The Analytic-Synthetic Dichotomy

[tjohnson]

How on earth do you know that "it just jumps"? I observe cats studying distance, target, etc., before they jump. They may not use numbers, but they sure as hell look and act like they use a primitive form of measurement.

Michael

Yes, it's primitve all right, it's called guessing how far to jump, and the more you do it the better you get at it. But what has this to do with the relationship beween mathematics and nature? Animals don't know any mathematics, do you think an animal could define a derivative of a function, for example?

[Michael Stuart Kelly]

Yes, it's primitve all right, it's called guessing how far to jump, and the more you do it the better you get at it. But what has this to do with the relationship beween mathematics and nature? Animals don't know any mathematics, do you think an animal could define a derivative of a function, for example?

GS,

That is the wrong order. Do you define mathematics as "derivatives of functions"? Don't you have to learn how to count first? Simple to complex, etc.?

Michael

[tjohnson]

GS,

That is the wrong order. Do you define mathematics as "derivatives of functions"? Don't you have to learn how to count first? Simple to complex, etc.?

Michael

I already expressed my opinion on that, counting is not 'knowing mathematics', it is doing arithmetic. Mathematics doesn't start until you define mathematical objects and start learning how to manipulate them.

[Michael Stuart Kelly]

GS,

If you do not consider arithmetic to be mathematics, we are using entirely different meanings. Rand's definition of mathematics (ITOE, 2nd ed., p. 7):

Mathematics is the science of measurement.

Michael

[tjohnson]

GS,

If you do not consider arithmetic to be mathematics, we are using entirely different meanings. Rand's definition of mathematics (ITOE, 2nd ed., p. 7):

Mathematics is the science of measurement.

Michael

Ah.. yes, that explains alot. I used to be a surveyor so I know a little about measurement in the topographical sense. There is a field of study called geodesy which is devoted to measurement and modelling of the shape of the earth and although it uses mathematics extensively it is, again, applied mathematics. Perhaps Rand did not distinguish between the two? I also have an Honours Mathematics degree in which I would go weeks or months without ever applying a single theorem to 'real' life, so I think I know the difference

[Michael Stuart Kelly]

GS,

It is not a matter of "knowing the difference." It is a matter of talking about the same thing. When two people use the same term with different meanings, but those meanings are not clear to each other, many unnecessary misunderstandings occur.

Rand's idea of mathematics was a foundation to her idea of concept formation. The way she concluded that categories (concepts) were formed was by an analogy with algebra. First she established what she called a "conceptual common denominator," which defined a basis of measurement of observed existents. This is a similarity (or similarities) that can be measured by the same unit. Then, once the measuring unit was defined, she eliminated the specific measurements.

So in order to get to her theory of concepts, she needed a theory of mathematics broad enough to serve as basis.

You really should read ITOE. It is a fascinating theory.

An interesting part of measurement is that all primary measurement units start out by being a standard that can be observed and/or used immediately without any instruments. Often a part of the body served for this until greater precision developed.

As pertains to our specific discussion, in Objectivism, when the term "mathematics" is used, it is to be understood in the widest meaning possible and it is based on the core idea of measurement.

Michael

[Dragonfly]

Mathematics is the science of measurement.

Michael

And if the great expert in mathematics, Rand, says so, it must of course be true.

[Michael Stuart Kelly]

And if the great expert in mathematics, Rand, says so, it must of course be true.

Dragonfly,

It's a definition. What is yours (in positive terms)?

Michael

[Brant Gaede]

And if the great expert in mathematics, Rand, says so, it must of course be true.

Dragonfly,

It's a definition. What is yours (in positive terms)?

Michael

Rand's definition ("science of measurement") is wrong, as Bob K. and I have already discussed. But I wonder if the math guys here would consider math to be a science? In my own very limited understanding, I don't think so. Logic, for instance, isn't, and logic is the manipulation of concepts. No?

--Brant

Edited July 5, 2007 by Brant Gaede

[Dragonfly]

It's a definition. What is yours (in positive terms)?

Why should I have a definition? Unlike Rand, I don't pretend to be able to give a simple all-encompassing definition of mathematics. Just look the term up in wikipedia for example, if you want to know more.

[Dragonfly]

Rand's definition ("science of measurement") is wrong, as Bob and I have already discussed. But I wonder if the math guys here would consider math to be a science? In my own very limited understanding, I don't think so.

Neither do I.

[Daniel Barnes]

Mike:

>I have a semantics quibble here. Maybe you would like to find a different term than "transcend"? How can you transcend anything without starting at the same place, i.e., the "grounds"?

Why is this a quibble? I wrote:

"To wit: even if it could be easily decided one way or the other, what does the genesis of the concept "triangle" matter? This is rather like saying there is no "dichotomy" between a bison and a jellyfish as we might speculate they all originated in some simple organism at the dawn of time. The cutting edge of the issue is...that "perfect triangles" do exist in our heads but do not exist in the physical world. Thus there is a clear dichotomy, or at the very least a highly useful distinction. If all you want to argue is that there is some kind of original connection between the abstraction and the real world, well so does Plato. But this seems to me to be beside the point."

[Dragonfly]

I think I've found a definition after all: mathematics is the gentle art of creating floating abstractions.

[tjohnson]

Definitions are a dime a dozen, it's undefined terms, or postulates that are important. I think (after Korzybski) of 'science' as the structured experience of the human race. Science is an attempt to explain what is going on (WIGO) around us and it is shared around the world with other scientists. Mathematics wouldn't really fit in there since it doesn't deal directly with WIGO, only indirectly. I would say mathematics is the main tool of science.

[Michael Stuart Kelly]

More definitions:

Measurement is the identification of a relationship in numerical terms...

Although I could not find a specific definition of "science" by Rand, I did see that she considered it to be something along the lines of "organized body of knowledge about a specific class of phenomena" (my paraphrase). Once again, she was using the widest definition possible. She considered science to be a concept of consciousness. Here is a small discussion (ITOE, 2nd ed., p. 35).

Certain categories of concepts of consciousness require special consideration. These are concepts pertaining to the products of psychological processes, such as "knowledge," "science," "idea," etc.

These concepts are formed by retaining their distinguishing characteristics and omitting their content. For instance, the concept "knowledge" is formed by retaining its distinguishing characteristics (a mental grasp of a fact(s) of reality, reached either by perceptual observation or by a process of reason based on perceptual observation) and omitting the particular fact(s) involved.

The intensity of the psychological processes which led to the products is irrelevant here, but the nature of these processes is included in the distinguishing characteristics of the concepts, and serves to differentiate the various concepts of this kind.

It is important to note that these concepts are not the equivalent of their existential content—and that they belong to the category of epistemological concepts, with their metaphysical component regarded as their content. For instance, the concept "the science of physics" is not the same thing as the physical phenomena which are the content of the science. The phenomena exist independent of man's knowledge; the science is an organized body of knowledge about these phenomena, acquired by and communicable to a human consciousness. The phenomena would continue to exist, even if no human consciousness remained in existence; the science would not.

I see nothing wrong with this. It is clear.

Some of the items I uncovered that she called "science" are: mathematics, physics, epistemology, metaphysics, ethics, medicine, grammar, psychology, and political economy.

She also used terms like "physical sciences," "science of method," "theoretical science," "applied sciences" (which she indicated was technology), "special sciences," "sciences that study man" (as a synonym for the humanities), and "normative science" (meaning ethics).

These explanations and lists are probably not complete, but they give an idea of what Rand meant by "science."

Michael

[Michael Stuart Kelly]

I think I've found a definition after all: mathematics is the gentle art of creating floating abstractions.

Dayaamm!!!

[BaalChatzaf]

Rand's definition ("science of measurement") is wrong, as Bob and I have already discussed. But I wonder if the math guys here would consider math to be a science? In my own very limited understanding, I don't think so.

Neither do I.

Right. Mathematics is a discipline and an art. It is not a science because, mathematics qua mathematics has no empirical content.

Ba'al Chatzaf

[BaalChatzaf]

GS,

If you do not consider arithmetic to be mathematics, we are using entirely different meanings. Rand's definition of mathematics (ITOE, 2nd ed., p. 7):

Mathematics is the science of measurement.

Michael

What in a non-metric topological space is being -measured-?

What is the permutation group of a set of objects -measuring-?

Mathematics is more about abstract structure and symmetry than it is about measurement, although there are some branches of mathematics that deal with measurement.

Ba'al Chatzaf

[Michael Stuart Kelly]

Daniel,

The quibble comes from saying that "grounded in reality" is without merit, then saying "transcends reality."

You can only transcend something if you use the same standard as a base to rise from. This is more than genesis.

Maybe "divorced from reality" is more along the lines of what you mean. That could include a common genesis, but then a divorce later occurs as complexity develops. (I do not endorse this idea. I am merely trying to clarify it for my own understanding.)

Michael

[Michael Stuart Kelly]

What in a non-metric topological space is being -measured-?

What is the permutation group of a set of objects -measuring-?

Mathematics is more about abstract structure and symmetry than it is about measurement, although there are some branches of mathematics that deal with measurement.

Measurement is the identification of a relationship in numerical terms...

Michael

[BaalChatzaf]

What in a non-metric topological space is being -measured-?

What is the permutation group of a set of objects -measuring-?

Mathematics is more about abstract structure and symmetry than it is about measurement, although there are some branches of mathematics that deal with measurement.

Measurement is the identification of a relationship in numerical terms...

Michael

Topology studies spaces in primarily non-numerical and non-measurement terms. Some wag once characterized topology as the geometry of a rubber sheet. Bottom line: Rand knew zilch about mathematics. Wittgenstein once said of that which we cannot speak we must (or should) remain silent.

Ba'jal Chatzaf

[Michael Stuart Kelly]

Bob,

Topology doesn't use math and measurement? Whatever.

Michael

[Dragonfly]

Topology doesn't use math and measurement? Whatever.

Topology is math, but doesn't use measurement. Without metric there is nothing to measure. Really, Rand didn't know what she was talking about.

[Michael Stuart Kelly]

I am still waiting to understand what we are talking about. Since nobody knows what math is, obviously Rand can't know either. (I find it strange that people know what math isn't and insist on it with strong rhetoric, but can't say what math is.)

According to Rand's definitions, measurement certainly falls nicely into place.

Michael

[Daniel Barnes]

Mike:

>The quibble comes from saying that "grounded in reality" is without merit, then saying "transcends reality." You can only transcend something if you use the same standard as a base to rise from. This is more than genesis.

You misunderstand my point, I think, which was: let's accept Rand and Peikoff's vague injunction that mathematics should properly be "grounded in reality." OK: then how does it change the present situation? What in current mathematical practice does this Objectivist doctrine specifically permit, and what does it specifically forbid? For example: are we now not supposed to use zero, as Objectivism holds that there is no such thing as nothing? Are infinities disallowed, because there are no infinities in reality? Are we not allowed to use negative integers, except where they relate to, say, electrons or bank accounts? What, exactly, difference does this make?

The answer is, AFAICS, is nothing at all, unless someone can show me different. Thus it seems to be mostly unproductive posturing - the philosophic equivalent of mere branding, not content - which obviously has little merit.

Further, should we accept this doctrine, it actually results in no practical objection to the central issue Rand and Peikoff rhetorically attack, which is that an abstract system like mathematics is very different from physical reality. After all, it is merely "grounded" in reality, it is not equivalent to it. The fact is that while some parts of mathematics can be applied, some cannot. And some that were once thought not to apply, turn out to apply, and vice versa. As Ba'al notes, there are no mathematical points in reality; no lines; no perfect triangles or circles. It can therefore be said to transcend physical reality, as try as we might, we cannot physically get closer than an approximation to any of these things. Note that this observation about mathematics' transcendence of physical reality is also quite plain to see, unlike the vaguer speculations about how we learn such a system, or how it develops epistemologically, which are for the most part simply your or my say-so. Nothing Rand or Peikoff says changes any of this in the least.

Thus, in what I find is a quite common pattern with Objectivism, this philosophic point is fiercely denied rhetorically but either tacitly accepted or is even untouched in practice. I am merely pointing out that in accepting the formulation that mathematics is "grounded" in physical reality you are simultaneously accepting that there is more to it than that.

Edited July 5, 2007 by Daniel Barnes