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Systems science is best seen as a science that consists of nested sub-sciences. It is presented most compactly using the notation of set theory. Let A represent a science of description. Let B represent a science of design. Let C represent a science of complexity. Let D represent a science of action (praxiology). Let…
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…
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…
The most singular, as well as the most excellent, of all his works, is that which is now the least read, and which is at the same time the most useful; I mean his "Novum Scientiarum Organum." This is the scaffold by means of which the edifice of the new philosophy has been reared; so that when the building was…
[Decimal currency is desirable because] by that means all calculations of interest, exchange, insurance, and the like are rendered much more simple and accurate, and, of course, more within the power of the great mass of people. Whenever such things require much labor, time, and reflection, the greater number who do…
Language is simply alive, like an organism. We all tell each other this, in fact, when we speak of living languages, and I think we mean something more than an abstract metaphor. We mean alive. Words are the cells of language, moving the great body, on legs. Language grows and evolves, leaving fossils behind. The…
Experiments may be of two kinds: experiments of simple fact, and experiments of quantity. ...[In the latter] the conditions will ... vary, not in quality, but quantity, and the effect will also vary in quantity, so that the result of quantitative induction is also to arrive at some mathematical expression involving…
I was just going to say, when I was interrupted, that one of the many ways of classifying minds is under the heads of arithmetical and algebraical intellects. All economical and practical wisdom is an extension or variation of the following arithmetical formula: 2 2=4. Every philosophical proposition has the more…
A casual glance at crystals may lead to the idea that they were pure sports of nature, but this is simply an elegant way of declaring one's ignorance. With a thoughtful examination of them, we discover laws of arrangement. With the help of these, calculation portrays and links up the observed results. How variable and…
Any experiment may be regarded as forming an individual of a 'population' of experiments which might be performed under the same conditions. A series of experiments is a sample drawn from this population. Now any series of experiments is only of value in so far as it enables us to form a judgment as to the statistical…
In the three-dimensional space in which we live, elementary particles are designated as fermions and bosons, depending on their spin. We associate with each variety of elementary particle a quantum number, which gives the value of its spin. This number can be an integer (0,1, 2,... ) or a half integer (1/2, 3/2…
There is a strange disparity between the sciences of inert matter and those of life. Astronomy, mechanics, and physics are based on concepts which can be expressed, tersely and elegantly, in mathematical language. They have built up a universe as harmonious as the monuments of ancient Greece. They weave about it a…
Among the multitude of animals which scamper, fly, burrow and swim around us, man is the only one who is not locked into his environment. His imagination, his reason, his emotional subtlety and toughness, make it possible for him not to accept the environment, but to change it. And that series of inventions, by which…
230(231-1) ... is the greatest perfect number known at present, and probably the greatest that ever will be discovered; for; as they are merely curious without being useful, it is not likely that any person will attempt to find a number beyond it.
There is another approach to the extraterrestrial hypothesis of UFO origins. This assessment depends on a large number of factors about which we know little, and a few about which we know literally nothing. I want to make some crude numerical estimate of the probability that we are frequently visited by…
On the title page of this chapter is shown the message. It is etched on a 6-inch by 9-inch gold-anodized aluminum plate, attached to the antenna support struts of Pioneer 10. The expected erosion rate in interstellar space is sufficiently small that this message should remain intact for hundreds of millions of years…
Science and technology multiply around us. To an increasing extent they dictate the languages in which we speak and think. Either we use those languages, or we remain mute.
[Learning is] the actual process of broadening yourself, of knowing there's a little extra facet of the universe you know about and can think about and can understand. It seems to me that when it's time to die, and that will come to all of us, there'll be a certain pleasure in thinking that you had utilized your life…