God Created the Integers: The Mathematical Breakthroughs That Changed History

5 min read

Author: Stephen Hawking

  • Kurt Gödel proved a theorem troubling to many philosophers, as well as anyone else believing in absolute truth: that in any sufficiently complex logical system (such as arithmetic) there must exist statements that can neither be proven nor disproven.

  • Over a millennium later, Frenchman René Descartes united the two fields: geometry and algebra, with his creation of analytic geometry.

  • Noah Webster, the great nineteenth-century lexicographer who gave his name to the leading American dictionary.

  • 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.

  • The Einsteinian space-time cosmos that we inhabit is curved. Euclidean geometry and Newtonian physics are only approximations.

  • 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,

  • 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.

  • Greeks speculated that there were not only an infinitude of prime numbers but also an infinitude of twin primes. But they weren’t able to prove that, nor has any mathematician since. Neither has any mathematician been able to disprove the existence of an odd 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.

  • 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.

  • Euclid posed the problem as follows: consider 4 magnitudes of length—a, b, c, and d. How can one determine whether the ratio of a to b is greater than, less than, or equal to the ratio of c to d?

  • The multiple he chose was the product of b and d. Multiplying the ratio of a to b by the product of b and d, gave the ratio of the product of a, b and d to b, or the area of the rectangle with sides of length a and d. Similarly, multiplying the ratio of c to d by the product of b and d gave the ratio of the product of c, b, and d to d, or the area of a rectangle with sides of c and

  • The multiple he chose was the product of b and d. Multiplying the ratio of a to b by the product of b and d, gave the ratio of the product of a, b and d to b, or the area of the rectangle with sides of length a and d. Similarly, multiplying the ratio of c to d by the product of b and d gave the ratio of the product of c, b, and d to d, or the

  • The multiple he chose was the product of b and d. Multiplying the ratio of a to b by the product of b and d, gave the ratio of the product of a, b and d to b, or the area of the rectangle with sides of length a and d. Similarly, multiplying the ratio of c to d by the product of b and d gave the ratio of the product of c, b, and d to d, or the area of a rectangle with sides of c and b.

  • Now the Egyptians must certainly be credited from a period at least as far back as 2000 B.C. with the knowledge that 42 + 32 = 52.

  • How then did Pythagoras discover the general theorem? Observing that 3, 4, 5 was a right-angled triangle, while 32 + 42 = 52, he was probably led to consider whether a similar relation was true of the sides of right-angled triangles other than the particular one.

  • The net results then of Bürks papers and of the criticisms to which they have given rise appear to be these. (1) It must be admitted that Indian geometry had reached the stage at which we find it in Āpastamba quite independently of Greek influence. But (2) the old Indian geometry was purely empirical and practical, far removed from abstractions such as the irrational. The Indians had indeed, by-trial in particular cases, persuaded themselves of the truth of the Pythagorean theorem and enunciated it in all its generality; but they had not established it by scientific proof.

  • According to the ancient Roman biographer Plutarch, it is in connection with this story that Archimedes uttered his famous remark, “Give me a place to stand on, and I will move the earth.”

  • Archimedes’s mathematical works fall into three groups: 1. Those that prove theorems concerning areas and solids bounded by curves and surfaces. These include On the Sphere and the and Circle, On the Measurement of the Circle, and The Method. 2. Works that geometrically analyze problems in statics and hydrostatics. 3. Miscellaneous works, especially ones that emphasize counting, such as The Sand Reckoner.

  • It is the first proposition that holds the most interest for us. It states that the area of a circle is equal to the area of a right triangle in which one of the sides about the right angle is equal to the radius and the other is equal to the circumference.

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