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[Expositions 9.2 (2015) 80–87]  Expositions (online) ISSN: 1747–5376 
	

Waiting for the Revolution 

 
DANIELLE MACBETH 

Haverford College 

 

 

In the summer of 1900 Bertrand Russell attended the first World Congress of Philosophy 

in Paris. There he met the Italian mathematician Giuseppe Peano and learned of Peano’s 

work in logic. In the fall of that year Russell extended Peano’s monadic predicate calculus 

to a complete logic of relations. Russell was euphoric. 

 

My sensations resembled those one has after climbing a mountain in a mist, 

when, on reaching the summit, the mist suddenly clears, and the country 

becomes visible for forty miles in every direction. For years I had been 

endeavoring to analyze the fundamental notions of mathematics, such as 

order and cardinal number. Suddenly, in the space of a few weeks, I 

discovered what appeared to be definitive answers to the problems which 

had baffled me for years. And in the course of discovering these answers, I 

was introducing a new mathematical technique, by which regions formerly 

abandoned to the vaguenesses of philosophers were conquered for the 

precision of exact formulae. Intellectually, the month of September 1900 

was the highest point of my life.1 

 

In 1905 Russell developed his theory of descriptions, the theory Frank Ramsey would later 

characterize as “that paradigm of philosophy”; and in 1910 the first volume of Russell and 

Whitehead’s Principia Mathematica appeared.2 Thus was begun the intellectual movement 

that would come to be known as analytic philosophy. 

The fundamental idea of analytic philosophy, at least at first, was to assume that 

philosophical difficulties arise because we fail adequately to understand the notions on 

which those difficulties are based and then to resolve those difficulties by providing 

conceptual analyses of the problematic notions, analyses that would reveal their underlying 

logical form. Mathematics provided the model. Much as the confusions and difficulties 

surrounding, for example, the notion of a limit in calculus were resolved by the epsilon-

delta definition due to Cauchy, Bolzano, and Weierstrass, so our philosophical confusions 

and difficulties were to be resolved by our coming to clarity about various non-

mathematical concepts. But although analysis is clearly central to the practice of 

mathematics—because it is the means by which mathematicians develop the conceptions 



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they need to support rigorous reasoning in their proofs—the role of analysis in the practice 

of philosophy has turned out to be much less clear-cut. First, as eventually became apparent, 

philosophy has nothing like the complex system of checks and balances that exists in the 

practice of mathematics and ensures that one has significant, though not complete, 

cognitive control over putative developments. And this lack of any system of checks and 

balances in philosophy has led many philosophers to question whether intuition—by 

appeal to which, it is supposed, one assesses the fruits of one’s philosophical analysis—

can have any role at all to play in the practice of philosophy. And if the success of any 

putative philosophical analysis is to be assessed by appeal to one’s intuitive sense of its 

success, then perhaps philosophers should investigate empirically, as (so-called) 

experimental philosophers do, what intuitions not only philosophers but also non-

philosophers actually have in regard to this or that question of analysis. 

A second problem the practice of analysis in philosophy has had to face—as Quine 

famously argues in “Two Dogmas of Empiricism”—is that it is not at all clear in the cases 

of concern to philosophy where meaning ends and fact begins. Because and insofar as the 

concepts of concern to philosophers are ineluctably caught up in the everyday world of 

which we speak, and are subject to the vicissitudes of our changing experiences, the 

traditional a priori methods of philosophy, including the method of philosophical analysis, 

have come to seem deeply problematic. And this has effected in turn just what Quine has 

urged, “a blurring of the supposed boundary between speculative metaphysics and natural 

science.”3 The upshot, philosophy naturalized, is armchair science, philosophy as little 

more than speculative science. 

The Quinean idea that there is no longer room for the practice of philosophy distinct 

from and in addition to the practice of empirical science has been hugely influential. And 

it has been so influential at least in part because it is reinforced by a conception of the 

history of philosophy and empirical science according to which the discipline of 

philosophy serves as little more than an incubator for the various other disciplines, other 

disciplines that, once they are sufficiently robust, can be separated off from philosophy to 

become autonomous fields of inquiry. Physics, for example, was at first a branch of 

philosophy but after the seventeenth century came to be contrasted with philosophy insofar 

as it is an experimental science. Psychology, the empirical study of the mind, and 

linguistics, the empirical study of language, similarly began life non-empirically, as 

branches of philosophy. And most recently the emergence of, first, cognitive science and 

then also neuroscience have seemed to many to have shown that philosophy has nothing 

left with which to concern itself. If consciousness, the last great mystery of the natural 

world, is now amenable to empirical investigation thanks to recent technological 

advances—for example, in brain imaging—then perhaps there really is nothing left for 



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philosophers to do. To this way of thinking, the sort of non-empirical inquiry that 

philosophers have traditionally engaged in had seemed viable as a form of intellectual 

inquiry only because, and so long as, we did not yet have the resources that are needed to 

engage in properly empirical investigations into the relevant phenomena. Because all 

questions have been revealed to be empirical questions, save, perhaps, for properly 

mathematical ones, there is nothing left for the philosopher to do. Philosophy no longer has 

any place in the intellectual culture. 

Of course there are philosophers who disagree. Indeed, there are those who do not 

merely reject the idea that the empirical sciences can and should take over whatever 

questions remain in philosophy; they hold that the empirical sciences have nothing at all to 

contribute to their philosophical work. This is true, for example, of those analytic 

metaphysicians who develop theories that have in principle no empirical significance or 

consequences. It is also true of at least some neo-Aristotelians aiming to recover the ancient 

Greek conception of ourselves as rational animals in the world that was eclipsed by the rise 

of modern science. Both philosophical movements have been criticized, the first as an 

intellectually irresponsible game of wits to no purpose, the second as an intellectually naïve 

attempt to pretend that modernity never happened, that the rise of modern science can be 

ignored. There is surely something right in these criticisms: philosophers cannot simply 

turn their backs on the extraordinary developments in the sciences that have transformed 

not only our everyday lives but our very conception of the universe. (For the record, I am 

nonetheless deeply sympathetic to the spirit of the neo-Aristotelian movement.) Suppose, 

then, that we do take the full measure of the practice of modern science. Is there, even so, 

a way forward for philosophy? 

Russell thought he had discovered “a new mathematical technique” that would enable 

philosophy finally to embark on the sure path of a science. Russell was wrong. Philosophy 

is not a science—not even, pace Kant, an a priori science alongside mathematics. Let us, 

for the purposes of argument, say that it is, as Bernard Williams characterizes it, a “general 

attempt to make the best sense of our life, and so of our intellectual activities, in the 

situation in which we find ourselves.”4 The philosopher, in other words, aims not so much 

to know as to understand: to understand how it all hangs together, what it all means, 

whether it is in fact true, and why we should even care. And because philosophy aims in 

this way to understand, it cannot in the nature of things be replaced or superseded by the 

empirical sciences. Its questions are not empirical questions. They are questions that will 

remain even after all the empirical questions have been answered. Simply put, we need 

both, both philosophy and the empirical sciences. The problem is to understand how, 

exactly, this is to work. 



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After more than a century of the efforts of our ablest philosophers, the practice of 

analytic philosophy has failed to deliver of its promise of clarity and insight. Despite an 

impressive array of technical results, the major problems of philosophy have remained 

essentially untouched. And nowhere is this more evident than in the philosophy of 

mathematics. Over the course of the twentieth century, the philosophy of mathematics 

came to be so detached from, and so completely irrelevant to, mathematical practice that 

practicing mathematicians are no longer willing even to talk to philosophers if they can 

possibly avoid it. In response, a small group of philosophers of mathematics has emerged 

calling for a radically new approach in the philosophy of mathematics, an approach one 

central feature of which is the wholesale rejection of logic as a tool for understanding 

mathematical practice. This is incredible. How could it possibly be that logic, the concern 

of which is reasoning, is irrelevant to our understanding of mathematical practice, at the 

core of which is reasoning? Surprisingly, the question has an answer. 

Although Frege is often cited as the father of modern logic, modern logic would have 

developed pretty much as it did had Frege never existed.5 And as I have argued, in fact 

Frege was doing something quite different from what Russell and the analytic tradition 

following him took him to be doing.6 Beginning with Russell, Frege’s strange two-

dimensional notation was systematically misread as a notation of the logic bequeathed to 

us by Russell, and because it was, Frege’s truly revolutionary work in logic remains to this 

day almost completely unknown. Fortunately, the general idea of Frege’s logical language, 

as it contrasts with our standard logical languages, is evident already in the positional 

system of Arabic numeration as that system contrasts with the system of Roman 

numeration. Reflecting on that latter system thus enables us to gain some insight into the 

former. 

Roman numeration is a system of written signs that was devised for recording how many 

things there are in a collection. It is based on a primitive tally system in which one makes 

one mark for each thing in the collection resulting in a collection of marks that is in a one-

to-one correspondence with the collection of things. The difference between such a tally 

system and the system of Roman numeration is only that in Roman numeration 

abbreviations are introduced: ‘V’ for a collection of five things, ‘X’ for a collection of ten 

things, and so on. Collections of such marks are then read additively: ‘XVII’, for example, 

is ten and five and one and one, that is, seventeen. (It is a late modification, and one that 

will not concern us here, to write, say, IX instead of VIIII, for nine, and so to distinguish 

IX, that is, nine, from XI, eleven.) In such a system, each sign means what it means 

independent of any context of use. The sign ‘V’ inevitably stands in for five things in the 

system of Roman numeration, ‘X’ for ten things and so on; it is merely a convention to 



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write the letters in a particular order. One could as easily write, for example, seventeen, 

XVII, as, say, VIXI. 

Arabic numeration is more interesting insofar as although one can read it as a system 

very like that of Roman numeration, one can also read it differently, as a radically different 

kind of numeration system. Consider, for example, the numeral ‘647’ of the positional 

system of Arabic numeration, and suppose, first, that just as in the case of Roman 

numeration each primitive sign of this system has its meaning independent of any context 

of use. The digit ‘6’, for example, just is the numeral ‘6’ on this reading; it is a name for 

the number six. And so for the other digits: each just is the relevant numeral, a name for 

some particular number, zero, or one, or two, and so on up to nine. What the position of 

the numeral tells one, on this reading, is what it is that one is counting, whether units, or 

tens, or hundreds, and so on. The position of the ‘6’ in ‘647’ tells us that it is counting 

hundreds: there are six hundreds. The position of the ‘4’ tells us that it is counting tens: 

there are four tens. And the position of the ‘7’ tells us that it is counting units: there are 

seven units. The whole is then to be read additively: there are six hundreds and four tens 

and seven units. On this reading, the system of Arabic numeration is essentially similar to 

Roman numeration. Obviously in this case, the order of the numerals matters because that 

is what is telling one what it is that is being counted. Nevertheless, the overall idea of the 

system is the same as that of Roman numeration: the collection of signs serves to record 

how many in a collection of things. 

But we can also read the system of Arabic numeration differently. On the alternative 

reading we take the individual digits, ‘0’ through ‘9’, to function as numerals, as names for 

numbers, only within a context of use. Independent of any context of use, the primitive 

signs of the language, the individual digits, only express what we can call Fregean senses; 

independent of a context of use the primitive signs do not designate or mean or name 

anything. Now we put some primitive signs together to form a complex sign for some 

number. We take the digits ‘6’ and ‘4’ and ‘7’, for example, to form the complex sign, the 

numeral, ‘647’, which is a name for the (one) number six hundred and forty-seven. In this 

way, the digits do not function independently as numerals, as names for numbers; it is only 

in combinations that collections of the primitive signs (in the limit, a “combination” of only 

one digit) function as numerals, as names for numbers. Because, on this reading of the 

language, the primitive signs only express Fregean senses independent of any context of 

use, because they do not designate except in a context of use, complexes of those signs 

such as ‘647’, although they do designate some one number (here, the number six hundred 

and forty-seven), do so through complex Fregean senses, senses that can serve as the basis 

for mathematical calculations precisely because they are complex. Because there are rules 

governing the manipulations of the primitive signs that together form complex signs for 



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numbers, reasoning according to those rules can reveal truths about the numbers that are 

designated by the relevant complex expressions. 

Roman numeration serves merely to picture collections of things. One cannot reason 

mathematically in such a language; one can only manipulate the signs mechanically, as one 

might manipulate the things collected, for instance, combining them or dividing them up 

into smaller collections. The positional system of Arabic numeration, by contrast, though 

it can be read merely mechanically, as essentially like the system of Roman numeration, 

can also be read as a properly mathematical language. It can be read, that is, as exhibiting 

the contents of mathematical ideas, what it is to be, say, six hundred and forty-seven, and 

exhibiting that content in a mathematically tractable way, in a way enabling rigorous 

reasoning in the system of signs. Exactly this distinction applies also to different logical 

languages, in particular to our standard logical language as it contrasts with the language 

Frege devised in his 1879 logic. The logic that is bequeathed to us by Peano, Peirce, Russell, 

and others is a language the primitive signs of which function just as the signs of Roman 

numeration do, to designate independent of a context of use. The language serves to record 

or picture states of affairs, truth conditions. Frege’s logic is different. It is a language within 

which the primitive signs only express senses independent of any context of use, and hence 

can be combined to form complex names for concepts on the basis of which to reason. 

What we need if we are to make real progress in philosophy is just such a Fregean logic 

because, as I show in Realizing Reason, it is this logic, as contrasted with standard logic, 

that radically transforms our understanding of the space of possibilities within which our 

thought can move. 

If analytic philosophy is to become the vibrant and productive discipline it once held 

out promise of being, we need to begin anew with Frege’s logic. We need to revisit 

developments in the nineteenth century, particularly in the practice of mathematics, so as 

to come to a better understanding of the great intellectual advances that were made in that 

century, and we need to recognize that those advances offer a new way forward, one we 

have hitherto failed to consider. It is in just this way that we will come to see that Frege’s 

logic is something essentially and radically new, in just this way that we will come to see 

that, unlike standard logic, Frege’s logic provides rich and powerful resources for 

addressing the traditional problems of philosophy, problems about, for example, our 

capacity for knowledge of the world around us, about the relationship of mind and body, 

and about the reality of objective values.7 

We also need sharply to distinguish between mathematical languages—

paradigmatically, the diagrammatic language of Euclidean geometry, the symbolic 

language of arithmetic and algebra, and Frege’s logical language Begriffsschrift—and the 

logics that govern them, on the one hand, and everyday language and reasoning (including 



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philosophical reasoning), on the other. We need to stop trying to apply standards of rigor 

and perspicuity that are appropriate to mathematics also to everyday life—including the 

everyday life of mathematicians—and to our attempts to achieve an adequate philosophical 

understanding of that life. And we need explicitly to recognize that the empirical sciences 

are neither the arbiter of all things nor merely something we do. We must take the full 

measure of the ways in which the rise of modern science has been transformative of our 

self-understanding while at the same time recognizing that the sciences are not and cannot 

be the wellspring of our deepest self-understanding. And we need, finally, to return to the 

history of philosophy, not because the answers we seek are there but because Hegel was 

right: we cannot understand where we are and need to be until and unless we understand 

how we came to be here. 

Philosophy is not a science among sciences. It has always been and at its best remains 

still today a rogue discipline within which anything can be called into question as reason 

sees fit, a discipline that has no particular subject matter and no particular method, a 

discipline powered only by a relentless, resolute, and passionate desire to understand. And 

there will always be practitioners of this discipline, thinkers who are conversant with the 

works of the great philosophers of the past, thinkers who are, in equal measure, 

intellectually curious and intellectually serious, thinkers who are willing to take risks and 

to question received wisdom and entrenched values. There are such thinkers, such 

philosophers, still today. Many of them one knows; some one does not. 

But one will. 

Come the revolution, one will. 

 

 

Notes 
 

1. Russell 1975, 147. 

2. Russell published his theory of descriptions in Russell 1905. Ramsey makes his 

remark about that theory in “Philosophy” (1929; repr. 1990). Cambridge University 

Press serially published the three volumes of Principia Mathematica in 1910, 1912, 

and 1913.  

3. Quine 1951, 20. 

4. Williams 2000, 479. 

5. Both Hilary Putnam and W. V. O. Quine make this point, Putnam in Putnam 1982, 

and Quine in Quine 1995. 



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6. See Macbeth 2005 and 2014.  

7. Again, see Macbeth 2014 for an extended elaboration and defense. 

 

Works Cited 
 

Macbeth, Danielle. 2005. Frege’s Logic. Cambridge, MA: Harvard University Press. 

_____. 2014. Realizing Reason: a Narrative of Truth and Knowing. Oxford: Oxford 

University Press. 

Putnam, Hilary. 1982. “Peirce the Logician.” Historia Mathematica 9: 290–301. 

Quine, Willard Van Orman. 1951. “Two Dogmas of Empiricism.” Philosophical 

Review 60: 20–43; repr. in From a Logical Point of View, Cambridge, MA: Harvard 

University Press, 1953, 20–46; rev. 2nd ed., 1961. 

_____. 1995. “Peirce’s Logic.” In Peirce and Contemporary Thought: Philosophical 

Inquiries, ed. Kenneth L. Ketner. New York: Fordham University Press, 23–31. 

Ramsey, F. P. 1990. “Philosophy.” Philosophical Papers, ed. D. H. Mellor. 

Cambridge: Cambridge University Press, 1–7. 

Russell, Bertrand. 1905. “On Denoting.” Mind 14.56 (October): 479–93. 

_____. 1975. The Autobiography of Bertrand Russell. London: George Allen and 

Unwin. 

Williams, Bernard. 2000. “Philosophy as a Humanistic Discipline.” Philosophy 75: 

477–96. 


