Here is a relatively short, and possibly simple, question that has just occurred to me. The fundamental concept necessary to understand mathematics, Rand has claimed, is measurement-omission. However, at least prima facie there are disciplines in mathematics which do not measure or claim to be able to measure anything. For instance, topology lacks a distance metric. It doesn’t measure anything, in any obvious sense, but studies the shapes of objects. (In topology, two objects are said to have the same shape if one of them can be stretched, bent, enlarged, or shrunk, such that it can be made to look just like the other one. What is forbidden is any tearing of the shapes, or gluing. Hence I have the shape of a sphere, and so does every other human being, since you can compress us into that shape. However, none of us have the shape of a donut, because that involves a tear in the center, or gluing your head to your foot. Likewise, a figure-eight donut is a distinct shape from a regular donut and a sphere, and so on.)
As another example, set theory does not seem to measure anything. It can certain be used to count things, but it is just the study of the basic relation of set membership, and without some restriction on the nature of the sets in a particular topic, doesn’t really say anything interesting or useful about the rest of the world (that’s not already known to a pre-schooler).
I guess there are a few obvious ways that an Objectivist could explain this: Either say that these are not genuine topics in mathematics, but perhaps their own topic, which is foundations of mathematics; or, these are genuine topics in mathematics, and the notion of “measurement” is a lot broader than is usually meant in casual conversation. Is there a third option, or a way to adjudicate between these?
A manifold surface such a torus, is similar in the sense you describe as a cup.
That being said, if you consider the orifices in the nostrils to the mouth, or if you want to take a trip thru the entrails, the sphere will not ‘reshape’ into a human being.
This is not your question though. Off the cuff, you may be discussing manifold topology
(In topology, two objects are said to have the same shape if one of them can be stretched, bent, enlarged, or shrunk, such that it can be made to look just like the other one. What is forbidden is any tearing of the shapes, or gluing. Hence I have the shape of a sphere, and so does every other human being, since you can compress us into that shape. However, none of us have the shape of a donut, because that involves a tear in the center, or gluing your head to your foot. Likewise, a figure-eight donut is a distinct shape from a regular donut and a sphere, and so on.)
This is actually a perfectly clear example of what measurement-omission is. In classifying the human shape as spherical by claiming some aspects of that shape are essential and not to be modified while others can be stretched, bent, enlarged, or shrunk to reach the shape of a sphere, there is a disregarding, discarding and omission of certain aspects of shape. Selective attention and selective disregard go hand-in-hand. Measurement-omission does not require actually having numbers in hand before you can do it.
As another example, set theory does not seem to measure anything. It can certain be used to count things, but it is just the study of the basic relation of set membership, and without some restriction on the nature of the sets in a particular topic, doesn’t really say anything interesting or useful about the rest of the world (that’s not already known to a pre-schooler).
Well yeah, measurement omission is involved in regarding a particular thing in the world as an element belonging to a set, as a unit. Once you have that perspective, working on consistent set of rules for set theory is work about method not content.
Edit: Set theory omits whatever context there may be between particular sets and what they refer to, and just concentrates on the sets themselves.
Could you give a refference to where she said mathematics is fundamentally measurement ommision? If I recall, she said mathematics was the science of measurement, as in, identifying quantitative relationships. Identifying these relationships using a standard (as in, “standard that serves as a unit”) that is easy to deal with cognitively is all part of the priciple of unit economy. Also, the things youre measuring, and the measurements youre ommitting arent neccessarily tangible.
This is actually a perfectly clear example of what measurement-omission is. In classifying the human shape as spherical by claiming some aspects of that shape are essential and not to be modified while others can be stretched, bent, enlarged, or shrunk to reach the shape of a sphere, there is a disregarding, discarding and omission of certain aspects of shape. Selective attention and selective disregard go hand-in-hand. Measurement-omission does not require actually having numbers in hand before you can do it.
I generally understand the point about measurement-omission as the claim that mathematics omits particular measurements, but that it is the science of measurements. If that’s a misconception–it’s been a while since I read ITOE–then the question is misguided.
Could you give a refference to where she said mathematics is fundamentally measurement ommision? If I recall, she said mathematics was the science of measurement, as in, identifying quantitative relationships. Identifying these relationships using a standard (as in, “standard that serves as a unit”) that is easy to deal with cognitively is all part of the priciple of unit economy. Also, the things youre measuring, and the measurements youre ommitting arent neccessarily tangible.
j..
That’s my understanding as well.
So the consensus thus far seems to be that the notion of “measurement” is much broader than is used in colloquial conversation.
So the consensus thus far seems to be that the notion of “measurement” is much broader than is used in colloquial conversation.
Right. The colloquial usage relies on an (arbitrary?) agreed upon standard, that serves as a unit of measure. (inch, mile, hour, etc.) Correct me if Im wrong, but she uses the term measurement to mean basically comparing similarities and differences between entities, so that things such as shape, smell, brightness can all be considered measurements. The measurements that are used to define the referrents, and the measurements to be ommitted depend on your knowledge of the referrent at a given time, hence, knowledge is contextual.
correct?
Edit: I should have said definitions are contextual, and concepts are open ended.
Using your examples, mathematics involves taking concepts such as donut, sphere, etc (which themselves were formed by measurement omission), and relating them to eachother regardless of measurement. That is, regardless of any of the dimensions of a donut, mathematics will show how it can or cannot be related to a sphere, regardless of its dimensions.
So the consensus thus far seems to be that the notion of “measurement” is much broader than is used in colloquial conversation.
Yes, Rand uses the term to apply to all sorts of things: for example to thoughts and emotions. By this reckoning, “anger” is a concept that we use for emotions with a certain type of attribute, even though the individual thoughts may have different “measurements” for that attribute. (Of course, this does not have to single-dimensional.)
Added: In case it interests you, somewhere on the web there’s a paper by someone (I think it was presented at some non-ARI/anti-ARI conference) that critiques Rand’s use of the term “measurement”.
Right. The colloquial usage relies on an (arbitrary?) agreed upon standard, that serves as a unit of measure. (inch, mile, hour, etc.) Correct me if Im wrong, but she uses the term measurement to mean basically comparing similarities and differences between entities, so that things such as shape, smell, brightness can all be considered measurements. The measurements that are used to define the referrents, and the measurements to be ommitted depend on your knowledge of the referrent at a given time, hence, knowledge is contextual.
correct?
Edit: I should have said definitions are contextual, and concepts are open ended.
j..
If this is the correct interpretation, then I suppose this is why I find her claims so unsatisfying. Why use the phrase “measurement-omission” rather than “abstraction” if all she claims is that mathematics is the science of reasoning about certain features of object(s), while omitting others? Measurement usually denotes a property of objects which can be quantified by rational numbers. Even quantity is not a measurement in the ordinary sense of the English word since it is only described by natural numbers, so if she wanted a very broad term she would have been a little bit better-served by using “quantity”. If there is no difference between measurement-omission and abstraction, then what is new about her philosophy of mathematics in light of Aristotle’s works on the subject?
Yes, Rand uses the term to apply to all sorts of things: for example to thoughts and emotions. By this reckoning, “anger” is a concept that we use for emotions with a certain type of attribute, even though the individual thoughts may have different “measurements” for that attribute. (Of course, this does not have to single-dimensional.)
Added: In case it interests you, somewhere on the web there’s a paper by someone (I think it was presented at some non-ARI/anti-ARI conference) that critiques Rand’s use of the term “measurement”.
Well this is not so unusual–this is just saying that some emotions come in degrees.
If this is the correct interpretation, then I suppose this is why I find her claims so unsatisfying. Why use the phrase “measurement-omission” rather than “abstraction” if all she claims is that mathematics is the science of reasoning about certain features of object(s), while omitting others?
Measurement ommision reffers to a conceptual process, she was trying to show the process as being mathematical, I dont think she was trying to define “mathematics”. Could you cite sources on this, it seems like youre focusing on her statements about mathematics, and missing her statements on conceptualization and concept formation as a mathematical process.
Measurement ommision reffers to a conceptual process, she was trying to show the process as being mathematical, I dont think she was trying to define “mathematics”. Could you cite sources on this, it seems like youre focusing on her statements about mathematics, and missing her statements on conceptualization and concept formation as a mathematical process.
j..
We can forget trying to define mathematics, since I don’t have the interest to pick through ITOE to find relevant quotes. I just want to know why she chose this phrase rather than “abstraction” to describe the mathematician’s activity, and how this account differs from Aristotle’s.
Thats just the thing, “this phrase” which I assume you to mean mathematics is fundamentally measurement ommision. I still dont know where thats from. She said mathematics is the science of measurement, and concept formation (measurement omission) is in large part, a mathematical process. Its about bringing quantities beyond our scope of perception, down to the percievable standard of measure, the decimal system for example is easier to deal with than some base 17, bizzare system. Same with tangible measurements, measuring 1/5,280th of a mile is easier when you say “foot”.
She used the term “measurement ommision” because thats exactly what youre doing, youre saying for entity to be referred to by this name (concept) “this attribute must exist in some quantity, but may exist in any quantity” within a range. measurements ommitted.
Well this is not so unusual–this is just saying that some emotions come in degrees.
Okay, then go the next step of considering the concept “emotion” itself. To combine “fear” and “anger” and “joy” under a single concept, one is omitting not just a single “dimension” (i.e the degree of fear, or the degree of joy), but one is also omitting the nature of the emotion. One is implicitly stating that all emotions have some attribute that distinguish them from (say) “thought” or “recollection” or some other non-emotional thing that goes on inside one’s head. So, it is no longer an obvious single scale along which we’re ignoring a measurement.
As for why “measurement-omission” rather than “abstraction”, I can’t say for sure, but the former seems to elucidate the process: i.e. when you are abstracting, what you are really doing is grouping together based on some factors, while ignoring the the differences that are still present along a range of measurements within that group.
Thats just the thing, “this phrase” which I assume you to mean mathematics is fundamentally measurement ommision. I still dont know where thats from. She said mathematics is the science of measurement, and concept formation (measurement omission) is in large part, a mathematical process. Its about bringing quantities beyond our scope of perception, down to the percievable standard of measure, the decimal system for example is easier to deal with than some base 17, bizzare system. Same with tangible measurements, measuring 1/5,280th of a mile is easier when you say “foot”.
She used the term “measurement ommision” because thats exactly what youre doing, youre saying for entity to be referred to by this name (concept) “this attribute must exist in some quantity, but may exist in any quantity” within a range. measurements ommitted.
j..
So let’s take this as the claim: Mathematics is the science of measurement. Presumably this means that, in a very broad reading of the term “measurement”, an element being in a set is a measurement of the set, or possibly of the element. I’m not really sure. And I suppose the claim would be that the study of sets omits every other possible measurement of the elements or sets. But why cast it in this language, rather than abstraction? How is this distinct from Aristotle?
Okay, then go the next step of considering the concept “emotion” itself. To combine “fear” and “anger” and “joy” under a single concept, one is omitting not just a single “dimension” (i.e the degree of fear, or the degree of joy), but one is also omitting the nature of the emotion. One is implicitly stating that all emotions have some attribute that distinguish them from (say) “thought” or “recollection” or some other non-emotional thing that goes on inside one’s head. So, it is no longer an obvious single scale along which we’re ignoring a measurement.
I think I’m missing the point that you’re making. What is it that you’re arguing?
As for why “measurement-omission” rather than “abstraction”, I can’t say for sure, but the former seems to elucidate the process: i.e. when you are abstracting, what you are really doing is grouping together based on some factors, while ignoring the the differences that are still present along a range of measurements within that group.
This seems like exactly the meaning of “abstraction” as Aristotle used it, when he provided his philosophy of mathematics.
So let’s take this as the claim: Mathematics is the science of measurement. Presumably this means that, in a very broad reading of the term “measurement”, an element being in a set is a measurement of the set, or possibly of the element. I’m not really sure. And I suppose the claim would be that the study of sets omits every other possible measurement of the elements or sets. But why cast it in this language, rather than abstraction? How is this distinct from Aristotle?
I think the element would be an attribute of the set, and yes in a broad sense a measurement. If you have the set 1-10, lets call it “bob”, the concept bob reffers to the elements 1,2,3,…10, but for purposes of unit economy, 1,2,3…10 are ommited and bob is the label on that file folder in your mind. When we focus on a particular element of the set, we ommit its particular measurements as well, knowledge is heirarchical. Like this, bob (the set 1-10) can be looked at as “furniture”, (1) is table, (1.5) is picnic table and (2) might be chair, (2.5) might be “lawn chair”, but (11) is a car, its fundamental characteristic is different, so its not included in the set. Am I unserstanding you or am I way off?
Yes, that sounds fine enough. But that’s basically the kind of picture Aristotle had, and he just used the term “abstraction” to describe it, without the more loaded term “measurement”. So was there a reason for departing from his language, or was she just repeating Aristotle’s view with her own jargon?
Ill have to read up on Aristotles epistemology, I cant remember where he deals with this specifically. Aristotle saw essence as metaphysical, Rand states that a things essence is not intrinsic to a it, but epistemological. She also states that consciousness is an active process of relating, whereas Aristotle with his signet stamp on wax view of consciousness reeks of intrinsicism. If concepts are passively absorbed by our surroundings, and not volitionally formed adhering to a specific method, that throws Rands whole theory out the window.
And, as for “abstraction”, I take that to mean starting with first level concepts (concretes), and moving up along a conceptual heirarchy to reach more complex, well… abstractions. Measurement ommision is a epistemic tool that allows us to do that without overloading the crow during the process of abstracion. (the principle of unit economy)
Yes, that sounds fine enough. But that’s basically the kind of picture Aristotle had, and he just used the term “abstraction” to describe it, without the more loaded term “measurement”. So was there a reason for departing from his language, or was she just repeating Aristotle’s view with her own jargon?
Aristotle’s view of abstraction was wrong because he held essences inhered in the objects. Instead, Rand affirms that something inheres in the objects (identity meaning its quantitative measurements) but essence is found in the subject. All of the attributes of an object are equally important when considering the object as an existent. To conceptualize an object, some attributes are recognized as more important than others within the mind of the subject.
Rand needed to get down to the level of identifying measurement-omission as a step within the process of abstraction in order to keep distinct the object and the idea of an object. Claiming abstraction involves identity-omission is alarmingly overbroad as a description, it immediately makes one wonder how abstraction could possibly be justified if it is no longer about the identities being abstracted from. Measurement-omission is just right is specifying how the details of a particular object’s identity are disregarded in classifying it as a type, a member of a set.
Observation is measurement in a broad sense, as there is a deterministic relation between what exists and what is observed.
With access to the Objectivism CD-ROM, I decided to search for mentions of mathematics. Here are some of the quotes:
Mathematics is the science of measurement. … …Measurement is the identification of a relationship—a quantitative relationship established by means of a standard that serves as a unit.
As to the actual process of measuring shapes, a vast part of higher mathematics, from geometry on up, is devoted to the task of discovering methods by which various shapes can be measured—complex methods which consist of reducing the problem to the terms of a simple, primitive method, the only one available to man in this field: linear measurement. (Integral calculus, used to measure the area of circles, is just one example.)
In this respect, concept-formation and applied mathematics have a similar task, just as philosophical epistemology and theoretical mathematics have a similar goal: the goal and task of bringing the universe within the range of man’s knowledge—by identifying relationships to perceptual data.
Mathematics is a science of method (the science of measurement i.e., of establishing quantitative relationships), a cognitive method that enables man to perform an unlimited series of integrations. Mathematics indicates the pattern of the cognitive role of concepts and the psycho-epistemological need they fulfill. Conceptualization is a method of expanding man’s consciousness by reducing the number of its content’s units—a systematic means to an unlimited integration of cognitive data.