121–140 of 262
Lojban is a predicate language, with no distinct nouns, verbs, or adjectives. What are the linguistic (communicative) properties of such a system? The answer has been partially explored through symbolic logic. But do people, when thinking linguistically, mimic in any way the processes of formal logic? What effects…
The proof is elegant and the result profound. Still, it is typical mathematics; so, it’s a good example to reflect upon. In doing so, we will begin to see the elements of the mathematical image, the standard conception of what mathematics is. Let’s begin a list of some commonly accepted aspects. By ‘commonly accepted’…
Theorem: There are infinitely many prime numbers. Proof: Suppose, contrary to the theorem, that there is only a finite number of primes. Thus, there will be a largest which we can call p. Now define a number n as 1 plus the product of all the primes: n = (2 X 3 X 5 X 7 X 11 X...X p) 1 Is n itself prime or composite?…
The simplest definition I can give for Big-O notation is this: Big-O notation is a relative representation of the complexity of an algorithm. There are some important and deliberately chosen words in that sentence: relative: you can only compare apples to apples. You can't compare an algorithm to do arithmetic…
actuarial escape velocity is defined as occurring when a year of medical research adds more than a year’s worth of longevity to the total population. Nothing even close to this has ever been achieved, and emerging signs of an asymptotic curve in progress suggest this velocity may never
More and more ordinary people are thrust into a winner-take-all economy. It is a 21st century reprise of the Horatio Alger stories from the 19th century. A token few will find success on Kickstarter or YouTube, while overall wealth is ever more concentrated and social mobility rots. Social media sharers can make all…
As we move from the "low" level to the "high", say from the domain of machine code all the way up to, for instance, a rich, expressive Ruby DSL, the question arises, as it does for all language: have we acquired a surplus of content that cannot be simply reduced to core rules? At this point things get less…
Rational mechanics soon became a vast and profound science. The true laws of the collision of bodies, respecting which Descartes was deceived, were at length known. Huyghens discovered the laws of circular motions; and at the same time he gives a method of determining the radius of curvature for every point of a given…
As the DM, the tools of your trade are dice — platonic solid-shaped or just about any other sort. The random numbers you generate by rblling dice determine the results based on the probabilities determined herein or those you have set forth on your own. In case you are not familiar with probability curves, there are…
One property of Bayes’s theorem, in fact, is that our beliefs should converge toward one another—and toward the truth—as we are presented with more evidence over time. In figure 8-8, I’ve worked out an example wherein three investors are trying to determine whether they are in a bull market or a bear market. They…
Bayes’s much more famous work, “An Essay toward Solving a Problem in the Doctrine of Chances,”24 was not published until after his death, when it was brought to the Royal Society’s attention in 1763 by a friend of his named Richard Price. It concerned how we formulate probabilistic beliefs about the world when we…
We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to…
I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our "creations," are simply the notes of our observations. * * * Let us suppose that I am giving a lecture on some system of geometry, such…
Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. 'Immortality' may be a silly word, but probably a mathematician has the best chance of whatever it may mean." [* * *] A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are…
Base 26 is one of two fairly natural ways of representing numbers as text using a 26-letter alphabet. The number of interest is expressed numerically in base 26, and then the 26 different base-26 digits are identified with letters as 0=A, 1=B, 2=C, ... 25=Z. Here are the first 100 digits of pi expressed in this way…
This flat conceptioning is manifest right up to the present in such every¬ day expressions as "the wide, wide world" and "the four comers of the Earth." As mentioned before, "up" and "down" are the parallel perpendicu¬ lars impinging upon this flat-out world. Only a flat-out world could have a Heaven to which to…
This meant, then, that the Greeks, in attempting to communicate their mathematical conceptioning, defined the circle as "an area bound by a closed line of equal radius from one point," the triangle as "an area bo)ound by a closed line of three angles, three edges, and three vertices." The Greeks talked only of the…
Trigonometry had to start with sea-people. It is the conclusion of British, German, and U.S.A. navies' experts that celestial—offshore—navigation be¬ gan with the South Pacific's island peoples. Much has been published on this subject. What is not as well pubHshed is the fact that the navigators on all those islands…
Environment-controlling buildings gain or lose their energy as "heat o cool" only through their containing surfaces. Spheres contain the most vololume with the least surface—i.e., have the least possible surface-to-volume ratio. Every time we double the diameter of a spherical structure, we in¬ crease its contained…
This simple equation, which states that the quantity 0.999, followed by an infinite string of nines, is equivalent to one, is the favorite of mathematician Steven Strogatz of Cornell University. "I love how simple it is — everyone understands what it says — yet how provocative it is," Strogatz said. "Many people don't…