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Were a man, such as Adam, created in the full vigor of understanding, without experience, he would never be able to infer motion in the second ball from the motion and impulse of the first. It is not anything that reason sees in the cause, which make us infer the effect. Such an inference, were it possible, would…
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 laws of thermodynamics decree that each quantity of en^rgy has a characteristic quaUty called entropy associated with it. The entropy measures the degree of disorder associated with the energy. Energy must always flow in such a direction that the entropy increases. Thus, we can arrange the different forms of…
Humanity is fortunate in having such a variety of energy resources at its disposal. In the very long run we shall need energy that is unpolluting; we shall have sunlight. In the fairly long run we shall need energy that is inexhaustible and moderately clean; we shall have deuterium. In the short run we shall need…
As to your first query, it seems to me that if the matter of our sun and planets and all the matter of the universe were evenly scattered throughout all the heavens, and every particle had an innate gravity toward all the rest, and the whole space throughout which this matter was scattered was but finite, the matter…
Some mathematicians are birds, others are frogs. Birds fly high in the air and survey broad vistas of mathematics out to the far horizon. They delight in concepts that unify our thinking and bring together diverse problems from different parts of the landscape. Frogs live in the mud below and see only the flowers that…
At any moment, we are either in a divergent phase of life, or in a convergent phase. During a divergent period, we’re usually focused on broadening our inputs. We go to school, take on internships, travel, try a career switch, and so on, exploring new paths. During a convergent time, we take what we’ve learned thus…
When dealing with a sequence of length N, the elements of which we wish to distinguish by subscript, the next vexing question is what subscript value to assign to its starting element. Adhering to convention a) yields, when starting with subscript 1, the subscript range 1 ≤ i < N 1; starting with 0, however, gives the…
This “algebraic” flexibility of a word encapsulates the essence of something while leaving unnecessary concretes out. A photo doesn’t and can’t. Further, a word offers enormous flexibility in terms of input/output. It can be spoken, thought, gestured (as in sign language), written, grammatically combined with other…
What are observations? Some philosophers have taken them to be sensory events: the occurrence of smells, feels, noises, color patches. This way lies frustration. What we ordinarily notice and testify to are rather the objects and events out in the world. It is to these that our very language is geared, because…
The world with its quarks and chromosomes, its distant lands and spiral nebulae, is like a vast computer in a black box, forever sealed except for its input and output registers. These we directly observe, and in the light of them we speculate on the structure of the machine, the universe. Thus it is that we think up…
Human beings live in ideas. That they were condemning their descendants to death and extinction did not occur to them, or if it did they repressed the thought, ignored it, and forged on anyway. They did not care as much about their descendants as they did about their ideas, their enthusiasms. Is this narcissism?…
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…
he one remaining area that reHgion can now lay claim to is the origin of the universe, but even here science is making progress and should soon provide a definitive answer to how the universe began. I published a book that asked if God created the universe, and that caused something of a stir. People got upset that a…
AI researchers have typically tried to get around the issues posed by by Montezuma’s Revenge and Pitfall! by instructing reinforcement-learning algorithms to explore randomly at times, while adding rewards for exploration—what’s known as “intrinsic motivation.” But the Uber researchers believe this fails to capture an…
The worm drives helically through the wood And does not know the dust left in the bore Once made the table integral and good; And suddenly the crystal hits the floor. Electrons find their paths in subtle ways, A massless eddy in a trail of smoke; The names of lovers, light of other days Perhaps you will not miss them…
There’s a continuing struggle between complexity and robustness in both evolution and human design. A kind of survival imperative, whether in biology, engineering, or business requires that simple, fragile systems become more robust. But the mechanisms to increase robustness will in turn make the system considerably…
Back in the early days of computers, one of the more popular methods of generating a sequence of random numbers was to employ the following scheme: 1. Choose a starting number between 0 and 1. 2. Multiply the starting number by 4 ("stretch" it). Subtract 4 times the square of the starting number from the quantity…