Russell’s Metaphysics and the Foundations of Mathematics

(revisiting an essay from 12 years ago)

There’s a good bit of writing I’ve been hoping to do during these months, but currently I seem to be in a bit of a rut.  So for the moment, I’m turning to the other thing I planned to do here sometime around this summer: revisit some of my writing from a long time ago.  In particular, this summer will be the 10th anniversary of my first short-lived attempt at blogging, and I intend to display and comment on (the least embarrassing) parts of those posts someday soon, but I’ve stumbled across other writings from even further back whose content seems appropriate for Hawks and Handsaws.  Today I want to put up and briefly comment on one of those.

In spring of 2006, I was a university student majoring in mathematics.  As one of my electives, that semester I took a course called “Foundations of Mathematics”.  This is actually the conventional name of a well-studied branch of mathematics which deals in what one might call meta-mathematics, more or less the field that logicians work in and arguably the bridge between the fields of mathematics and philosophy.  (The course I took was in my university’s mathematics department, but I was aware of a very similar course being taught in the philosophy department.)  In this class, we were required to do a research project which would be given as both an essay and an in-class presentation at the end of the semester.  Being a major fan at the time of Bertrand Russell, one of the giants in the realm of mathematical logic, and having recently bought a book of his essays on metaphysics from the campus bookstore, I decided to base my project on connections I’d noticed between Russell’s philosophical writings and the material we had been learning in the course.

Below is the essay that I wrote for that final project.  Everything in blue font is completely unchanged; there are a few replacements and insertions in normal font.  The essay is dated April 21st, 2006, and probably I gave my presentation in class around that time.  Even though this was fairly early in my career as a college student, both the essay and the presentation stood out in my mind in subsequent years as some of the university work that I was prouder of, and as far as I remember, I received an A for both.  (I was inexperienced with giving talks at the time without the help of other project group members or powerpoint; all I can remember is that I somehow managed to be entertaining, particularly by picking up different pieces of chalk and displaying them as examples of the one-place relations [color] and not-[color].)

Anyway, here goes.

Bertrand Russell was one of the foremost philosophers and mathematicians of the early 1900’s, and he spent much of his career studying both the nature of reality and the nature of the world of mathematics.  Therefore, it is not surprising that his metaphysical ideas are extremely analogous to some of the concepts that lie at the heart of modern mathematics.  Russell’s discussions of atomic facts and propositions, the distinction between universals and particulars, individuation, and vagueness are relevant not only to metaphysics, but also to the foundations of mathematics.  Furthermore, these essays show how the tenets of first-order logic are applicable to the nature of the real world in which we live.

Bertrand Russell was born in 1872 and lived to the ripe old age of 98.  Although he spent much of his life trying to put forth his radical political and religious views and taking part in pacifist demonstrations, he also did a great deal of research on philosophy and the foundations of mathematics.  He devoted much time and energy trying to prove that mathematics is entirely reducible to pure logic, but many thinkers believe that his arguments were not conclusive.  However, he also wrote numerous essays on the subject of metaphysics, which is the area of philosophy that deals with the nature of existence and essence, the definition of truth, and what the universe is made of.  I have found that many of his essays on metaphysics also give insight to the foundations of mathematics in a way that is consistent with John C. Boolos’ model of first-order logic as presented in his textbook Logic and Computability.

One of the earliest writings by Russell that touches on some of these ideas is The Principles of Mathematics, which was written in 1903.  Interestingly, the technical discussions in Chapter 4 of The Principles of Mathematics contain little in the way of numbers or mathematical symbols.  Instead, Russell concentrates more on language and grammar, relating aspects of the English language to the language of mathematics.  Early on in the excerpt that is included in Russell on Metaphysics, Russell defines the word term as “[w]hatever may be an object of thought, or may occur in any true or false proposition, or can be counted as one” and calls it “the widest word in philosophical vocabulary”.  Examples include “a man, a moment, a number, a class, a relation”.  Then he divides terms into two classes, which he calls things and conceptsThings are each indicated by what he calls “proper names,” while concepts are indicated by “all other words”.  Although these definitions initially may seem vague, it quickly becomes clear what he means.  His notion of “proper names” is broader than our usual idea of proper names, and involves a theoretical naming of “all particular points and instances”.  For instance, they would include the usual names of people, such as Smith and Socrates, but would also include denotations of a particular room, piece of paper, or cloud in the sky.  The concepts, however, are divided into those indicated by adjectives and those indicated by verbs (which he believes are a type of relation analogous to the relations that are part of a mathematical language L.)  It is clear that this notion of things and concepts is simply a forerunner to Russell’s later notion of particulars and universals, which he develops more completely in later essays and which I will later show to be analogous to constant terms and relations in a language L.  Furthermore, this excerpt from The Principals of Mathematics makes an interesting point that our English language has simple ways of distinguishing between things and concepts.  For instance, in the sentence “I met a man in the street,” the phrase “a man” refers to a particular thing, and not the general concept of “man”, and it does this by including the indefinite article “a”.  In the sentence “life is short,” however, “life” refers to the concept of life in general, and the listener know this because the word “life” is not preceded by “a”.

In 1911, Russell announced that he was an analytic realist in his essays “The Basis of Realism” and “Analytic Realism”.  Realism is the view that reality does not depend on what we experience with our senses, and Russell’s brand of realism is said to be “analytic” because he believes that the existence of the complex depends on the existence of the simple.  Note that the analytic viewpoint is especially crucial for Boolos’ view of the foundations of mathematics, because complex mathematical sentences must be constructed using simpler ones or atomic ones.  In the latter of these two essays, Russell introduces new terms for things and concepts, which are “universals” and “particulars”.  He introduces them as “[u]niversals which are known are called concepts, while particulars which are known are called sense-data”.  Thus, the concept of “chair” is a universal, but the chair that I am sitting in now is a particular, or instance of the universal “chair”.  Similarly, the concept of “three” is a universal, but the three cars that I see driving down [main street bordering campus] is an instance of the universal “three”.

This distinction between particulars and universals certainly did not begin with Russell in 1911.  In fact, it has been an issue in philosophy for over two thousand years.  Aristotle described “substances” as independent things, which “universals” depend on.  Plato famously proposed a world in which particulars lie (our world) and a world in which universals lie (a sort of afterlife where one might find the “ideal” chair or “ideal” three-ness).  During the early Renaissance, general interest among philosophers began to shift away from this division, and nominalism, the view that there are no universals outside of the mind, became popular.  However, in modern times, nominalism is usually rejected.  Roger Penrose has proclaimed that the foundations of mathematics are inconsistent with the nominalist viewpoint and depend on the Platonic viewpoint.  Nino Cocchiarella said that realism is the best response to the paradoxes produced by nominalism.  Russell clearly followed the more modern trend of rejecting nominalism and believing in a distinction between particulars and universals.

In Russell’s 1918 essay, “The Philosophy of Logical Atomism,” a number of new terms are introduced, compared, and contrasted.  He defines a proposition as an assertion which may be true or false, such as “Socrates is dead” or “Two and two are five”.  He “do[es] not propose to attempt an exact definition” for the term fact, but explains that it is “the kind of thing that makes a proposition true or false”.  When one makes such a proposition as “Socrates is dead,” there are certain facts, such as the historical fact of Socrates’ death in Athens, which will render that proposition true.  Interestingly, Russell does not consider facts themselves to have a truth value; their main function is to determine the truth value of a proposition.  Russell goes on to differentiate between what he calls “particular facts” and what he calls “general facts”.  A particular fact is one that concerns a particular thing, such as “This [particular thing] is white.”  A general fact is one that makes a general statement such as “All men are mortal.”

By making an analogy between these concepts and Boolos’ discussion in “A Précis of First-Order Logic,” one can come up with a much clearer way to define these terms.  First, one must think of the English language (or a modified, more logical version) as a mathematical language L, and the universe in which we live as a particular model or interpretation M of that language.  (Another model could represent some hypothetical universe.)  The domain of M is the collection of all particular things that exist in our universe.  Each particular in our universe can be represented by some constant term in L, and each sentence in the English language can be represented by a sentence in L.  The model M will have a diagram (a set of all atomic and negated atomic sentences in L that it shows true), and from that diagram, a set of sentences in L shown true by M can be generated.  These are analogous to Russell’s notion of facts (except for the fact that Russell does not consider facts to have a truth value).  We can assume that the diagram of M is complete; that is, M evaluates the truth value of all possible sentences of L.  Then, a proposition is a sentence in L which is shown either true or false in M.  It is easy from here to define the distinction between particular and general facts: particular facts are sentences which use constant terms in L, and general facts are sentences which use quantifiers over variable terms instead.  For instance, the sentence “This [particular thing] is white” can be written in our language as “white([this particular thing])”, where [this particular thing] is a constant symbol in L.  However, the sentence, “All men are mortal” must be written as “x(man(x) → mortal(x)),” where x is a variable symbol in L.

Russell spends an entire section of this essay discussing negative facts.  A negative fact states that something is not the case, and is clearly analogous to a sentence in L that includes a “not” symbol (~).  Apparently, many thinkers in Russell’s time did not believe in the existence of negative facts.  In fact, Russell says, “When I was lecturing on this subject at Harvard I argued that there were negative facts, and it nearly produced a riot: the class would not hear of there being negative facts at all.”  One of the students, Mr. Demos (Russell does not give a first name) wrote a paper afterwards arguing that they do not exist.  His argument was that what Russell would think of as a negative fact, such as the statement “This is not red,” (which Russell would rephrase as “not: this is red”), is actually a positive assertion that “this is not-red,” where “not-red” is a quality incompatible with redness.  It would be symbolized as “not-red([this])”.  However, Russell rebuts this argument by saying that the fact would be meaningless without specifying that the quality called “not-redness” is incompatible with redness.  This specification requires another sentence, a general fact, which must be added on to the fact “This is not red”.  The fact becomes, “This is not-red, and not-redness is incompatible with redness.”  Isn’t this much more complicated than simply allowing negative facts?  Furthermore, to write the latter part of the sentence in our logical language L, I have found that there is no way to avoid using the “not” symbol.  Therefore, negative facts should be allowed, and analogously, our language must contain the “not” symbol.

Russell tackles the subject of universals directly in his essay “The Problem of Universals”.  In this essay, he touches on the topic of “atomic sentences” as well as “certain words which are called ‘logical words’; such as ‘not’ ‘or’ ‘and’, ‘if’, ‘all’, ‘some’.”  These are, of course, analogous to atomic sentences in L, as well as its logical symbols.  At this point, it also becomes clear that universals (which Wikipedia defines as types, qualities, quantities, or relations) are simply relations in L.  The universal relations are clearly n-place relations in L, where n is greater than one[1].  Russell obviously sees this and explains that such relations each require a certain number of nouns to have meaning.  For instance, the relation “loves” requires two nouns, a and b, so that “loves(a, b)” means “a loves b”.  He points out that although it would be grammatically correct in the English language to simply say, “a loves,” this is really an abbreviation for the sentence, “For some b, a loves b” or “b (loves(a, b))”.  However, it seems clear to me that types, qualities, and quantities are simply one-place relations (Russell does not make this simplification.)  For instance, “[Liskantope] is a student” can be symbolized by “student([Liskantope]),” and “Einstein was not stupid” is “~stupid(Einstein)”.  Russell also confuses the order of terms within the relations, saying that a relation above, as used in the sentence, “A is above B,” must also contain an inherent “sense or direction” to distinguish “A is above B” from “B is above A”.  This problem, which Russell remedies by adding a “sense or direction,” is easily solved by specifying an order for the arguments of the relation “above” in our logical language L.

Russell also proposes a method of determining individuation in this essay.  Russell is concerned with how to determine whether two particulars in a universe are the same or not (or in our analogy, whether two closed terms in L are equal).  In this essay, he argues that two particulars in our universe are the same object if and only if they have the same properties and relations to all other objects (equivalent to saying that two closed terms a and b are equal if for every sentence f(a) in L, M shows f(b) true if and only if M shows f(a) true.)  However, one of the properties of every object is its spatial location.  Therefore, two objects that are physically identical are still not the same if they occupy the same position in space[2].

In one of Russell’s later essays, “Vagueness,” Russell delves into the subject of vague language, which is a famous subject in philosophy.  Two adjectives in the English language that can be considered vague are “red” and “bald”.  There are many shades of redness and many degrees of baldness.  There are situations in which we can look at an object and not be sure whether or not it should be considered red.  Similarly, baldness is ambiguous: is a man bald if he has only twenty hairs on his head?  The answer is almost certainly yes.  However, the same question can be asked for thirty hairs or forty hairs, until we come to a point where we are no longer sure whether or not we can claim that the man is still bald.  Where is the cut-off point?  If no cut-off point is given in our language, then there will be sentences in our language that can be shown neither true nor false by the facts of our universe.  In other words, there is some sentence f in L such that M shows f neither true nor false.  However, by our assumption that our diagram of M is complete, this is unacceptable.  Therefore, although vague language exists, we cannot allow it in our logical language L.  Russell argues that all human language is to some degree vague, and that this causes all human propositions to have an inconclusive truth value.  However, in most cases, the vagueness is minimal, and so we can at least approximate a purely logical language by employing a clear usage of the English language.

I have discovered a constructed language called Lojban that provides an instance of Russell’s metaphysical ideas in action.  Lojban was constructed in 1987 by a group of people who wanted to invent a language that is purely logical.  Part of their motive was that they hypothesized that language influences thought, and that therefore, if children were raised on a purely logical language, they would grow up to think more clearly and logically[3].  As with all languages (except for computer languages), Lojban is used as a medium of communication between humans discussing real events and situations.  In other words, it can be used to state facts and make propositions of the kind that Russell describes (as all languages do).  However, it is grammatically constructed in a most interesting way.  The grammar revolves around the verbs, each of which is treated as an n-place relation.  One example of a Lojban verb is vecnu, which means “to sell”.  This verb vecnu will appear in a dictionary as “x1 vecnu x2 x3 x4,” where the x’s are the required nouns (analogous to the arguments of a four-place relation).  The dictionary definition would explain that x1 is the seller, x2 is the good sold, x3 is the buyer, and x4 is the price.  This is like writing “sells(x, y, z, w)” in our mathematical language to mean, “x sells y to z for w.”  Also, what we think of as a descriptive or qualitative word, such as “beautiful,” is a verb in Lojban that is treated as a one-place relation.  The verb melbi, which means “to be beautiful,” would be found in the dictionary as “x1 melbi,” where x1 is the thing that is beautiful.  This is like writing “beautiful(x)” in our mathematical language to mean, “x is beautiful.”  The nouns in Lojban are easily distinguished from the verbs (or relations) by being preceded by an article, such as le or la.  If Lojban nouns are equivalent to particulars, and verbs are equivalent to universals, then we can see that by using the articles le and la in front of nouns, Lojban grammar highlights a very fundamental distinction between particulars and universals.  Lojban speakers use these specially constructed sentences to make statements about everyday life, showing that Russell’s metaphysical ideas about propositions and relations are indeed applicable to the real world, and not just to the world of pure mathematics.

After reading Bertrand Russell’s various essays on metaphysics, it becomes clear that the nature of the real world is closely related to the nature of pure mathematics, and vice versa.  Therefore, the logic taught by Boolos is useful not only in mathematics, but in real life.  Russell was a rationalist and a strong proponent of believing things only if they are derivable by logic, reason, and evidence.  Perhaps our understanding of the link between the nature of mathematical models and the nature of the universe in which we live can lead to a more logical and rational society.


[1] Here I think I meant “n is greater than or equal to 1″.

[2] Here perhaps I accidentally left out the negative and should have written “if they do not occupy the same position in space”.

[3] This is, of course, a particular case of the famed Sapir-Whorf hypothesis.  It seems unlikely that I was unaware of this name at the time I wrote the essay, but I find it a little strange that I didn’t choose to name it.



Russell on Metaphysics: selections from the writings of Bertrand Russell edited by Stephen Mumford.  New York, New York: Routledge, 2003.  The essays used as sources are:

The Principals of Mathematics (Extracts 1903)

“Analytic Realism” (Extract 1911)

“Philosophy of Logical Atomism” (Extracts 1918)

“The Problem of Universals” (1946)

“Vagueness” (1923)


Boolos, George S., Burgess, John P., Jeffrey, Richard C.  Computability and Logic.  New York, NewYork: Cambridge University Press, 2002.  “The Problem of Universals from Wikipedia, the free encyclopedia”.  “Lojban from Wikipedia, the free encyclopedia”.  “Diagrammed Summary of Lojban Grammar Forms with Example Sentences”.  Copyright 1990, 1991, 1992 – The Logical Language Group, Inc.

It’s been fun for me to look back on this project and revisit some of the material I was thinking about 12 years ago, which involves a little re-learning and occasional lapses in my memory of foundations of mathematics material.  I don’t have all that much to add to this essay, but there are a few off-the-cuff comments I want to make.

First of all, it’s interesting to me to see just how little my writing style for philosophical essays has changed over the last 12 years.  A careful reread of the above essay yields only a handful of very subtle changes that I would make (plus two apparent typos that I mention in footnotes).  Hmm well on the one hand, I wrote pretty darn well for a college freshman, but on the other hand, I would like to think that I have made and am continuing to make progress rather than to feel stuck pressed up against a ceiling.  It is also telling that at the time, I considered that essay a fairly major project which took serious time, but it is shorter than the average essay I write for Hawks and Handsaws, a blog that I run as (presumably) a fun hobby I do in my spare time.

Secondly, I can’t entirely recall everything I alluded to when discussing material we learned from Boolos’ book, which was our main textbook for the class but which I got rid of many years ago.  Despite being a mathematician, I actually have relatively little interest in mathematical logic and foundations and such.  I think I first realized this about myself from taking that course, prior to which I had sort of assumed that my intellectual passion was proportional to the generality and abstractness and purity of logic in any branch of academic studies.  In fact, I find deep investigations into mathematical logic to be dry and rather a headache.  (I have a similar opinion on category theory, which I now consider a useful tool in small doses but overall another example of a branch of mathematics that’s too general and abstract for my tastes.)  The crowning result we learned at the end of the “Foundations of Mathematics” course was the celebrated Gödel’s Incompleteness Theorem, which I’ve retained only on a very fuzzy level, much to my own embarrassment and the shock of non-mathematicians who assume that I must be a solid source of knowledge on it.  I do remember that class fondly — we did a lot of fun group projects, and the professor’s personality was endearing (I think she won me over on the first day by saying, as an aside, “We have to ask ourselves, just what is the number five?”).  But I don’t think I acquired much permanent knowledge, outside of the ideas which relate to my chosen project as presented above.

Thirdly, I’m not so sure I take Bertrand Russell’s side in regard to the existence of negative facts.  To be fair, Russell had a relatively low level of confidence in his opinion on this: he said in the same lecture, “It is a difficult question.  I really only ask that you should not dogmatize.  I do not say positively that there are [negative facts], but there may be.”  I don’t want to mount a full-blown counterargument here as I’ve only considered this briefly in the last few days, and I have have an even lower level of confidence than Russell anyway.  But I will say, first of all, that it bothers me a little that there should be an asymmetry between the one-place relations X and not-X.  It’s not always so clear which one of them should be designated within the language as the original relation symbol (for something like “red” and “not-red” it feels like there’s a natural answer, but what about “married” versus “single”?).  Also, I’m not so sure that there is so much of a cost to having relations exist in pairs of X and not-X accompanied by general facts saying “~∃ x (X(x) ∧ not-X(x))”.  Yes, as I pointed out in 2006, such a general fact requires the “not” (~) symbol to write down.  But I think perhaps my reasoning was faulty when I took this as an argument against allowing the “not” symbol to be used to modify propositions, since there’s a difference between that and adding “~∃” to our very small collection of quantifier symbols.  It’s entirely possible that I’m forgetting about some limitations of the kind of logical language we studied in class which renders these suggestions invalid.

And finally, I find the auxlang Lojban just as fascinating today as I did 12 years ago.  Since that time, I’ve read descriptions of Lojban conferences where enthusiastic learners make use of the rare opportunity of meeting other enthusiasts of this rather obscure project for the purpose of practicing their skills.  The result, as I understand it, usually looks like two people painstakingly stumbling phrase-by-phrase through a conversation, constantly correcting each other and themselves and sometimes resorting to whatever learning materials are on hand in order to determine which out of half a dozen choices is the correct translation of “and” in a particular context.  While Lojban may be on some objective level the most logical conlang ever built, it certainly doesn’t come remotely naturally to most who attempt to use it.  The idle fantasies of those who long for a society of almost completely rational thinkers are severely limited by certain parameters of human minds, and the failure of Lojban to succeed on practically any measure as a fluently spoken language appears to be an intriguing demonstration of that reality.

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