The Definition of Points
in [Math]
|
Prev: Guide to presenting Lemma, Theorems and Definitions
Next: Density of the set of all zeroes of a function with givenproperties
From: Sam Wormley on 17 Mar 2007 11:19 SucMucPaProlij wrote: > Isaak (sic) Newton: Philosophiae Naturalis Principia Mathematica > > or "academic style bullshit" > May I suggest: "Newton's Principia for the Common Reader" by S. Chandrasekhar (1995) Clarendon Press . Oxford ISBN 0 19 851744 0 Quoting from "Great Physicists: The life and times of leading physicists from Galileo to Hawking: by William H Cropper. 'For his final study, Chandra chose a remarkable subject--Isaac Newton. Chandra was a student of science history and biography, and he had a wide acquaintance among his contemporaries in physics and astrophysics. But for him one scientist stood above all those of the past and present, and that was Newton. He decided to pay homage to Newton, and try to fathom his genius, by translating "for the common reader" the parts of Newton's Principia that led to the formulation of the gravitational law. 'Newton relied on the geometrical arguments that are all but incomprehensible to a modern audience. To make them more accessible, Chandra restated Newton's proofs in the now conventional mathematical languages of algebra and calculus. His method was to construct first his own proof for a proposition and then to compare it with Newton's version. "The experience was a sobering one," he writes. "Each time, I was left in sheer wonder at the elegance, the careful arrangement, the imperial style, the incredible originality, and above all the astonishing lightness of Newton's proofs, and each time I felt like a schoolboy admonished by the master."'
From: Tony Orlow on 17 Mar 2007 11:42 Lester Zick wrote: > The Definition of Points > ~v~~ > > In the swansong of modern math lines are composed of points. But then > we must ask how points are defined? However I seem to recollect > intersections of lines determine points. But if so then we are left to > consider the rather peculiar proposition that lines are composed of > the intersection of lines. Now I don't claim the foregoing definitions > are circular. Only that the ratio of definitional logic to conclusions > is a transcendental somewhere in the neighborhood of 3.14159 . . . > > ~v~~ Hi Lester - How's it going? Yes, the relationship between points and lines is rather codependent, isn't it? I looked at some of the responses, and indeed, one can define points as tuples of coordinates, but of course, that all depends on defining a set of dimensions as a space to begin with, each dimension constituting an infinite line along which that coordinate is defined. In language, both points and lines are taken as primitives, since their properties are not rooted in symbols and strings, but geometry. So, we may be left with the question as to what the primitives of geometry really are, sets of points, or sequences of lines. That's the conundrum right, that differences and differences between differences are lines, and not points? :) Well, here's a thought I had that I think relates to your derivative perspective. I was considering the number line and geometrical representation of basic arithmetic operations, and in considering whether addition or subtraction was more basic, I noticed something interesting. In order to represent a-b, all we need to know is the locations of a and b. In order to represent a+b, we need to know the location of 0 as well, form a vector from 0 to a or b, and apply it to b or a, respectively, to find the sum point. So, in that respect, the difference between the two is more basic than the sum, since it requires less information. That sits well with your differences between differences. The specification of a given point on this line is the same as the difference between that point and 0. A number is a vector from the origin, a difference, not just a point. The elements of the tuple specifying the point represent offsets from the origin, in specific directions, defined by lines. Have a nice day. 01oo
From: Randy Poe on 17 Mar 2007 11:51 On Mar 17, 10:22 am, "SucMucPaProlij" <mrjohnpauldike2...(a)hotmail.com> wrote: > "Bob Kolker" <nowh...(a)nowhere.com> wrote in message > > news:5629arF26ac36U1(a)mid.individual.net... > > > SucMucPaProlij wrote: > > >> I don't want you to expect too much because this is not mathematical proof, > >> it is philosophical proof (or discussion). This is just the way how I explain > >> things to myself. > > > If it ain't mathematics and it ain't physics, it is bullshit. Philsophy, by > > and large, is academic style bullshit. > > Isaak Newton: Philosophiae Naturalis Principia Mathematica > > or "academic style bullshit" What was called "Natural Philosophy" in Newton's time is what is now called "physics" and is not what is currently called "philosophy". - Randy
From: Bob Kolker on 17 Mar 2007 11:57 SucMucPaProlij wrote: > > > Isaak Newton: Philosophiae Naturalis Principia Mathematica Natural Philosophy, the old name for Science. It was not metaphysics. "When we run over libraries, persuaded of these principles, what havoc must we make? If we take in our hand any volume of divinity or school metaphysics, for instance, let us ask, Does it contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it then to the flames, for it can contain nothing but sophistry and illusion." An Inquiry Concerning Human Understanding by David Hume, one of the few philosophers that ever made any sense. Bob Kolker
From: Bob Kolker on 17 Mar 2007 11:59
SucMucPaProlij wrote: > > > And I agree but can you tell me does point exist? > How do you explain it? Point is an idea or a notion. It has no physical existence. Neither does the integer 1. Point is a place holder for an intuition about space. Nothing more. Along with line, plane and a few other place holders they constitute the undefined terms of geometry. Intuitive notions are useful guides for finding logical proofs, but they have not probatory or logical standing. Bob Kolker |