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Bacon s adaptation of traditional methods was to begin with observation of discrete facts, and then from observed similarities in many separate events, to arrive at generalizations—and thus he formulated his inductive method. This method ofinduction enabled him to derive rational generalizations from his observations…
The English words "man," "fish," "star," "Hower," ·run," "grow," all denote classes of objects or events which may be recognized as members of their class by very simple attributes, abstracted from the total complexity of the things themselves. Abstraction is thus almost a necessity for communication, since it enables…
Algorithms for finding prime numbers date back at least as far as ancient Greece, where mathematicians used a straightforward approach known as the Sieve of Erastothenes. The Sieve of Erastothenes works as follows: To find all the primes less than n, begin by writing down all the numbers from 1 to n in sequence. Then…
Let’s consider Arabic numerals as compared with Roman numerals. With a positional notation system, such as the one created by Arabic numerals, it’s easier to perform multiplication and division; if you’re competing in a multiplication contest, Arabic numerals provide you with an advantage. But I wouldn’t say that…
The idea that space and time can be curved or warped is fairly recent. For more than 2,000 years the axioms of Euclidean geometry were considered to be selfevident. As those of you who were forced to learn geometry at school may remember, one of the consequences of these axioms is that the angles ot a triangle add up…
As an example of the power of the Anthropic Principle, consider the number of directions in space. It is a matter of common experience that we live in three-dimensional space. That is to say, we can represent the position of a point in space by three numbers. For example, latitude. longitude and height above sea…
Whenever we communicate, rules everywhere restrict our freedom to choose the next letter and the next pineapple.I Because these rules render certain patterns more likely and certain patterns almost impossible, languages like English come well short of complete uncertainty and maximal information: the sequence “th” has…
"If video games had been around in 350 BC, Euclid would have made a video game," Devlin told me. The thirteen books of Euclid's Elements would have been the supplemental material, a PDF file that you could read if you wanted to. "People think I'm joking—I absolutely mean that. Euclid would not have written a textbook…
A game, toy or article of manufacture includes a set of rules and at least one model. The model has multiple movable parts, where under the rules of play, the model begins in an assembled configuration. As the model loses points under the rules of play, at least some of the parts are removed from the model, or…
“Do not squander time,” said Benjamin Franklin, “for that is the stuff life is made of.” Our consciousness, even more than it is posted in space, unrolls in time. I can imagine abolishing space from my awareness—if, say, I were floating in a sensory deprivation tank or became blind and paralyzed—while still continuing…
“In One Dimensions, did not a moving Point produce a Line with two terminal points? In two Dimensions, did not a moving Line produce a Square wit four terminal points? In Three Dimensions, did not a moving Square produce - did not the eyes of mine behold it - that blessed being, a Cube, with eight terminal points? And…
Imagine a vast sheet of paper on which straight Lines, Triangles, Squares, Pentagons, Hexagons, and other figures, instead of remaining fixed in their places, move freely about, on or in the surface, but without the power of rising above or sinking below it, very much like shadows—only hard with luminous edges—and you…
What and where is Flatland? A Square gives us several interesting answers, many of th contradictory. We know that it’s flat, big (but how big?), and very thin, the most important question of all is “how thin?” A lot depends on the answer… A Square himself eliminates the version that’s easiest for three-dimensional…
We humans haven't changed much since our remote ancestors roamed the African savannah. Our brains evolved to cope with the human-scale environment. So it is surely remarkable that we can make sense of phenomena that confound everyday intuition: in particular, the minuscule atoms we're made of, and the vast cosmos that…
The problem is that the laws of addition and multiplication you are using hold for natural numbers, but infinity is not a natural number, so these laws do not apply. If they did, you could use a similar argument that multiplying anything by infinity, no matter how small, gives infinity, thus ∞×0=∞. More sophisticated…
Name any event in history and you will find a star out there whose light gives you a glimpse of something happening during the year of that event. Provided you are not a very young child, somewhere up in the night sky you can find your personal birth star. Its light is a thermonuclear glow that heralds the year of…
The spectrum depends upon light of different colours being slowed by different amounts: the refractive index of a given substance, say glass or water, is greater for blue light than for red. You could think of blue light as being a slower swimmer than red, getting tangled up in the undergrowth of atoms in glass or…
‘The achievement of the Kaminari-zoku implies that there are other spacetimes. Certainly other regions of the Universe beyond our causal horizon. If rational actors have evolved in them, they will have broken their Planck locks – or worse, evolved natively in an environment with no restrictions on computational…
It is time now to try to describe what a superstring really is. Here I run into the same difficulty which the geometer Euclid encountered 2,200 years ago. Euclid was trying to convey to his readers his idea of a geometrical point. For this purpose he gave his famous definition of a point: "A point is that which has no…
no matter what the odds are at one point in spacetime probability materialize