Quote for Yesterday, 12-02-2010
I was finishing my reread of Neverness last night (well, almost finishing - I have yet to revisit the Alaloi) and this bit grabbed me.
"We create mathematics as surely as we create a symphony. We manipulate our axioms with logic as a composer arranges musical notes, and so the holy music of our theorems is born. And in a different sense we also discover mathematics: The ratio of a circle's circumference to the diameter remains the same for human minds and for aliens of the Cetus cloud of galaxies. All minds discover the same mathematics for that is the way the universe is. Creation and discovery; discovery and creation - in the end I believe they are the same. We create (or discover) undefined concepts such as point, line, set and betweenness. We do not seek to define these things because they are as basic as concepts can be. (And if we did not try to define them, we would make the mistake of The Euclid and say something like: A line is breadthless length. And then, using other words we would have to define the concept "breadthless" and "length." And so on, and so on, until all the words in our finite language were eventually used up, and we returned to the simple concept: A line is a line. Even a child, after all, knows what a line is.) From our basic concepts we make simple definitions of mathematical objects we believe to be interesting. We define "circle"; we create "circle"; we do this because circles are beautiful and interesting. But still we know nothing about circles. Ah, but some things are obviously true (or it is fun to treat them as if they were true), and so we create the axioms of mathematics. All right angles are congruent, parallel lines never intersect, parallel lines always intersect, there exists at least one infinite set - these are all axioms. And so we have lines and circles and axioms, and we must have rules to manipulate them. These rules are logic. By logic we prove our theorems. We may choose the natural logic where a statement is either true or not, or one of the quantum logics where a statement has a degree or probability of trueness. With logic we transmute our simple, obvious axioms into golden theorems of stunning power and beauty. With a few steps of logic we may prove that in hyperbolic geometry rectangles do not exist, or that the number of primes is infinite, or that aleph null is the smallest infinity that exists, or that... we may prove many wonderful things which are not obvious at all; we may do this if we are very clever and if we love the splendor of the number-storm as it rages and consumes us, and if we are filled with the holy fire of inspiration." - Neverness, David Zindell
Rather lengthy, but it reminded me how much I enjoy falling into the flow and pattern of the main character's thoughts sometimes. <3
"We create mathematics as surely as we create a symphony. We manipulate our axioms with logic as a composer arranges musical notes, and so the holy music of our theorems is born. And in a different sense we also discover mathematics: The ratio of a circle's circumference to the diameter remains the same for human minds and for aliens of the Cetus cloud of galaxies. All minds discover the same mathematics for that is the way the universe is. Creation and discovery; discovery and creation - in the end I believe they are the same. We create (or discover) undefined concepts such as point, line, set and betweenness. We do not seek to define these things because they are as basic as concepts can be. (And if we did not try to define them, we would make the mistake of The Euclid and say something like: A line is breadthless length. And then, using other words we would have to define the concept "breadthless" and "length." And so on, and so on, until all the words in our finite language were eventually used up, and we returned to the simple concept: A line is a line. Even a child, after all, knows what a line is.) From our basic concepts we make simple definitions of mathematical objects we believe to be interesting. We define "circle"; we create "circle"; we do this because circles are beautiful and interesting. But still we know nothing about circles. Ah, but some things are obviously true (or it is fun to treat them as if they were true), and so we create the axioms of mathematics. All right angles are congruent, parallel lines never intersect, parallel lines always intersect, there exists at least one infinite set - these are all axioms. And so we have lines and circles and axioms, and we must have rules to manipulate them. These rules are logic. By logic we prove our theorems. We may choose the natural logic where a statement is either true or not, or one of the quantum logics where a statement has a degree or probability of trueness. With logic we transmute our simple, obvious axioms into golden theorems of stunning power and beauty. With a few steps of logic we may prove that in hyperbolic geometry rectangles do not exist, or that the number of primes is infinite, or that aleph null is the smallest infinity that exists, or that... we may prove many wonderful things which are not obvious at all; we may do this if we are very clever and if we love the splendor of the number-storm as it rages and consumes us, and if we are filled with the holy fire of inspiration." - Neverness, David Zindell
Rather lengthy, but it reminded me how much I enjoy falling into the flow and pattern of the main character's thoughts sometimes. <3