Getting back to a thing I said about how a line could be represented by an elastic string. Elasticity is a fundamental thing that you see everywhere, collisions are inelastic and elastic (in terms of momentum), and things have a tension, or springiness. Anyway, an elastic string can be inextensible along its length (I think there's some word for this), but still be elastic in an orthogonal sense. You can have a string that doesn't stretch lengthways, but stretches sideways. A projection from some point along this elastic string can go in any direction (and both ways), or the projections (with definite lengths, say like vectors), in series, might only have a slight, directional rotation, say, to the left; a ribbon-like structure emerges, with "elastic" edges, that is also twisted. I think there's some model around, courtesy of a Leonard Euler, that describes something similar. The reals are like a string that has a "sideways" elasticity, because it can be stretched indefinitely, between any two points (say, 0 and 1), to "map" every real value on the string between the points.
Numbers are a result of algebra? What is that suppose to mean? That we cannot learn about numbers until we reach the math level of Algebra? Gee...wiz...I think I started to learn things like 2+2 in elementary school and that that was a prelude to algebra. Can you make your mind up? First you say "no, numbers have nothing to do with shapes", then "well off course they do", then "That does not mean numbers have 'much' to do with shape and geometry---Numbers exist independent of geometry". Please be consistent if you can. Make up your mind! Does a cube (a geometric shape!!!) have six corners or not? Intentionally??? No, I am having difficulty following you, because you keep changing your mind! I am unintentionally misunderstanding where you stand! I will get back to your bold question in a bit...it looks like fun! Does that mean our universe has no shape? Huh? What does Euclidean geometry have to do with the question of the shape of the universe? You are bringing things in that are either unconnected or you are failing to make the connection obvious! Stating that the Universe has no shape is absurd. Even nothing has a dinstinguishing quality relative to shape. I am setting this one up...it looks like I am saying something funny...the statement in this paragraph is bait for the fish...be careful!!! Oh...wow...news to me!!! Ehh...don't you think I don't already know this? What is your point in stating the obvious? I can say something too, with out making the connection clear...women have babies and men don't!!! My point is that your failed to tie your point into a valid statement regarding the topic. You are just stating the obvious with out saying why or how this adds to your own point. Heh???????? Are you kidding me? Can you give any proof once and for all to deal the death blow? That is, proof it!!! What is a REAL number, then? Have fun with this one...take your time and give us a REAL answer! Thanks!
Did I ever say there is a largest Integer? Maybe you should study the English Language a little more!!! You obviously haven't been paying attenton to a word I said...your too busy trying to look smart! I am carefully looking at your words...word for word...and searching for the flaws if they occur. ??? Please learn English!!! The point I was making was that misunderstanding occurs frequently due to the way logic works differently from one mind to the next. You fail to understand how my mind perceives logic, thus you are unable to convince me of hardly anything. If you can get into my way of thinking...understand why I see logic where you don't...you stand a chance to correct me IF I am wrong. Dor dee dor...Mee noot knuww matmateacs...wat it porof? Dora dee dor!! Yes...I went to school and got many A's in math (Especailly Algreba!!!). I completed Algebra and left off at Trig. I understand how proof works. I could of probably go much, much further...but I have a busy life (maybe later). My father is a math wiz...he is so freaky good at it that he could of gone to any school in the world based on his work in math...solving a problem no one thought could be solved. I think I get that from him. Once he made me learn a whole school year of math in 5 days...5 DAYS!!! I was pounding out problems 8-10 hours a day at full speed...head throbbing and all. It was...ALGEBRA!!! The schools tested me the next week and jumped me up a level so as to not waste my time learning stuff I already knew. Maybe you are getting arsey when you fail to make your logic clear in a way that has no weakness. What we are talking about is both simple and complex (sorta like an equation with two or more answers). I am not getting arsey...I am teasing you for overlooking the obvious!!!
I have consistently said that numbers per se have nothing to do with shape. Yes, we do use numbers in analyzing shapes. That does not mean numbers per se have anything to do with shape. We also use numbers to count the number of sheep in a flock (one of the first uses of numbers). Numbers per se have nothing to do with sheep either, or with dollars in a bank account, or with any other application where we use numbers. Numbers are a very useful tool. So are hammers. Don't confuse the tool (a number or a hammer) with the things we build with those tools (analytic geometry or a house). Your insistence on confusing the tool with the use of the tool is one cause of your difficulties in understanding. You are the one who insists on numbers having something to do with shape, so you keep putting words in my typing that are not there. Here is an example of you putting words in my typing: Where did I ever say our universe doesn't have a shape. Stop putting words in my typing. Perhaps I misunderstood you when said "geometric shapes are a fundamental feature of our universe" and I am likewise guilty of putting words in your typing. I took you to mean that the shape of our universe has influence on geometry. I never said the universe has no shape. You did. First, an elementary explanation: the reals are the set of all numbers expressed in decimal notation. I already did answer your question in post #73, repeated here: The real numbers are the set of all Cauchy sequences of the rationals. Another short answer: The real numbers are the set of all Dedekind cuts on the rationals. Yet another answer: The real numbers are the complete ordered field. That one can say something as bold as the complete ordered field is quite deep. That all four answers are equivalent is also quite deep. For example, it takes 450 pages of development in my algebra text (post-calculus algebra) before they even start delving into the reals. I am not going to write a 500+ page exposition for you. I gave you some links. I suggest you read them. These web pages will only take you so far. There are some old-fashioned things called libraries, bookstores, and colleges where you can get these deep results.
My "view" of the shape of numbers goes like this: 1 is an important number (a primary shape), and so is 0. "One" is smooth (like a sphere), zero has no shape, or it's the shape of a circle or sphere with no radius. Two is the first number with "pointy bits"; a line (wire segment), or a shape with two "ends" (two circles or spheres, connected by projecting the circumference of one, is what Two looks like (it isn't necessarily a nice straight shape, it might be curved, like a horseshoe shape, or something). Three has three projections or points, a triangle comes to mind. You can construct a triangle with line segments (Twos), but you need three of them; you could use just two (three minus one), like a wire with a sharp bend, or two bits of (not necessarily straight) wire joined at an apex, so it looks something like a "v" Four is a square, or three "line segments", with two joins, like a square with a missing edge. Five is a pyramid, with perhaps equal area faces, or four segments, like a "w". And so on. These are just some of the shapes; the more points a number has, the more kinds of shape (geometry) are possible. None of these shapes is necessarily "flat" (a sphere certainly isn't, a circle is). All numbers have a dimensionality, a form, or a shape (at least I think they do, but then I am kind of crazy, or something).
Daniel Tammet: http://www.boingboing.net/2007/03/21/daniel-tammet-amazin.html http://www.youtube.com/watch?v=7bVVQ0FZeys Truthfully, I think Daniel has a better grasp of numbers than you do. The way you visual numbers has a STRONG basis on learned concepts and memorized facts. Is this how you see Pi? Please Register or Log in to view the hidden image! Of course, this is a product of the mind. Similarly with numbers. By themselves, numbers have no meaning. Meaning must be assigned to them to do anything useful. Maybe someone will say that because of the nature of numbers, 2 apples added to 3 apples is always 5 apples. This isn't because of numbers but because of how we look at it. If I shift my mental paradigm a bit, I can make myself see only one apple instead of five. It all depends on how you look at things. It's not like God gave Moses, Pythagoras, or anyone else a tablet with THE rules of mathematics and the universe. Therefore, we are free to explore the results and consequences of the rules we choose. The beauty of mathematics is that we can make the rules so basic that they describe everything, both realistic and unrealistic. Just how realistic the results are depend on the accuracy of your assumptions. Gabriel's Horn, for example. Totally unrealistic. But the math that gives us GH is the same that gives us space exploration (among other things). It takes a lot of mathematics study to understand what DH means by numbers coming from algebra. Mathematics is sort of taught in a backwards manner, similar to the way mathematics was developed. We started out by counting and eventually doing things similar to high school algebra and basic calculus and number theory. However, we would always run into problems or have questions we could not answer. For example, we can easily calculate \(2^4\). What do we do for \(2^{2.4}\)? What about \(2^\pi\)? Or even \(i^i\)? What we do is develop axioms and work old problems again, this time with better rules and assumptions.*** From these axioms we CONSTRUCT basic arithmetic. With abstract algebra, we can construct our own number systems. So, if you like, we can use abstract algebra to create numbers (or set-theoretic definition). It's a pain in the ass though because you have to define what "+" means and what "*" means. Subtraction and division are definitions, depending on addition and multiplication respectively. Just imagine if we started out first graders with basic set theory. It's the same in college mathematics. In many schools (mine included) they teach linear algebra with just real numbers (and sometimes polynomials). They don't address the field. Why? In general, linear algebra is easier than abstract algebra so they teach linear algebra first. Once you understand both concepts, you can go back and generalize it even more. And this is how research is conducted, too. Researchers start off very basic. They lay out the general framework. Later on, they work on tweaking it and turning specific statements into general ones. I think I rambled on long enough. I'm supposed to be studying. ***Referring back to the part about running into problems and changing our assumptions. Many people say that multiplication is just repeated addition. \(2*3 = 3 + 3 = 6\). But what about \((-1)*(-3)\)? You can't do this knowing only first grade math. To PROVE it equals 3, you must know two rules: distribution and additive inverse cancellation. We add the latter rule to our list of axioms when we ask questions like what x gives \(2 + x = -3\)? Of course, we must have already discovered negative numbers, too. This is an example of.. what? Oh right, algebra. We aren't using arithmetic and basic counting to equate \((-1)*(-3)\).
That would be more like a way someone might think "seeing" a number would "look". Here's the thing though: I don't "see" pi, at least not the way you seem to imply with a picture. I disagree, algebra is actually quite straighforward, or the basic ideas are. I don't see any problem with approaching algebra via geometry and through combinatorics at the same time. Doing it in small steps is a good idea, just where to start looks like the problem. Like I say, numbers have a shape. The integers aren't just marks on bits of paper or a whiteboard. Introduce the notion of numbers having "extended" bits, or connections to other numbers, and to "spaces", rather than being discrete members of some set, should start a lot earlier than it does now, I reckon. Also numeration and denomination should be explored more than just "division" and "repeated subtraction". Multiplication of two "negative" numbers, is multiplication of two "same kind" of things. Explaining positive and negative (reverse) counting is simple enough. You can explain what roots are and what i is using (2) x (2) and (-2) x (-2), pretty simply. Two points are, or define a line, and convey the idea of points on a line, and extension or projection. Algebra isn't a separate math from other math, why treat it like it is? The idea of counting single things comes straight from the idea of "one" (things) and "not-one". Is there a simpler connection? Increase and decrease, things we see happening all around, explain "real" or continuous variables. Why does a youngster need years of math to understand these basic ideas? The symbology and rigid theorems take some of the fun out, but that doesn't have to happen either. Like most things, mathematics is based on essentially simple ideas, or rules, but a lot of complexity emerges (like in a chess game). Numbers aren't "by themselves". This is what young people get told, maybe, but it's something someone made up. The problem with it is you need to explain what it means for numbers to be "by themselves", and why meaning is important. Despite all the attempts and hundreds of pages of theorems, lemmas and proofs, numbers are not a "fixed" shape. Not even a little bit, except when we think about them.
It occurs to me that education is largely a series of encounters with ideas. First you get told the "easy" stuff, then they tell you to forget all those ideas you just learned, because actually the story is so-and-so; you get to high school, and they tell you there's a "lot" to learn, and some of the stuff you know already is actually wrong, it's really so-and-so. At uni, they say things like "all that stuff you learned about so-and-so, isn't actually correct so you need to forget all about what you know", and so on. So the dogma seems to be that you never actually learn the "story", because the teachers have to tell porkies, and half-truths (so young minds can grasp the difficulties). If kids were told about QM, would it be "over their heads"? Do they really have to be led gently down the path of knowledge, and get told at each step that they haven't been told the "real story" yet? Why the need to instil notions of "difficulty" and "hard to understand", into children's minds? Isn't science supposed to be about essentially simple ideas?
To be quite honest, the universe will only have a shape if it is observed to have a shape. This could be a number of things... from a flat-like universe, to some spherical shape. Or perhaps, just as i have speculated, some shape with lobes sticking out.
Kids are introduced to basic concepts of QM such as wave-particle duality in elementary school. They aren't taught what that really means because (a) they don't have the requisite mathematical background to do so, and (b) even the teachers at that level don't have the requisite mathematical background. Yes. How is this different from any other human endeavor? This isn't even true of animals, let alone humans. Wolf cubs have to be taught, in stages, how to hunt. A wolf cub does not instantly go from chasing its own tail to taking down a moose. A human cub similarly cannot instantly go from one+one=two to the Schrödinger wave equation or the eightfold way. These things must be done delicately and in stages. Maybe because things like the Schrödinger wave equation and the eightfold way are both difficult and hard to understand, even for quantum physicists? To quote Feynman, "I think I can safely say that nobody understands quantum mechanics." No. Science is about understanding the nature of the universe. "Essentially simple ideas" are the domain of philosophy and religion. Quoting Feynman again, "Philosophers say a great deal about what is absolutely necessary for science, and it is always, so far as one can see, rather naive, and probably wrong."
Nor would hunters give spears and clubs to children and tell them to go and get dinner for the tribe. Usually this happens somewhere near puberty, as an initiation into the "adult" world, but that's culture and anthropology, or something. My point is, despite all the "tricky" mathematical ideas, the world looking complicated, and full of strange things; fundamentally there are simple "explanations" for much of it. Mathematical games (board games, e.g. go, chess, draughts) are based on a handful of simple rules. Einstein tried to show "everyone" that the relation between matter and energy is simple (not necessarily easy to understand, though). You can explain his ideas with math and symbology, or with words and pictures. Words are symbols too; math is a language that's constrained, exacting, precise (as you've pointed out, you can construct hundreds of pages of equations, just to connect algebra with "numbers"). I don't see that kids need to understand hundreds of pages of math, or the entire contents of an encyclopedia, to "get the hint". I think we give them a lot less credit for insightfulness than we should, for starters.
Quantum mechanics is not draughts. The quantum mechanical world is very strange and definitely is not simple. Physicists try to explain at least parts of quantum mechanics in simple words, such as "wave-particle duality". For those who lack the requisite mathematical understanding, these words remain little different from fairy tales. If you understand the math it is quite wonderful, but it is not simple. If you want simple answers, go to philosophy or religion. They have simple answers. Wrong, but simple.
Please. Quantum electrodynamics, quantum field theory, the eightfold way are far from statistical theories. They are very, very deep -- and very, very accurate. If QM were merely "a statistical theory at best" it would be chock full of ad-hoc relationships and chock full of magic numbers. Instead, there are but fourteen magic numbers (aka fundamental constants) for which we currently do not have an explanation. Fourteen fundamental constants to describe everything to the incredible accuracy that QM attains is not a "statistical theory at best." The LHC is expected to reduce the number of requisite magic numbers by means of the Higgs.
But it is full of ad-hoc theories, despite it's own predictions. Dr. Balentine shares my own opinions. You take a particle from the past t=0... it's past rallies on past variables that can only be summed statistically. If we could map put all of reality, we would have a set of binary digits that described everything.
Notwithstanding that QED is "the most accurate theory we have to date", Richard Feynman was fairly categorical (in the lectures I went to at Waikato) about the probabilistic nature of the world (the behaviour of light, was his favourite example). I'd say I agree with what he said (except the bits I don't really understand). The world and quantum behaviour is statistical, unless someone can come up with a non-statistical meaning for "probability-amplitude", we're stuck with it. I didn't say QM is draughts, I said mathematical games (or call them sets of transforms on a 2-d surface, maybe), have simple rules; complexity nonetheless emerges. This is still a guiding principle in science, despite thousands of math equations and a lot of complex systems that still defy analysis (but we're "getting there").
The probabilistic character of world still doesn't mean, it cannot have the deterministic meaning as well in analogy with particle-wave duality. After all, the only way, how the statistical system can be realized in physics is the dense inertial gas. Only the fermionic, mutually colliding particles can realize tangible "statistics" for us.
Firstly, anyone who misuses 'your' when they mean 'you're' shouldn't be taking cheap shots at people's language skills. Secondly, I'm well aware you didn't say "There is a largest integer". However, you said "Numbers can only get so big...before they cease to be numbers." By proving there is no largest integer I demonstrated your statement is false. If 'numbers can only get so big' then they are bounded. If they are bounded then integers are bounded. Thus there is an upper bound to integers and since the integers are not dense in the Reals there must be a largest integer if your statement were true. Since there isn't, your statement is incorrect. The logic is correct because you said the statement P, "Numbers can only get so big". This implies statement Q, "There is a largest integer". I've just proven ¬Q and if P=>Q, ¬Q=>¬P. Thus P is false. Numbers can increase and increase and still remain 'numbers'. No, the misunderstandings are the result of sloppy terminology and mistakes. Logic, if done carefully and precisely, is absolute and not a matter of interpretation. Just look at the work of people like Russell and Whitehead in 'Principia Mathematica' from about 100 years ago. They used logic to make rigorous, from the uttermost axioms of mathematics, as much of analysis as they could do. You are basically implying that 1+1=2 is a matter of how the reader's logic works. No, given the axioms of basic arithmetic (say the Peano axioms) 1+1=2 is true. It cannot be false from the axioms, there is no "I read it differently from you" about it. You're welcome to choose other axioms upon which to build a mathematical or logical system, mathematicians do it all the time when developing new ideas. That wouldn't alter the fact that given the Peano axioms 1+1=2. Given the axioms of arithmetic, my statement (or rather, my disproof of your statement) stands. Given the notion of 'numbers' everyone is familiar with (ie the field of Reals, which are the Cauchy completion of the rationals, which are in turn constructed via equivalence relations on integers, which are constructed from the Peano axioms), the statements "Numbers can only get so big...before they cease to be numbers. This also applies to the lengths of decimals. They can only get so small...before they cease to be numbers." Are all false. Of course if you're referring to an entirely different number system to everyone else you should have said so. But you didn't, so the assumption "He's talking about the Reals" is a decent one. Particularly since you mention decimals and they are the Reals. So if you accept the construction and notion of the Real numbers, you have made several incorrect statements. That isn't a matter of opinion, it's logic. Otherwise it's like saying "1+1=3 because I feel like it". Consistency is out the window and any discussion of mathematics is then pointless. And my dad has a maths degree, a maths PhD, is a professor of fluid mechanics and I've got a Cambridge maths degree and masters and am doing a string theory PhD. So what? The fact of the matter is your, mine or anyone else's qualifications are irrelevent if your statement is proved to be false. If I stood on a street corner and yelled out "2+2 equals 5" and a 6 year old walked up to me and said "No, 2+2 is 4" the fact I've done a lot more maths than them doesn't make me right and them wrong. As I said, logic is logic. 2+2 is 4 under normal arithmetic and the aforementioned statements of yours are incorrect under normal arithmetic. I'm not getting arsey. A few people on these forums know what I'm like when I'm arsey. My corrections of you were correct. No, I didn't construct them from the ground up because that would have been pointless. Did I need to prove that the Reals are a field or that the rationals' completion is the Reals? No. The fact I can add 1 to a number as many times as I like is enough, just as dividing a number as many times as I like by 2 is enough. Of course if you think your statements are correct can you provide examples to prove them? You said : Numbers can only get so big...before they cease to be numbers. How big? Give me a decimal I can't go larger than. They can only get so small...before they cease to be numbers. How small? Give me a decimal I can't go smaller than. And if you think solving equations with 2 roots is 'complex' (other than the i*i=-1 way) I'd be less hasty in trying to make boasts about mathematical ability. Even the 1st year geography students I sometimes do marking for have mastered that particular area Please Register or Log in to view the hidden image!
