Robert Knapp makes his case that mathematics springs from and is shaped by requirements of indirect measurement in a new book:
Mathematics Is about the World: How Ayn Rand’s Theory of Concepts Unlocks the False Alternatives Between Plato’s Mathematical Universe and Hilbert’s Game of Symbols
Robert E. Knapp (2014)
Thanks for posting this. I have a fair amount of math/engineering from college, but I’ve recently become interested in mathematical foundationalism, and more specifically the analog nature of thought. From the abstract:
Its theme is that mathematics, however abstract, arises from and is shaped by requirements of indirect measurement. Eratosthenes, in 200 BC, demonstrated the power of indirect measurement when he estimated the circumference of the earth by measuring a shadow at noon, in Alexandria, on the day of the summer solstice. Establishing geometric relationships, solving equations, finding approximations, and, generally, discovering quantitative relationships are tools of indirect measurement: They are the core of mathematics, the drivers of its development, and the heart of its power to enhance our lives.
This is a description of how knowledge is not “metaphorically” analog but is ACTUALLY Analog i.e. varying amplitude over time, stored in/on a media with granularity. Our brains are analog machines - and not just “metaphorically”.
Thanks!
I should add that there is a dedicated site for Robert Knapp’s book here: http://mathematicsisabouttheworld.com/
NB,
Is your idea that the brain is an analog machine in its activities supporting perception and in its activities supporting our conceptual activities as well? Is your idea that the brain’s analog activities supporting our conceptual activities are experienced as analogies, hence the pervasiveness of analogies in our conceptual operations?
Is your idea that the brain is an analog machine in its activities supporting perception and in its activities supporting our conceptual activities as well? Is your idea that the brain’s analog activities supporting our conceptual activities are experienced as analogies, hence the pervasiveness of analogies in our conceptual operations?
These are all ideas that I’m working on.
You and I are of an age when we can remember life without access to binary calculators. I think our brains work more along the lines of PopSci Calc than the Texas Instruments LED calculator my father bought in the 1970’s. There are also multiple ways of performing mathematical computations.
Robert Knapp’s book aims to make the case that mathematics, all of it, is best characterized as the science of measurement, direct or indirect. In that general outlook, as well as in its general outlook that mathematics is about the world, it seems to fit comfortably with Ayn Rand’s theory of concepts in terms of measurement-omission. The fit is not good when examined more closely.
To characterize mathematics as the science of measurement, we need to integrate such a perspective with modern theory of measurement as lain out in the three-volume work Foundations of Measurement. Therein one learns the ordered, hierarchical relations of the various measurement structures, which for single-dimension measurement includes these plateau: absolute (counting), then ordinal, then ratio measurement. That middle one is extremely important for Rand’s measurement-omission analysis of concepts. She mistakenly supposed that all magnitude structures in the world or in consciousness possess the suit of traits making ratio measurement appropriate to them, but that we have ordinal measurement to make do when we have not yet learned to apply ratio measurement to a domain (such as to value relations and to states of consciousness). That mistake is easily remedied, and does not undermine her measurement-omission way of analyzing concepts: There are magnitude structures in reality to which these various forms of measurement are appropriate, including structures for which ordinal measurement is appropriate, but ratio measurement is not.*
Counting is often thought of as a way of measuring, and that is also the way it is analyzed by the authors of Foundations of Measurement. In Rand’s Introduction to Objectivist Epistemology, counting was not what Rand had in mind, in topic, as measurement. To have a theory of concepts in which counting was the type of measurement being omitted in conceptual abstraction from instances would not be a novel theory, for that much is true of any theory of concepts. But to say that conceptual abstraction can be understood as not only that which-one sort of suspension of specifics, but further, as suspension of particular measure-value along shared dimension(s) of the particular instances falling under the concept, now that, that is a distinctive theory. Let measure-value be so little as relative places in a linear ordering, even then the theory is substantial and original.
Yet in Dr. Knapp’s book, I’m finding no treatment of ordinal measurement, hyperordinal measurement, or in the case of multidimensional magnitude structures, such geometries as affine (which is the measurement structure appropriate to spacetime in the situations for which special relativity applies). Our author goes with a definition of measurement stated by Rand, one that (unfortunately for her theory of concepts) implies that all measurement is ratio-scale measurement. I say broaden your definition of measurement. Knapp does portray counting as a form of measurement, in addition to ratio-scale measurement; the forms of measurement between them in the hierarchy of measurement is neglected. Oddly, for a treatment of mathematics aiming for concordance with Rand’s epistemology, there is no consideration of ordinal measurement in this book. Please correct me if I’m wrong. Odd too is the treatment of groups as measurement of symmetry taken as related to the broader category similarity, yet without assimilation of Rand’s measurement analysis of similarity into the account. Again, please correct me if I’m just missing it.
What are “ordinal measurement” and “ratio/scale measurement” and “hyperordinal measurement”?
Please concretize the concepts with reference to reality, I’m a layperson when it comes to classes of measurement.
can you give examples where measurements of concretes in reality require these as distinct concepts?
How are measurement classes fundamentally different? [i’ve studied General Relativity in university, why is affine geometry (insofar as it is applicable to reality…) an issue?]
Also, years ago I played around with things like fractal dimension (Hausdorff) which are fascinating and can be used to classify physical systems (up to a point) and mathematical fractals.. but that is a special mathematical curiosity of mine which may have nothing to do with the subject of measurement.
SL,
The definition of measurement I follow is the one on the first page of the first volume of Foundations of Measurement: the association of numbers (include characters like vectors and quaternions as well) to the attributes of some class of objects or events “in such a way that the properties of the attribute are faithfully represented as numerical properties.”
What I call a magnitude structure are concrete relations in the world (or in our perceptual and thinking operations) to which application of a measurement scale is appropriate. Where I say magnitude, some others say quantity. Same difference. By appropriate, I mean, in this context, as in my 2004 Universals and Measurement: All of the mathematical structure of the measurement scale is needed to capture the magnitude structure of concretes under consideration. It means as well that all the magnitude structure is describable in terms of the mathematical structure of the measurement scale. (As noted [12] in the paper, I take this norm from Robert Geroch’s Mathematical Physics and adapt it for our broader context.)
Some conceptions of measurement are so loose that they would rate numbers on football jerseys as a measurement, which they call nominal measurement. I exclude that under the definition of measurement as I mean in the definition above. Mere distinct individuality, and even mere individuality with mere membership, does not a magnitude structure make, at least not in my book. I’d say the concepts same, different, and membership are logically prior to the concepts magnitude or measurement. The technical analysis of the various types of measurement, the various types of scales, is cast in terms of logically prior elements of logic (such as the logical connectives and, or, . . .) and elements of set theory (such as membership). So I do not go along with Rand’s attempt to analyze connectives such as and in terms of measurement omission. But I stay with her idea that all concretes stand in magnitude relations, at least to the level of ordinal structure, with other concretes. Though I exclude the logical and set-theoretic presuppositions of measurement from the measurement-omission model, it remains with me that all concretes can be brought under some concept(s) or other under the measurement-omission model. That remainder is large and a substantive conjecture. I’m not talking about the genesis of concepts here, only possibilities of analysis of concepts (and similarity relations) in terms of measurement omission. Because of the logical priority (however might stand genetic priority) of logic and set theory in relation to theory of measurement, I approach a book aiming to analyze such things as sets in terms of measurement—as I gather, so far, is an aim in Robert Knapp’s book—with much wariness.
SL, I’ll have to pass on physical magnitude structures, such as studied in relativity, to which affine geometry, affine measurement, is appropriate. Digging into my relativity books would take me too far from other studies in which I’m immersed for a book I’m writing.
I’ll neglect the additional relations and axioms that are progressively added to logical and set-theoretic relations and axioms for analysis of the hierarchy of measurements from ordinal to ratio, but here are examples for the salient types of measurement we actually go about. I take these from Patrick Suppes’ Representation and Invariance of Scientific Structures (2002, 118), without further comment, although with hyperlinks to my pertinent previous expositions. These are one-dimensional scales; a similar story goes for hierarchies of geometry.
Ordinal – Mohs hardness scale; Beaufort wind scale; qualitative preferences.
Hyperordinal – perceived pitch and loudness; utility.
Interval – temperature;* potential energy; cardinal utility.
Ratio – mass, distance, electric current, voltage.
So are you saying Rand’s “measurement omission” is somewhat a misnomer although the concept it refers to is correct? Should it simply be called something else?
No. It is not a misnomer. Ordinal measurement is a kind of measurement. Suspension of particular measure values on an ordinal scale along dimensions shared by instances falling under a concept, where ordinal scaling is appropriate to the magnitude character of those dimensions, is measurement omission. The idea of measurement omission—suspension of particular measure value—can be applied in the case of any of those measurement scales, ordinal to ratio.
Could you explain what you meant by:
“She mistakenly supposed that all magnitude structures in the world or in consciousness possess the suit of traits making ratio measurement appropriate to them, but that we have ordinal measurement to make do when we have not yet learned to apply ratio measurement to a domain (such as to value relations and to states of consciousness). That mistake is easily remedied, and does not undermine her measurement-omission way of analyzing concepts: There are magnitude structures in reality to which these various forms of measurement are appropriate, including structures for which ordinal measurement is appropriate, but ratio measurement is not.”
Take the property of solids: scratch hardness. They have other types of hardness, such as dent hardness, and levels of dent hardness may be ordered with all the structure of ratio scales. However, when it comes to scratch hardness, we order the levels of hardness only with the less elaborate structure of the ordinal scale. To my knowledge, no one has ever been able to analyze scratch hardness such that measures of it would meaningfully have the full structure of ratio scales. It could well be that the property of scratch hardness is a magnitude structure that simply has no more structure than has ordinal scale. The ordinal scale we use for scratch hardness is called the Mohs scale. The instances of scratch hardness of a solid must have some measure-value of scratch hardness to fall under the concept scratch hardness, but may have any measure-value of scratch hardness. The measurement-omission key to concepts works fine for all measurement types, from ordinal to ratio.
Similarly it stands with with rankings of preferences in utility theory. The ordinal case of the Beaufort scale of winds for sailing is especially interesting to me. Looking at the linked chart, consider it without the parenthetical (Wind Speed) and the column for speed. Wind speed is a more recent addition, changing the meaning of the scale (made possible by our later ability to measure wind speeds). By the original descriptions of conditions by which a given level on the scale (first column or third) is to be identified, it is clear the scale was originally an measure (merely ordinal) of strength in effects of wind. That is a pretty complex property, and the later correlations with ranges of wind speed (speed is a ratio measure, both numerator and denominator being ratio measure themselves, allowing ratio of them to be taken) is a correlation with, and a partial cause of, that complex property. But wind speed is not that complex property itself and does not transmute the original ordinal measure of that complex property into a ratio one.
Do you believe Rand’s “any value” of her measurement omission is contextual? i.e. do you take her to mean, for example in the context of the concept, “integer number” to mean that measurement omission means the concept “number” is any INTEGER quantity, but no particular integer quantity? or a concept like “truth of statement” being a concept in a context providing a binary “any value” i.e. true or false, omitted when referring to the concept prior to evaluation.
I’m really trying to understand the mistake you are identifying Rand made re. ordinality as regard to her views of concept formation and “measurement omission”.
As related to the OP, I found an interesting book:
.
SL (#12)
No. Rand did not make an error peculiar to ordinality. In my view, she had an incorrect view of the different kinds of measurement, ordinal to rational, in that she supposed use of a measurement scale weaker than ratio scale (which she called extensive measurement and which is a type of ratio measurement) can eventually be supplanted with ratio-scale measurement once we know enough about the nature of the area to which the weaker (less structure) sort of scale is presently being applied, viz., in the areas of personal value rankings and in psychophysical measurements (Those two areas were her examples, and they are appropriate examples. She may have been unaware of the physical examples of ordinal measurement, used in my 2004 paper and in Binswanger 2014.) As I have shown, that misunderstanding of Rand—a misunderstanding of all sorts of other smart folks I’ve met—does not undermine her measurement-omission model for analyzing concepts one bit.
To say the concept counting number allows in its scope positive integers of any value, though each must be of some value is indeed a case of measurement omission, and that type of measurement scale (the counting sequence) is known as absolute. Rand did not use an example from that category of measurement. Her premier example was length, which is in the category of ratio scale. That was the right way to start, because it is the prototypical type of measurement people are familiar with, and of course, it is a very important type of measurement. From there she went on to give a measurement-omission analysis of the concept _shape. _It turned out she did not actually know how to measure shape and her allusions to integral calculus in that regard really was a derailment into how to measure area and volume, when shape is really independent of size. Not to worry. In my paper, I gave the correct way of measuring shape and showed how it fits just fine with a measurements-omitted analysis of the concept shape. That too is ratio-scale measurement.
I think I threw you off in #5 in saying that the type of measurement involved in counting is not novel to Rand’s theory. That was a poor statement of what I was getting at. What I meant is that the membership relation in a collection being counted is not novel to Rand’s theory of universal concepts as opposed to any other theory of universal concepts, realist or nominalist. They all take instances to be substitutable with each other as instance of the concept, and that sort of substitutability is part of what the child has to grasp in grasping one of the principles of counting, namely the principle that items in a collection may be counted in any order. See further “Universals and Measurement.”
My 2004 essay has been online for most of the last ten years since its original hardcopy publication in The Journal of Ayn Rand Studies. Online it has had thousands of reads, and they never stop, pleased to say.
Thank you for your answers. That helped to clarify things. I might have to check out your paper one day. I did not know Rand had anything to say about the formal study/science of measurement.
Boydstun,
I’m only partially into the second chapter of Knapp’s book. I may be giving this a broad brush here, but the Axiom of Archimedes seems to deal with ordinals of sort. As in the example of the scratch test, comparing the property of hardness between materials, until the Rockwell Hardness established a numerical method for comparing relative hardness (at least in metals), the scratch test provided a means of ordering materials according to which material scratched which.
Prior to settling on standard units of length for measurement, the method outlined by Knapp does accommodate an ordinal approach. Lengths could be ordered from shortest to longest. At this point, the longest length could be determined to be comprised of X times the shortest length where the longest length may be equal to, slightly more, or slightly less than the multiplicity of the shortest length invoked.
I may be jaded by repetitive exposure to Corvini’s works, available at the ARI e-store (esp. the 2, 3, 4 sequel, with regard to this matter.) Knapp may not delve as deeply into it as Corvini, but from the citations offered by Knapp, it appears he too, is influenced by her work as well.
Thanks Greg. I look forward to learning more about the Axiom of Archimedes. I’ve had some exposure to it from measurement theory, but need to learn more.
Scratch hardness of a solid is one physical property; dent hardness of a solid is another physical property of a solid. The Rockwell hardness test is a measure of dent hardness, not scratch hardness.
Dream Weaver said: I may be giving this a broad brush here, but the Axiom of Archimedes seems to deal with ordinals of sort.
I don’t wish to intrude on a private discussion, but the proof of the Axiom of Archimedes (For each real number x there exists a natural number n such that x<n) uses the completeness property of the real numbers. Since the real numbers are the unique complete ordered field (up to isomorphism), the Axiom of Archimedes refers to more than just ordinals. It requires not only the ratio property of a field and an ordering, but the completeness property of the real numbers. Completeness is necessary in order to have a “nice” notion of limits. One may object to it philosophically as an extrapolation too far. The axiom of completeness comes after the axiom of infinity, and so there are philosophical objections that may be made prior to this point in the development of mathematics.
At the top of pg. 106 he does identify an issue with the ordinal approach not lending itself to stating a relationship of diamond being say twice as hard as a quartz crystal per the scratch test. Given your stated interest in theory of moral value (in another thread), I think I see a potential connection to the interest in ordinal ranking. In this sense, we can rank things, such as what concepts of consciousness deal with, as being more or less, but until a commensurable axis is discovered that can be quantized, we cannot establish mathematical relationships among them.
The Axiom of Archimedes is a step in measurement that uses one length (in this case) to measure another in a cardinal approach. By dividing the lengths, increasingly accurate counts could be made. I think this, then, deviates from what you are looking for. With regard to Corvini, the referenced presentation looks into the notion of how the counting numbers lend themselves to using number to measure number with.
aleph_1, I was simply directing those comments more toward Boydstun’s content in this thread, rather than trying to initiate a private discussion.
Dr. Knapp’s approach to the Axiom of Archimedes is stated as dealing with magnitude rather than number. One of his contentions is that the actual meaning of the Axiom of Archimedes is that all magnitudes are measurable. He also posits, with a footnote attached, that mathematics cannot offer a proof of the Axiom of Archimedes. Knowledge of the axiom is something one brings to mathematics.
In the footnote he points out that we do find such proofs in the mathematical literature, but that only means that the axiom has been already introduced in some other form, citing Euclid’s inclusion of it as Definition 4 in Book V. I don’t really know how to tie this into the other points you bring up.
Robert Knapp’s book is erroneous in the relation it puts forth between quantity and measurement. It represents falsely both Aristotle and Rand on this issue.
Knapp holds measurement as more fundamental than quantity (also called magnitude). That is like holding objectivity as more fundamental than existence.
Knapp represents Aristotle as holding that mathematics is the study of quantity and not the relations of quantity. Wrong. Aristotle took geometry to be the “the science of spatial magnitudes” (EN 1143a3). Aristotle repeatedly uses the 2R result of geometry (and its proof) as a splendid example of knowledge. That result and its proof is about relations of magnitudes (relations of quantities).
Ayn Rand recognized that magnitude and relations of magnitude, or quantity (the same thing)— which is in the world without our presence, acts, and consciousness—is the foundation for the objectivity attainable by us in our measurements. See the exchange between Peikoff and Rand in the Appendix of ITOE, pp. 199–200.
Knapp misses the real contribution Rand’s measurement-omission analysis of concepts can have for philosophy of geometry. A real one is here.