Principia Mathematica, Vol 1

by Alfred North Whitehead, Bertrand Russell

An Unabridged, Digitally Enlarged Printing Of Volume I of Part I - MATHEMATICAL LOGIC - The Theory Of Deduction - Theory Of Apparent Variables - Classes And Relations - Logic And Relations - Products And Sums Of Classes - Part II - PROLEGOMENA TO CARDIANL ARITHMITIC - Unit Classes And Couples - Sub-Classes, Sub-Relations, And Relative Types - One-Many, Many-One, And One-One Relations - Selections - Inductive Relations

Genres: Philosophy, Mathematics, Science, Nonfiction, Logic, Reference, Textbooks

680 pages, Hardcover

First published January 1, 1910

About the author

Alfred 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.

Reviews

[Paul St Clair Terry · Rating 4 out of 5]

I first encountered this work by Russell and Whitehead when I was aged sixteen. At that stage, it was incomprehensible because I had no knowledge of mathematical logic. I felt that the work must be significant, partly because the cryptic content seemed so arcane. I came back to it in my late teens/early twenties and read it selectively. Now, much older, I have my own three-volume set. I know very few other people who have ventured into this work. To the casual reader, it is unreadable. This is not so much an edifying read as an exposition of a metamathematical thesis. The work was an attempt to found mathematics on the bedrock of logic. It is now understood that, while finite and discrete mathematics, e.g. number theory, can be so derived, non-finite, continuous mathematics, e.g. real analysis, cannot. Principia Mathematica is principally of its time and our metamathematical perspective and understanding have moved on. Thus, this profound text is mostly of historical significance. Nevertheless, it remains an instructive read in terms of its noble undertaking.

[Laura · Rating 4 out of 5]

I formatted and post-processed this book for DistributedProofreaders and Project Gutenberg will publish pretty soon.

Principia Mathematica, the landmark work in formal logic written by Alfred North Whitehead and Bertrand Russell, was first published in three volumes in 1910, 1912 and 1913. A second edition appeared in 1925 (Volume 1) and 1927 (Volumes 2 and 3). In 1962 an abbreviated issue (containing only the first 56 chapters) appeared in paperback. In 2011 a digest of the book's main definitions and theorems, originally transcribed by Russell for Rudolf Carnap, was reprinted in The Evolution of Principia Mathematica, edited by Bernard Linsky.

Written as a defense of logicism (the thesis that mathematics is in some significant sense reducible to logic), the book was instrumental in developing and popularizing modern mathematical logic. It also served as a major impetus for research in the foundations of mathematics throughout the twentieth century. Along with Aristotle's Organon and Gottlob Frege's Grundgesetze der Arithmetik, it remains one of the most influential books on logic ever written.

[William Bies · Rating 5 out of 5]

Once in a lifetime, every self-respecting mathematician ought to have a look at Alfred North Whitehead and Bertrand Russell’s celebrated but, presumably little known masterpiece, the Principia Mathematica, published in three volumes by the Cambridge University Press over the years from 1910 to 1913. For one has a right to expect from it the epitome of rigor, as the authors’ considered and mature response to the crisis of foundations in logic and set theory that raged in the early decades of the twentieth century. Here, their theory of types finds its fullest expression. Probably never since has anyone tried as stupendously hard to expel the paradoxes that beset the carefree user of naïve Cantorian set theory.

Indeed, Whitehead and Russell’s text is admirably clear, for which reason it may be recommended as required reading along with beginning graduate-level analysis. A few comments off the bat for those intrepid enough to wade into this monumental opus. To begin with, the proofs are tedious to follow by and large. As recommended in the preface by the authors themselves, survey them first without attempting to follow the derivations in detail, and later come back to them. Granted, many of the propositions look easy enough on first sight, but appearances can be deceptive when one gets into statements involving several variables and nested quantifiers. Thus, one will want to exercise his skill by simultaneously using the formalism to reason about something non-trivial. Logic and set theory, on the one hand, and real analysis at the graduate level on the other, constitute two fields that offer ample scope to an undertaking such as this. Therefore, we would recommend simultaneously going through at least the first several chapters of Thomas Jech’s Set Theory and Gerald Folland’s Real Analysis (to be reviewed by us in a moment).

Let us now remark on what is novel about the authors’ point of view in vol. I. First, they venture a theory of descriptions that has some novelty. The two main conceits here are, first, a thorough-going logical atomism and second, a concern about predicativity. Roughly speaking, a property is predicative if it can be defined non-circularly. Now, one might be tempted to rejoin, of course no competent logician would resort to a circular definition! But the problem of concept formation is not quite so simple as this in view of what Hans Georg Gadamer will call the hermeneutical circle: in practice, we always find ourselves alternating back and forth between the part and the whole – that is, understanding the part in light of the whole to which it belongs and also, at the same time, comprehending the whole in terms of the parts it comprises. The reason for this is that we human beings are constrained to reason discursively. Gadamer’s hermeneutical circle is not vicious as the terms our minds employ continue ever to be incomplete. If instead we were angelic intelligences capable of an intellectual intuition, we could transcend the hermeneutical circle and have at our disposal completely specified terms. To be a mathematician, then, rests upon an arrogant pretension already to have closed the hermeneutical circle, in the ideal!

Now, Whitehead and Russell maintain that mathematics is concerned only with extensional functions, where ‘a function of a function is called extensional when its truth-value with any argument is the same as with any formally equivalent argument….A function of a function is called intensional when it is not extensional’ [p. 76] The succeeding paragraph outlines their reason why: in their analysis, to arrive at any intensional properties presupposes an invocation of extra-mathematical considerations, such as the beliefs of a human being which can cast, so to speak, an inflection on the proposition being entertained (for instance, it is possible for a human being to believe something without its being true).

In the authors’ account, what is distinctive about extensional as opposed to intensional properties is that the former ‘may be regarded as a function of the class determined by the argument-function, but that an intensional function cannot be so regarded’ [p. 78]. This raises the problem of predicativity. As the authors resolve it, ‘in virtue of the axiom of reducibility...every class can be defined by a predicative function’ [p. 79].

The authors subscribe to a rather expansive view of what properties of a function can be! [p. 88]: obviously problematic if one allows among properties of a function those having to do with beliefs of human beings, but what if the same went for physics as well? Are there physical properties of a system that are not definable just as extensional functions of some mathematical model of its configuration space? This reviewer would be inclined to judge the matter affirmatively. If so, hence, it indicates the possibility of a logic germane to physics itself that goes beyond mere mathematical logic (pace Russell). But we digress.

The second major topic for which the Principia Mathematica is noted is a theory of relations. These days, mathematicians in their train are so accustomed to reasoning in terms of relations that one might fail to appreciate the significance of the concept and of the departure the authors instigate here, for the first time. Yet Rudolf Carnap, for one, picked up on the importance of Whitehead and Russell’s theory of relations early on when he was studying as a university student under Gottlob Frege and went on to enshrine it as basic to his entire program of a logical construction of the world from sensory data alone in his Der logische Aufbau der Welt (1928), a landmark of the logical empiricism of the Vienna Circle.

One might be disposed to suspect a theory of relations of being trivial, in that they can be viewed as sets of ordered pairs, but in fact there are some properties of relations which have no analogues for classes (of the greatest utility in mathematics, by the way), as the authors point out [cf. p. 92]. Indeed, they explain there why the calculus of classes amounts to more than the calculus of propositions and assert the same for the calculus of relations vis-á-vis the calculus of classes, without justification (see part I, section D). The ensuing sections make good on this claim: the propositional calculus in Part IA [pp. 94-131] and the predicate calculus incorporating a theory of types in Part IB [pp. 132-195].

The key section is that on the axiom of reducibility [pp. 168-175]. There follows a section on the theory of descriptions, which basically says that in mathematics every property can be defined extensively. This bold claim is embodied in the concept of an intensional function [p. 196]. Maybe the axiom of reducibility is sufficient for mathematics, after all, but one would have to think more carefully about it in physics when temporality and causality are introduced. At this stage, it remains unclear why one needs a theory of relations as distinct from the theory of classes. In theory it isn’t needed but, just as the way our minds work means that it is convenient to introduce an existential quantifier even though not a basic notion, so too with relations for some purposes (our minds tend to think relationally as well as causally and in terms of existence for some).

A few general comments on vol. I up to this point: none of the theorems proved is very deep. The ratio of new notation to actual results tends to be much too large, one has to have a large memory bank and trust that the concepts being introduced will eventually prove their worth when it comes to analysis in later parts. All of the attention focuses on niceties having to do with types and logical conditions, e.g., when we speak of the x satisfying ϕ it might not exist or might not be unique, or a relation xRy is defined only for such x and y as are themselves significant (i.e., belong somewhere in the hierarchy of types). The authors, however, do manage a much more efficient way of writing out proofs than does Frege in his Grundgesetze der Arithmetik (see our review here), on which the Principia Mathematica is modeled.

In the second half of vol. I, we finally get down to a good definition of number. Whitehead and Russell’s definitions of 0,1,2 appear to be a lot more intuitive than Frege’s! Let’s hope their development of cardinal arithmetic in vol. II will be as well! This reviewer wonders, though, whether we are defining zero in terms of itself, one in terms of itself etc.? Maybe a more fundamental philosophical analysis along the lines of Edmund Husserl would throw light on this nagging question, though Husserl himself never followed up on his early work in the Philosophie der Arithmetik of 1891 (see our review here). Husserl did later publish meditations on time and space, though.

Another question that should surface for the reflective reader concerns whether descriptive set theory constitutes an interesting subject? Certainly it would be if one allows for non-extensively defined properties; i.e., if the theory is in accord with the way people really think in natural languages. Otherwise, one suspects it of being somewhat artificial and lifeless, or merely a field for logicians to exercise their cleverness in without much relevance to actual mathematical problems.

This second half culminates in two substantial results, Zermelo’s axiom [p. 407] and the Schröder-Bernstein theorem [p. 498], The problem at least this reviewer encounters is that there is such a welter of minor propositions that it can be hard to pick out when something important is being said. In particular, pp. 569-575 would repay close study because key to the theory of induction and to characterizing finite versus infinite cardinals; on p. 574, so to speak, an itemized account of the equality proved by the Schröder-Bernstein theorem. A nice feature later on is the explanation with diagrams of what is really going on in Zermelo’s versus Bernstein’s proofs of the Schröder-Bernstein theorem [see pp. 618-619].

Prospective: what to look for in vols. II and III will be the employment of the foundational concepts introduced in vol. I in arithmetic and analysis. In keeping with their preferences, Whitehead-Russell will want to approach these subjects in the most general conceivable manner, which explains why they have to be so methodical in building up the theory step by step over three volumes.

[Arya Tabaie]

Too hard and technical for a non-mathematician as myself

[Alexa Daskalakis · Rating 5 out of 5]

There exists a rare breed of intellectual undertaking so audacious, so staggeringly meticulous, that it does not merely document knowledge—it constructs the very architecture of thought itself. Bertrand Russell and Alfred North Whitehead’s Principia Mathematica is not a book; it is the blueprint of rationality, the attempt to reduce all of mathematics to logic with a precision so exacting it verges on the divine.

To turn these pages is to witness the deconstruction of arithmetic at a molecular level—to see the integers themselves, those seemingly indivisible building blocks of numerical reality, reduced to the raw machinery of symbolic logic. Before Principia Mathematica, numbers were taken as axiomatic; after it, they were proven into existence. The simple equation 1 + 1 = 2 is not assumed—it is constructed across 362 pages of relentless, diamond-hard logical scaffolding.

This is not a work meant to be read; it is meant to be endured and interrogated. Each theorem, each lemma, each proof unfolds with a rigor so suffocating that comprehension feels less like an intellectual exercise and more like an act of mental alpinism, scaling the Everest of pure logic with nothing but the raw muscle of human reason.

Yet, buried beneath its cold, precise, almost alien notation, Principia Mathematica is also a philosophical monolith. It raises terrifying questions: Is mathematics truly universal, or is it a construct of our symbolic systems? Can logic ever fully describe reality, or does it collapse under the weight of its own incompleteness? It is no coincidence that this work, so confident in its quest to ground all of mathematics in logic, would be indirectly dismantled by Kurt Gödel, who proved that any sufficiently complex formal system will contain truths that cannot be proven within itself. In this way, Principia Mathematica becomes an intellectual tragedy—a monument to human ambition that was both triumphant and doomed.

This book is a crucible. To engage with it is to press against the very limits of human cognition, to see the world as an elegant lattice of inference and deduction, to glimpse, for a fleeting moment, the pure and terrifying machinery of mathematical truth. It is not an easy read, nor a book that will offer quick revelations. But for those who are drawn to the deepest questions of logic, mathematics, and the very nature of truth itself, Principia Mathematica remains one of the most significant intellectual achievements in history.

[Dolf van der Haven · Rating 5 out of 5]

I had no idea what to expect when starting this book, but it contains miracles.
The principles of mathematics are apparently based in formal logic, of which the first one hundred or so pages of this book define the “language”. Once you have learnt the language, the remainder of the book sets out to proof basic statements such as “if p implies q, then q implies p”. I had somehow expected basic definitions of arithmetic and calculus, but this book is all about abstract classes (i.e. collections of elements with certain properties), rather than specific operations.
The proofs re tedious, but once you know the language of formal logic, it can be followed.
Not sure if this was a life changing experience, but it was memorable to go through this once in my life.

[Marv · Rating 3 out of 5]

Ok. so I did not finish this book. Although I did learn a great deal about logic from the first thirty or so pages I did manage to read.

[Rachel Willis]

I'm not actually going to read this, but I can pretend I'll get to it one of these days.

[Bob Kaufman · Rating 2 out of 5]

I’m a maths fan but this was a lot. I think I bit off too much.

[pierre · Rating 5 out of 5]

diciamo che il fascicolo sul primo volume che ho trovato di circa 20 pagine mi perpetra violenza esistenziale, devo trovare il secondo però

[A YOGAM · Rating 5 out of 5]

„Principia Mathematica“ ist nicht einfach nur ein Buch; es ist ein intellektuelles Monument, dessen schiere Ambition noch heute Ehrfurcht gebietet. Alfred North Whitehead und Bertrand Russell unternahmen hier den Angst einflößend komplexen Versuch, die gesamte Mathematik aus einer Handvoll rein logischer Axiome abzuleiten. Über Tausende von Seiten erstreckt sich ein Geflecht aus Symbolen und Beweisen von legendärer Dichte, das den Gipfel des logizistischen Traums darstellt und das Fundament für die analytische Philosophie und die moderne Logik legte.
Der wahre, unsterbliche Stellenwert dieses Werkes bemisst sich jedoch ironischerweise an seinem vermeintlich größten Kritiker. Dass sich ein Logiker vom Kaliber eines Kurt Gödel berufen fühlte, seine revolutionären Unvollständigkeitssätze ausgerechnet in direkter Auseinandersetzung mit dem System der „Principia Mathematica“ zu entwickeln, ist die ultimative Verbeugung. Gödel widerlegte nicht irgendein System; er wählte den am weitesten entwickelten, rigorosesten und einflussreichsten Versuch der Geschichte. Gödels Arbeit dokumentiert daher weniger das Scheitern der „Principia Mathematica“ als vielmehr deren unangefochtene historische Dominanz als der Maßstab, an dem sich selbst die größten Geister messen lassen mussten.

[Richard Marney · Rating 5 out of 5]

One doesn’t “finish” this book (nor the next two volumes....) but rather “visits”, knowing a return trip is coming out of necessity or perhaps just curiosity.

This collection has been on my reading bucket list since university.

Happily I have been asked to help on a book project. I will be spending much “visiting time” in the complex but beautiful world of Bertrand Russell.

Volume II awaits.

[Svetoslav Ivanov · Rating 4 out of 5]

Very well-structured I-st volume of Principia Mathematica. Seemed unintelligible initially, but it most perfectly encompasses logics and it allows for understanding the deductive propositions and lemmas without linguistical misconstructions.
NB: It was funny how one of the Theorems states that if a certain Mathematical object exists=>it exists. And this was proved, not taken as a Philosophical/Quasi-Logical Axiom.

[Witty Loser · Rating 5 out of 5]

mathematics are indeed based

[Dil Shrestha]

i want to buy this book