Mathematics and Philosophy

[Boydstun]

At the entrance of Plato’s Academy was the inscription: LET NO ONE IGNORANT OF GEOMETRY ENTER HERE! I quite agree. No general epistemology having no competent epistemology of mathematics is a real competitor in general epistemology. Mathematical knowledge is crucial in the conceptual power of humans. These books are some of those I study in my quest for formulating a competent epistemology of mathematics, and some may be excellent for interest of some readers here.

 

THE JOY OF ABSTRACTION

Eugenia Cheng

 

CATEGORIES FOR THE WORKING PHILOSOPHER

Elaine Landry, editor

 

MATHEMATICS: FORM AND FUNCTION

Saunders MacLane

 

THE FOUNDATIONS OF GEOMETRY AND THE NON-EUCLIDEAN PLANE

George E. Martin

 

ALGEBRA

Saunders MacLane and Garrett Birkhoff

 

MATHEMATICAL PHYSICS

Robert Geroch

Edited October 13, 2022 by Boydstun

[StrictlyLogical]

"Mathematics is About the World" by Robert E. Knapp may be of interest to you.

[Boydstun]

Yes. I did not think that book (2014) successfully advanced philosophy of mathematics or what Rand's epistemology might contribute to philosophy of mathematics. However, it is a good entry place that makes some areas of mathematics accessible, and for that, it can be added to the list above of places to get some grasp of those areas. As far as a philosophy of mathematics tied to a more general philosopher goes, I think the book that same year by James Franklin (2014) AN ARISTOTELIAN REALIST PHILOSOPHY OF MATHEMATICS – Mathematics as the science of Quantity and Structure is better.

(On the ontology side, I expect my own philosophy of mathematics to have taken for definition at the outset: mathematics is the discipline studying the formalities of situation, where situation is one of my categories as presented in my fundamental paper "Existence, We", and the formal is divided between the foundational formalities which in that paper I introduced as belonging-formalities (in the world regardless of our discernment) and tooling-formalities (our set-theoretic [or better, perhaps, categoric-theoretic {in the sense of categories in mathematics, as in some of the books in my list above}]; formalities of situation would cover both of those formalities. Formalities of my other two categories that are not entity—character and passage—would belong to logic, rather than mathematics. If this allotment to these disciplines can indeed be shown appropriate, it would show a big advantage of my category-division of existence over Rand's category-division: entity, action, attribute, relationship. Although, whatever I am able to come up with for using my catergories in ontology of mathematics, I could also probably mimic using Rand's categories, though that would be less tidy. It is important that I amend Rand's measurement-omission analysis of concepts, expanding it to give theory of mathematical concepts beyond kind-concepts in order to bring forth for her a serious epistemology of mathematics.) 

On 11/3/2014 at 11:20 AM, Boydstun said:

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.

 

Edited October 13, 2022 by Boydstun

[Doug Morris]

1 hour ago, Boydstun said:

absolute (counting), then ordinal, then ratio measurement.

What about interval measurement?

 

[Boydstun]

1 hour ago, Doug Morris said:

What about interval measurement?

 

Oh yes. I think we should insert interval measurement between ordinal and ratio measurement. Thanks.

Suppes et al. discuss this in chapter 4 of FOUNDATIONS OF MEASUREMENT(vol.1), which is what Suppes is presenting in lecture in the video. He covers representation of different measurement structures, more briefly than in FM, in 3.4 of his final book REPRESENTATION AND INVARIANCE OF SCIENTIFIC STRUCTURES (2002).

Patrick Suppes was a wonder. In the American Philosophical Association meetings in the Pacific division, I had the great experience of hearing Suppes both in lecture and as an audience participant in presentations by others. By then his hair was all white. His mind and knowledge and memory still fantastic.

Robert Nozick wades into this sort of measurement in his paper "Interpersonal Utility Theory" which I heard him read at the University of Chicago in 1982. The large room was packed. This paper is in his book SOCRATIC PUZZLES (1997). 

Edited October 13, 2022 by Boydstun

[Boydstun]

The links in the root post did not hold firm. That's no fun.

Academy

The Joy of Abstraction

Robert Geroch

[Boydstun]

Universals and Measurement

I argue in U&M that Rand's measurement-omission analysis of concepts implies a distinctive magnitude structure for metaphysics. This is structure beyond logical structure, constraint on possibility beyond logical constraint. Yet, it is structure ranging as widely as logical structure through all the sciences and common experience. I uncover this distinctive magnitude structure, characterizing it by its automorphisms, by its location among mathematical categories, and by the types of measurements it affords. I uncover a structure to universals implicit in Rand's theory that is additional to recurrence structure. 

Several years after writing U&M, I developed my own metaphysics, akin to Rand's, but significantly different from hers. For the future, on the ontology side, I expect my own philosophy of mathematics to have taken for definition at the outset, as mentioned above: mathematics is the discipline studying the formalities of situation, where situation is one of my categories as presented in my fundamental paper Existence, We, and the formal is divided between the foundational formalities which in that paper I introduced as belonging-formalities (in the world regardless of our discernment) and tooling-formalities (our set-theoretic [or better, perhaps, categoric-theoretic {in the sense of categories in mathematics; sets being one such category}] characterization of belonging-formalities.) Formalities of situation would cover both of those formalities. Formalities of my other two categories that are not entity—character and passage—would belong to logic, rather than mathematics. If this allotment to these disciplines can indeed be shown appropriate, it would show a big advantage of my category-division of existence over Rand's category-division: entity, action, attribute, relationship. Although, whatever I am able to come up with for using my categories in ontology of mathematics, I could also probably mimic using Rand's categories, though that would be less tidy. It is important that I amend Rand's measurement-omission analysis of concepts, expanding it to give theory of mathematical concepts, beyond kind-concepts, in order to bring forth for her a serious epistemology of mathematics—one competitive, notably, with Kant's epistemology of mathematics.

Edited October 22, 2022 by Boydstun

[Boydstun]

I had recently written:

“Ratios are in the magnitude structure of the world, independently of discernment by intelligent consciousness (with its devised measurement scales, coordinate systems, and so forth). However, there is no such thing as the proportionate in a world not faced by the organizations that are living beings."*

To which SJ responded: “So Kant was right.”*

Wrong. How disappointing to Kant were he to hear such a report of his view on space, after all his argumentation and reiterations that the magnitude structure of the world (i.e. Euclidean geometry) is not independent of intelligent consciousness.

Edited April 29, 2024 by Boydstun

[Boydstun]

On 11/3/2014 at 11:20 AM, Boydstun said:

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.

 

On 10/13/2022 at 2:13 PM, Boydstun said:

Yes. I did not think that book (2014) successfully advanced philosophy of mathematics or what Rand's epistemology might contribute to philosophy of mathematics. However, it is a good entry place that makes some areas of mathematics accessible, and for that, it can be added to the list above of places to get some grasp of those areas. As far as a philosophy of mathematics tied to a more general philosopher goes, I think the book that same year by James Franklin (2014) AN ARISTOTELIAN REALIST PHILOSOPHY OF MATHEMATICS – Mathematics as the science of Quantity and Structure is better.

(On the ontology side, I expect my own philosophy of mathematics to have taken for definition at the outset: mathematics is the discipline studying the formalities of situation, where situation is one of my categories as presented in my fundamental paper "Existence, We", and the formal is divided between the foundational formalities which in that paper I introduced as belonging-formalities (in the world regardless of our discernment) and tooling-formalities (our set-theoretic [or better, perhaps, categoric-theoretic {in the sense of categories in mathematics, as in some of the books in my list above}]; formalities of situation would cover both of those formalities. Formalities of my other two categories that are not entity—character and passage—would belong to logic, rather than mathematics. If this allotment to these disciplines can indeed be shown appropriate, it would show a big advantage of my category-division of existence over Rand's category-division: entity, action, attribute, relationship. Although, whatever I am able to come up with for using my catergories in ontology of mathematics, I could also probably mimic using Rand's categories, though that would be less tidy. It is important that I amend Rand's measurement-omission analysis of concepts, expanding it to give theory of mathematical concepts beyond kind-concepts in order to bring forth for her a serious epistemology of mathematics.) 

 

I've now found a case in which what is distinctive about Rand's theory of concepts can contribute to the philosophy of mathematics. It is here, the following portion.

 

Quote

Ayn Rand gave an original analysis of concepts in terms of what she called measurement-omission. The dimensions of traits shared by members in a class under a concept are considered in terms of appropriate measure scales for the traits. The suspension of which particular member in thinking the concept is also a suspension of particular measure-values had by any particular member along the shared measurable traits. Rand did not work out any extension of her measurement theory of concepts from concretes such as PENCIL with its trait of length to geometric concepts such as LINE SEGMENT or CIRCLE, but those two in Euclid are easily seen to fit tightly with her general measurement-omission model.

 

Postulate 1 of Book I of The Elements states our ability to draw a straight line from any point to any point. Today we might put that as: any two distinct points in the Euclidean plane determine a unique straight line between them. The connection with Rand’s general model is that the understanding of what is the concept LINE SEGMENT is just that any line segment falling under that concept must have SOME particular length, but may have ANY length. The particulars under the concept are variable in their particular lengths. Postulate 3 relates to Rand’s general model also. Postulate 3 states our ability to draw with a compass a circle with any center and with any radius. CIRCLE in Randian concept-analysis is just that any member under the concept must be a locus of points having some single non-zero distance from a point.

 

One is not in position to do the proofs for the constructions (proofs that the construction procedures do necessarily effect the desired configuration, exactly so) and proofs for the theorems of Euclid (such as that the three interior angle of any triangle sum to two right angles) simply by having the Randian way of holding the concepts LINE SEGMENT and CIRCLE (or any other way of analyzing those discursive concepts). We must draw the lines, our capability of which is what is being is being stated in Postulates 1 and 3. We do not need to be able to draw with straightedge a line so long as the radius of the earth nor with compass a circle so large as the equator. What we prove, being enabled by only lines we can draw on a paper before us, will suffice to prove the constructions and theorems for all such figures in the Euclidean plane however large or small (above zero) the figures.

 

The lines we draw for accomplishing our proofs in Euclid’s Elements are iconic signifiers of invisible lines having no breadth and which are perfectly straight or circular. In any line we draw, there are an infinite number of such lines within its bounds. Iconic representations enable reference and meaning through their similarity with their object (the invisible lines that are our subject with Euclid). Spoken or written words are not iconic representations, but symbolic ones. Symbolic representations, such as LINE SEGMENT do not rely on similarity between the signifiers (words) and intended object. The symbols that are words can be strung with other words to state relationships between items in the world, concrete or formal. Both the drawings (icons) and the arguments in language, working together, are required for the exquisite body of truths attained in Euclid’s geometry in The Elements.

 

 

Edited January 17 by Boydstun