I carried this problem around in my head basically the whole time. I would wake up with it first thing in the morning, I would be thinking about it all day, and I would be thinking about it when I went to sleep. Without distraction I would have the same thing going round and round in my mind.(Recalling the degree of focus and determination that eventually yielded the proof of Fermat's Last Theorem.)
The Professor never really seemed to care whether we figured out the right answer to a problem. He preferred our wild, desperate guesses to silence, and he was even more delighted when those guesses led to new problems that took us beyond the original one. He had a special feeling for what he called the "correct miscalculation," for he believed that mistakes were often as revealing as the right answers.
Shebna scraped the tablet clean and began drawing circles in the soft clay. "Suppose you had six figs and you ate two. How many would--""Four." Hezekiah answered before Shebna finished, and the tutor's thick black eyebrows rose in surprise."And suppose I had five figs. How many would we--""Nine.""Have you done this before?"Hezekiah thought the question was ridiculous. "I've eaten figs lots of times.
Further, the same Arguments which explode the Notion of Luck, may, on the other side, be useful in some Cases to establish a due comparison between Chance and Design: We may imagine Chance and Design to be, as it were, in Competition with each other, for the production of some sorts of Events, and many calculate what Probability there is, that those Events should be rather be owing to the one than to the other.
On a plaque attached to the NASA deep space probe we [human beings] are described in symbols for the benefit of any aliens who might meet the spacecraft as “bilaterly symmetrical, sexually differentiated bipeds, located on one of the outer spiral arms of the Milky Way, capable of recognising the prime numbers and moved by one extraordinary quality that lasts longer than all our other urges—curiosity.
He calculated the number of bricks in the wall, first in twos and then in tens and finally in sixteens. The numbers formed up and marched past his brain in terrified obedience. Division and multiplication were discovered. Algebra was invented and provided an interesting diversion for a minute or two. And then he felt the fog of numbers drift away, and looked up and saw the sparkling, distant mountains of calculus.
From the age of 13, I was attracted to physics and mathematics. My interest in these subjects derived mostly from popular science books that I read avidly. Early on I was fascinated by theoretical physics and determined to become a theoretical physicist. I had no real idea what that meant, but it seemed incredibly exciting to spend one's life attempting to find the secrets of the universe by using one's mind.
So what were your favorite subjects in school?""School?" He leaned back in his chair as though he needed the extra space to think about it. "Probably math. It always made sense. Unlike English, economics, and girls.""And exactly how do you plan on taking over the free world if you don't understand economics?""I'll hire advisers. I'll hire you, in fact.""Okay. Let me know when your army of junior high zombies is ready.
In this moment I’m reminded of Thor tutoring me math in the second grade. Thor was a math major in college, and in fact, Thor has an Erdős number of three. This means that Thor studied with someone who studied with someone who studied with Paul Erdős. Erdős was a brilliant mathematician who was as famous for his eccentric lifestyle as his mathematical theorems. I guess this give me an Erdős number of four.
Thor again tutored me in the seventh grade, when I was learning Algebra. Algebra was quite an experience for me. Thor spent countless hours with me, because it was very hard for me to grasp how to pull out a number from a letter. I thought it was an exercise in the absurd. Now I know how to get a number from a letter. The letter is K, as in K Street—where all the lobbyists are in Washington, and the number is billions of dollars.
Primary causes are unknown to us; but are subject to simple and constant laws, which may be discovered by observation, the study of them being the object of natural philosophy.Heat, like gravity, penetrates every substance of the universe, its rays occupy all parts of space. The object of our work is to set forth the mathematical laws which this element obeys. The theory of heat will hereafter form one of the most important branches of general physics.
Philosophy [nature] is written in that great book which ever is before our eyes -- I mean the universe -- but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.
Like Molière’s M. Jourdain, who spoke prose all his life without knowing it, mathematicians have been reasoning for at least two millennia without being aware of all the principles underlying what they were doing. The real nature of the tools of their craft has become evident only within recent times A renaissance of logical studies in modern times begins with the publication in 1847 of George Boole’s 'The Mathematical Analysis of Logic'.
Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.
The integrals which we have obtained are not only general expressions which satisfy the differential equation, they represent in the most distinct manner the natural effect which is the object of the phenomenon... when this condition is fulfilled, the integral is, properly speaking, the equation of the phenomenon; it expresses clearly the character and progress of it, in the same manner as the finite equation of a line or curved surface makes known all the properties of those forms.
