thomtg

The Opposite of Nothing Is/Isn't Everything

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thomtg    0
Thom,

You proposed, in #244 , classifying nothing under the derivative concept absence, which is a privative. What do you mean by the qualification "derivative"?

A magnolia leaf is greener and shinier on the light-catching side of the leaf than on the under side. The opposition of greenness to brownness and the opposition of shininess to dullness are plausible privative oppositions. But the opposition of shape of the leaf to the shininess of the leaf is simply difference, and it is labored to squeeze flat difference into a privative opposition. (Compare with paragraphs 58 in the "Analytic Constraint" subsection of Genesis.)

Further, it seems natural to take all privative oppositions to be types of difference-relations; while it seems contrived to take all difference-relations to be types of privative opposition. A or not-A, exists or not, these are straightforwardly difference-relations.

In #241 you remarked that "anything that is utilized [sic] by man can be counted (i.e. measured), not just entities." It is true that things besides entities can be counted; events can be counted. But counting is not the only type of measurement, and the type of measurement appropriate to a given magnitude structure is determined by that structure. Counting is the form of measurement known as absolute measurement. There are others (A, B, C).

On whether zero and the imaginaries are numbers, I go with the mathematicians. There is a history of the discipline of mathematics, and it has its splendid reasons.

In addition to histories of mathematics, I can recommend The Nature of Mathematical Knowledge by Philip Kitcher and Naturalism in Mathematics by Penelope Maddy.

Stephen,

What do I mean by "derivative" in my classification of "absence"? I mean by the qualification that the concept "absence" cannot be formed without having first formed the concept "presence." Here is Ayn Rand's explanation: "One can arrive at the concept "absence" starting from the concept "presence," in regard to some particular existent(s); one cannot arrive at the concept "presence" starting from the concept" absence," with the absence including everything." (ITOE 58c) In my view, both "zero" and "nothing" are species of "absence"--the former in regard to quantity, the latter in regard to everything.

---1

I disagree that the pair greenness and brownness, and the pair shininess and dullness are privative oppositions. They are contraries, not privative oppositions. Separately, I consider the shape of the leaf and the shininess of the leaf as incommensurables; they are different at the basic level, but they are not related in any opposition. An opposition presumes a commensurable standard of unit. (For example, a contradiction is an opposition in which the relata share the standard unit of truth.)

I do agree, however, that the difference relation is more fundamental than the positive-privative opposition.

---2

The links you give in ROR show that there really are other forms of measurement. Would you agree with Rand that mathematics is the science of measurement?

---3

Speaking of math, I too would defer to mathematicians to determine whether zero and imaginaries are numbers--provided that they adhere to proper epistemological criteria of mathematical knowledge. David Ross, for instance, is a mathematician who takes a stand on an epistemological standard that disqualifies imaginaries as numbers. Is his standard the same as other mathematicians' in your Post #279 (above) and Post #286? I think not. If not, which set of criteria is proper for mathematics? The answer can only be provided by epistemologists, not mathematicians. (ITOE 78b, also root post and my Post #198)

The present question is whether "zero" is a quantity or an absence of any quantity. The second choice is the one found in dictionaries. (Merlin's Post #188) And given my understanding of positives and privatives, just as blindness pertains not to the eye but to the state of privation of someone with eyes, so "zero" pertains not to a quantity but to the absence of any quantity.

---

On a related point, recall that I implied in #244 that a privative is a pole/relatum in an oppositional relation. I also stated that absence (and presence by implication) is categorized as a relation in Aristotle's categories. What in reality is this absence relation? This is a question of ontology that should pique anyone with interests in "zero" and "nothing."

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Guyau    0

Thom,

Nothing is no thing at all. I do not see how it can be a species of anything with identity. Absence has identity, consistent with your remarks, in the sense that particular possible existents can be found absent. But as with possibility, so absence; they have specific connections to some of existence. Not so with nothing.

I thought that positive state and privation were the principles of contrariety and contraries. Contrariety is a type of relation that permits comparisons to be made, the fundamental contraries are positive state and privation, and this fundamental one is the principle and cause of all others. (I’m taking this from a fourteenth century Dominican, whom I happen to be studying at this time. He takes himself to be true to Aristotle, but maybe not all Aristotelians have shared his understanding of these relationships.)

I was thinking of greenness as something one could obtain from brownness by removing some pigments that make up the latter. I was thinking of the shiny surface as restricting the angles of scattered light in comparison to the angles from the dull surface. I agree, however that the contrast of the shape of the leaf with the shininess of one side of the leaf is not a contrast along a common genus, other than the genus attribute (of a leaf in this case). That broad genus does not yield a commensurable dimension between shape and shininess. I think we agree on that (see the subsection “Superordinates and Similarity Classes” here. On this we are in contradiction of Rand’s essay (ITOE 23), but in oral exchange, she seemed to lean our way at this high level of genus (ITOE 275–76).

I cannot go your way on two sides of an issue in contradiction having an opposition that has a commensurable dimension, specifically the goal of truth shared by both sides. That is like the leaf, were it taken as the commensurable dimension of all its attributes. The attributes are radically dependent on the particular entity, but I would not take the opposition of shininess of the entity and shape of the entity to be opposition along a commensurable dimension. The opposition of contradictories seems even less likely to be an opposition along a commensurable dimension: at least an entity can have shape and shininess at the same time. Not so with the sides of a contradiction.

Rand’s conception of mathematics as the science of measurement is a classical one, and it remains a good first approximation. But mathematics has progressed and expanded. There are parts of mathematics that are not measurement theory. Even the part of mathematics known as measure theory (Lebesque integration, etc.) is not theory of measurement. It would be a watering down of the concept of measurement to the point of uselessness to take all of mathematics to be theory of measurement. (We physics folks—at least the children of Physics at Chicago—do not refer to mathematics as a science, but as a theoretical discipline. We are referring to the same discipline as those who follow the ancient tradition of calling mathematics a science.) Rand’s measurement-omission analysis of concepts has punch only because she there uses measurement in a sufficiently narrow sense to be saying something substantial and novel. Rand’s knowledge of mathematics and measurement theory was limited. It is my understanding that she was trying to learn more about them. I should mention that very important discoveries in theory of measurement were made a few years after her death. I learned the pertinent mathematics for her measurement analysis of concepts. My essay “Universals and Measurement” is Chapter 2 to her Chapter 1.

I’ll rest now.

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thomtg    0
Thom,

Nothing is no thing at all. I do not see how it can be a species of anything with identity. Absence has identity, consistent with your remarks, in the sense that particular possible existents can be found absent. But as with possibility, so absence; they have specific connections to some of existence. Not so with nothing.

I thought that positive state and privation were the principles of contrariety and contraries. Contrariety is a type of relation that permits comparisons to be made, the fundamental contraries are positive state and privation, and this fundamental one is the principle and cause of all others. (I’m taking this from a fourteenth century Dominican, whom I happen to be studying at this time. He takes himself to be true to Aristotle, but maybe not all Aristotelians have shared his understanding of these relationships.)

I was thinking of greenness as something one could obtain from brownness by removing some pigments that make up the latter. I was thinking of the shiny surface as restricting the angles of scattered light in comparison to the angles from the dull surface. I agree, however that the contrast of the shape of the leaf with the shininess of one side of the leaf is not a contrast along a common genus, other than the genus attribute (of a leaf in this case). That broad genus does not yield a commensurable dimension between shape and shininess. I think we agree on that (see the subsection “Superordinates and Similarity Classes” here. On this we are in contradiction of Rand’s essay (ITOE 23), but in oral exchange, she seemed to lean our way at this high level of genus (ITOE 275–76).

I cannot go your way on two sides of an issue in contradiction having an opposition that has a commensurable dimension, specifically the goal of truth shared by both sides. That is like the leaf, were it taken as the commensurable dimension of all its attributes. The attributes are radically dependent on the particular entity, but I would not take the opposition of shininess of the entity and shape of the entity to be opposition along a commensurable dimension. The opposition of contradictories seems even less likely to be an opposition along a commensurable dimension: at least an entity can have shape and shininess at the same time. Not so with the sides of a contradiction.

Rand’s conception of mathematics as the science of measurement is a classical one, and it remains a good first approximation. But mathematics has progressed and expanded. There are parts of mathematics that are not measurement theory. Even the part of mathematics known as measure theory (Lebesque integration, etc.) is not theory of measurement. It would be a watering down of the concept of measurement to the point of uselessness to take all of mathematics to be theory of measurement. (We physics folks—at least the children of Physics at Chicago—do not refer to mathematics as a science, but as a theoretical discipline. We are referring to the same discipline as those who follow the ancient tradition of calling mathematics a science.) Rand’s measurement-omission analysis of concepts has punch only because she there uses measurement in a sufficiently narrow sense to be saying something substantial and novel. Rand’s knowledge of mathematics and measurement theory was limited. It is my understanding that she was trying to learn more about them. I should mention that very important discoveries in theory of measurement were made a few years after her death. I learned the pertinent mathematics for her measurement analysis of concepts. My essay “Universals and Measurement” is Chapter 2 to her Chapter 1.

I’ll rest now.

Stephen,

It is too bad you don't want to continue with the thread's discussion, which I think is only beginning to get interesting.

On the chance you feel like continuing in the future, ...

I wonder why "nothing" cannot according to you be a species of anything with identity. If "nothing" is a word and if it is not a proper name, what do you think it is? And if you define it (whatever it is or isn't), would you define it ostensively or with a verbal definition? And if it's with the latter, what is the genus?

---

According to Aristotle, positives and privatives are oppositions of a different kind from oppositions of contraries. (Categories Ch. 10, GBWW1952) Paper cup sizes, such as small, medium, large, can be said to be contraries to each other; and for the extremes, small and nonsmall cups are also contrary opposites. In no way is each cup size a privation of the other cup sizes. Contraries cannot be in the self-same singular subject. A 7-11 cup is either small, medium, or large. Positives and privatives, by contrast, may have reference to the very self-same singular subject. At one time someone was blind and now she sees.

---

As for the reference to ITOE 23, I do not think my view of commensurable characteristics in forming intensive qualified instances of a concept contradicts Ayn Rand's. I will study your view in the cited post after my biweekly study group meeting. Meanwhile you may peruse my understanding of qualified instances of a concept here.

---

When I previously cited contradiction as an example of opposition that shares a standard unit, I meant that on either side of this relation each must be at least a proposition. That is the standard unit from which contradiction can be judged.

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thomtg    0
[...]

[...] That broad genus does not yield a commensurable dimension between shape and shininess. I think we agree on that (see the subsection "Superordinates and Similarity Classes" here. On this we are in contradiction of Rand's essay (ITOE 23), but in oral exchange, she seemed to lean our way at this high level of genus (ITOE 275-76).

[...]

[...]

As for the reference to ITOE 23, I do not think my view of commensurable characteristics in forming intensive qualified instances of a concept contradicts Ayn Rand's. [...]

Stephen,

Because we are comparing my view to your view about whether they match and whether the one or the other contradicts Ayn Rand's view, I am placing my comments here (rather than at where your article is posted).

I think your conception of strength of a solid as a superordinate concept is flawed ontologically. (See the subsection "Superordinates and Similarity Classes" here.) If so, your claim that Rand's supposition--that there is always some same, common measurable dimension supporting the conceptual common denominator for any superordinate concept--is false, is invalidated.

Let us first compare yours with a conception of shape of a solid as a superordinate concept. The species of shape may include cube, tetrahedron, sphere, etc. The species of strength according to the subsection may include "[h]ardness, fatigue cycle limit, critical buckling stress, shear and bulk moduli, and tensile strength."

As I understand it, a concrete referent when classified abstractly in a superordinate concept must fall under one and only one subordinate concept. A bowling ball when classifed in terms of its shape is considered as a sphere. It cannot be a sphere and cube at the same time. By contrast, the strength of the bowling ball--i.e., its " resistance to degradations under stresses"--can be various and simultaneously classified under your concept strength in all the "species." That is, when asked what is the shape of the bowling ball, I can only give you one answer. But from your view about its strength, you will give me no less than five simultaneous answers. Your answers can only mean there is an error on the superordinate-subordinate relation of classification. It is an ontological error.

As I see it from my current reading of ITOE, what your proposal is doing is an attempt to classify categories on the basis of incommensurables. Going back to shape, what would be the equivalence of this attempt? Shape as a superordinate concept would instead have for subordinates, for example, vertex, edge, area, volume, etc. On this basis, I cannot blame you for seeing these concepts as not having anything commensurable that may somehow be subclassed under shape.

The error is in not conceptualizing that attributes such as shapes and strengths of solid objects may themselves be discovered metaphysically to have attributes--that attributes too have identities. Entities such as solid objects are not the only aspects of reality that can be identified conceptually. As Ayn Rand states, a definition of a concept of some existents encompasses not only their distinguishing essential characteristic but all their characteristics, including those not yet discovered. (ITOE 27a-b, 42b, 65d) Although the shape and strength of a bowling ball are not the entity ball but merely its attributes, shape and strength can be individually discovered to have many other characteristics in themselves, and among which only one or few distinguishing characteristics will be stated in their definitions.

If nothing else, the ontological error may have affected your identification of "nothing."

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Guyau    0

Following on #286, a neat note on zero is here.

In #289 Thom remarked:

Speaking of math, I too would defer to mathematicians to determine whether zero and imaginaries are numbers--provided that they adhere to proper epistemological criteria of mathematical knowledge. David Ross, for instance, is a mathematician who takes a stand on an epistemological standard that disqualifies imaginaries as numbers. . . .

I will address the remarks of David Ross now. This is a bit of an ordeal for me. In the years I was making Objectivity, because I was the editor, I set a fairly rigid heuristic for myself: Do not write commentaries on what contributors have said. There were about three occasions upon which it was natural to note in a piece of my own that something I was expounding was at odds with something another writer had said. For the most part, I was opaque on the contributions to the journal, and that was intentional. There were in fact almost no compositions in Objectivity, other than mine, with which I entirely agreed. I usually had extensive correspondence with the writers over the ideas in their composition as we moved towards acceptance. The writers often learned some of the points on which I disagreed with them. But no one else knew.

I met David Ross at a colloquium of the Institute of Objectivist Studies in New York in April 1991. I learned that he was a mathematician, and he mentioned that he had enjoyed Merlin’s article “The Nature of Numbers.” I suggested that he write to Merlin with any thoughts he had about the article. David wrote a letter to Merlin the following month, I asked David if I could publish those comments in the Remarks section of Objectivity, and I asked him questions about his comments. He responded with elaborations and a Yes for publication, I cut-and-pasted a bit, sent it to him, he returned tweaks, and it went into V1N3.

Dr. Ross’ remarks include the following:

Number is not a technical term; it is an everyday word. Wave too. If specialists want to use these terms, or others, in a technical way to denote something other than what the man in the street denotes by them, specialists must keep in mind that they are doing this. When I specify and define the concepts with which such words are associated as simple English words, I adhere to two criteria. One is based on the need to communicate; the definition must coincide as much as possible with common usage. The other is based on objectivity; the concept must conform to our cognitive need to ‘cut reality at the joints’. . . .

“I think that mathematicians have done an excellent job of classifying mathematical entities; the technical categories of natural number, integer, positive real, complex number, and so on, are quite useful to the specialist. My second criterion is met. The only remaining question is, what subclass of the mathematical entities correspond to the entities known to laity as numbers? I should say that this subclass is the non-negative reals. I think, however, that a good case can be made for saying that it is the non-negative rationals. The basic point is, when the layperson is told that the square root of minus-one is a number, and he balks, he is right. Of the entities he integrated under the concept he calls number, none has the property of becoming negative upon being squared. He has identified an objectively valid cognitive category. It seems to me that given this, to insist, over the protests of zillions of laypeople, that the square root of minus-one is a number is either to hold an intrinsicist view of concepts or to refuse to communicate out of stubbornness. . . .

“Similar considerations apply to the concept wave. My definition of wave is: a disturbance in a medium that moves through the medium due to its effect on the medium. In this case, I should say that scientists, including mathematicians, have not done a good job of parsing up reality. . . . Not only do they apply the term wave to a much larger class of phenomena than my definition allows; they have no term to denote disturbances in media. I think that the phenomena specified by my definition are sufficiently important to warrant their own concept; they define a joint at which reality ought to be cut. . . . I have no objection to the terms electromagnetic wave or probability wave, as long as it is borne in mind that these are not actually species of the genus wave; that they are quite similar to waves in certain respects (they interfere, they are described by hyperbolic equations), but they are different in a crucial respect (no medium).” (105–7)

In his remarks, Dr. Ross mentioned also:

“There are applications of complex analysis to other branches of mathematics proper, but I am less familiar with these: differential equations is my field. The most famous connection, within mathematics proper, is probably between complex analysis and number theory, via the Riemann zeta function; the Riemann conjecture, which is about the roots of this analytic function, is one of the greatest unsolved problems of mathematics.” (108)

I would like to provide the nice links for those terms now available at Springer:

Number Theory

Function

Differential Equations, Ordinary

Differential Equations, Partial

Differential Equations, Partial, Complex-Variable Methods

Complex Analysis (Wikipedia)

Zeta Function

Riemann Hypothesis (WolframMathWorld)

There are many different levels of mathematical education among laypersons. My Mom’s second husband was born into a cattle ranching family and went to school only through the third grade. He was a paratrooper in North Africa and in Italy during WWII, and after the war he returned to ranching. I found him perfectly intelligent to talk to. He had a good memory, kept up on current affairs, knew some fine points of theology, and could do the arithmetic necessary for raising and selling cattle.

He had zero, the positive integers, some arithmetic operations, and dollars-and-cents. That was enough. Those span to a natural joint. There are some other kinds of numbers that he had never heard of, numbers known to more educated people. He would not have known about negative integers or irrational numbers, let alone real numbers. Most people I know outside of engineering and the sciences don’t know what is a real number. Their education in mathematics did not go that far, and their work and life does not require they know about such numbers. Their numbers end at another natural joint.

As we construct these numbers from zero and the positive integers on up to the reals, at which joint should the name number stop when talking to laypeople? Which laypeople?

David mentioned that he thought mathematicians have done an excellent job in forming their classifications: natural number, integer, positive real number, and complex number. This suggests he thinks it is fine for mathematicians to think of complex numbers (that is what they call them) as numbers in the ways mathematicians usually do. They show that the real numbers R are a field in the sense of abstract algebra (and whisper to trusting laypersons, “Hey, the reals are a kind of field”). They show that the Cartesian space RxR is a field and that no higher Cartesian product spaces of R such as RxRxR are a field. They then show that the complex number system C is a field, having the same addition operation as the field RxR, but having as well a multiplication operation. RxR has no multiplication operation, and that might seem an inferior way of being, but C, unlike R, lacks the order properties. Give them each a ribbon.

Were one to learn what a real number is and what being a field is, I bet one would learn something about real numbers when learning that it is a field. Then one could get a better grasp of the relationship between real numbers and complex numbers.

The cuttings of the mathematicians are at sensible joints. If a layperson wants to learn about more numbers than he presently knows about, let him. If a layperson wants to insist that the numbers they know about are the only numbers there are, let him. In any event, the epistemology of mathematics does not depend on the public relations of mathematicians with laypeople.

David’s definition of a wave would not be accepted by any of my physics professors and is not accepted by me. His definition is at first brush a definition of the concept of an important subclass of the class wave in media. Electromagnetic waves, of course, are waves sometimes in media, sometimes not—but enough on that already (Objectivity V2N6, 143–48).

Edited by Stephen Boydstun

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thomtg    0
[...]

Dr. Ross’ remarks include the following:

Number is not a technical term; it is an everyday word. Wave too. If specialists want to use these terms, or others, in a technical way to denote something other than what the man in the street denotes by them, specialists must keep in mind that they are doing this. When I specify and define the concepts with which such words are associated as simple English words, I adhere to two criteria. One is based on the need to communicate; the definition must coincide as much as possible with common usage. The other is based on objectivity; the concept must conform to our cognitive need to ‘cut reality at the joints’. . . .

[...]

There are many different levels of mathematical education among laypersons. My Mom’s second husband was born into a cattle ranching family and went to school only through the third grade. He was a paratrooper in North Africa and in Italy during WWII, and after the war he returned to ranching. I found him perfectly intelligent to talk to. He had a good memory, kept up on current affairs, knew some fine points of theology, and could do the arithmetic necessary for raising and selling cattle.

He had zero, the positive integers, some arithmetic operations, and dollars-and-cents. That was enough. Those span to a natural joint. There are some other kinds of numbers that he had never heard of, numbers known to more educated people. He would not have known about negative integers or irrational numbers, let alone real numbers. Most people I know outside of engineering and the sciences don’t know what is a real number. Their education in mathematics did not go that far, and their work and life does not require they know about such numbers. Their numbers end at another natural joint.

As we construct these numbers from zero and the positive integers on up to the reals, at which joint should the name number stop when talking to laypeople? Which laypeople?

David mentioned that he thought mathematicians have done an excellent job in forming their classifications: natural number, integer, positive real number, and complex number. This suggests he thinks it is fine for mathematicians to think of complex numbers (that is what they call them) as numbers in the ways mathematicians usually do. They show that the real numbers R are a field in the sense of abstract algebra (and whisper to trusting laypersons, “Hey, the reals are a kind of field”). They show that the Cartesian space RxR is a field and that no higher Cartesian product spaces of R such as RxRxR are a field. They then show that the complex number system C is a field, having the same addition operation as the field RxR, but having as well a multiplication operation. RxR has no multiplication operation, and that might seem an inferior way of being, but C, unlike R, lacks the order properties. Give them each a ribbon.

Were one to learn what a real number is and what being a field is, I bet one would learn something about real numbers when learning that it is a field. Then one could get a better grasp of the relationship between real numbers and complex numbers.

The cuttings of the mathematicians are at sensible joints. If a layperson wants to learn about more numbers than he presently knows about, let him. If a layperson wants to insist that the numbers they know about are the only numbers there are, let him. In any event, the epistemology of mathematics does not depend on the public relations of mathematicians with laypeople.

David’s definition of a wave would not be accepted by any of my physics professors and is not accepted by me. His definition is at first brush a definition of the concept of an important subclass of the class wave in media. Electromagnetic waves, of course, are waves sometimes in media, sometimes not—but enough on that already (Objectivity V2N6, 143–48).

I think David Ross is not advocating that the epistemology of mathematics be dependent on the public relations of mathematicians with the man on the street. What he is advocating is merely what Ayn Rand has stated in the ITOE workshop. (ITOE 237) Specifically, a concept, such as "number," should have the same identical meaning for the man on the street as for the mathematician. The latter may know more than the former--not by means of inductive evidence but by sheer deductive integration--but that is the extent of the difference.

Now granted that your mother's second husband may not have used or thought of negative numbers or irrational numbers, his concept of "number" would admit them, in my view, to be numbers. Given the opportunity to observe that aspect of reality, his limited understanding of numbers should still "pick out" numbers from reality. I may not have seen every kind of birds there is, but I should think that what I mean by "bird" should mean the same as what is meant by an expert ornithologist.

This is basically a criterion of objectivity in mathematics.

But as I understand your objection, you want to make out the distinction that mathematicians understand numbers radically differently and cannot be held down to the same meaning of "number" as held by laypeople, that the former may at will define "number" compositionally as "field" to allow for the inclusion of "complex numbers" and presumably "vectors." (See Ross's remark, p. 104, here not quoted).

To me, this is a prescription to corrupt mathematics. Mustn't a concept mean the same for all time and place, past, present, and future--even if its definition changes contextually? It is not a matter of public relations but a matter of unit economy that the concept "number" must mean its open-ended units, including those not yet discovered, and only those units. To allow mathematicians to take liberties at changing the meaning of "number" is to admit that knowledge of reality at one level cannot transition gradually to a new level--that scientific war is a fact of reality--that every advance in science calls for the destruction of old knowledge. (ITOE 67)

Finally, I find David's definition of waves (p. 106) to be exceptionally clear. It is not a subject of any public relations to win over physics professors; it is merely an accurate identification of some existents in reality that may be studied in more depth by physicists. What you have stated at Objectivity V2N6, 143–48 about the history of "the wave model of light" is at best a metaphorical use of "wave," not wave per se, if "wave" is to be understood to mean identically by anyone in or out of physics.

By the way, thanks for the insight to the behind-the-scenes struggles of editors.

Edited by Thom T G

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tjohnson    0
To me, this is a prescription to corrupt mathematics.

How on earth could you presume to decide when mathematics is corrupt or not? What does that even mean?

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BaalChatzaf    0
To me, this is a prescription to corrupt mathematics.

How on earth could you presume to decide when mathematics is corrupt or not? What does that even mean?

You see quite a bit of this in Objectivist Circles. Mathematics is corrupt, Physics is corrupt. What it comes down to is this: Anything that does not conform to their Objectivist philosophical prejudices is corrupt. In Objectivist Circles some of the very abstract aspects of mathematics far removed from everyday experiences is dismissed as (mere) method. This dismissive attitude goes hand in glove with the notion that words have a True Meaning.

The only corruptions I know of in mathematics are begging the question (a Cardinal Sin) and putting together an inconsistent system.

Ba'al Chatzaf

Edited by BaalChatzaf

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BaalChatzaf    0
To me, this is a prescription to corrupt mathematics. Mustn't a concept mean the same for all time and place, past, present, and future--even if its definition changes contextually?

Not at all. If this were the case then new concepts could not be invented.

Some changes are not "contextual". Some new concepts are invented to deal with properties not heretofore experienced or known. Sometimes old words are pressed into service to denote new kinds of things. Sometimes we have to invent new systems of numbers to make certain equations generally solvable. For example x^2 + 1 = 0. Mathematicians set out to construct a system of numbers such that all polynomials with (so-called) real coefficients have a solution. A new kind of number is invented to provide the generality. This new kind of number (complex number) is not like the so-called real numbers. The complex numbers do not form a dense locally compact linearly ordered topological space. They are different beasts. It so happens that the field of complex numbers contains a subfield that can be linearly ordered and is isomorphic to the so-called real numbers. In this subfield multiplication is restricted so that the product of a number with itself is positive or zero.

In physics what is a "wave"? The most general answer is any process that can be described by a certain kind of partial differential equation - the wave equation. (see http://en.wikipedia.org/wiki/Wave_equation). In that sense a wave need not be a periodic oscillation in a viscoelastic medium. A "non-waving" wave is defined by analogy, to wit something that satisfies the same sort of hyperbolic partial differential equation as a water wave or a wave in jelly or a wave that travels down a rope. Electromagnetic waves are non-waving waves. There is no elastic medium in which electromagnetic fields wave. Insisting that waves "wave" in some jiggly medium held back physics over one hundred years. (In short there is no aether).

I sense (i.e. I have the hunch or feeling) that you believe that some words, at least, have True Meanings. (I apologize if I am mistaken in this supposition) Not so. Words are conventions. They mean what the users of the words declare them to mean. This is how we have progressed from languages that consist of a few dozen grunts to languages that have 500,000 words (like English, for example). New experiences, new facts, new requirements, new words. That is how it goes.

Ba'al Chatzaf

P.S. added. I just had an inspiration. Your difficulty is that you think concepts are the result of measurement omission. Some concepts are. Some are not. Some concepts come about by analogy and some by metaphor. For example: What on the surface of a sphere is analogous to a straight line in space? It turns out that it is a great circle which is not straight at all. It is analogous to a straight line in that the great circle path connecting two points on a spherical surface is the shortest path that one can travel and still be on the spherical surface. What makes this analogous to a straight line in space is that a straight line is the shortest path (in space) between two points. In this sense a great circle path is analogous to a straight line. But the analogy is not perfect. On a sphere there are no parallel great circles. Given a great circle and a point (on the spherical surface) which is not on the great circle there is no great circle through that point which does not intersect the given great circle. In short, the surface of sphere where great circles are "the straight lines" is not a Euclidean space, i.e. it does not obey all the postulates of Euclidean geometry.

Edited by BaalChatzaf

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Guyau    0

Thom,

There is a careful account of conceptual change in mathematics in Philip Kitcher’s The Nature of Mathematical Knowledge. (Click on the link in #279 for a fairly detailed review of this work.) His conception of concepts and how they change is consistent with Rand’s idea of the open-endedness of concepts. His account of conceptual change is harmonious with the example Rand gave in ITOE of the changes in one’s concept man as one develops and comprehends more about human beings. One of Prof. Kitcher’s examples of conceptual change in mathematics is the introduction of the square root of negative one.

In the late sixteenth century, mathematicians began to use compact symbols (such as the familiar square-root sign with a minus-signed number inside it) that stood for the expression in words “the square root of a negative number.” Did such an expression with a minus-signed number one in them refer to i? That hypothesis “has the advantage of making it clear how reference to complex numbers became possible. The language of mathematics always had the resources to refer to these numbers. But the hypothesis fails to explain the deep and long-lasting suspicion of complex numbers and the strenuous efforts which were made to understand them.” Did such an expression with a minus-signed number in it fail to refer? This second hypothesis “enables us to account for the resistance to complex numbers at the cost of making it mysterious how we ever came to be in a position to refer to them.” Such expressions failed to refer, “we might say, because, in the way in which ‘number’ was used at the time of the alleged introduction of complex numbers, there is no number whose product with itself is -1. The opposition to the numbers was so intense because mathematicians were all acquainted with a theorem to this effect” (175).

Neither the first nor the second hypothesis is correct, “but both have captured part of the story. One way to fix the referent of ‘number’ is to use the available paradigms—3, 1, -1, sqrt 2, pi, and so forth—to restrict the referents to the reals. Given this mode of reference fixing, the theorem that there is no number whose product with itself is -1 is almost immediate. . . . Given a different way of fixing the referent of ‘number’, numbers are entities on which arithmetical operations can be performed. Here, from the point of view of medieval and renaissance mathematics, it is an open question whether one can find recognizable analogs of the paradigm operations which allow for the square of a ‘number’ to be negative. In effect Bombelli and the other mathematicians who allowed [such expressions] to enter their calculations were fixing the referent of ‘number’ in this second way, and were referring to complex numbers. What needed to be done to show that the more restrictive mode of reference fixing should be dropped from the reference potential of ‘number’ was to allay fears that recognizable analogs of ordinary arithmetical operations could not be found. During the seventeenth and eighteenth centuries, algebraists, analysts, and geometers responded successfully to such fears. Gradual recognition of the parallels between complex arithmetic and real arithmetic led to repudiation of the more restrictive mode of reference fixing, so that the reference potential of [a square-root sign with a -1 in it] came to include only events in which i was identified as the referent.” (175– 76)

That last clause is unclear unless one has read some of the earlier part of the book. Kitcher is conceiving of concepts as having a certain potential for reference to be filled out as the concept advances. One had been using numbers in arithmetical operations. But until certain new discoveries in mathematics, there was not so much pressure to bring that feature to the fore as defining feature of numbers.

There is more to the story of how this particular conceptual change came about, which is also related by Kitcher. “One special feature of the concern about complex numbers was the felt need for a concrete interpretation of them. . . . An important episode in the acceptance of complex numbers was the development, by Wessel, Argand, and Gauss, of a geometrical model of the numbers. . . . Prior to the work of Bombelli and his succcessors, the referent of ‘number’ was fixed with respect to paradigms of number operations. Each of the paradigm number operations could be given a construal in concrete physical terms: natural number operations obtained their physical interpretation in the process of counting; real number operations found theirs in the process of measurement. Bombelli can be regarded as suggesting that there is a type of number operation which had not hitherto been recognized. To eliminate from the reference potential of ‘number’ the restrictive mode of fixing the reference to the familiar kinds of number operation, it was not sufficient to show that the new operations would submit to recognizably arithmetical treatment. Proponents of complex numbers had ultimately to argue that the new operations shared with the original paradigms a susceptibility to construal in physical terms. The geometrical models of complex numbers answered to this need, construing complex addition in terms of the operation of vector displacement and complex multiplication in terms of the operation of rotation.”(176)

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Brant Gaede    1

reprise

I'm a little embarrassed to have posted anything on this thread considering my lack of mathematical knowledge and skill.

At least I didn't post much.

--Brant

500 minus 501 equals negative one (?)

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BaalChatzaf    0
5 hours ago, Peter said:

Draw me a negative one without using the representing number and minus sign.

Positive and negative refer to directions.   If one is standing on the number line,  a move to the right is positive and a move to the left is negative (this is the standard convention). A move of one length unit to the right adds 1 to the position at which you stood.  Where you end up is the original position + 1.  A move of one unit to the left  subtracts 1 from the position at which you stood.  Where you end up is the original position - 1.  We do that measuring temperatures with a mercury thermometer.   A mark is made when the thermometer is immersed in frozen brine.  That mark is 0 degrees celsius.  Now if you use the marked thermometer to measure the temperature of (say) warm soup you will find the mercury above the merk.  If you use the thermometer to measure the temperature of dry ice  you will find the mercury below the mark.  That indicate negative temperatures on the Celsius or Centigrade scale.

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Brant Gaede    1
On 5/11/2009 at 1:29 PM, BaalChatzaf said:

Do you mean the contrary or do you mean the negation. The negation of Nothing is Something. Any particular in the world is something. In set theory we talk about sets and their complements. Let A be a set. Then A-comp is the set of elements not in A. If A is empty then any element is a member of A-comp. The general practice when using sets is to identify some non-empty set as the universal set. Let is call it U. Then the complements are taken relative to U. Let A be a subset of U. The A-comp relative to U is usually written U - A which is the set of elements in U not in A. The (relative) complement of 0 is U. The relative complement of U ( U - U) is 0. So when we are dealing sets of real numbers, the universal set is taken to be the set of all real numbers. That pins the meanings of the sets down better.

Have a look here for some background:

http://www.britannica.com/EBchecked/topic/...9/universal-set

Not all set theories have universal sets. For example Zormelo-Frankel set theory does not, but set theory based on Quine's New Foundations does.

You also might want to look at this:

http://en.wikibooks.org/wiki/Discrete_mathematics/Set_theory

and

http://en.wikipedia.org/wiki/Set_theory

Ba'al Chatzaf

If you negate something then "something" was always nothing.

If you negated nothing then nothing was always nothing--there was nothing to negate, only a false proposition. Negation is epistemological. Non negation is metaphysical.

These are semanticals.

You can't negate something and you can't negate nothing. You can try. You can only negate propositions.

--Brant

 

 

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BaalChatzaf    0
6 hours ago, Brant Gaede said:

If you negate something then "something" was always nothing.

If you negated nothing then nothing was always nothing--there was nothing to negate, only a false proposition. Negation is epistemological. Non negation is metaphysical.

These are semanticals.

You can't negate something and you can't negate nothing. You can try. You can only negate propositions.

--Brant

 

 

You are ignoring the mathematical details.  Obviously these notions of complementarity apply to mathematical objects,  not physical objects.  The mathematical objects are patterns of neural activity, not hard free standing  material objects.  Abstract entities live and multiply in our heads. 

You are misapplying your philosophical notions.  Which is no surprise.  Philosophical notions are notoriously easy to misapply.  Aristotle and Plato made careers out of misapplying ideas.  Which is unfortunate because they held up the development of real physical science between 1000 and 2000 years. 

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On 7/20/2017 at 0:19 AM, Brant Gaede said:

If you negate something then "something" was always nothing.

If you negated nothing then nothing was always nothing--there was nothing to negate, only a false proposition. Negation is epistemological. Non negation is metaphysical.

These are semanticals.

You can't negate something and you can't negate nothing. You can try. You can only negate propositions.

Hmmm, this sounds like a (proposed?) Law of Conservation of Existence!

REB

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On 7/20/2017 at 7:19 AM, BaalChatzaf said:

Aristotle and Plato made careers out of misapplying ideas.  Which is unfortunate because they held up the development of real physical science between 1000 and 2000 years. 

 

Three comments - take your pick over which to distort or misrepresent:

1. Aristotle and Plato held up the development of real physical science between 1000 and 2000 years. Isn't that an awfully wide range of values? Is Heisenberg screwing with us again? :wink: 

2. Aristotle and Plato defined the laws of logic, including the law of identity, which some claim to not be able to find in Aristotle or Plato. This held up the development of science? This sounds more like a Sophist argument.

3. Only 1/4 of Aristotle's works survived to the Renaissance and the Modern Era. Would "real physical science" be better off or worse off today if ALL of Aristotle's works had survived? If NONE of them had survived? Do you have candidates for "wish they hadn't survived"?

REB

P.S. - I suppose you have a point. I see a similar problem with how Rand herself held up the development of Objectivism, perhaps by 50 years or more by various of her practices and policies. (And I'm not talking about Closed Objectivism.)

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BaalChatzaf    0
1 hour ago, Roger Bissell said:

Three comments - take your pick over which to distort or misrepresent:

1. Aristotle and Plato held up the development of real physical science between 1000 and 2000 years. Isn't that an awfully wide range of values? Is Heisenberg screwing with us again? :wink: 

2. Aristotle and Plato defined the laws of logic, including the law of identity, which some claim to not be able to find in Aristotle or Plato. This held up the development of science? This sounds more like a Sophist argument.

3. Only 1/4 of Aristotle's works survived to the Renaissance and the Modern Era. Would "real physical science" be better off or worse off today if ALL of Aristotle's works had survived? If NONE of them had survived? Do you have candidates for "wish they hadn't survived"?

REB

P.S. - I suppose you have a point. I see a similar problem with how Rand herself held up the development of Objectivism, perhaps by 50 years or more by various of her practices and policies. (And I'm not talking about Closed Objectivism.)

Purging Aristotle's errors  was a gradual process.  Around 650 C.E.  Philliponus  falsified Aristotle idea that the motion of a body requires a force.  He took the first step to identifying momentum. Other scholars  picked apart Aristotle's  work in The Physics.  The big breaks came with Copernicus who postulated a moving earth and about 100 years later Galileo.  But even Galileo did not get rid of circular motion entirely  Kepler and Newton completed the "de-Aristotlization"  of celestial mechanics.   

Even today we still retain some of Aristotle's thinking.  For example in biology part of our understanding of organisms in the context of a natural environment is stated in terms of function  which is an instantiation of Aristotle's Final Cause.   We cannot think of how a heart is constructed or functions apart from the idea that the heart -is for- pumping blood.  It is not merely that the heart moves blood through the blood vessels  but that is -why- and -what for-  we have hearts.  So even today  Aristotle is not totally purged from our thinking about nature.  Mostly we have (1) got rid of his worst mistake, the failure to identify inertia  and (2) Aristotle's bad habit of not checking his conclusions empirically in a thorough manner. 

But we still use, for some purposes,  his categorical logic (the logic of syllogisms)  which is a correct system of logic in a restricted domain of application. Also Aristotle's  view of ethics is still with us. We still recognize virtue and excellence.  And Aristotle's view of politics still is applicable.  Aristotle worked in so many fields  that it is not surprising that he was right as often as he was.  But he did not check thoroughly  so it is not surprising that he was as wrong as often as he was.   Aristotle's  view of motion and matter (which was erroneous)  took a long time  to purge, because they are intuitively plausible and appealing. 

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Peter    0
On ‎7‎/‎19‎/‎2017 at 11:18 PM, Roger Bissell said:

Negative One.jpg

It took me a while to get it, plus I needed to enlarge my screen to 250 percent to read it. Very good! The number one is a negative personality. Brilliant!

Peter  

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