“People are impressed with numbers, but the mere existence of data that can be quantified and manipulated is no guarantee of valid results.” – John J. Lentini, Scientific Protocols for Fire Investigation, Second Edition
“An algorithm is defined by a sequence of steps and instructions that can be applied to data. Algorithms generate categories for filtering information, operate on data, look for patterns and relationships, or generally assist in the analysis of information. The steps taken by an algorithm are informed by the author’s knowledge, motives, biases, and desired outcomes. The output of an algorithm may not reveal any of those elements, nor may it reveal the probability of a mistaken outcome, arbitrary choice, or the degree of uncertainty in the judgment it produces. So-called ‘learning algorithms’ which underpin everything from recommendation engines to content filters evolve with the datasets that run through them, assigning different weights to each variable. The final computer-generated product or decision—used for everything from predicting behavior to denying opportunity—can mask prejudices while maintaining a patina of scientific objectivity.” – Executive Office of the President, Big Data: Seizing Opportunities, Preserving Values
“Never try to put too much into any single piece. There is always a point beyond which it cannot be improved, and further attempts to improve it will in fact destroy it.” – Douglas R. Hofstadter, Gödel, Escher, Bach
“A problem is determinate if it has a definite number of solutions, indeterminate if it has an indefinite number of solutions, and impossible if it has no solution.” – G. A. Wentworth, Plane Geometry (emphases in original)
“In the Valley of Youth, through which all wayfarers must pass on their journey from the Land of Mystery to the Land of the Infinite, there is a village where the pilgrim rests and indulges in various excursions for which the valley is celebrated. There also gather many guides in this spot, some of whom show the stranger all the various points of common interest, and others of whom take visitors to special points from which the views are of peculiar significance. As time has gone on new paths have opened, and new resting places have been made from which these views are best obtained. Some of the mountain peaks have been neglected in the past, but of late they too have been scaled, and paths have been hewn out that approach the summits, and many pilgrims ascend them and find that the result is abundantly worth the effort and the time.
“The effect of these several improvements has been a natural and usually friendly rivalry in the body of guides that show the way. The mountains have not changed, and the views are what they have always been. But there are not wanting those who say, ‘My mountain may not be as lofty as yours, but it is easier to ascend’; or ‘There are quarries on my peak, and points of view from which a building may be seen in process of erection, or a mill in operation, or a canal, while your mountain shows only a stretch of hills and valleys, and thus you will see that mine is the more profitable to visit.’ Then there are guides who are themselves often weak of limb, and who are attached to numerous sand dunes, and they say to the weaker pilgrims, ‘Why tire yourselves climbing a rocky mountain when here are peaks whose summits you can reach with ease and from which the view is just as good as that from the most famous precipice?’ The result is not wholly disadvantageous, for many who pass through the valley are able to approach the summits of the sand dunes only, and would make progress with greatest difficulty should they attempt to scale a real mountain, although even for them it would be better to climb a little way where it is really worth the effort instead of spending all their efforts on the dunes.
“Then too, there have of late come guides who have shown much ingenuity by digging tunnels into some of the greatest mountains. These they have paved with smooth concrete, and have arranged for rubber-tired cars that run without jar to the heart of some mountain. Arrived there the pilgrim has a glance, as the car swiftly turns in a blaze of electric light, at a roughly painted panorama of the view from the summit, and he is assured by the guide that he has accomplished all that he would have done, had he laboriously climbed the peak itself.
“In the midst of all the advocacy of sand-dune climbing, and of rubber-tired cars to see a painted view, the great body of guides still climb their mountains with their little groups of followers, and the vigor of the ascent and the magnificence of the view still attract all who are strong and earnest, during their sojourn in the Valley of Youth.” – David Eugene Smith, The Teaching of Geometry (1911)
“Just as a being may be imagined as having only two dimensions, and living always on a plane surface (in a space of two dimensions), and having no conception of a space of three dimensions, so we may think of ourselves as living in a space of three dimensions but surrounded by a space of four dimensions. The flat being could not point to a third dimension because he could not get out of his plane, and we cannot point to a fourth dimension because we cannot get out of our space. Now what the flat being thinks is his plane may be the surface of an enormous sphere in our three dimensions; in other words, the space he lives in may curve through some higher space without his being conscious of it. So our space may also curve through some higher space without our being conscious of it.” – David Eugene Smith, The Teaching of Geometry (1911)
“An interesting application of the theorem relating to similar triangles is this: Extend your arm and point to a distant object, closing your left eye and sighting across your finger tip with your right eye. Now keep the finger in the same position and sight with your left eye. The finger will then seem to be pointing to an object some distance to the right of the one at which you were pointing. If you can estimate the distance between these two objects, which can often be done with a fair degree of accuracy when there are houses intervening, then you will be able to tell approximately your distance from the objects, for it will be ten times the estimated distance between them.” – David Eugene Smith, The Teaching of Geometry (1911)
“The natural temptation in the nervous atmosphere of America is to listen to the voice of the mob and to proceed at once to lynch Euclid and every one who stands for that for which the ‘Elements’ has stood these two thousand years. This is what some who wish to be considered as educators tend to do; in the language of the mob, to ‘smash things’; to call reactionary that which does not conform to their ephemeral views. It is so easy to be an iconoclast, to think that cui bono is a conclusive argument, to say so glibly that Raphael was not a great painter,—to do anything but construct. A few years ago every one must take up with the heuristic method developed in Germany half a century back and containing much that was commendable. A little later one who did not believe that the Culture Epoch Theory was vital in education was looked upon with pity by a considerable number of serious educators. A little later the man who did not think that the principle of Concentration in education was a regula aurea was thought to be hopeless. A little later it may have been that Correlation was the saving factor, to be looked upon in geometry teaching as a guiding beacon, even as the fusion of all mathematics is the temporary view of a few enthusiasts to-day.” – David Eugene Smith, The Teaching of Geometry (1911)
“When the time comes that knowledge will not be sought for its own sake, and men will not press forward simply in a desire of achievement, without hope of gain, to extend the limits of human knowledge and information, then, indeed, will the race enter upon its decadence.” — Charles Evans Hughes (quoted by David Eugene Smith in The Teaching of Geometry)
“In the present utilitarian age one frequently hears the question asked, ‘What is the use of it all?’ as if every noble deed was not its own justification. As if every action which makes for self-denial, for hardihood, and for endurance was not in itself a most precious lesson to mankind. That people can be found to ask such a question shows how far materialism has gone, and how needful it is that we insist upon the value of all that is nobler and higher in life.” — Sir Arthur Conan Doyle (quoted by David Eugene Smith in The Teaching of Geometry)
“One of the essential properties of intelligence is flexibility. What does it mean to be an expert at something? What makes someone an expert is her ability to respond to a completely novel situation—to solve original problems, for example. Expertise does not only involve having command of a huge amount of factual knowledge—it does not mean being a human data bank. It involves the capacity for flexible response. It is a form of creativity. This description applies equally to the notion of ‘understanding.’ Understanding is a kind of expertise. A true measure of intelligence is this capacity for flexible and original response.” – William Byers, How Mathematicians Think
“Gödel’s incompleteness theorem is one of the great intellectual accomplishments of the twentieth century. Its implications are so far reaching that it is difficult to overestimate them. Gödel’s result puts intrinsic limitations on the reach of deductive systems; that is, it shows that given any (sufficiently complex) deductive system, there are results that are beyond the reach of the system—results that are true but cannot be proved or disproved on the basis of the initial set of axioms. The new result might be proved by adding new axioms to the system (for example, the result itself) but the new strengthened system will itself have unprovable results.” – William Byers, How Mathematicians Think
“Mathematics is not a body of facts arranged and justified by a stringent logical structure. To give a dynamic metaphor that stems from the related field of fluid flow, it is like a turbulent river. The flow of the water is extremely complex, but every now and again you see the appearance of stable structures, eddies and whirlpools. These stable structures correspond to the propositions and theorems of mathematics. It is the logical structure of mathematics that gives these theorems their stability, but when we look at things in this way we see that the stability is not absolute. The data can be structured in many different ways corresponding to what we are interested in and in the mathematical ideas that arise to do the structuring. These ideas arise in response to the question, ‘What is going on here?’—that is, from the attempt to make sense of the phenomena in question.” – William Byers, How Mathematicians Think