called rectilineal.
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A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another;
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Thus Bhāskara (born 1114 A.D.) simply draws four right-angled triangles equal to the original one inwards, one on each side of the square on the hypotenuse, and says “see!”,
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The formula of Pythagoras amounts, if m be an odd number, to the sides of the right-angled triangle being m, .
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Cantor, taking up an idea of Röth (Geschichte der abendländischen Philosophie, II. 527), gives the following as a possible explanation of the way in which Pythagoras arrived at his formula.
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Pythagoras must have used some method which would produce his rule only; and further it would be some less recondite method, suggested by direct observation rather than by argument from general principles.