KQ: Yes. I am sitting about 10 feet from my kitchen. My apartment is some specific distance from the town hall. etc etc
JG: You got the idea! :-)
KQ: 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.
JG: You are not thinking properly. :-) The concept of density means: "the quantity per unit area, volume or unit length". Since you can't count points, the notion of point density is rot. You can't just say things like "every coordinate" without having pre-established "real numbers."
As I've said, essential to the topic is the fact that *no valid construction of irrational numbers exist*, and hence real numbers don't exist either. So you can't talk about R^2 in terms of anything but magnitudes. It is meaningless nonsense, unless understood in terms of magnitudes. But magnitudes are NOT numbers.
It's easy to reify points on a number line if rational numbers are being used. Before I continue, please take a moment to study the following comments, because they are of utmost importance to the discussion. I know from experience that not even one professor of mathematics has ever been able to tell me the difference between a magnitude and number. In other words, all of the professors of mathematics I have known, do not know what is a number.
The first link shows you how a point can be reified on the number line as an invisible marker (essentially just a point!).
The second link explains the difference between magnitude and number.
KQ: Once you've established the shortest distance between these two points, there is a path, i.e. a line, between them.
KQ: 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?
JG: If you are exactly halfway, then you can reify the midpoint as an invisible marker, that is, 1 unit. If you have travelled a magnitude of sqrt(2), then you cannot reify the invisible marker because there is no way for you to arrange two line segments adjacently such that an invisible marker corresponding to a measured magnitude, can be placed between them. What this means is that you cannot mark off sqrt(2) on the number line by means of a point. Very important to understand this! Sure, you can construct a line of magnitude sqrt(2), but you cannot place a marker between sqrt(2) and another line segment that has a corresponding measured magnitude (a number). The first abovementioned link explains these details.
KQ: Here's a perfectly good set of an infinite number of points (I would say uncountable you would say countable) lying between them.
JG: I say there is no such thing as an infinite number of points which can be called a set. You would first have to name every single one of those points in order to call it a set. Since there is no way of doing this, the elements themselves are not defined, so I don't see how you can even begin to speak of a set. It's absurd.
KQ: 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.
JG: How can anything contain a point? It has no dimensions. I can say that on a number line marked with units, that certain points signifying units can be placed inbetween. But in order to talk about other points, one would first have to show that they are reifiable, that is, you can create an invisible marker for every distance from the origin on the number line. You can't!!!
KQ: 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?
JG: The line you imagine is only a visualisation. There is NOTHING between the two points. NOTHING.
KQ: The one dimension establishes a length on the line, say, from some other point.
JG: Only if the point corresponds to a rational number.
KQ: So the line is THERE at that length, and how can that THERE be anything than what we call a point?
JG: It is an incommensurable magnitude for which there is no corresponding invisible marker (point).
KQ: There IS 'there' there. A point is just a place on a line, right?
JG: That's like saying, "There is a *nothing* on the line". A line is a concept. It is not a visualisation.
KQ: All I mean by continuity is that all the points that SEEM to lie on the line actually DO lie on the line.
JG: Nothing lies on the line. It has no width. There is nothing between the endpoints of a line except a path describing the distance. The path itself is not a tangible object. It cannot consist of *nothings* (points).
KQ: Do you object to points considered as places on a line?
JG: Not if they can be reified. This means that each point corresponds to a rational number.
KQ: Continuity then, is just that all the places on a line that seem to lie on the line, do lie on the line.
JG: You're beating a dead horse.
KQ: 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.
KQ: This seems to imply that the geometric objects exist independent of visualisations (or visualizations - I'm going to waffle between these spellings).
JG: Of course!!
KQ: This is partly my point in wondering what OTHER visualizations the purely verbal/logical relationships among the propositions would be conducive to.
JG: A visualisation is in some sense a reification of the geometric object. It is a form, but not the ideal. :-)
KQ: 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.
JG: Not if they think like I do! :-)
KQ: 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.
JG: Not so.
KQ: 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.
JG: My notion of path has *nothing* to do with continuity. Just think about the word *continuity* for a moment, because I don't think you understand it very well. In order to say *anything* about continuity, you must have well-defined objects. You can talk about a row (path) of men holding hands as being continuous. If they do not hold hands, then the row (path) is not continuous. You must establish how to determine continuous and discontinuous in terms of the objects that are its reason for existence.
If I have two points, you cannot tell me how you will distinguish continuity because there is no way for you to know if there are any points inbetween or infinitely many. There are no points inbetween 1 and 3 on a number line calibrated with natural numbers. Since there are no real numbers, you can't calibrate the real number line.
KQ: He in fact specifies that a point has no dimensions, but yet it exists.
JG: As the concept of location or place only. Other than that it is meaningless.
KQ: He also makes reference to points on lines throughout his work.
JG: Reifiable points usually through the intersection of paths.
KQ: 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).
JG: Again, reifiable points.
KQ: Therefore the line is full of points, which are the end-points of all the line segments that a line contains.
KQ: I suppose you would grant the existence of these points for the rationals, but not what you would call incommensurable quantities?
KQ: Even though for example sqrt(2) is a constructible quantity? But that's a side issue.
JG: Not at all, it's a very valid example. Sqrt(2) is a constructible magnitude, but not reifiable as a point between two known line segments.
Important: It is insufficient to place a marker between two line segments. That marker must correspond to the measure of the magnitude (an established number).
KQ: 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.
JG: Only reifiable points.
KQ: How then can points not exist in a formal geometry?
JG: But of course they do! Every geometric object is defined from the concept of point.
KQ: 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.
JG: Only if they correspond to a measurable magnitude.
KQ: 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.
JG: Descartes is my favourite French mathematician. He discovered the Cartesian plane because he wanted to clarify the thought paradigms of Ancient Greeks. The Cartesian plane is only a visualisation of what the Greeks already knew existed. Bear in mind that one of Descartes goals in the C. plane was to establish a simple means of constructing the symptoms of conics.
KQ: 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?
JG: Of course it will intersect the circle because both the circle and the line are *paths*. If paths cross, then they intersect, don't they? :-)
KQ: Without invoking the continuity of the line, you can't prove this intersection.
JG: You can. Paths cross.
KQ: Imagine that the line is missing one point - that at which it intersects the circle.
JG: I can't, because a point has no dimension. So much so, that it can be used as an invisible marker to calibrate a straight edge!!!!!!
KQ: When you define a line, if you don't know that the line is continuous, how do you know it isn't missing points?
JG: A line does not consist of points. It is the concept of distance described by a path. The ability to determine distance came much later with the Pythagoreans. It was a monumental accomplishment.
KQ: 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?
JG: It's possible only if I think of sqrt(2) as a measurable length, which of course it's not. Therefore, it's not possible.
KQ: 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.
JG: First and foremost, *intersection* refers to the crossing or meeting of paths. It is not influenced whether a point is there or not.
KQ: 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.
JG: Of course it does. A path's chief attribute is describing the distance between two points, NOT THE POINTS BETWEEN THE TWO ENDPOINTS!
KQ: 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?
JG: He is not talking about points, but about distance.
KQ: Don't you mean well-defined visualizations?
JG: No. Do you think Euclid imagined the visualisations first and then developed the theory? Absurd.
KQ: 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?
JG: Correct. sqrt(2) does not exist as a number.
KQ: So you discard all the lengths of hypotenuses that are irrational in length even though constructible?
JG: No. I realise they can be approximated.
KQ: 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.
JG: A magnitude is not a number. :-)
KQ: In fact all circles unless their radii are some a/pi. But then the radii can't exist since they're irrational.
JG: A magnitude is not a number. :-)
KQ: So your universe contains no circles. All your lines are porous.
JG: I don't think the term "porous" applies, because lines aren't made up of points.
KQ: The thread I'm referring to is called 'The New Calculus - The first and only 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 in NewCalculus-Abstract-Part-1.pdf
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.
JG: You mean this:
KQ: I must be misunderstanding your usage of the term 'ordinate'.
Wikipedia gives the meaning I usually associate with the word, namely, y-coordinate of a point on the plane. Is that your use of the term?
KQ: If so, and in the case of the Trapezium you present, that is, where the variable side is tangent to some curve, you're saying this tangent line intersects the curve at exactly two points?
JG: NO. That is not a tangent line. It is a secant line that is parallel to the tangent line whose gradient is the mean value.
The variable side of the trapezium is the parallel secant line.