Posted 22 Sep 2005 · Report post When a given group of existents has more than one characteristic distinguishing it from other existents, man must observe the relationship among these various characteristics and discover the one on which all the others (or the greatest number of others) depend, i.e., the fundamental characteristic of the existents involved, and the proper defining characteristic of the concept. Metaphysically, a fundamental characterisitc is that distinctive characteristic which makes the greatest number of others possible; epistemologically it is the one that explains the greatest number of others. I know of three definitions of parabolas (listed below), based on three different characteristics of the parabola. I don't see how to apply the criteria that AR mentioned to determine the proper, or most fundamental, definition. I believe that each characteristic mentioned could be used to show that the other two must be true. Therefore it seems that each characteristic leads to the other two, and each can be thought of as depending equally on the others. Therefore which is the most fundamental?1) A parabola is the curve formed by the border of the region generated when a (right-circular) cone is intersected by a plane parallel to the slope of the cone. (See here.)2) A parabola is the set of all points in a plane equidistant from a fixed line (the directrix) and a fixed point (the focus).3) A parabola is the curve generated by the set of all points graphed on a Cartesian plane satisfying a quadratic equation, i.e, an equation that can be written in the form y=ax^2 + bx + c (where a does not = 0). Share this post Link to post Share on other sites

Posted 23 Sep 2005 · Report post From what I know, I think your implication that each of these characteristics is equally fundamental is correct. So I think they would all be valid. Therefore, which definition you choose would be dependent on the context. That is, what is your reason for needing a definition of a parabola? What are you trying to differentiate parabolas from? What is your context of knowledge? (Or: what is the context of knowledge of the person to whom you'll be explaining the concept of parabola; i.e., the person who will have to use the concept.) A good definition should serve to connect a new concept to what one already knows.I remember in my education, I first learned about parabolas (in first year algebra) as curves describing the solutions to quadratic equations: probably close to definition (3). At that time, this definition was appropriate for us, because we had just learned about some quadratic equations. So it showed me why the concept of parabola was important. If I had been presented with one of the other two definitions, I'm afraid the concept would have been a floating abstraction to me. I'd have understood the definition, but my thought would have been: "So what? Why is this important, and what does it have to do with what else we're learning?"I next remember learning the focus/directrix definition (2) of a parabola in second year algebra. We were at the time learning about some other plane figures too (circle, ellipse, hyperbola), and so this definition served to diffentiate parabolas from the others, as well as show their similarities. (But back in first year algebra, we had not had this context yet.) This definition also might be useful in certain physical situations in which you have a parabolically shaped reflector that focuses sound or light onto a point (the focus).I can't remember where I heard definition (1). It might have been to distinguish parabolas from other conic sections: circles, ellipses and hyperbolas again (which are also intersections with cones). Note that this one is more complicated than (1) or (2), since it requires one to visualize a three-dimensional figure (cone) being cut by a plane. Because of this, it seems this definition would be harder to fit into a conceptual hierarchy: it depends on one already having a definition of a cone, some knowledge of three-dimensional figures, and what happens when a plane intersects one. But if one already has definitions of those things, I don't see that this definition of a parabola is less fundamental than the other two offered.Somebody with more knowledge of mathematics than I have could better say whether one of these definitions is more fundamental than the other two. Share this post Link to post Share on other sites

Posted 25 Sep 2005 · Report post Nate:In the quote from ITOE I see no prohibition against the existence of multiple definitions, each which must be equally fundamental.In your example, however, I believe some distinctions can be made. Definition #3 relies upon a Cartesian coordinate system, a rather late development in the history of geometry. Furthermore, it is an analytic approach to the definition, requiring algebraic notation. A large set of non-trivial concepts are involved.Definition #1 defines a 2-dimensional figure using higher-order 3-dimensional concepts. In addition, the non-trivial concepts of parallelism and slope are involved. This is an unnecessary complication of the definition.Definition #2, therefore, is the preferred definition. It uses only concepts that must be grasped prior to defining a parabola (a point, a line, and distance). Share this post Link to post Share on other sites

Posted 26 Sep 2005 · Report post In the quote from ITOE I see no prohibition against the existence of multiple definitions, each which must be equally fundamental.←Remember what the function of a definition is: it is to mark out which units a concept refers to and differentiate them from the things it doesn't refer to within an individual's context of knowledge. As a person's context of knowledge expands, his definitions may have to change. A definition of "man" as "something that moves and makes noises" may be a perfectly good definition for an infant because it differentiates Mom and Dad from the crib and the pacifier. When he is an adult, defining "man" as "the rational animal" is more appropriate. Share this post Link to post Share on other sites

Posted 3 Oct 2005 · Report post Remember what the function of a definition is: it is to mark out which units a concept refers to and differentiate them from the things it doesn't refer to within an individual's context of knowledge. As a person's context of knowledge expands, his definitions may have to change. A definition of "man" as "something that moves and makes noises" may be a perfectly good definition for an infant because it differentiates Mom and Dad from the crib and the pacifier. When he is an adult, defining "man" as "the rational animal" is more appropriate.←I agree entirely and wholeheartedly. The question is this: which definition comes first and which are the more expanding definitions. This question is separate from which one is more fundamental, which generally comes with the more expansive definition. To continue with the example of "man" (which you cite), the most fundamental definition comes last ("rational animal"). Doesn't this raise a more foundational question? Which is the proper order of foundation for algebra and geometry? In other words, do algebra and geometry arise simultaneously in the hierarchical acquisition of knowledge? I think so. I also think that Cartesian (or analytic) geometry comes pretty clearly after algebra & geometry, so the analytic-geometric formulations would be hierarchically after the discrimination of parabolas versus hyperbolas and other conic sections. But I think that the maximum explanatory power comes from the "equations of the form" definition, which most precisely *measures* the parabola. So I think that the "equations of the form" definition is the most fundamental, as it has the greatest explanatory power. And I think that it is an artifact of our modern approach to math education, which places algbra prior to geometry, when they really arise hierarchically at the same point, that results in Jay's assumption that the conic-sections definition is later in the expansion series of definitions. I think that grasping the conics is prior to all the rest. Every child of 6 has graped a cone [literally!, as that is what ice-cream goes on! :-) ] and a plane is one level of abstraction up from 3-D objects. (3-D shapes are the given, perceptually; and 2-D surfaces are an abstraction from them; and lines and points are yet another level of abstraction) So these three levels of abstraction have to be reached and are well before algebra or synthetic geometry. So, these curves are graspable by a learner using the conic-sections approach well before the analytic geometry definitions are graspable. The "locus of points equidistant from a point and a directrix" definition is so comparatively subtle, that I don't see how a learner could grasp it without having grasped the conic-sections elements. Share this post Link to post Share on other sites

Posted 5 Oct 2005 · Report post I agree entirely and wholeheartedly. The question is this: which definition comes first and which are the more expanding definitions. This question is separate from which one is more fundamental, which generally comes with the more expansive definition. To continue with the example of "man" (which you cite), the most fundamental definition comes last ("rational animal"). ←But "moves and makes noises" is just as fundamental in a baby's context as "rational animal" is in ours. Both keep units of the concept clearly distinct from non-units of the concept within a particular context of knowledge. As our context of knowledge expands, let's say we discover intelligent life on other planets, a more fundamental definition might be a rational animal of terrestrial origin.Doesn't this raise a more foundational question? Which is the proper order of foundation for algebra and geometry?Any order that proceeds logically from sense perception and gains and keeps values is "proper." In going from perception to higher order abstractions in mathematics, there may be many options. Mathematics is a science of method, and there may be several ways of achieving the same result. In other words, do algebra and geometry arise simultaneously in the hierarchical acquisition of knowledge? I think it depends on whose hierarchy you are talking about. People have different goals and mental abilities. One child might be sensitive to quantities, another to shapes, and another to processes. One might use math to keep track of his money and other to build something to precise measure. They may all end up with the same knowledge, but have learned it in completely different orders. Share this post Link to post Share on other sites

Posted 6 Oct 2005 · Report post I think it depends on whose hierarchy you are talking about. People have different goals and mental abilities. One child might be sensitive to quantities, another to shapes, and another to processes. One might use math to keep track of his money and other to build something to precise measure. They may all end up with the same knowledge, but have learned it in completely different orders.←We aren't discussing the hierarchy of values. Nor are we discussing the accidental, chronological order of acquisition. There is only one objective hierarchy of knowledge for everyone. From ItOE:Prof. E: ....Then is it true that ... once the conceptual apparatus has been developed and you establish a logical hierarchy, that hierarchy is invariant for human beings, being dictated by the nature of the concepts, with no option as to which concept is higher-level and which is lower-level?AR: Correct.Now, our question is this: given that we have three definitions of the ellipse, which one is the most fundamental? And what is the hierarchical order of acquisition of those principles is a very different question from that. Hierarchical dependence is shown by the need to grasp one principle in order to logically grasp the other. Fundamentality is determined by explanatory power (or metaphysical dependency). Share this post Link to post Share on other sites

Posted 7 Oct 2005 · Report post Now, our question is this: given that we have three definitions of the ellipse, which one is the most fundamental? And what is the hierarchical order of acquisition of those principles is a very different question from that. Hierarchical dependence is shown by the need to grasp one principle in order to logically grasp the other. Fundamentality is determined by explanatory power (or metaphysical dependency).←Which definition is the most fundamental is still an issue of context and motivation. For instance, my main experience with ellipses involves the shape as a graphic design element. How I arrive at a fundamental definition that adequately explains what I mean by the word "ellipse" would differ from that of a mathematician. The one thing that my definition does have in common with a mathematician's is that is that it separates "ellipses" from everything else I know. Since a mathematician knows more mathematical concepts than I do, he might have to be more precise. As a result, we could come up with two quite different definitions that function well as definitions and may very well refer to the exact same units.For instance, I would get my definition by looking at ellipses and seeing how they differ from non-ellipses like circles, or rectangles. Ellipses are like circles in that they are not flat anywhere, but, like non-square rectangles, they are regular top to bottom and right to left but are longer on one axis than on the other. A good beginning definition might be ostensive: something that looks like (point to three different ellipses) as distinguished from (point to a circle) and (point to a rectangle). That's a good fundamental definition for 90% of the population because the fundamental, in their context, is the perceived shape.Now, let's say I want to DRAW an ellipse. I know how to draw a circle with a compass. This works because all the points on the circle are the same distance from the center. Then I learn I can draw an ellipse with a pencil and a string fastened at two points. This works because all the points on the ellipse have the same sum of the distance between the two points. My old ostensive definition still works, but defining it as "set of all points whose sum of distances from two given points (focal points) is fixed" is a more fundamental definition. That is because it is causal. It not only says that an ellipse has a certain distinct shape, but why it has that shape.This is where I stop. This is as good as it gets, or needs to get, for me. Anything else, in my context would just be words until I acquired the mathematical knowledge necessary to understand what the definition was saying and whether it provided any additional causal insight into ellipses. Share this post Link to post Share on other sites

Posted 7 Oct 2005 · Report post Which definition is the most fundamental is still an issue of context and motivation. .... A good beginning definition might be ostensive: something that looks like (point to three different ellipses) as distinguished from (point to a circle) and (point to a rectangle). That's a good fundamental definition for 90% of the population because the fundamental, in their context, is the perceived shape.Now, let's say I want to DRAW an ellipse. I know how to draw a circle with a compass. This works because all the points on the circle are the same distance from the center. Then I learn I can draw an ellipse with a pencil and a string fastened at two points. This works because all the points on the ellipse have the same sum of the distance between the two points. My old ostensive definition still works, but defining it as "set of all points whose sum of distances from two given points (focal points) is fixed" is a more fundamental definition. That is because it is causal. It not only says that an ellipse has a certain distinct shape, but why it has that shape.This is where I stop. This is as good as it gets, or needs to get, for me. Anything else, in my context would just be words until I acquired the mathematical knowledge necessary to understand what the definition was saying and whether it provided any additional causal insight into ellipses.←Hi Betsy. Sorry for the typo where I switched from parabolas to ellipses. I'll start referring back to parabolas and assume your comments would (generally) apply, except for the production mechanism. (You would have to do some other tricks to draw a parabola, involving a fixed line and a fixed point and something that kept your distance the same between both)With all due respect I don't think you are responding to the question. The question was assuming a fully mathematical context of knowledge. The question is wondering what is the most fundamental defnition of the three described within the entire context of knowledge. That is one question. The other question is: what is the hierarchical relation of these three definitions? Does one of these definitions have to be grasped in order to reach the other one? If so, which one? Are you saying that these aren't valid questions? I agree that at a given context of knowledge the definition has to be the most fundamental wrt that context. But the question is, after having learned all that math (and that assumes selecting such a purpose of learning the mathematics of conic sections), which concept has to be grasped first in order to grasp or communicate the other. I guess that I would say this: you are right that "each dfn is fundamental" because you are saying "each dfn is fundamental in it's own given context." But we want to know which one is fundamental given the widest context. And, I raised the point that if there is a definite, non-optional order to grasping these definitions, then we can also determine that the most fundamental one will be the one most posterior hierarchically, following the example of "man." Share this post Link to post Share on other sites

Posted 8 Oct 2005 · Report post The question was assuming a fully mathematical context of knowledge. The question is wondering what is the most fundamental defnition of the three described within the entire context of knowledge. ←Even granted we are talking about "a fully mathematical context of knowledge," the question remains, whose a fully mathematical context of knowledge.A context, like a right, only exists in individuals and you can't talk about a "context" in the abstract without referencing the individual(s) whose context it is. If you do, you lose the tie to reality and you can't answer the question, "Which is the most fundamental definition?" So let's say we are talking about some learned math professors with state of the art mathematical knowledge and concepts.Then it becomes an issue of purpose. Since the purpose of a definition is to distinguish units of a concept from everything else in reality it matters very much what someone seeks to distinguish the concept from and why. Again, we must reference individuals, because purposes, like rights and contexts, are properties of individuals. Someone trying to distinguish parabolas from figures like X may legitimately have a different definition than someone trying to distinguish them from figures like Y.OK, let's say all our professors want to distinguish parabolas from Xs. Now what? The first thing to do is to look at the properties of parabolas and Xs and note in which respects they are the same and in which respects they differ. The former will form the genus and the latter will be the differentia. If you do that, you may come up with two very long lists and that's where fundamentality comes in. Fundamentality is the guide for paring down the list of characteristics to the genus and differentia that do the best job of distinguishing parabolas from Xs.First you look at one list, the list of differences for example, and eliminate similar characteristics. Then you look for causal connections. If a characteristic is caused by another characteristic, eliminate it. You will end up with a small list of fundamental (causal) characteristics that distinguish parabolas from X's. You would do something similar for the similarities to get the genus.And, I raised the point that if there is a definite, non-optional order to grasping these definitions, then we can also determine that the most fundamental one will be the one most posterior hierarchically, following the example of "man."Fundamentality has nothing to do with order and everything to do with how successfully a definition does what a definition is supposed to do: clarify the difference(s) between units of a concept and non-units of a concept. Share this post Link to post Share on other sites

Posted 13 Oct 2005 · Report post Even granted we are talking about "a fully mathematical context of knowledge," the question remains, whose a fully mathematical context of knowledge.A context, like a right, only exists in individuals and you can't talk about a "context" in the abstract without referencing the individual(s) whose context it is. If you do, you lose the tie to reality and you can't answer the question, "Which is the most fundamental definition?" It doesn't seem like we are on the same page here. Ayn Rand most certainly talks about a context for the whole of mankind. This is a widest context. Let me try a different tack. In ItOE, p. 46, AR says, "If definitions are contextual, how does one determine an objective definition valid for all men? It is determined according to the widest context of knowledge available to man on the subjects relevant to the units of a given concept." That is what we want to do with the three definitions of parabola. We want to determine "an objective defintion valid for all men." We want to determine it "according to the widest context of knowledge available to man on the subject." I think this is the formulaic definition, because it reduces the parabola to a mathematical law.I agree that this has nothing to do with the chronological order. However, I would say that generally the search for fundamental law comes much later than an early or an initial grasp of how to distinguish something. This was the pattern of man, where the most fundamental definition comes at the end of the chronological progression. Share this post Link to post Share on other sites

Posted 14 Oct 2005 · Report post It doesn't seem like we are on the same page here. Ayn Rand most certainly talks about a context for the whole of mankind. This is a widest context. Let me try a different tack. In ItOE, p. 46, AR says, "If definitions are contextual, how does one determine an objective definition valid for all men? It is determined according to the widest context of knowledge available to man on the subjects relevant to the units of a given concept." .←As "the widest context of knowledge available to man on the subjects relevant to the units of a given concept" expands when men acquire more knowledge about the units of a concept, the definition may have to be changed.That is what we want to do with the three definitions of parabola. We want to determine "an objective defintion valid for all men." We want to determine it "according to the widest context of knowledge available to man on the subject." I think this is the formulaic definition, because it reduces the parabola to a mathematical law.Does it? Or does an objective, fundamental definition integrate all that is currently known about the units of a concept that best distinguishes them from other existents.Correct me if I am wrong, but I get the impression that you are looking for a "law" -- an unchanging correct definition that can always be used as a standard of a "real" parabola.I agree that this has nothing to do with the chronological order. However, I would say that generally the search for fundamental law comes much later than an early or an initial grasp of how to distinguish something. This was the pattern of man, where the most fundamental definition comes at the end of the chronological progression.I don't understand what you mean by a "fundamental law" and what it has to do with definitions. A "fundamental law," as I use the term, is a wide-ranging identification that explains or describes a large class of phenomena in basic causal terms. Examples would be Newton's Laws of Motion or the Law of Identity. A definition, on the other hand, is a cognitive tool used to keep one's concepts well organized so that the units they reference can be identified and used unambiguously. Share this post Link to post Share on other sites

Posted 14 Oct 2005 · Report post As "the widest context of knowledge available to man on the subjects relevant to the units of a given concept" expands when men acquire more knowledge about the units of a concept, the definition may have to be changed.Yes. And we want to know, given the widest context available today what the most fundamental definition is. That is what the question is. They can't all be the most fundamental. So which one is. Does it? Or does an objective, fundamental definition integrate all that is currently known about the units of a concept that best distinguishes them from other existents.It does. It specifies formulaically the exact characteristics of the units. Their measurements are omitted, but the relationship they must all bear is retained in the formula: they can be any of the values that render the formula true. Furthermore, the other definitions, can all be derived as consequences of the formulaic definition.Correct me if I am wrong, but I get the impression that you are looking for a "law" -- an unchanging correct definition that can always be used as a standard of a "real" parabola.Wrong here. It never ocurred to me to look for a non-open ended law. However, it is an interesting point. Doesn't Ayn Rand specify that mathematics is unusual in that the definitions of it's objects never change? [i don't have my ItOE at hand]. I don't understand what you mean by a "fundamental law" and what it has to do with definitions. A "fundamental law," as I use the term, is a wide-ranging identification that explains or describes a large class of phenomena in basic causal terms. Examples would be Newton's Laws of Motion or the Law of Identity. A definition, on the other hand, is a cognitive tool used to keep one's concepts well organized so that the units they reference can be identified and used unambiguously.The parabola is defined by a formula, which both identifies the nature of it's units and consequently discriminates it from other two-dimentional shapes, but also explains a large class of phenomena in basic causal terms. So this formula is not only definitional, but specifies the nature of the units as an implicit proposition: "all parablolas are equations of the form ... &ct.Perhaps it was unfortunate to use the term law. It is a formula. The formula retains only the relationships and omits the particular values held by any parabola. The formula discriminates between equations which are non-parabolic and those which are, so in that sense it is definitional. It also has the greatest explanatory power of what constitutes a parabloa. In that sense it is most fundamental. Share this post Link to post Share on other sites

Posted 15 Oct 2005 · Report post Perhaps it was unfortunate to use the term law. It is a formula. The formula retains only the relationships and omits the particular values held by any parabola. The formula discriminates between equations which are non-parabolic and those which are, so in that sense it is definitional. It also has the greatest explanatory power of what constitutes a parabola. In that sense it is most fundamental.←If that's what you mean, it sounds like that might be a good definition. Share this post Link to post Share on other sites