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I read an article entitled, "Why Seven?" The author's (Charles Leroux) theory was that the number seven resonates with people for some very good reasons. I have never thought about it before, but here are some of them: "Seven" was held to be sacred by most ancient people-- the Egyptians, Phoenicians, Persians, etc.--which he says likely stems from the seven visible planets. Still, he says, there are the seven gates of heaven (I haven't heard of that), seven deadly sins (pride, envy, anger, lust, greed, sloth and gluttony), and the Catholic Church's seven sacraments (from baptism to annointing the sick). But then there is the magic seven. Roll a lucky seven in craps, and you win. He says that seven has staying power. Sailors will sail the seven seas even though that number has changed over time. And women read a whole range of magazines, but the "Seven Sisters" of service magazines still persist: Redbook, McCalls, Ladies Home Journal, Good Housekeeping, Family Circle, Women's Day, and Better Homes and Gardens. [I wonder where the Cosmopolitan is!  ]He mentions other things, like "Snow White and the Seven Dwarfs" and "the film "The Magnificant Seven." Is there something special about seven or is he just picking out isolated examples? Are there other numbers that have special meanings? I know my favorite number is 23, for a very special reason!  | ||
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Seven is mentioned continually in the bible, beginning with Genesis, which says that God created the world in six days and rested the seventh. Wikipedia has an article about the number seven. Tinman | |||
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Aren't there seven virtues as well as seven sins, or is that a fallacy? I seem to remember being told that faith, hope, charity, prudence and grace were five of them. There are seven main chakras: the base, the sacral, the solar plexus, the heart, the throat, the third eye and the crown. All the opposite faces of six-sided dice add up to seven. If you break a mirror some people believe you get seven years bad luck. I've heard that that's because centuries ago people thought that the soul renewed itself completely every seven years (presumably because at the ages of 0,7,14,21 etc one is a very different person), so because they also thought that the soul was contained in the mirror, breaking one damaged it, and it needed the full seven years to recover. I have no idea if that's really how that particular superstition came about (it's not verified on Snopes), but it would make sense if it were true. | |||
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My old biology master once told us, "Beware of any scientific 'fact' you see quoted (especially in the popular media) that contains the number seven. It is almost bound to be untrue". I believed him than and I still believe him now, half a century later. Richard English | |||
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Fun game to play. Let's see. "There are seven objects visible to the naked eye in the solar system, (i.e., the Sun, Mercury, Venus, the Moon, Mars, Jupiter, and Saturn)." "Seven is a Mersenne prime." (2^3 - 1) "Seven is the fourth prime number." "Seven is the atomic number of hydrogen." "There are seven noble gases." I'm sure there are more. —Ceci n'est pas un seing. | |||
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These are not the facts that you see in the popular press. To start with, they are not tittilating or even interesting. The sorst of "facts" I refer to are those like: London's water has already been drunk seven times before you get to drink it. The seventh wave is always the biggest. It takes SEVEN times worth of repetition of your name and your product before it takes hold in a buyer's mind. It takes seven times more acreage for a pound of beef than a pound of milk. It takes seven times more effort and money to attract new customers than it does to hold on to existing ones. It takes seven times more energy to transport a ton of cargo by truck than it does by train. It takes seven times as much nutrition to heal, as to maintain a healthy body. It takes seven times as far to stop on an icy road as on a dry one. And so on. All plausible - almost certainly all complete cobblers! Richard English | |||
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"Seven is the largest single digit prime number in base ten." I feel stimulated all over by the tintinnabulation of it all. Back to the question that started the thread. There is a famous paper by sociologist George Miller called The Magical Number Seven, Plus or Minus Two: Some Limits on Our Capacity for Processing Information; see here for an online copy of the paper. (It always seems to be quoted in AI literature.) Oh, yes, and the Seven Types of Ambiguity by Wm Empson.This message has been edited. Last edited by: zmježd, —Ceci n'est pas un seing. | |||
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Yes. The funniest number is 22. | |||
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And there's always 69 and 77. Tinman | |||
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Then, of coures, there's always 42. Build a man a fire and he's warm for a day. Set a man on fire and he's warm for the rest of his life. | |||
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And the named numbers, pi, e, i, and Aleph-naught. | |||
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I understand the significance of 69. But why 22, 77, 42? Or isn't there a significance? Cat, thanks for that great theory for why breaking a mirror is supposed to cause bad luck for 7 years. I had always wondered what the background of that superstition was. I remember, as a child, breaking a mirror, and my grandmother said I'd have bad luck for 7 years. I was petrified! Zmj, what a great article! I learned a new word from the Wikipedia explanation, "subitising." I could only find it in the OED, and here is what it says: "1949 E. L. KAUFMAN et al. in Amer. Jrnl. Psychol. LXII. 520 A new term is needed for the discrimination of stimulus-numbers of 6 and below... The term proposed is subitize... We are indebted to Dr. Cornelia C. Coulter, the Department of Classical Languages and Literatures, Mount Holyoke College, for suggesting this term. Ibid., If no discontinuities had appeared in the results, no distinction between subitizing and estimating could have been drawn. 1971 Jrnl. Gen. Psychol. Jan. 121 The number of items in an array capable of being subitized. 1981 Nature 15 Oct. 569/2 Judgements of ‘small’ numerosities..are ordinarily attributed to subitizing." | |||
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That's the The Answer to Life, the Universe, and Everything in The Hitchhiker's Guide to the Galaxy. Not sure about 22, which is one less than 23, which is a big number in Illuminatus!. 77 Sunset Strip was a TV show. —Ceci n'est pas un seing. | |||
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Catch-22 was originally titled Catch-18, but before it was published Leon Uris published Mila 18. So Heller changed the title to Catch-22 because he thought 22 was a funnier number. 22 comes up in comedy a lot. Start listening for it and you'll be surprised how often you hear it. | |||
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Neveu, I don't think that aleph-naught is a number, and you can't add/subtract/multiply with it. Aleph-naught is defined as the size of the natural numbers, or more correctly, the size of a set of numbers for which there is a bijection to the natural numbers. This means that aleph-naught is not a number, but a cardinality, made confusing by the fact that all natural numbers are cardinalities. In mathematics, the concepts are often misunderstood, especially since they are counterintuitive, i.e., the cardinality of the positive integers is equal to the cardinality of all the integers. Of course, in mathematics, we replace cardinality with "size", and it is understood what we mean. | |||
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Cantor thought so, though. Aleph-null + Aleph-null = Aleph-null, etc. 2^Aleph-null = Aleph-one. —Ceci n'est pas un seing. | |||
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That isn't accurately portraying what is going on. Take a set A with size aleph-null and a set B with size aleph null and the size of A union B is aleph-null. This is what is described by the notation you use above, and isn't really addition.
Again, this is shorthand for a more complicated mathematical process. Furthermore, this is not correct. More precisely, this is unknown, and Godel proved that in our number system, it is impossible to prove whether these two things are equals. This is known as the continuum hypothesis. What you probably want to say is 2^aleph-null > aleph-null, which means the cardinality of the set containing all subsets of a set S, is greater than the cardinality of S. I suppose this all gets into quibbling over what the definition of number is, but almost any definition I can come up is failed by infinite quantities and cardinalities. It is difficult to consider aleph-null a number because two sets with clearly different size can both have cardinality aleph-null. For example, the set of all positive integers and the set of all integers both have cardinality aleph-null, but clearly the former is a subset of the latter. To me, this defies any notion one has of number. | |||
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Well, neveu and zmjezhd already told you about 42 and 22, and you know about 69. And 77 is just like 69, except you get 8 more! You don't like that? OK, 4 square + 5 square + 6 square = 77, and the sum of the first eight prime numbers = 77. No, I didn't know that; I got it from Wikipedia. Wikipedia has entries on most any number. Tinman | |||
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What you want to say is 2^A > A, whether the set A is finite or infinite. Cantor proved that. Cantor also proved that 2^aleph-nought = (lower-case German)c, or that the set of all sets of the natural numbers is equivalent to the continuum or the set of all points on a line segment. The continuum hypothesis is that there is no set with cardinality between Aleph-nought and c, that is, there is no infinite set of points on a line segment that is not equivalent to the set of natural numbers, and also not equivalent to the whole line segment.
But clearly they are the same size, since I can set up a correspondence relation such that if you give me any number from one of the sets, I can give you exactly one number from the other that corresponds to it. Or at least that's what my high-school math teacher told me. These kinds of paradoxes are at the core of a lot of modern set theory. I included aleph-nought in that list to be provocative, and clearly I succeeded. When we were teaching math to our kids the question arose whether infinity was a number or not. My wife, who has a degree in math, her sister (a professor of education) and a host of others said no. I was in grad school at the time and raised the question to roomful of engineers and physicists. There was a moment of thought, then everyone, simultaneously, said "Yes!" as though it were perfectly obvious. You can do math with infinity. The first calculus course I took in college was based on hyperreals, the real numbers plus infinitesimals and infinity. This was the original intuitive approach used to develop calculus, but questions and controversies not unlike those in this discussion led Weierstrass to make calculus rigorous using the notion of limits instead. In 1960 Abraham Robinson found a way to make infinitesimals rigorous using nonstandard analysis. In this system a number is said to be positive infinitesimal if it is smaller than any positive real number yet greater than zero. They are introduced into the system with an axiom. Similarly a number is positive infinite if it is greater than any real number (this is not the same thing as aleph-nought, of course). It was a great way to learn calculus -- I've still got the book, Elementary Calculus by H. Jerome Keisler. | |||
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Was Cantor driven mad by his theory of transfinite numbers? Fnord! —Ceci n'est pas un seing. | |||
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Um, you see to have a strange definition of the word clearly. Take two sets with equals size, and add something to one of them, and they can still have equal size. This isn't really a paradox either, since we're not saying the sizes are equal, we're saying, "both have a bijection to the natural numbers"
I can see how mathematicians would go one way and physicists would go the other. I have degrees in both engineering and mathematics, and once you've taken the right math courses, you'd probably agree with me, unless you'd taken the right physics courses, of course. I've done just about every proof mentioned in this thread, except for Godel's proof, which was presented in class, and I have a very good idea of what cardinality is, and how you can do math with infinite sets, and my intuitive notion of number doesn't extend out that far. Of course, there may be mathematicians whose definition of number is related to enumerability, meaning that aleph-null is a number, but not any higher order of infinity. | |||
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But that is how I'd always heard aleph-null described: as the cardinality of the set of natural numbers. So, is the cardinality of a set not its size? Or, better yet, the number of elements in the set? What is the cardinality of the set of primes less than ten? Is four not a number? —Ceci n'est pas un seing. | |||
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My intuitive notion of number is something that represents a quantity, that is, something that is greater or less than some other quantity. I was surprised you jumped on Aleph-nought; to me i is much less number-like than Aleph-nought. | |||
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Yes, aleph-null is defined as the cardinality of the set of the natural numbers. However, cardinality is referred to as such because size doesn't really work for infinite things. One can discuss the size of the Earth, or of the galaxy, but not the size of the universe. I am fine with imaginary numbers because you can do math with them. You can add them, subtract, multiply, and do cool things in exponents with them. Without i, you can't have e^(i*pi) + 1 = 0. | |||
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So is i greater than, less than or equal to zero? | |||
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I can answer your question, but you won't like it. It is "above" it, at least, in the Cartesian sense. The thing is, i exists as the root of a polynomial, making me it a number in my book. Of course, there are an infinite amount of imaginary numbers which exist as the root of polynomials, however, this number is countable(cardinality aleph-null), and these roots are called "algebraic numbers". There are more imaginary numbers of a higher cardinality which can not be expressed as the root of a polynomial, stretching my definition of imaginary number quite a bit. Still, all imaginary numbers have a clear definition in a geometric sense. The computer you use to read this involves some high level electrical engineering which is based on math and imaginary numbers. While the name "imaginary" is accurate in the sense that i is not a number, these are quantities that show up in all sorts of things, like signals, quantum mechanics, etc. You can't do any of that stuff with aleph-null. | |||
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you are all nuts | |||
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Welcome to the nuthouse!  | |||
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