Measure and Quantity and Rand's notion of mathematics

[tjohnson]

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.

You know, measurement (in Merlin's and general use of the term) does come into play with qualities at some level if we delve deep enough into the issue, as we do in science. A simple example is colours. One might argue that 'green' is a qualitative characteristic of leaves that is not measurable but within the context of EM wave theory it in fact is measurable, but we don't include any description of the wavelength of the light emanating from leaves in our definition. In this sense, there is 'measurement omission' in the definition. Ultimately, all of our knowledge about the world can be reduced to relations which explains the fundamental importance of a language devoted to relations of all sorts; mathematics.

[merjet]

What do you mean by 'conceptual knowledge'? I interpret that to mean simply perceptions of some sort.

I suggest reading Rand's ITOE. If you don't want to do that, I suggest Chapters 3-11 here:

http://objectivism101.com/Lectures/

[merjet]

You know, measurement (in Merlin's and general use of the term) does come into play with qualities at some level if we delve deep enough into the issue, as we do in science. A simple example is colours. One might argue that 'green' is a qualitative characteristic of leaves that is not measurable but within the context of EM wave theory it in fact is measurable, but we don't include any description of the wavelength of the light emanating from leaves in our definition. In this sense, there is 'measurement omission' in the definition. Ultimately, all of our knowledge about the world can be reduced to relations which explains the fundamental importance of a language devoted to relations of all sorts; mathematics.

Strictly speaking we can't measure green. We can, or some people can, with the right instrument, measure the wave frequency of light we call green. Of course, the concept green is formed long before, and without the aid of, such measurement. So it's irrelevant to form the concept.

The point you make about 'measurement omission' is mildly interesting with respect to Ayn Rand's theory of concepts. Did she mean only measurements we are aware of, or also ones we are completely ignorant of? I suspect only the former, but given her example of a child's "implicit measurement" of length in ITOE, who knows? Heck, maybe she even thought a MLB baseball pitcher "implicitly solves" partial differential equations when planning the trajectory of his next pitch.

Edited October 20, 2007 by Merlin Jetton

[Michael Stuart Kelly]

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?

Merlin,

This is a very good set of questions. However one question comes to mind. How do you propose to identify qualities? To my mind, this is done by identification (noting similarities and differences to other things, and integrating these observations). What standard do you use for noting similarities and differences, or do you have another form of identifying qualities? Or is it just simply a given, like a percept?

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.

For the record, I do not consider Rand's theory of concept formation to be anywhere near "junk science," "junk math" or "the likes of Orren Boyle and certain Enron executives."

Michael

[merjet]

How do you propose to identify qualities? To my mind, this is done by identification (noting similarities and differences to other things, and integrating these observations). What standard do you use for noting similarities and differences, or do you have another form of identifying qualities? Or is it just simply a given, like a percept?

For the most part, no different than Rand says. Similarities, differences - qualitative and quantitative - and integration provide the foundation of concept formation. Clearly many qualities are given via perception.

For the record, I do not consider Rand's theory of concept formation to be anywhere near "junk science," "junk math" or "the likes of Orren Boyle and certain Enron executives."

For the record I don't either, nor did I say it was. As a whole it's excellent. I even said so in Omissions and Measurement. But it does have some flaws. One flaw is the exaggerated role of "measurement omission", which I've provided plenty of explanation for. There she blew it. I recognized it years ago and for a while gave her the benefit of the doubt. That tactic didn't last. I eventually corrected it to my own satisfaction, and moved on. Now "measurement omission" is in my mental file folders by 'phlogiston' and by 'mantra.'

Why did she latch onto the uncommon notion of measurement she did rather than the traditional one? She made her choice, but what good reason was there for it? (Aware of its origin, I have a hunch how she got it.) It's not the only case where she had an uncommon meaning for a common word. 'Selfishness' is another. In its case she went to great lengths to make her meaning and purpose for it clear. Not so for 'measurement'.

P.S. In post #49 I said I can't stop you from using "measure" as a synonym for "identify". I amend that to: use "measure" to mean "identify using numbers that might be used in nearly any way."

[merjet]

Tensors do not decompose in the manner that you suggest.

Why not?

Maybe my concept is a little narrow. Maybe it's a borderline case. Maybe such things are more like mere computation without a unit of measurement in the usual sense. If the first then it doesn't kill the concept. It only weakens it, and I'm open to making it stronger.

Edited October 21, 2007 by Merlin Jetton

[Newberry]

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

Was that a sneeze? Carry on.

Michael

[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.

Ba'al Chatzaf

Was that a sneeze? Carry on.

Michael

The particles from a sneeze are subject to gravity and follow curved trajectories. But a manifolded hanky or kleenex frustrate calculation of the particular paths in terms of vectors or tensors.

[Newberry]

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

Was that a sneeze? Carry on.

Michael

The particles from a sneeze are subject to gravity and follow curved trajectories. But a manifolded hanky or kleenex frustrate calculation of the particular paths in terms of vectors or tensors.

Merlin, I got all the way through until vectors and tensors. What about wind?

Michael

[merjet]

Merlin, I got all the way through until vectors and tensors. What about wind?

Well, that's a good question, since wind would complicate it more. Or did you mean flatulation?

[Newberry]

Merlin, I got all the way through until vectors and tensors. What about wind?

Well, that's a good question, since wind would complicate it more. Or did you mean flatulation?

Hot air from above and flatulation from below, could intermingle and create a tornado of snot particles that could overrun the continent and infect everyone with Ba'al Chatzaf's cold?

Merlin, I apologize for being gruff on the sculpture thread, though, perhaps, I wonder if I were a scientist if I would take science as personally as I do art. Hong Zhang, from RoR, seemed to be of like minds with me though. What do you think?

Michael

[merjet]

Merlin, I apologize for being gruff on the sculpture thread, though, perhaps, I wonder if I were a scientist if I would take science as personally as I do art. Hong Zhang, from RoR, seemed to be of like minds with me though. What do you think?

Apology accepted. I didn't take your post on the sculpture thread very personally or for long. I concluded I was probably one of the least of your targets. I also apologize for my sarcasm.

If you were a scientist, I hope it would be with enthusiasm. I don't know Hong very well. But she is a scientist and likely a better source to ask about taking art versus science personally.

[UncleJim]

There are two related notions that occur in a mathematical context: measure and quantity.

Let us examine the notion of quantity. There are roughly three kinds of quantity

a. Ordinal. An ordinal quantity indicates where in a sequence something is. First, Second, etc.

b. Cardinal. A cardinal quantity answers the question -- How Many.

c. Measure. A measure quantity answers the question -- How Much.

Ordinals and Cardinals are represented by positive and non-negative integers (respectively) in the finite case. So we have the first item, the second item etc. for ordinals. Or zero items, one item, two items etc. for cardinals.

Measure is different. First of all Measure is generally compact and continuous and not discrete. Measure is handled in physics by the use of -real numbers-. The set of real numbers is linearly ordered, is compact and dense (no holes, no gaps, no jumps).

That are other mathematical objects that are not quantities but have quantities associated with them. For example vectors. Vectors have a magnitude (represented by a non-zero real number) and a direction (conventionally a real number between 0 and 2*pi in the radian system for measuring angles.

When Rand spoke of measurement omission she implied the underlying quantities were measures and involved linearly ordered, dense and compact systems, the simplest of which is the real number systems. If compactness is not absolutely required (i.e. we do not require sequences to converge), then the rational numbers will do. They form a dense but not compact system. When Rand held that mathematics is the science of measurement she was effectively restricting mathematics to the system of real numbers or rational numbers. Mathematics is much broader than that. For example there are mathematical systems that deal with symmetries of all sorts. Group theory is the kind of mathematical system used to express and describe symmetries. Groups, qua groups are not bound to systems of quantities.

One could say mathematics is the discipline or art of abstract structures and relations. For the more technically informed, mathematical systems can be expressed as categories of one kind or another. [Category her is not the general term meaning class; in a math context, category is a particular kind of mathematical system, and I do not intend to discuss categories here]

Ba'al Chatzaf

Under Objectivism mathematics is merely 'another' language. It is a resultant of mans need to explain what his observes.

Man needs to explain what he observes because his continued survival is dependent on his understanding of what it is that he is able to observe. In other words: Under Objectivism, mans survival depends on what he understands about his-self, where he lives and what these say about how he must act to remain what he is where he is living.

[BaalChatzaf]

Under Objectivism mathematics is merely 'another' language. It is a resultant of mans need to explain what his observes.

Man needs to explain what he observes because his continued survival is dependent on his understanding of what it is that he is able to observe. In other words: Under Objectivism, mans survival depends on what he understands about his-self, where he lives and what these say about how he must act to remain what he is where he is living.

There is nothing "merely" about mathematics. It is not an easy discipline to master. Rand certainly did not master it. She dismissed it. Most people do not have the knack for it since it is a system of abstractions generally far removed from everyday experience. It isn't everyday that one meets a one sided closed surface (Klein Bottle) but there it is. Fortunately for most people, it can be practiced by a relative few with no great loss to the public. The public is largely innumerate. They are also generally clueless when it comes to abstraction. Most people cannot understand why there is basically no difference between something shaped like a teacup and something shaped like an anchor ring. (I am referring to the shapes, not the materials). One can be continuously deformed into the other.

Ba'al Chatzaf

[UncleJim]

Under Objectivism mathematics is merely 'another' language. It is a resultant of mans need to explain what his observes.

Man needs to explain what he observes because his continued survival is dependent on his understanding of what it is that he is able to observe. In other words: Under Objectivism, mans survival depends on what he understands about his-self, where he lives and what these say about how he must act to remain what he is where he is living.

There is nothing "merely" about mathematics. It is not an easy discipline to master. Rand certainly did not master it. She dismissed it. Most people do not have the knack for it since it is a system of abstractions generally far removed from everyday experience. It isn't everyday that one meets a one sided closed surface (Klein Bottle) but there it is. Fortunately for most people, it can be practiced by a relative few with no great loss to the public. The public is largely innumerate. They are also generally clueless when it comes to abstraction. Most people cannot understand why there is basically no difference between something shaped like a teacup and something shaped like an anchor ring. (I am referring to the shapes, not the materials). One can be continuously deformed into the other.

Ba'al Chatzaf

This is much the same way the Pope looks at how people, more normal than he, view religion.

Under Objectivism if the language being used does not describe what is know to exist then it is of little value. This is the way Ayn Rand referred to mathematics. She was not a trained mathematician nor did she ever pretend to be.

The Pope's religion goes on at considerable length talking about how a dead man, after three days of rotting, got up, walked about, spoke to his friends and then ascended into heaven to sit at the right hand of his true father; the God of all that is and will ever be. This nonsense has little value to the more normal persons of the earth.

Its the same with mathematics. If you can show me the one-sided closed surface of which you speak then I can at least consider whether its a value or not. Otherwise I'm not much interested.

Edited March 11, 2008 by UncleJim

[tjohnson]

Under Objectivism mathematics is merely 'another' language. It is a resultant of mans need to explain what his observes.

Man needs to explain what he observes because his continued survival is dependent on his understanding of what it is that he is able to observe. In other words: Under Objectivism, mans survival depends on what he understands about his-self, where he lives and what these say about how he must act to remain what he is where he is living.

Yes, mathematics is another language but it is very different language - one in which we speak about ideals that have no existence outside our nervous system. Math is not about what man observes, it is about what man imagines. We can, however, apply math with great success to what we can observe.

[UncleJim]

Under Objectivism mathematics is merely 'another' language. It is a resultant of mans need to explain what his observes.

Man needs to explain what he observes because his continued survival is dependent on his understanding of what it is that he is able to observe. In other words: Under Objectivism, mans survival depends on what he understands about his-self, where he lives and what these say about how he must act to remain what he is where he is living.

Yes, mathematics is another language but it is very different language - one in which we speak about ideals that have no existence outside our nervous system. Math is not about what man observes, it is about what man imagines. We can, however, apply math with great success to what we can observe.

When man observes reality he is observing individual units of it. Under the language of math this is denoted by the audio/visual symbol '1'. In this way math is used to describe what man observes.

He may also denote the same thing with the audio/visual symbol 'man' or 'self' or 'I'.

When language is used to describe what is imagined this gets into a discussion of the theoretical rather that the real. This is OK as long as that is understood to be what is happening. The trouble we get into is when we attempt to substitute the theoretical into the place where reality is as if it one and the same.

The theoretical needs to be proved before it can be understood; the real does not. The real only needs to be observed to be understood. As has been discussed; understanding the real may be very difficult.

But proving a theory which has no basis in reality is impossible. The offering of such proofs; anyway, is what I term "fancy math tricks."

[tjohnson]

But proving a theory which has no basis in reality is impossible. The offering of such proofs; anyway, is what I term "fancy math tricks."

Do you have an example of this?

[BaalChatzaf]

But proving a theory which has no basis in reality is impossible. The offering of such proofs; anyway, is what I term "fancy math tricks."

What do you mean by "proving"? There have been plenty of scientific theories which were supported by some facts, but later on were shown to be defective or deficient. For example, Newtonian Mechanics aka Classical Physics. This has been thoroughly falsified by scientific observation which is why we now have quantum physics and relativity theory.

Evidence indicating Newtons Law of Gravitation is not generally true was found in the middle of the 19th century. The anomalous precession of the perihelion of Mercury indicated a problem with Newtonian Gravitation. Einstein supplied the "fancy mathematics" which showed why Newtonian Gravitation fails.

Ba'al Chatzaf

[UncleJim]

But proving a theory which has no basis in reality is impossible. The offering of such proofs; anyway, is what I term "fancy math tricks."

What do you mean by "proving"? There have been plenty of scientific theories which were supported by some facts, but later on were shown to be defective or deficient. For example, Newtonian Mechanics aka Classical Physics. This has been thoroughly falsified by scientific observation which is why we now have quantum physics and relativity theory.

Evidence indicating Newtons Law of Gravitation is not generally true was found in the middle of the 19th century. The anomalous precession of the perihelion of Mercury indicated a problem with Newtonian Gravitation. Einstein supplied the "fancy mathematics" which showed why Newtonian Gravitation fails.

Ba'al Chatzaf

To prove something means to demonstrate its existence in reality. When a math statement describes natural phenomenon that natural phenomenon is the proof that the math statement actually describes it.

The issue with Newtonian Gravitation seems to be that it didn't actually describe what it was proposed to describe. Did Newton lie? No, he just did have all the data. With out all the data its impossible to describe it.

Did Einstein use fancy math. Well..... you seem to think he did! You are claiming that Einstein's math has no basis in reality and yet you are using it to refute "Newtonian Gravitation"?

You're confusing me.

[tjohnson]

To prove something means to demonstrate its existence in reality. When a math statement describes natural phenomenon that natural phenomenon is the proof that the math statement actually describes it.

You can't prove a law of physics, you can only prove a theorem in mathematics. 'Proof' has no meaning in physics. In science we can propose things, we can't prove them.

When a math statement describes natural phenomenon that natural phenomenon is the proof that the math statement actually describes it.

I have no idea what this statement even means.

[BaalChatzaf]

The issue with Newtonian Gravitation seems to be that it didn't actually describe what it was proposed to describe. Did Newton lie? No, he just did have all the data. With out all the data its impossible to describe it.

Did Einstein use fancy math. Well..... you seem to think he did! You are claiming that Einstein's math has no basis in reality and yet you are using it to refute "Newtonian Gravitation"?

You're confusing me.

The General Theory of Relativity which is a theory of gravitation correctly predicts the bending of light around a massive body, the redshift of light in a strong gravitational field and the precession of the perihelia of planets, non of which which predicted by Newtonian Gravitation correctly. Newton predicted light bending, but was off a factor of two. His gravitation law did not predict the perihelion precession of Mercury correctly. This was not known until sufficiently accurate telescopes were developed. By the middle of the 19th century, Fraunhoffer developed telescopes that were able to detect an anomalous precession of Mercury that could not be accounted for by the gravitational actions of the sun and other planets (particularly Jupiter, which is the most massive planet). Newton was not aware of spectral shift for a number of reasons, not the least of which was his particulate theory of light and the crudeness of his prisms.

The success of Einstein's theory in making these predictions showed why Newton's laws didn't work.

1. Newton assumed instantaneous gravitational interaction between massive bodies.

2. Newton assumed physical space is "flat" and is properly described by Euclidean geometry. This is not the case. Mass alters the curvature of the space-time manifold.

3. Newton assumed both space and time were absolute. The observed rate at which clocks "tick" depends are on their relative velocity wtr to an observer and the strength of the gravitational field in which the move. Clocks "tick" slower in stronger gravitational fields.

This last is the basis for the clock corrections which enable the GPS to give positions correct to within ten meters.

Given the anomalies of rotations curves of stars in galaxies (the move faster than one might normally expect) one of two things is true.

1. General Relativity is off for strong gravitational fields

or

2. There is unseen matter surrounding galaxies, the so-called Dark Matter (or non baryon matter).

or perhaps something else.

The matter is being studied, even as we speak.

Classical physics is very close for weak gravitational fields and regimes where velocities are small compared to the speed of light. Which is why it took so long to detect the failure of Newtonian mechanics. For "normal" conditions (i.e. low speeds, low mass) classical physics is still close enough for government work. The trajectories of space probes to the outer solar system are reckoned using classical orbital mechanics and the orbits of binary stars are still estimated well enough using Kepler's laws. On the other hand the behavior of clocks is best reckoned using the theory of relativity.

Ba'al Chatzaf

[UncleJim]

The issue with Newtonian Gravitation seems to be that it didn't actually describe what it was proposed to describe. Did Newton lie? No, he just did have all the data. With out all the data its impossible to describe it.

Did Einstein use fancy math. Well..... you seem to think he did! You are claiming that Einstein's math has no basis in reality and yet you are using it to refute "Newtonian Gravitation"?

You're confusing me.

The General Theory of Relativity which is a theory of gravitation correctly predicts the bending of light around a massive body, the redshift of light in a strong gravitational field and the precession of the perihelia of planets, non of which which predicted by Newtonian Gravitation correctly. Newton predicted light bending, but was off a factor of two. His gravitation law did not predict the perihelion precession of Mercury correctly. This was not known until sufficiently accurate telescopes were developed. By the middle of the 19th century, Fraunhoffer developed telescopes that were able to detect an anomalous precession of Mercury that could not be accounted for by the gravitational actions of the sun and other planets (particularly Jupiter, which is the most massive planet). Newton was not aware of spectral shift for a number of reasons, not the least of which was his particulate theory of light and the crudeness of his prisms.

The success of Einstein's theory in making these predictions showed why Newton's laws didn't work.

1. Newton assumed instantaneous gravitational interaction between massive bodies.

2. Newton assumed physical space is "flat" and is properly described by Euclidean geometry. This is not the case. Mass alters the curvature of the space-time manifold.

3. Newton assumed both space and time were absolute. The observed rate at which clocks "tick" depends are on their relative velocity wtr to an observer and the strength of the gravitational field in which the move. Clocks "tick" slower in stronger gravitational fields.

This last is the basis for the clock corrections which enable the GPS to give positions correct to within ten meters.

Given the anomalies of rotations curves of stars in galaxies (the move faster than one might normally expect) one of two things is true.

1. General Relativity is off for strong gravitational fields

or

2. There is unseen matter surrounding galaxies, the so-called Dark Matter (or non baryon matter).

or perhaps something else.

The matter is being studied, even as we speak.

Classical physics is very close for weak gravitational fields and regimes where velocities are small compared to the speed of light. Which is why it took so long to detect the failure of Newtonian mechanics. For "normal" conditions (i.e. low speeds, low mass) classical physics is still close enough for government work. The trajectories of space probes to the outer solar system are reckoned using classical orbital mechanics and the orbits of binary stars are still estimated well enough using Kepler's laws. On the other hand the behavior of clocks is best reckoned using the theory of relativity.

Ba'al Chatzaf

Theory is developed from observation. Since we are unable to observe everything then theory, by its nature, is limited in its ability to accurately predict that which has not; yet, been observed.

This being the case then is it helpful to say that Newton's theories are wrong? I don't think so! Isn't it more accurate to say that Newton's theories have been adjusted to more closely describe what has been observed since his theories were offered for consideration?

[BaalChatzaf]

This being the case then is it helpful to say that Newton's theories are wrong? I don't think so! Isn't it more accurate to say that Newton's theories have been adjusted to more closely describe what has been observed since his theories were offered for consideration?

Mr. Moderator. Kindly let this reply go through.

Einstein's theory is of a completely different character from Newton's theory or classical physics.

a, It is based on a different set of invariant quantities, i.e it is locally Lorentzian, and not Galilean invariant.

b. It assumes interactions cannot occur instantenously.

c. It is based on a completely different conception of space and time. In fact, in Einsteinian theory, space and time are one manifold.

d. In Einstein's theory gravitation is not a force. It is curvature of the spacetime manifold.

Einstein's theory is not a "fix" to classical physics. It is a replacement.

Ba'al Chatzaf

[John Dailey]

~ It is not a 'replacement'; it is an 'encompassing.'

~ Classical Physics did view Space and Time as separate items ('entities'?) to be dealt with on their own, and, integrated with each other (heh, 'somehow'!), BUT, this was a 'scientific'-ideological view (akin to too many so-called 'theories' today re Quantum Physics), and, not inherent in the math; the math implied nothing about the ideology. Were the latter not true, Einstein's math would not 'derive' Classical Physics as a 'special case.'

2Bcont

LLAP

J:D

Edited April 21, 2008 by John Dailey