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Principia Mathematica; Volume 2
An Unabridged, Digitally Enlarged Printing Of Volume II Of III With Additional Errata To Volume I: Part III - CARDINAL ARITHMETIC - Definition And Logical Properties Of Cardinal Numbers - Addition, Multiplication And Exponentiation - Finite And Infinite - Part IV - RELATION ARITHMETIC - Ordinal Similarity And Relation-Numbers - Addition Of Relations, And The Product Of Two Relations - The Principle Of First Differences, And The Multiplication And Exponentiation Of Relations - Arithmetic And Relation-Numbers - Part V -SERIES - General Theory Of Series - On Sections, Segments, Stretches, And Derivatives - On Convergence, And The Limits Of Functions
808 pages, Hardcover
First published January 1, 1912
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About the author
Alfred North Whitehead
101 books426 followersAlfred North Whitehead, OM FRS (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found application to a wide variety of disciplines, including ecology, theology, education, physics, biology, economics, and psychology, among other areas.
In his early career Whitehead wrote primarily on mathematics, logic, and physics. His most notable work in these fields is the three-volume Principia Mathematica (1910–13), which he co-wrote with former student Bertrand Russell. Principia Mathematica is considered one of the twentieth century's most important works in mathematical logic, and placed 23rd in a list of the top 100 English-language nonfiction books of the twentieth century by Modern Library.
Beginning in the late 1910s and early 1920s, Whitehead gradually turned his attention from mathematics to philosophy of science, and finally to metaphysics. He developed a comprehensive metaphysical system which radically departed from most of western philosophy. Whitehead argued that reality was fundamentally constructed by events rather than substances, and that these events cannot be defined apart from their relations to other events, thus rejecting the theory of independently existing substances. Today Whitehead's philosophical works – particularly Process and Reality – are regarded as the foundational texts of process philosophy.
Whitehead's process philosophy argues that "there is urgency in coming to see the world as a web of interrelated processes of which we are integral parts, so that all of our choices and actions have consequences for the world around us." For this reason, one of the most promising applications of Whitehead's thought in recent years has been in the area of ecological civilization and environmental ethics pioneered by John B. Cobb, Jr.
Isabelle Stengers wrote that "Whiteheadians are recruited among both philosophers and theologians, and the palette has been enriched by practitioners from the most diverse horizons, from ecology to feminism, practices that unite political struggle and spirituality with the sciences of education." Indeed, in recent decades attention to Whitehead's work has become more widespread, with interest extending to intellectuals in Europe and China, and coming from such diverse fields as ecology, physics, biology, education, economics, and psychology. However, it was not until the 1970s and 1980s that Whitehead's thought drew much attention outside of a small group of American philosophers and theologians, and even today he is not considered especially influential outside of relatively specialized circles.
In recent years, Whiteheadian thought has become a stimulating influence in scientific research.
In physics particularly, Whitehead's thought has been influential, articulating a rival doctrine to Albert Einstein's general relativity. Whitehead's theory of gravitation continues to be controversial. Even Yutaka Tanaka, who suggests that the gravitational constant disagrees with experimental findings, admits that Einstein's work does not actually refute Whitehead's formulation. Also, although Whitehead himself gave only secondary consideration to quantum theory, his metaphysics of events has proved attractive to physicists in that field. Henry Stapp and David Bohm are among those whose work has been influenced by Whitehead.
Whitehead is widely known for his influence in education theory. His philosophy inspired the formation of the Association for Process Philosophy of Education (APPE), which published eleven volumes of a journal titled Process Papers on process philosophy and education from 1996 to 2008. Whitehead's theories on education also led to the formation of new modes of learning and new models of teaching.
In his early career Whitehead wrote primarily on mathematics, logic, and physics. His most notable work in these fields is the three-volume Principia Mathematica (1910–13), which he co-wrote with former student Bertrand Russell. Principia Mathematica is considered one of the twentieth century's most important works in mathematical logic, and placed 23rd in a list of the top 100 English-language nonfiction books of the twentieth century by Modern Library.
Beginning in the late 1910s and early 1920s, Whitehead gradually turned his attention from mathematics to philosophy of science, and finally to metaphysics. He developed a comprehensive metaphysical system which radically departed from most of western philosophy. Whitehead argued that reality was fundamentally constructed by events rather than substances, and that these events cannot be defined apart from their relations to other events, thus rejecting the theory of independently existing substances. Today Whitehead's philosophical works – particularly Process and Reality – are regarded as the foundational texts of process philosophy.
Whitehead's process philosophy argues that "there is urgency in coming to see the world as a web of interrelated processes of which we are integral parts, so that all of our choices and actions have consequences for the world around us." For this reason, one of the most promising applications of Whitehead's thought in recent years has been in the area of ecological civilization and environmental ethics pioneered by John B. Cobb, Jr.
Isabelle Stengers wrote that "Whiteheadians are recruited among both philosophers and theologians, and the palette has been enriched by practitioners from the most diverse horizons, from ecology to feminism, practices that unite political struggle and spirituality with the sciences of education." Indeed, in recent decades attention to Whitehead's work has become more widespread, with interest extending to intellectuals in Europe and China, and coming from such diverse fields as ecology, physics, biology, education, economics, and psychology. However, it was not until the 1970s and 1980s that Whitehead's thought drew much attention outside of a small group of American philosophers and theologians, and even today he is not considered especially influential outside of relatively specialized circles.
In recent years, Whiteheadian thought has become a stimulating influence in scientific research.
In physics particularly, Whitehead's thought has been influential, articulating a rival doctrine to Albert Einstein's general relativity. Whitehead's theory of gravitation continues to be controversial. Even Yutaka Tanaka, who suggests that the gravitational constant disagrees with experimental findings, admits that Einstein's work does not actually refute Whitehead's formulation. Also, although Whitehead himself gave only secondary consideration to quantum theory, his metaphysics of events has proved attractive to physicists in that field. Henry Stapp and David Bohm are among those whose work has been influenced by Whitehead.
Whitehead is widely known for his influence in education theory. His philosophy inspired the formation of the Association for Process Philosophy of Education (APPE), which published eleven volumes of a journal titled Process Papers on process philosophy and education from 1996 to 2008. Whitehead's theories on education also led to the formation of new modes of learning and new models of teaching.
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Displaying 1 - 3 of 3 reviews
May 1, 2024
At last, we prove that 1+1=2! The reason we do not get around to a seemly simple proposition such as this until well into vol. II, of course, is that before doing so one has to have defined one’s terms. Whitehead and Russell want to work with absolute rigor and in extreme generality. Thus, it takes the better part of vol. I to build one’s way up to a satisfactory and purely logical definition of 0,1,2. Here in vol. II, we define the concept of number itself and the arithmetical operations that can be performed on them. The basic approach draws on Frege’s concept of equinumerosity. For Whitehead and Russell, though, it is not quite so easy since, first, we want to embrace the entire hierarchy of infinite quantities in Cantor’s transfinitum and second, we want to take especial care to preclude any possible contradictions by means of a theory of types.
All this complication leaves this reviewer wondering whether there is a useful simplified concept of number so that we can employ ordinary arithmetic without worrying about all the complexities of type theory. The answer is yes: this is what Whitehead and Russell call formal numbers [see pp. 292-298].
At this juncture, one may well raise the overall question as to whether Whitehead and Russell do anything interesting in vol. II: their relation arithmetic reduces to Cantor’s ordinal arithmetic for well-ordered relations but works in principle for any relations, yet Whitehead-Russell do not demonstrate that the generalization leads to anything new and interesting, e.g., for non-serial relations. For instance, it is not even clear that anything very deep has ever been done with Cantor’s ordinal arithmetic itself. If so, then what cause have we of generalizing it? In a similar vein, it is far from evident that all the complications of type lead to anything of interest. Whitehead and Russell’s propositions are usually trivial, which means one might as well derive them on the fly as needed rather than build up a huge stock of such cases. Thus they purvey a wealth of new concepts but hardly any results, none of which is deep.
Later sections in Part V get into interesting concepts such as limits, convergence and continuity but note: treated here in full generality, we won’t get to the reals until Part VI in vol. III. To get a handle on what the authors are up to here, let us remark that what general topologists now refer to as a net seems to be just a transfinite series in Whitehead-Russell’s terminology? Now, what the authors go on to do may be of interest as it should show equivalence of Cauchy and Dedekind’s methods of defining the reals [p. 651ff]. Maybe all the material about segments becomes interesting for series involving transfinite ordinals! Also, we know that derivatives in the set-theoretic sense are interesting because of Cantor’s work [p. 700ff]. So maybe all the formalism here is not altogether useless. Unfortunately, it is not spelled out with any examples of situations where, as functional analysts are aware, sequences are insufficient and one needs to avail oneself of the full machinery of nets.
Now the authors discuss convergence and limits of functions without topology! [p. 715ff.] They seem here really to be doing functional analysis in nuce along the lines of Kelly’s textbook on general topology. Their serial relations amount to what we call nets. But everything here is too abstract as again not illustrated by means of any substantial examples. It remains doubtful whether anything is gained by working in such generality, for topological language avails itself of spatial intuition and hence will be easier to comprehend than Whitehead-Russell’s relation language, in which the basic concept is an order relation. For instance, the conventional designations of lim sup and lim inf are easier to understand than Whitehead-Russell’s ‘ultimate oscillation’ though they seem to be essentially the same notion, though with respect to an arbitrary order and not just to the real number line [p. 727]. It will be observed that practically nothing in the basic theory of continuous functions requires the use of numbers themselves (hence Whitehead-Russell recognize that they are covertly doing functional analysis here, cf. p. 755).
A handful of questions for reflection:
1) Whitehead-Russell’s way of defining things is reminiscent of functors in homological algebra (which itself was criticized as empty formalism when first introduced but has since shown its itself productive of good ideas). If someone were to follow up on the Principia Mathematica in earnest, would it play out into something worthwhile as homological algebra seems to have, despite its naysayers?
2) Is the Whitehead-Russell approach to foundations fruitful for the rest of mathematics or just a rabbit hole they like to go down? Along the same lines, is anything Whitehead-Russell do in Principia Mathematica deep? This reviewer is not so sure; for depth, one would want to see something pleasing and unexpected come out, not just an array of over-refined technical distinctions that are never pursued into any interesting consequences.
3) The whole problem of extensive versus intensive definition of properties comes down to this: is their concept of mathematics = everything extrinsically definable adequate? For what purpose, that is? As a bare and formal language game one could so restrict oneself, but what about for physics, viz., for comprehending the world? We know that natural language is a powerful instrument, maybe we need another kind of logic to describe the real world? In other words, might we have proper inclusions: mathematics < physics < life-world, where physics would rest upon a more expressive language than Whitehead-Russell type mathematics that is nevertheless simpler than natural language, but self-contained and useful for the analysis of natural processes (not involving intervention of human beings)?
To summarize our evaluation after reading vol. II, let us ask whether a closer study of the Principia Mathematica would facilitate comprehension of any major branch of mathematics other than set theory and logic? Probably not, a good textbook on real analysis such as Folland or papa Rudin should be sufficient for most purposes. The present approach of Whitehead and Russell may be said to be hampered by 1) an excessive accumulation of special notations; 2) outdated terminology different from what has since become standard; 3) lack of perspective: too many propositions are labeled as important; and 4) a focus almost exclusively on treating things in far more generality than is needed for any substantial applications. Hence, one gets bogged down in minutiae that have no bearing on any interesting examples.
Thus, unless one wants to become a logician, it is doubtful that devoting more than a cursory attention to the Principia Mathematica would repay the effort. It may be well that somebody tried to work out the theory of types in minute detail as far as Whitehead and Russell did but it seems that any major advance in logic would be more likely to stem from an original idea (namely, a new basic concept beyond that of set or class membership, such as may be found in abundance in medieval supposition theory) or philosophical criticism of the construction of existing concepts (as in the age-old problem of universals) than from attempting to take the framework of the present Principia Mathematica itself any further. For there is little evidence that any deep ideas in a conventional field of mathematics, say in functional analysis or algebraic topology, have been stimulated by Whitehead and Russell’s work as we find it here.
Therefore, this reviewer’s judgment: spend as much time as one can stand on the Principia Mathematica for the sake of culture but don’t get mired in the minor details which will probably never lead to anything of substantial interest, then move on to more arresting ideas taken from somewhere else in the vast domain of the rest of mathematics.
All this complication leaves this reviewer wondering whether there is a useful simplified concept of number so that we can employ ordinary arithmetic without worrying about all the complexities of type theory. The answer is yes: this is what Whitehead and Russell call formal numbers [see pp. 292-298].
At this juncture, one may well raise the overall question as to whether Whitehead and Russell do anything interesting in vol. II: their relation arithmetic reduces to Cantor’s ordinal arithmetic for well-ordered relations but works in principle for any relations, yet Whitehead-Russell do not demonstrate that the generalization leads to anything new and interesting, e.g., for non-serial relations. For instance, it is not even clear that anything very deep has ever been done with Cantor’s ordinal arithmetic itself. If so, then what cause have we of generalizing it? In a similar vein, it is far from evident that all the complications of type lead to anything of interest. Whitehead and Russell’s propositions are usually trivial, which means one might as well derive them on the fly as needed rather than build up a huge stock of such cases. Thus they purvey a wealth of new concepts but hardly any results, none of which is deep.
Later sections in Part V get into interesting concepts such as limits, convergence and continuity but note: treated here in full generality, we won’t get to the reals until Part VI in vol. III. To get a handle on what the authors are up to here, let us remark that what general topologists now refer to as a net seems to be just a transfinite series in Whitehead-Russell’s terminology? Now, what the authors go on to do may be of interest as it should show equivalence of Cauchy and Dedekind’s methods of defining the reals [p. 651ff]. Maybe all the material about segments becomes interesting for series involving transfinite ordinals! Also, we know that derivatives in the set-theoretic sense are interesting because of Cantor’s work [p. 700ff]. So maybe all the formalism here is not altogether useless. Unfortunately, it is not spelled out with any examples of situations where, as functional analysts are aware, sequences are insufficient and one needs to avail oneself of the full machinery of nets.
Now the authors discuss convergence and limits of functions without topology! [p. 715ff.] They seem here really to be doing functional analysis in nuce along the lines of Kelly’s textbook on general topology. Their serial relations amount to what we call nets. But everything here is too abstract as again not illustrated by means of any substantial examples. It remains doubtful whether anything is gained by working in such generality, for topological language avails itself of spatial intuition and hence will be easier to comprehend than Whitehead-Russell’s relation language, in which the basic concept is an order relation. For instance, the conventional designations of lim sup and lim inf are easier to understand than Whitehead-Russell’s ‘ultimate oscillation’ though they seem to be essentially the same notion, though with respect to an arbitrary order and not just to the real number line [p. 727]. It will be observed that practically nothing in the basic theory of continuous functions requires the use of numbers themselves (hence Whitehead-Russell recognize that they are covertly doing functional analysis here, cf. p. 755).
A handful of questions for reflection:
1) Whitehead-Russell’s way of defining things is reminiscent of functors in homological algebra (which itself was criticized as empty formalism when first introduced but has since shown its itself productive of good ideas). If someone were to follow up on the Principia Mathematica in earnest, would it play out into something worthwhile as homological algebra seems to have, despite its naysayers?
2) Is the Whitehead-Russell approach to foundations fruitful for the rest of mathematics or just a rabbit hole they like to go down? Along the same lines, is anything Whitehead-Russell do in Principia Mathematica deep? This reviewer is not so sure; for depth, one would want to see something pleasing and unexpected come out, not just an array of over-refined technical distinctions that are never pursued into any interesting consequences.
3) The whole problem of extensive versus intensive definition of properties comes down to this: is their concept of mathematics = everything extrinsically definable adequate? For what purpose, that is? As a bare and formal language game one could so restrict oneself, but what about for physics, viz., for comprehending the world? We know that natural language is a powerful instrument, maybe we need another kind of logic to describe the real world? In other words, might we have proper inclusions: mathematics < physics < life-world, where physics would rest upon a more expressive language than Whitehead-Russell type mathematics that is nevertheless simpler than natural language, but self-contained and useful for the analysis of natural processes (not involving intervention of human beings)?
To summarize our evaluation after reading vol. II, let us ask whether a closer study of the Principia Mathematica would facilitate comprehension of any major branch of mathematics other than set theory and logic? Probably not, a good textbook on real analysis such as Folland or papa Rudin should be sufficient for most purposes. The present approach of Whitehead and Russell may be said to be hampered by 1) an excessive accumulation of special notations; 2) outdated terminology different from what has since become standard; 3) lack of perspective: too many propositions are labeled as important; and 4) a focus almost exclusively on treating things in far more generality than is needed for any substantial applications. Hence, one gets bogged down in minutiae that have no bearing on any interesting examples.
Thus, unless one wants to become a logician, it is doubtful that devoting more than a cursory attention to the Principia Mathematica would repay the effort. It may be well that somebody tried to work out the theory of types in minute detail as far as Whitehead and Russell did but it seems that any major advance in logic would be more likely to stem from an original idea (namely, a new basic concept beyond that of set or class membership, such as may be found in abundance in medieval supposition theory) or philosophical criticism of the construction of existing concepts (as in the age-old problem of universals) than from attempting to take the framework of the present Principia Mathematica itself any further. For there is little evidence that any deep ideas in a conventional field of mathematics, say in functional analysis or algebraic topology, have been stimulated by Whitehead and Russell’s work as we find it here.
Therefore, this reviewer’s judgment: spend as much time as one can stand on the Principia Mathematica for the sake of culture but don’t get mired in the minor details which will probably never lead to anything of substantial interest, then move on to more arresting ideas taken from somewhere else in the vast domain of the rest of mathematics.
January 25, 2025
I spent the better part of 3 months in a pure fixation over the revised vol 1 & 2. For my money it’s the perfect antidote to Gödel - though I have great admiration for Gödel.
August 15, 2021
Two down, one to go.
See comments from Volume 1. A one week break, then on to volume III.
Having time and quiet time at that to explore this series is thanks to being locked-down on and off for 16 months!
A proper review will come when the full set is completed and the project work for which the reading undertaken published.
See comments from Volume 1. A one week break, then on to volume III.
Having time and quiet time at that to explore this series is thanks to being locked-down on and off for 16 months!
A proper review will come when the full set is completed and the project work for which the reading undertaken published.
Displaying 1 - 3 of 3 reviews