On Monday, February 17, 2014 6:51:09 AM UTC-5, John Gabriel wrote: > On Monday, February 17, 2014 12:31:03 PM UTC+2, Ken Quirici > > > You said that constructions are merely visualizations. Of what are they > > > visualizations? > > > > It should first be stated that Ancient Greek philosophers were preoccupied with questions regarding space and time. They wanted to understand these difficult concepts not only in relation to geometry, but with the purpose of clarifying puzzling questions about the universe (Astronomy). > > > > What is a location? How can you tell one location from another? Imagine for one moment that you and another person (let's say your friend) are alone in a void (empty) universe. Your friend can move, but you cannot. You can't see each other. You can't hear each other. You can only communicate telepathically. There is no light, there is absolutely nothing besides the two of you. How would you communicate to your friend where you are so that he can meet you at your location? > > > > You will soon realise after sufficient thought that it is impossible to communicate your position. Now, if you could switch on a bright light which your friend can see, do you think that you can communicate your position? Well, you would increase his chances because he can tell you that the light is above or below him. But it will still be impossible for you to guide him to your location. I don't know if the Ancient Greeks thought of these things exactly as I described, but I am certain they did. > > > > Back to the topic: Any geometric object other than the location (point) consists of paths. If you and your friend are at two points in the universe, there are many paths between you, but there is only one path that is the shortest. In fact, a location is meaningless without prior life/existence, and yet locations exist independently of life. A location only takes on meaning once an origin and frame of reference is established.
Yes. I am sitting about 10 feet from my kitchen. My apartment is some specific distance from the town hall. etc etc
> > > > Points have no dimension. You can't even begin to discuss *density* with regards to points, because density only applies to objects that contain space/volume.
yes you certainly can begin to discuss density with regards to points. The points on a line are dense, in the sense that, once you have that coordinate system you were talking about just above, every coordinate on the line exists as a point on the line - at least in R^2.
> > > > So, the only way to make sense of location/s is through means of paths which have the attribute of length. The measurement of such lengths did not occur until after a result concerning areas (Pythagoras) was established. The immediate problem at hand was how to relate two different points in the universe. Obviously, the answer is in the shortest distance. It would be futile to attempt relating these points by non-reifiable points in between them.
Once you've established the shortest distance between these two points, there is a path, i.e. a line, between them. So suppose the points are 2 units apart. Travel one unit along the path. You are at a point in between them. Travel sqrt(2) between them. &c &c. So what do you mean by non-reifiable points in between them? Here's a perfectly good set of an infinite number of points (I would say uncountable you would say countable) lying between them.
So again, I'm not sure what you're saying. I would be obliged if you could provide arguments that lines don't contain points. Since the line acc. to Euclid, and I think you said it somewhere yourself, has length but no width, i.e. is one-dimensional, what does it contain except points? The one dimension establishes a length on the line, say, from some other point. So the line is THERE at that length, and how can that THERE be anything than what we call a point? There IS 'there' there. A point is just a place on a line, right? Marking some distance from another place on the line. All I mean by continuity is that all the points that SEEM to lie on the line actually DO lie on the line.
Do you object to points considered as places on a line? Continuity then is just that all the places on a line that seem to lie on the line do lie on the line.
> > > > Thus, to cut to the chase, a straight line is the path with the shortest distance between two points. That's the second geometric object to be realised after the point. The next object concerns the path with the shortest distance between three distinct points, that is, the triangle. The square or rectangle or any other 4-sided figure, is the path with the shortest distance between four points. A circle describes the path from which the shortest distance to the circle centre is always the same, wherever you are on that circular path. > > > > Now, imagine investigating these geometric objects by thinking solely in terms of paths with the shortest distance. It is very difficult to perform mental geometry. Thus, the visualisations are a form of the geometric object. I agree with Plato that there are perfect forms. But the visualisation is never a perfect form and not intended to be anything else besides a tool used to investigate the properties of geometric objects. > > > > So, to answer your question, the constructions are merely visualisations of intangible geometric objects whose chief attribute is the path. > > > > > Your proof is 'merely' a sequence of visualizations. This is > > > exactly my problem with Euclid. It depends on visualizations. > > > > As you should be able to see now, this is clearly not true. It would be very difficult to describe the properties without reifying the geometric objects through visualisations.
So you're saying that Euclid doesn't depend on visualizations, but that it's just difficult to understand without reifying the geometric objects through visualisations. This seems to imply that the geometric objects exist independent of visualisations (or visualizations - I'm going to waffle between these spellings). This is partly my point in wondering what OTHER visualizations the purely verbal/logical relationships among the propositions would be conducive to. > > > > > The words themselves, the constructions, have no meaning except through our > > > own human intuitive understanding of concepts like continuity they depend on > > > these concepts to have a concrete 'real' (no pun intended) meaning. > > > > Not so. These concepts (geometric objects) exist entirely outside the human mind or any other mind. An alien in a distant galaxy could arrive at our knowledge through these same ideas. The concepts are not human or alien.
I misspoke. Certainly I am sympathetic to the notion that there are aliens in the universe with at least our level of intelligence, and sentient. They would probably have a similar intuitive sense of continuity as we do. However my point was really that Euclid's Elements depends on the notion of continuity - geometric objects depend on our notion of continuity. The abstract circle contains all its points.
In fact, your notion of 'path' is saturated with the notion of its continuity, and requires, as it did to Euclid, the notion of points on the path. He in fact specifies that a point has no dimensions, but yet it exists. He also makes reference to points on lines throughout his work. Every proposition refers to points on lines. Yes, sometimes it's just endpoints of line segments, but nevertheless, these are points on the extended line (which he posits can be extended indefinitely in either direction).
Therefore the line is full of points, which are the end-points of all the line segments that a line contains.
I suppose you would grant the existence of these points for the rationals, but not what you would call incommensurable quantities? Even though for example sqrt(2) is a constructible quantity? But that's a side issue.
My point about the intuition people (and other sentient beings probably) have about the continuity of a line, even that abstraction of physical lines that we do in geometry, and Euclid did in his Elements, is that these abstract lines contain certainly an infinitude of points. In fact Euclid describes points as lying on lines.
How then can points not exist in a formal geometry? They are not quantities, although they can lie at a certain distance from another point on that line, and can therefore be associated with the distance to that line. But that is an artificial association. Just like doing analytic geometry in two dimensions is an arbitrary positioning of a coordinate system, and assigning numerical coordinates to various points in the coordinate system (which we're here considering to be two dimensional).
Euclid didn't 'have' analytic geometry - Descartes is credited with inventing it, and the two-dimensional coordinate system. But he did have a very concrete and specific 'awareness' of points lying on a line.
> > > > As far as being real, the concepts are *very real* indeed. They exist independently. > > > > > It would be an interesting exercise (at least it seems to me) to imagine this > > > Euclidean axiomatic system INDEPENDENT of ANY pictures. > > > > It can be done as I showed with some simple geometric objects such as the straight line, but even the Great Archimedes used visualisations. I do too when I lie awake at night. :-) I have what one might call a "photographic memory". I can see complex geometrical structures clearly in my mind's eye - almost as if they were drawn on paper. > > > > > How many 'realities' would this system model? I suspect there are many > > > that have absolutely nothing to do with geometry. > > > > That is simply not possible. Every concept is related and has its origins in geometry - the study of location and space. Remove geometry and you have NOTHING. > > > > > That is, take the axioms as simply words that relate to each other in logical > > > ways and see what they can represent. > > > > Well, words convey "meaning". A meaning takes on different forms, but the main vessel in geometry for conveying meaning is visualisation. > > > > >> As previously stated, that the two circles intersect depends on showing that > > > a line between two points, one outside the circle and one inside the circle, > > > meets the circle. > > > > Not at all. Which point do you imagine is outside and inside the circle? And which circle are you talking about?
I was a little incoherent there. Imagine an abstract circle with a point inside and a point outside and a line connecting them. Any kind of line, straight or curved. This line will intersect the circle. Why? Without invoking the continuity of the line, you can't prove this intersection. Imagine that the line is missing one point - that at which it intersects the circle. When you define a line, if you don't know that the line is continuous, how do you know it isn't missing points? Are your 'incommensurable quantities' on that line, in the sense that given point A on a line, and given that the line is say two units long, is there a point B that is sqrt(2) away from it? If not, then suppose the line intersected the circle at a point that was sqrt(2) away from one of the points, either inside the circle or outside? No intersection.
> > > > As for a line (any path), this is established long before the proposition in the definitions. There is no question that many lines exist between the centres of the circles, and that many lines exist between the intersections. > > > > Any line is simply a path. >
The notion of path has no meaning if it doesn't include the notion that points lie on it, and that ALL points lie on that are between the two points the path connects.
> > > > Doesn't it? It seems obvious to the intuition, but that intuition is > > > based on prior concepts of the continuity of the plane and of lines on the > > > plane. > > > > Has nothing to do with continuity of *anything*. > > > > > How so? Are you saying that because the real number line is 'only' points > > > then it can't 'hold together'? > > > > I have answered this question in previous paragraphs. No line consists of points. The chief attribute of any line is the distance described by the path.
And even Euclid would say that a distance is a distance between two points. If they don't lie on the line, which even Euclid grants, then what points is he talking about?
> > > > > If you held the real number line at both ends then it would fall apart > > > because there's no 'glue' holding the points together? > > > > It does not need to be held anywhere. It does not consist of points. Points can be reified on the line after measurement is established, but until then, points don't determine the path. Points are a consequence that arises when a path is described and not vice-versa. > > > > > This is in fact the whole point of the real number line. You've got it! The > > > real number line is NOTHING but a set of points that is so dense as to give > > > MEANING to the concept of continuity. > > > > No. :-) > > > > > This density is the glue holding the real number line 'together'. This is the > > > beauty of mathematics - it lives in its own world. > > > > Oh I could not disagree more! The beauty of mathematics lies in well-defined concepts.
Don't you mean well-defined visualizations?
> > > > > I hate to invoke the Platonic ideals because I don't really believe in them, > > > but they do seem to push themselves into the discussion. > > > > The Platonic ideals are very relevant. I think you forget that were it not for Plato, we probably would never have had the knowledge of mathematics, or any other knowledge. As mathematics is the queen of sciences, philosophy is the king of all knowledge. Without philosophy, no knowledge could be known. > > > > > This seems to be you saying what I picturesquely described above as holding a > > > real-number line - that is, a line 'built' only from real numbers (which > > > include of course rationals) - at both ends and it falls apart because it has > > > no glue. > > > > But real numbers do not exist. In order to understand this, you have to start with a magnitude, which is the idea of size, dimension or extent. The magnitudes the Greeks dealt with are related to paths. Numbers cannot be known without magnitudes. A magnitude need not be a distance, it can be an area, volume, mass, speed, etc. The measurement of a magnitude gives rise to the number. You can understand this from my 5 point derivation of number from scratch.
One of Euclid's famous constructions is the sqrt(2). The hypotenuse of a right triangle with sides of lengths 1 and 1 has a hypotenuse whose length is sqrt(2). So you can construct sqrt(2) as the length of a path, and yet sqrt(2) doesn't exist? So you discard all the lengths of hypotenuses that are irrational in length even though constructible? You must then discard all crcles which have unit radii, since their circumferences have a length of pi, which is demonstrably (so I've been told) irrational. In fact all circles unless their radii are some a/pi. But then the radii can't exist since they're irrational. So your universe contains no circles. All your lines are porous.
> > > > > YOUR conceptualization depends on the existence of glue on the line, glue > > > that is not points - what is it then? > > > > Has nothing to do with glue. :-) > > > > > It's like phlogiston or the ether - some intuitive superfluity. > > > > Oh no, it's very, very real. Rational Numbers are very real and exist independently of any mind. > > > > > I started reading your unpublished book and had a question about some of your > > > definitions in another thread. > > > > Nope. You have not read my unpublished book. :-) No one has. You read an article on the New Calculus. That was intended to communicate some of the ideas. My book is over 2000 pages long. It contains three sections: The New Elements, The New Calculus and a new kind of mathematics based entirely on tangent objects. > > > > > > > This question arose very early in my reading. It's possible that, even though > > > my not understanding some of your definitions is preventing me from finding > > > purchase on your presentation, that what you're saying early in the book - > > > your very first illustration and the text that follows - is really, even if > > > flawed, not necessary to the heart of your argument. > > > > I have no idea exactly which illustration you are referring to. Could you be more specific? Oh wait, you mean the illustration demonstrating some of the paths between two points? That's in the document called Calculus for Dummies. > > > > It is absolutely essential to the understanding of the New Calculus. Therefore, it's an important illustration! :-) > > > > > It would nevertheless be nice to be able to advance, from the beginning, in a > > > step-by-step manner with each step agreeably understandable to me. > > > > Well, I'll be glad to answer your questions whenever I can. > > > > > So if you could find that thread and my question I would appreciate a > > > clarification. It may not work but it may. The thread was the one titled The > > > New Calculus. > > > > I really don't know which "thread" you are referring to. Sorry.
The thread I'm referring to is called 'The New Calculus - The first and onlhy rigorous formulation of calculus in human history' and was last updated on Feb. 16. I have only one post there. I would appreciate it if you would do me the favor of reading my post. If you get a chance.
The document in question was
The diagram in question was Fig. 1 and showed a curve with one point having the upper-left corner of a Trapezium touching it, so that what you call the variable side is tangent to the curve.
The paragraph explaining it is immediately following Fig. 1.