Richard Jernigan -> RE: The Beauty and Mystery of Mathematics (Nov. 20 2013 21:40:02)
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Another look at Platonism. We communicate via shared concepts. If we didn't have shared concepts, how could we understand one another? But concepts can be fuzzy around the edges and that fuzziness can get us into trouble. The classical Greeks, while guilty of introducing the Platonic forms, also came up with a workaround for the fuzziness of concepts: the axiomatic method. "Color" is a fuzzy concept. More specifically, the color "blue" is a fuzzy concept. I sat on the balcony of a fancy hotel in Nusa Dua on Bali, enjoying the tropical breeze and a second cup of coffee after a late breakfast. My companion was a pretty young Javanese girl. (Don't miss your cue here, Vega. Doesn't this make you thirsty?) She was teaching me bahasa indonesia the Malay dialect that is the official language of polyglot Indonesia. Pointing to the border of the menu cover she pronounced hijak. Hijak is the color of green beans in Malay, but the menu border had a decided bluish cast. Noting the blue-green color of the band bordering the sleeve of my polo shirt, I pointed to it and responded, "Hijak." "Tidak! Tidak!" she objected. "Biru." Blue, not blue-green. She held the menu up to my shirt, "Can't you tell the difference?" Indeed the shirt sleeve was that tiny bit more blue. "Yes, but to me they are the same color. I can see the difference now, but if you had not pointed it out, and you asked me at dinner, I would say they were the same color." Then I started laughing. "What's funny?" "The same thing happened in the third grade, only in reverse." The teacher was a red-haired gringa with the build and manner of a Marine drill sergeant. In Spanish class she held up a card and asked what color it was. No one answered. People either didn't know, or some of the quicker Spanish speaking members of the class foresaw a difficulty I did not. I raised my hand and said, "Alazán." "No," said the teacher, "marrón." "But, Mrs. DeV., marrón is the color of chestnuts. Alazán is the color of a sorrel horse, like the card." On the ranch where I spent every summer from the age of four, these were important distinctions. Some of the horses had names, but most of the steeds in the remuda were described simply by their physical characteristics, "The big chestnut mare, the piebald sorrel gelding…" And 21 of the 22 families who lived on the ranch spoke Spanish at home. I persisted. Mrs. DeV. assigned me extra work for defying the teacher. Fuzzy concepts can get you in trouble. The axiomatic workaround is this. You set out a certain set of statements about a concept you wish to discuss like "line" or "point". These statements are called "axioms". Everyone agrees that everything said in the future about the concept will be deduced from the axioms by certain rules of logic. This eliminates large areas of disagreement. There is at present still a bit of discussion about the rules of logic to be employed, but most mathematical logic agrees with ordinary patterns of thought. This approach has one of the characteristics of Platonism. The mathematical objects are accessible only to thought. Indeed they are thoughts. What else could a concept be? Concepts exist, else how could we communicate? But they do not transcend the human mind. They are the shared products of human minds. Concepts can be taught. They can be invented. Invented concepts can be consistent with other, previously tested concepts, or they can be wrong. Useful, striking, consistent concepts are often said to be "discovered." Indeed, that's just how it feels sometimes. Concepts have fallen into and out of favor over the centuries. For example Leibnitz, in his development of calculus, spoke of infinitesimals: numbers smaller in magnitude than any positive number, yet still not zero. Newton did too, at first, but later abandoned infinitesimals. The trouble with infinitesimals is that they violate the rules or ordinary arithmetic. In ordinary arithmetic no matter how small a number may be, if it isn't zero, there's another number, still smaller, between it and zero. In arithmetic there are no nonzero numbers smaller than all the others. Infinitesimals were ridiculed in such a withering and effective manner by Cardinal Newman ("ghosts of departed quantities") that mathematicians of the generations after Newton and Leibnitz abandoned them. Finally in the 19th century Cauchy and others described the idea of "limit" in sufficiently clear and logical terms that calculus was declared to be a logical pursuit after all. When I took calculus in 1956, infinitesimals were still being ridiculed, then by mathematicians, not clergymen. But they still appeared in profusion in physics texts and the lower orders of calculus texts. Then along came Abraham Robinson, the inventor of Non-Standard Analysis. Using sophisticated topological and logical concepts, he extended the idea of the set of real numbers and the rules for manipulating them, to include things that behaved just like Leibnitz's infinitesimals, if you used the same symbols for the new operations as the ones used for ordinary arithmetic. I'm pretty sure Leibnitz's ideas of "infinitesimal" were quite different from Robinson's, yet….who "discovered" infinitesimals? What does "existence" mean to mathematicians? Most would evade the question. Euclid said you could draw a line between any two points. Hilbert in his more rigorous presentation, 2,400 years later, said, "If A and B are two points, there exists a line containing them." This is a statement about conceptual existence. The concept "line" has the property ascribed to it by Hilbert's axiom. It doesn't mean that any physical objects have the properties mentioned, nor that something exists in never-never land. It means that the concepts "line" and "point" abstracted from our physical experiences can safely be said to behave this way. This paper from 1962 http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1359523/pdf/jphysiol01247-0121.pdf cited at least 8700 times, indicates that there are specific brain responses to things we would call lines and points in the visual field, at least for cats. I don't know whether anyone has wired up humans to test us for these responses. This suggests that the concepts "line" and "point" may describe fundamental operations of our brains, honed by evolution over the ages to deal with our environment. Who wouldn't be excited to discover new relations between such fundamental brain responses? The responses (not the concepts, as far as we know) are indeed likely to transcend the human mind, extending to cats, dogs, porcupines…. Ernest Thompson Seton argued that crows could count to at least ten… And the last words of the famous African Grey Parrot Alex, before he died unexpectedly in the night from unknown causes, were his usual, "You be good. See you tomorrow. I love you." http://en.wikipedia.org/wiki/Alex_(parrot) RNJ
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