Measure and Quantity and Rand's notion of mathematics

[BaalChatzaf]

I am not aware of any "rules of a good definition"

They are given in many introductory logic texts, e.g. David Kelley's The Art of Reasoning.

There are 5 rules here: http://en.citizendium.org/wiki/CZ:Definitions

Kelley's book has these 5 plus one more: A definition should include a genus and a differentia.

There are other modes of definition besides general-special. See http://en.wikipedia.org/wiki/Definition

Ba'al Chatzaf

[tjohnson]

One of the rules of a good definition is that it not be vague or obscure. I note that you misspelled his name, omitting the "y". If you were to correct it, would that be doing math?

OK, so you think Korzybski's definition is vague or obscure? How so? From wikipedia, italics mine.

The definition must not be obscure. The purpose of a definition is to explain the meaning of a term which may be obscure or difficult, by the use of terms that are commonly understood and whose meaning is clear. The violation of this rule is known by the Latin term obscurum per obscurius. However, sometimes scientific and philosophical terms are difficult to define without obscurity.

[merjet]

OK, so you think Korzybski's definition is vague or obscure? How so? From wikipedia, italics mine.

The definition must not be obscure. The purpose of a definition is to explain the meaning of a term which may be obscure or difficult, by the use of terms that are commonly understood and whose meaning is clear. The violation of this rule is known by the Latin term obscurum per obscurius. However, sometimes scientific and philosophical terms are difficult to define without obscurity.

The terms "linguistic scheme" and "multiordinal" are both vague and obscure. It makes an encryption scheme in cryptography sound like a referent of "mathematics." I ask again, would correcting the spelling of Korzbski be doing math?

The word "mathematics" is far too common and familiar to warrant the exception for scientific and philosophical terms. Candidates like "substance" and "form" (as used in Greek philosophy), "supervenience", "auto-correlation", some chemical terms, some medical terms, probably some terms in quantum mechanics, yes. "Mathematics", no.

[BaalChatzaf]

The terms "linguistic scheme" and "multiordinal" are both vague and obscure. It makes an encryption scheme in cryptography sound like a referent of "mathematics." I ask again, would correcting the spelling of Korzbski be doing math?

The word "mathematics" is far too common and familiar to warrant the exception for scientific and philosophical terms. Candidates like "substance" and "form" (as used in Greek philosophy), "supervenience", "auto-correlation", some chemical terms, some medical terms, probably some terms in quantum mechanics, yes. "Mathematics", no.

Encryption and decryption schemes are objects of mathematical analysis. In fact encryption can be reasonably thought of as a branch of applied mathematics.

The problem with the term mathematics used without qualification is that it applies to too many different subjects at once. Qualified terms like combinatorial mathematics, the mathematics of manifolds, etc., ... are better able to be defined.

It is as difficult as defining the term philosophy. Maybe more so.

Ba'al Chatzaf

[merjet]

Encryption and decryption schemes are objects of mathematical analysis. In fact encryption can be reasonably thought of as a branch of applied mathematics.

The first sentence is true. Not knowing enough about it, I hesitate to say the second is. In any case, the apparent principle used can be taken too far. Grocery shopping, gambling, economics, and investing are subject to mathematical analysis. That doesn't make grocery shopping, gambling, economics, and investing branches of mathematics. They are cases of already recognized branches of mathematics applied to grocery shopping, etc.

[Michael Stuart Kelly]

Off-hand, I think a better way of saying it is that one measures magnitudes.

A measurement is a ratio of one magnitude to another of the same type. For example, the length of a football field is 100 times the length of one yard.

Merlin,

Doesn't that apply to all features that are measured? One measures length. One measures scope. One measures intensity. And so on.

So maybe my original question (""Isn't magnitude a kind of measurement?") would have been better as: "Isn't magnitude a kind of commensurable characteristic?" or "Doesn't magnitude automatically come with the idea of unit of measurement as a fundamental part of it?"

This is as given in ITOE, 2nd, p. 12:

All conceptual differentiations are made in terms of commensurable characteristics (i.e., characteristics possessing a common unit of measurement).

Michael

[BaalChatzaf]

Encryption and decryption schemes are objects of mathematical analysis. In fact encryption can be reasonably thought of as a branch of applied mathematics.

The first sentence is true. Not knowing enough about it, I hesitate to say the second is. In any case, the apparent principle used can be taken too far. Grocery shopping, gambling, economics, and investing are subject to mathematical analysis. That doesn't make grocery shopping, gambling, economics, and investing branches of mathematics. They are cases of already recognized branches of mathematics applied to grocery shopping, etc.

Almost all advanced encryption technology is mathematically formulated. The days of simple substitution encryption is gone forever. Pick up any book on advanced encryption and it will open up with a discourse on number theory and group theory. The Enigma encryption was initially cracked by Polish ( primarily Rejewski ) mathematicians applying group theory. Alan Turing based his Enigma decryption on mathematical logic and even developed computers to carry out the steps quickly.

Ba'al Chatzaf

[Roger Bissell]

One problem in some discussions of this kind is that a certain key term is used in both a metaphysical or mind-independent sense and in an epistemological or mind-dependent sense. ("Or" is not meant to claim that the two terms on either side are synonymous, just that the some cognitive process is involved in some usages and not in others.

"Magnitude" is such a term. My visit to Wikipedia shows that it is used to refer both mind-independently to a kind of quantity and mind-dependently to a measure of that quantity.

Quantity is the fundamental and general term, referring to "any type of quantitative properties or attributes of things"; and it includes two main kinds: amount (aka magnitude in the mind-independent sense) and number (aka multitude). "Quantity is a property that exists in a range of magnitudes and multitudes."

When we mentally grasp the amounts and numbers of things, we do so in the form of measurements along a range of quantitatively graspable properties -- and these are also called magnitudes/amounts and multitudes/numbers. Hence, the source for some amount of confusion when philosophers, scientists, and paraphilosophers (such as myself) argue about such things.

REB

[merjet]

I'm responding to post #31 by Michael Stuart Kelly.

Many magnitudes have a unit of measurement, but many don't. (Hardness, for example, does not. The Mohn scale does not have a unit of measurement. It is an ordinal scale.) A measurement has a magnitude in both the numerator and denominator of the ratio, both of the same type, and the one in the denominator is the standard.

As I explain in "Omissions and Measurements" regarding concept formation, differences in instances that fall under the same concept are not all magnitudes, or even quantitative. For example, two paragraphs in my JARS article are:

Consider the concept book. Some attributes of books are measurable, e.g. height, width and thickness. However, some important ones are not. Consider the language in which it, or part of it, is written – English, Spanish, German, C++ (the computer programming language), musical notation, mathematical notation. The last three are not ordinary languages. Yet all are congruous – all are ways of conveying ideas in writing. However, these languages are not commensurable. The differences between them that need to be omitted to form the concept book are qualitative, not measurable.

Consider also the content of the book. Some books are fiction and others are non-fiction. Types of fiction are mystery, romance, children’s stories, etc. Types of nonfiction are history, science, mathematics, music, food recipes, etc. These various contents are congruous but not commensurable. The differences between them that need to be omitted to form the concept book are qualitative, not measurable.

As this suggests, (1) "congruous" is far more accurate and comprehensive than "commensurable" and (2) "qualitative" is far more accurate and comprehensive than "measurable."

Edited October 19, 2007 by Merlin Jetton

[tjohnson]

The terms "linguistic scheme" and "multiordinal" are both vague and obscure. It makes an encryption scheme in cryptography sound like a referent of "mathematics." I ask again, would correcting the spelling of Korzbski be doing math?

I admit 'multiordinal' is a term used in GS exclusively AFAIK and in all fairness to Korzybski he describes this as a semantic definition of mathematics. The term 'linguistic scheme' could easily be replaced with simply 'language' I think. A more common definition might be " language of structures and relations capable of exact treatment at a given date". Also, no, correcting a mis-spelled word would not be doing math.

[Michael Stuart Kelly]

Merlin,

In Rand's writing, the term "measurement" includes ordinal scales. For instance, ITOE, 2nd, p. 33:

Measurement is the identification of a relationship—a quantitative relationship established by means of a standard that serves as a unit. Teleological measurement deals, not with cardinal, but with ordinal numbers—and the standard serves to establish a graded relationship of means to end.

For instance, a moral code is a system of teleological measurement which grades the choices and actions open to man, according to the degree to which they achieve or frustrate the code's standard of value. The standard is the end, to which man's actions are the means.

There is no reason to presume that she would exclude ordinal scales from physical measurements. See this excerpt also from the ITOE workshops (p. 223):

All measurements in regard to concepts of consciousness are only approximations. With that in mind, the common denominator would be intensity—the intensity of a given mental state and its hierarchical importance to you, measured by the ordinal numbers.

So far, I see your fundamental disagreement with Rand being on what "measurement" means. As to your examples about features of books, I see measurement in them (using Rand's broader meaning of measurement, not the restricted one you use), but I want to wait a bit until I have mulled this over more to make a comment. This question does deserve a serious thinking effort.

Michael

[Michael Stuart Kelly]

Roger,

I like the terms mind-dependent and mind-independent to express epistemology and metaphysics. As regards concept formation, I cannot imagine a mind-independent concept. The whole purpose of measurement (and measurement omission) is to form concepts. That is why I find the unit of measurement an intrinsic part of something like magnitude.

There is no way to think conceptually about something without using concepts. And there is no way to use concepts without using the mind. In a very real sense, for human beings, there is no such thing as mind-independent magnitude other than some unidentified quality that is just "out there" somewhere.

In other words, we cannot identify magnitude without implicitly establishing a unit of measurement. Part of everything expressed in conceptual terms involves measurement in some form or another.

Michael

[merjet]

Merlin,

In Rand's writing, the term "measurement" includes ordinal scales. For instance, ITOE, 2nd, p. 33:

I know that. Please read "Omissions and Measurement" or "The Corruption of Measurement" for my explanation that "ordinal measurement" is an oxymoron and should not be confused with true measurement.

So far, I see your fundamental disagreement with Rand being on what "measurement" means. As to your examples about features of books, I see measurement in them (using Rand's broader meaning of measurement, not the restricted one you use), but I want to wait a bit until I have mulled this over more to make a comment. This question does deserve a serious thinking effort.

It is a significant disagreement, but my fundamental disagreements are (1) there are qualitative differences besides differences of measurement and (2) qualitative differences are more common and basic. Rand insists that all differences are measurable.

It would save both of us much time if you were to read "Omissions and Measurement" before proceeding. There is little difference between the JARS version and the one here - http://www.objectivistcenter.org/events/ad.../JettonOaM3.PDF. The major difference is the book example in JARS versus the occupation example in the other.

[Michael Stuart Kelly]

It would save both of us much time if you were to read "Omissions and Measurement" before proceeding.

Merlin,

Agreed. I even mentioned that a few times myself. Although I do remember mentioning that I skimmed both your articles (which I did).

Michael

[merjet]

In other words, we cannot identify magnitude without implicitly establishing a unit of measurement.

Not so. A young child, knowing nothing about measurement or even numbers, can perceive that "daddy is bigger than me", or daddy is bigger than a sibling, or similar phenomena. "Implicitly" is mere hand-waving.

Part of everything expressed in conceptual terms involves measurement in some form or another.

Not so. "Some things", not "everything".

[Michael Stuart Kelly]

Merlin,

Once again, I think the root of disagreement is over the difference in meaning of the word "measurement." I use Rand's version and that implicitly includes the question "measure what?" (i.e., qualities).

So if your meaning of the word measurement is used, your observations in Post 40 are correct. If Rand's meaning is used, your observations are incorrect.

I was using Rand's meaning.

Michael

[tjohnson]

It would save both of us much time if you were to read "Omissions and Measurement" before proceeding. There is little difference between the JARS version and the one here - http://www.objectivistcenter.org/events/ad.../JettonOaM3.PDF. The major difference is the book example in JARS versus the occupation example in the other.

I read your article and there is something that bothers me in this whole discussion of concept formation. The term 'concept formation' seems a misnomer because the process you are describing looks like 'definition formation'. What do you mean by 'concept'?

[BaalChatzaf]

It would save both of us much time if you were to read "Omissions and Measurement" before proceeding. There is little difference between the JARS version and the one here - http://www.objectivistcenter.org/events/ad.../JettonOaM3.PDF. The major difference is the book example in JARS versus the occupation example in the other.

Where do vectors and tensors fit into you scheme? In advanced manifold theory, curvature is not a scalar quantity (such as with curvature of curves and surfaces). This makes a major ueber difference in such theories as Einstein's General Theory of Relativity.

Ba'al Chatzaf

[merjet]

I read your article and there is something that bothers me in this whole discussion of concept formation. The term 'concept formation' seems a misnomer because the process you are describing looks like 'definition formation'. What do you mean by 'concept'?

I used 'concept' the way Rand did. Other might call it 'universal' or 'general term'. 'Concept' encompasses all the known characteristics of the referents. (Somebody might quibble with this -- that it encompasses all the characteristics, known or unknown. However, that is not important to answering your question.) The definition of a concept encompasses only the fundamental characteristics of the referents.

[tjohnson]

In General Semantics, Korzybski asserts (also see Sapir-Whorf ) that our 'concepts' are affected or influenced by our language.

When we look at an object, like a tree, for example, our nervous system creates an "image" (abstraction) of some sort. We then create a language of 'trees' by further abstraction of the initial abstraction which involves leaving out certain characteristics and including others. This sounds somewhat similar to 'measurement omission' but Korzybski's view is that we basically ignore differences while noticing similarities, in fact, that can be taken as an approximate definition of 'abstracting'. So in creating our definition for 'tree' we ignore perceived characteristics that are different from one tree to another but we look for similarities instead and decide among these which ones should be included in the definition.

Now that we have a working definition when we look at trees we "look for" these characteristics even more and so a cycle has been initiated, like this;

abstract => define => abstract again => possibly refine definition => abstract again => etc. So after looking at trees for awhile we start noticing differences that we left out in the definition of 'tree' but we may want to use to create a subset of 'trees' like 'spruce trees', for example. Someone who has educated themselves in silviculture may be able to pick out 10 kinds of trees from a sample while a "regular" person may only recognize 4 kinds, for example. This illustrates the 'circularity of human knowledge', meaning our knowledge affects the way we perceive the world.

[merjet]

Where do vectors and tensors fit into you scheme? In advanced manifold theory, curvature is not a scalar quantity (such as with curvature of curves and surfaces). This makes a major ueber difference in such theories as Einstein's General Theory of Relativity.

Wow, you're really taxing my memory here. It's been decades since I studied those things. It seems your concern is multi-dimensional measurement, so I think it would fit my scheme similar to other kinds of multi-dimensional measurement. Area = length x width. It's 2-dimensional and the dimensions are of the same kind. Velocity = distance divided by time. It's 2-dimensional but the dimensions aren't the same kind. Take volume (length x width x height). It's 3-dimensional with each dimension the same kind. Of course, there are a lot of multi-dimensional measurements in physics. But extrapolating one could have a measurement with even more dimensions and even more kinds of dimensions. It seems that would cover non-scalar curvature.

[merjet]

Responding to post #45 by general semanticist.

I don't have major problems with any of that. But I'll comment, considering what Rand might have said, too.

Instead of "our concepts are affected or influenced by language", I'd say our conceptual knowledge is contextual. It depends on the particular experiences we've had and our conceptualization of them.

Concepts for Ayn Rand are based on similarities and differences (as they were for John Locke). However, she insisted that all the differences among the referents of a concept were "measurements" (until she got around to addressing concepts of consciousness).

Instead of "our knowledge affects the way we perceive the world", I'd say our knowledge affects what we notice when we perceive the world. (Of course, we don't all perceive the world in exactly the same way, color-blind versus normal color vision for example.)

[BaalChatzaf]

Where do vectors and tensors fit into you scheme? In advanced manifold theory, curvature is not a scalar quantity (such as with curvature of curves and surfaces). This makes a major ueber difference in such theories as Einstein's General Theory of Relativity.

Wow, you're really taxing my memory here. It's been decades since I studied those things. It seems your concern is multi-dimensional measurement, so I think it would fit my scheme similar to other kinds of multi-dimensional measurement. Area = length x width. It's 2-dimensional and the dimensions are of the same kind. Velocity = distance divided by time. It's 2-dimensional but the dimensions aren't the same kind. Take volume (length x width x height). It's 3-dimensional with each dimension the same kind. Of course, there are a lot of multi-dimensional measurements in physics. But extrapolating one could have a measurement with even more dimensions and even more kinds of dimensions. It seems that would cover non-scalar curvature.

Tensors do not decompose in the manner that you suggest.

Ba'al Chatzaf

[merjet]

Once again, I think the root of disagreement is over the difference in meaning of the word "measurement." I use Rand's version and that implicitly includes the question "measure what?" (i.e., qualities).

Then you missed the point of my post #38. Maybe that's partly my fault due to inadequately making my point. But the root is wider and deeper. Ironically your second sentence touches the issue. The root extends into the fundamentals of metaphysics and epistemology, and qualities are more fundamental than measurements. Consider the following questions.

1. What is it?

2. What qualities does it have?

3. Is a given quality quantifiable?

4. Is a given quality measurable?

5. If the answer to #4 is 'yes', how is it done?

I've ordered these by decreasing fundamentality. Of course, one cannot answer question #1 without delving into #2 to some extent. In some circumstances one can go beyond #2 to bolster the answer to #1. One might insert other questions in the schema, but these are enough for this context.

On the other hand, to exaggerate the role of measurement is like trying to "make a mountain out of a molehill" and to reverse the order of fundamentality. It suggests abandoning Aristotle's metaphysics for Pythagoras'. It suggests abandoning entity as the primary existent for number as the primary existent.

I can't stop you from using "measure" as a synonym for "identify", and obviously I could not have stopped Rand from doing so. But I use "measure" as it's been handed down by Aristotle, Newton, other scientific giants and the engineers who have vastly improved the conditions of human life. It's not simply a semantic choice between their use and yours or Rand's. It's a matter of precise identification of what measurement is. To measure is to identify in a specific way using an objective standard. To approve of others using "measure" in whatever arbitrary way they please is an open invitation to subjectivity, "junk science" and "junk math". It's analogous to approving of others using "businessman" for the likes of Orren Boyle and certain Enron executives.

Edited October 20, 2007 by Merlin Jetton

[tjohnson]

Instead of "our concepts are affected or influenced by language", I'd say our conceptual knowledge is contextual. It depends on the particular experiences we've had and our conceptualization of them.

What do you mean by 'conceptual knowledge'? I interpret that to mean simply perceptions of some sort. If we had no language, our perceptions could not be influenced by language and "what you see is what you get" but with language this is not the case because we are "filtering" or noticing what we see via the structure of our language.

Instead of "our knowledge affects the way we perceive the world", I'd say our knowledge affects what we notice when we perceive the world. (Of course, we don't all perceive the world in exactly the same way, color-blind versus normal color vision for example.)

I don't think your statement is substantially different than mine. Also, you're correct, we do begin with different sensory input but whatever input we have is subject influence of our language.