On Logic, Language and Numbers: Sanford Shieh Discusses Gottlob Frege’s Enduring Mathematical Legacy

On June 16, 1902, Gottlob Frege received a letter from a young Bertrand Russell that began with generous praise and ended with a bombshell. Russell had been working through the logical system Frege had spent years constructing, the grand attempt to show that all of arithmetic could be derived from pure logic alone, and had found a contradiction at its heart. The problem, now known as Russell's paradox, was elegant and devastating: it concerned the set of all sets that do not contain themselves, and it exposed a flaw that no patch could easily fix. Frege had just sent the second volume of his Basic Laws of Arithmetic to the printer. He wrote back to Russell within days, acknowledging the problem with a candor that remains one of the more dignified moments in the history of philosophy. "Your discovery of the contradiction," he told Russell, "has surprised me beyond words." He added a hastily written appendix to the volume acknowledging the flaw. The project he had devoted his working life to was, in its original form, finished.

What makes the story remarkable is not the collapse but what survived it. Friedrich Ludwig Gottlob Frege, born in 1848 in the coastal town of Wismar, stands today as one of the essential architects of how we think about logic and meaning. His work did not merely refine the tools philosophers were using; it reinvented them. Before Frege, logic largely rested on the classical syllogistic structures inherited from Aristotle. After Frege, it emerged as a formal and precise language capable of expressing complex mathematical and philosophical truths. His innovations in predicate logic laid the groundwork for vast stretches of twentieth-century thought, from mathematical philosophy to linguistics and analytic philosophy, and his concepts of sense and reference, developed to solve highly technical problems about identity and meaning, today roll easily off the tongue in philosophical circles as though they had always been there.

Speaking about Frege's intellectual legacy requires a guide who not only understands the terrain but can navigate its sharper philosophical edges. Sanford Shieh, Professor of Philosophy at Wesleyan University, is one such guide. He has devoted his career to tracing the evolution of ideas about logic and mathematics, how they took the form we now recognize, and what assumptions shaped them along the way. Alongside Juliet Floyd, he co-edited Future Pasts (2001), one of the early and influential collections that helped establish the history of analytic philosophy as a serious field of inquiry, tracing how thinkers like Frege, Russell, and Wittgenstein came to define a philosophical era. His most recent work, the first volume of Necessity Lost, is a sustained and ambitious investigation into the concepts of necessity and possibility in early analytic thought.

In the conversation that follows, Shieh brings the precision of a logician and the sensibility of a historian to bear on why Frege's work continues to command attention nearly a century after his death, and what it means that a project could fail so decisively and yet change everything anyway.

Charles Carlini: What initially sparked your interest in exploring the work of Gottlob Frege, a little-known German mathematician and logician?

Sanford Shieh: The first thing I read by Frege was The Foundations of Arithmetic (Die Grundlagen der Arithmetik). At that time, I had studied a lot of math, but no philosophy whatsoever. Frege began the book with a question that had never occurred to me: What is the number 1? For me, it was a striking question. What was more striking was the fact that, not only had I never thought of that question, but I also had no idea what I would say in answer.

Now, perhaps it would seem to some that this is the kind of question that would only grip mathematicians and other people interested in math. For most of us, who are interested in concrete things or who have to worry about the realities of making a living, of what use are the airy-fairy abstractions of math?

But in fact, math is perhaps the most concrete thing there is in the world. Recently, for example, several news headlines have been about a number, 20,000, that the Dow had reached. The relation of this number to the stock offerings of 30 companies is the subject of excitement, and maybe also fear. Numbers also permeate the realities of making a living; what do we worry about if not a bunch of numbers: the number related to your wages, the number related to the sum of your mortgage or your rent, your car or property taxes, your food bill, etc. And, the numbers with which engineers and scientists work are, ultimately, responsible for, among other things, our ability to travel from New York to Paris in three and a half hours, the existence of machines that can defeat the best human chess players, and, alas, the traffic congestion of mid-town Manhattan. Of course, numbers are not the only objects of math that concern us. We are often preoccupied with the shapes of things: we worry about the shape and size of a table relative to the shape and size of our dining room, baseball teams worry about the shape of Fenway Park, and some of us, sometimes, spend quite a bit of money and time to alter the shapes of parts of our bodies.

But what exactly are these things, numbers, and shapes, that we bump into all the time, and obsess over sometimes? That’s Frege’s question, and his attempts to answer it got me interested in philosophy in general, and Frege’s philosophy in particular.

CC: For over two millennia, Aristotle’s deductive logic reigned supreme until it was superseded by Frege’s discovery of a much more powerful propositional and predicate logic. In what important ways did Frege’s logic differ from Aristotle’s?

SS: I should say first, just so that my colleagues don’t jump on me, that the ancient Greek philosophers now known as the Stoics (yes, the very same Stoics who influenced Marcus Aurelius and are the subject of books like The Obstacle is the Way) had already discovered propositional logic. Frege’s revolutionary advance in logic consists in going beyond both Aristotelian and Stoic logic.

Let’s first say a word about what logic is. It’s the study of the distinction between correct and incorrect arguments. Logicians call these, respectively, valid and invalid arguments.

The system of logic first formulated by Aristotle identified a set of correct arguments. For example,

This argument has a structure or form that it shares with lots of other arguments. For example,

We can display this common form as

All Ss are Ms

All Ms are Ps

Therefore, all Ss are Ps

Aristotle recognized that there are a number of other forms of valid argument. For example,

No idealists are empiricists

Some politicians are idealists

Therefore, some politicians are not empiricists

Note that all the forms of argument we’ve looked at involve two premises and a conclusion, and contain only 3 terms, e.g., the terms in the first are ‘dog’, ‘mammal’, and ‘vertebrate’. In his logical theory, Aristotle concentrated on arguments with these features, which are called syllogisms.

The Stoic logicians noticed that there are valid arguments that don’t fit into this scheme, for instance:

Either I stay in academics, or I re-train as a lawyer.

I will not re-train as a lawyer (too old)

Therefore, I will stay in academics.

The Stoics observed that the validity of these forms of argument depended on the fact that the premises are statements that are built up from shorter sub-statements, and not on the terms that occur in statements.

The Stoic logicians formulated their theories of valid argument some 250 years after Aristotle did. From that point all the way to the 19th century, what was taken to be logic was fundamentally a combination of Aristotelian and Stoic logic. In the 19th century, before Frege, there were attempts to go beyond this traditional Aristotelian-Stoic combination, starting with the work of George Boole. But these attempts did not question the assumption that all valid reasoning had an Aristotelian or a Stoic form.

Frege made an entirely fresh start. To begin with, he discerned a number of failures of traditional logic. One of these is the fact that traditional logic can give no explanation for logical connections between Aristotelian and Stoic statements. For example,

Either all amphibians are four-legged or all amphibians are six-legged

by itself, without the help of any other premise, implies

All amphibians are either four-legged or six-legged,

but not vice versa. More important for Frege is the fact that traditional logic fails to do justice to certain forms of reasoning that occur both in mathematics and in non-mathematical thinking. Here’s an example:

Some one soprano is admired by every tenor

implie

Every tenor admires some soprano or the other.

But not vice versa.

Frege came to see that, in order to account for the validity of these arguments, one has to break completely with the traditional conception of logical structure. In particular, one has to see that a single statement has more than one logical structure. For example,

Alagna admires Netrebko

says that Alagna has a property, the property of admiring Netrebko. But it also says that Netrebko has a property, the property of being admired by Alagna. It also says that Netrebko and Alagna stand in a relation to one another, the relation of admiration. Finally, it says that the relation of admiration has a property, that of connecting Alagna to Netrebko. The systems of logic that Frege formulated, starting with Begriffsschrift (Concept-Script), are based on this insight and succeed in accounting for the validity of forms of inference beyond the reach of traditional logic.

Let me emphasize one important upshot of Frege’s innovation in logic. Traditional logic operated with a very narrow conception of the logical structure of statements: they are either made up of terms or of other statements. This goes with a very restricted conception of the creativity of human thought and language: we start with some pre-given terms and basic statements, and combine them in a few fixed ways to represent the world. The basis of Frege’s logic makes room for an expanded role for creativity. To fully discern valid reasoning, we have to have the capacity to take a thought or statement that we understand and imagine how it may say a variety of different things. Only in this way can we discover the spectrum of logical structures that it has, and thereby the valid forms of inferences in which it may figure.

CC: In his Begriffsschrift, Frege set out to establish mathematics as part of logic. Why was it important for Frege to reduce mathematics to logic?

SS: To answer this question, we have to know a little bit about the philosophical background in Germany at the time Frege wrote. The most important aspect of this background is the transcendental idealist philosophy of Immanuel Kant. For our purposes, let me give a simple and, I hope, relatively unobjectionable characterization of this philosophy: for the world to be knowable by us human beings, there have to be certain objective features of the world that derive from the ways in which the human mind works.

Kant arrived at this position starting from his view of math; the philosophy of mathematics is the foundation of Kant’s philosophy. Kant thought that there is something unique about mathematical truths, such as 7+5=12 or the sum of angles of every triangle is 180º. On the one hand, they are not like truths such as ‘some swans have black feathers’, which we can know to be true only by actually seeing some black swans; what we actually see, hear, touch, or smell has no relevance to these mathematical truths. On the other hand, they are also not like truths such as ‘all spinsters are unmarried’, which we can know to be true simply by knowing that the concept of a spinster is defined as an unmarried female, so that the idea of a married spinster is the idea of someone who is both married and unmarried, which is a logical contradiction. But, Kant argues, you can’t know that 7+5=12 in this way: the concepts of seven, five, twelve, and addition do not enable you to show that it is a logical contradiction to suppose that 7 and 5 do not add up to 12.

Kant concludes that our knowledge of 7+5=12 must be based on something else. This something else is how our senses operate: whatever we sense we sense as located in space and time. Now, if 7+5=12 is to be true of the world, then this way in which our mind works has to be an objective feature of the world.

A crucial assumption of this line of argument by Kant is that there is no logical contradiction in supposing that 7 and 5 do not add up to 12. This is the assumption that Frege rejects. He sees two problems with it.

One problem is that it’s really not that clear how we can coherently take arithmetical equations like 7+5=12 to be false. We can see this by considering a really simple arithmetical equation, 1+1=2. Let’s try to suppose that it’s false. Well, if it’s false, then 1+1 must equal some other number. Which number? Maybe 3? So then 1+1=3. But then we can subtract 1 from both sides of the equation: \strikeout off\uuline off\uwave off1+1–1=3–1\uuline default\uwave default, and conclude that 1=2. But now, consider what it means to say that the Empire State Building is different from the World Trade Center. Doesn’t it mean that the Empire State Building and the World Trade Center are two buildings, not one? But we have just concluded that 2=1. So, the Empire State Building and the World Trade Center are 1 building, which means that they are not different. So we get a logical contradiction after all: the Empire State Building is both different and not different from the World Trade Center.

This is Frege’s most fundamental motivation for demonstrating that arithmetic is really based on nothing more than logic and the analysis of concepts, a position known as logicism. Now, the reasoning we’ve just looked at doesn’t amount to a demonstration of logicism. For one thing, this reasoning depends on the general claim that if one subtracts the same number from both sides of an equation, the equation remains true. How is this general claim justified? Kant would insist that something like his pure intuition is involved. For another thing, the reasoning depended on going from the claim that X is a different building from Y to the claim that they are two buildings. Why should we accept this? So, all this line of reasoning establishes is the plausibility of logicism. Actually, to demonstrate logicism requires showing how all arithmetical truths are provable from logic and conceptual analyses.

This brings us to the other problem that Frege sees in Kant’s assumption: what Kant takes to be logic is traditional Aristotelian-Stoic logic. But, as we have seen, traditional logic is not all of logic. Logic must contain at least Frege’s logic. So, starting with Frege’s logic, there is a chance of demonstrating logicism. For this demonstration, Frege needs to provide an analysis of the concept of number and demonstrate how any truth of arithmetic can be proven on the basis of this analysis and Frege’s logic. He also needs some way of being sure that, in giving a proof, only logical forms of reasoning are used. In order to meet this last requirement, Frege in Begriffsschrift constructed we would nowadays call a formal language and a formal system of specifying precisely whether a sentence of this language follows logically from another. Frege came up with a first draft of the analysis of numbers in Grundlagen. Finally, in his magnum opus, Grundgesetze der Arithmetik (Basic Laws of Arithmetic), Frege refined the analysis and attempted to carry out all the required proofs.

I should note that Frege did not reject all of Kant’s philosophy of mathematics. In particular, Frege accepted Kant’s view of geometry. So Frege didn’t think that all of mathematics is based on logic, only arithmetic, and all mathematics is based on arithmetic alone.

CC: Another important work of Frege’s, Die Grundlagen der Arithmetik, sought to investigate the philosophical foundations of arithmetic. Perhaps because of the nature and complexity of its subject, it was met with a similarly lukewarm reception as that of David Hume’s Treatise on Human Nature: “It fell,” to use Hume’s phrase, “still-born from the press.” Why do you suppose his work, despite its importance, was largely ignored? And when did he begin to receive wider notice?

SS: Actually, in contrast to Begriffsschrift, the Grundlagen survived its birth from the press and went on to live a little. But not much. A few German philosophers, such as Carl Stumpf and Edmund Husserl, took notice of Frege’s work, but that’s about it as far as philosophy is concerned. Some of the reasons are how foreign Frege’s work is to the prevailing philosophical climate. German philosophy in the second half of the 19th century was reacting to the legacy of the idealism of Kant and Hegel. Some philosophers were trying to pick out parts of idealism to preserve and parts to reject; others were trying to inject more empirical science—psychology or biology—into philosophy. Frege’s philosophical thinking, based as it is on a revolutionary system of logic, was too far out of the mainstream even for logicians to understand and appreciate.

The failure of mathematicians to notice Frege’s work is quite interesting. Around the time Frege wrote Grundlagen, mathematicians were preoccupied with conceptual issues concerning the foundations of mathematical analysis and geometry. Through the works of Cauchy and Weierstrass, analysis had become increasingly rigorous, and, in pursuit of rigorous proof, Georg Cantor and Richard Dedekind in the 1870s both attempted to give an account of the nature of the real numbers. One might think that they would have been interested in Frege’s account of the concept of natural numbers, but, perhaps because Frege’s account seems to have little to do with the mathematical issues they were trying to address, they never paid much attention to Frege’s work. The case of geometry is even more interesting. Frege’s own work in mathematics was in geometry, and it is plausible that some of the technical moves he made in attempting to carry out logicism are closely connected to innovations in projective geometry. But, perhaps, again, it was not clear that Frege’s logicism had anything to do with the mathematical issues that geometers faced, and so they ignored Frege.

It should also be said that a little of this neglect is no doubt due to Frege’s style. He was not a charitable reader of others; he didn’t give other doctrines the benefit of the doubt or try to see what truth may underlie erroneous views.

Frege began to receive notice when Giuseppe Peano read volume 1 of Grundgesetze. From Peano, Russell learned of Frege’s logicism. Russell had independently arrived at a more ambitious logicist project, to reduce all mathematics, not just arithmetic, to logic, and he looked to the Grundgesetze to see what progress may have already been made. Sadly, this attention from Russell, as we will see in a bit, led to the downfall of Frege’s logicism

Through Russell, Wittgenstein came to read and revere Frege. Peter Geach has a story of how Wittgenstein knew long passages of Frege by heart. In many ways, Frege’s philosophy, as much as Russell’s, is what made Wittgenstein’s Tractatus Logico-Philosophicus possible. The Tractatus’ most important immediate influence was on logical positivism, which came to dominate academic philosophy in England and America after World War II. The positivists acknowledged Frege as an important logician, but did not read or emphasize his philosophical writings. So in this period, Frege didn’t have much philosophical influence. One problem was linguistic; not many of Frege’s works had been translated into English. It was only starting in the early 1960s that a group of philosophers in England—J. L. Austin, Peter Geach, and Michael Dummett began to read Frege seriously. They also came up with English translations. Austin translated the Grundlagen, Geach and some collaborators produced a collection of translations of Frege’s papers and selections from Begriffsschrift and Grundgesetze. From this point on, Frege’s writings began to have increasing influence on philosophizing about language in the analytic tradition. This influence is mainly due to Frege’s distinction between sense and reference, which we’ll talk about more later. Frege’s philosophy of mathematics, in contrast, for reasons that we’ll get to next, was seen as a failure, albeit a magnificent one. But in recent years, starting in the 1990s, philosophers have started to pay increasing attention to the actual details of Frege’s proofs in the Grundgesetze, and thereby to recognize that there is much more of interest and relevance to contemporary philosophy of mathematics than had been realized.

CC: While Frege was completing his Grundgesetze der Arithmetik, he was delivered an unexpected blow to his whole project of reducing arithmetic to logic by Bertrand Russell. How so?

SS: Frege thought that associated with every concept is something called the extension or the value-range (Wertverlauf) of that concept. This is, roughly speaking, the set of objects that fall under that concept. Thus, under the concept of being a Republican senator in 2017 fall Mitch McConnell, John Cornyn, Lisa Murkowski, and so on. The extension of this concept, then, is the set of these legislators, and each of them is a member of this extension. Frege’s answer to the question of what numbers are is that they are the extensions of certain special concepts. What makes these concepts special is that the properties of their extensions are determined by logic alone. So, take the concept of not being identical with itself. It’s a law of logic that every object is identical to itself. So logic determines that nothing belongs to the extension of this concept. We would, of course, say that this extension has zero members. Now, Frege’s move is to take the number zero to be this extension. The properties of zero are then set by logic alone. Naturally, Frege has to tell us what the other numbers are, but that gets complicated, so let’s not go there. What’s important is that numbers for Frege are extensions.

Russell’s Paradox begins with the observation that the members of most sets are not sets: the members of the class of 1981 of Cornell, for instance, are not sets of entities but human beings. Thus, the class of 1981 is not a graduate of Cornell, and so isn’t a member of itself. Since it seems that most sets are like this, let’s call sets that are not members of themselves “normal.”

Now consider the concept of being a normal set. According to Frege, it has an extension, the set of all normal sets. Russell, in effect, asked: Is this extension a member of itself? Let’s call this set R, for Russell. Now it seems obvious that there are only two possible answers to Russell’s question: R is a member of itself or R is not a member of itself. Let’s consider these answers in turn.

Suppose we answer: R is a member of itself. Then R isn’t normal. But R is the set of all normal sets, so if R is not normal, then R isn’t a member of R. So this answer implies that R is not a member of itself.

Suppose we answer: R is not a member of itself. Then R is normal. But R is the set of all normal sets, so if R is normal, then R is a member of R. So this answer implies that R is a member of itself.

Since these are the only possible answers, it follows that R both is and is not a member of itself. This is, of course, a logical contradiction. There must, then, be something wrong with Frege’s notion of the extension of concepts. And if there’s something wrong with this notion, then all cannot be well with Frege’s identification of numbers as extensions of concepts.

CC: Aside from his contributions to mathematical logic, Frege made important contributions to the philosophy of language. Perhaps his most influential contribution in this area is his distinction between sense (Sinn) and reference (Bedeutung). Can you briefly tell us what he was trying to convey?

SS: The best way to see what Frege’s distinction amounts to is by an example. An explorer, A, discovers a mountain peak, decides to name it ‘Sagarmāthā’. Another explorer, B, discovers a mountain peak, decides to name it ‘Zhūmùlǎngmǎ’. Later, a team of other explorers found out that A and B had come upon the same mountain peak. Now, a natural way to communicate this later discovery is to send a telegram saying:

Sagarmāthā is Zhūmùlǎngmǎ

Suppose, though, that the telegram said:

Sagarmāthā is Sagarmāthā

It’s hard to see how the second telegram could possibly communicate the new discovery. Indeed, if the folks at home got the second telegram, they may well wonder whether it’s a joke, or start wondering whether the hardships of the expedition may not be getting to the team.

The question is: what explains this difference in the communicative effectiveness between the telegrams? It’s not because ‘Sagarmāthā’ and ‘Zhūmùlǎngmǎ’ name different mountain peaks, since they don’t. Frege answers that these names express different senses; they “present” a single object in different ways. Perhaps ‘Sagarmāthā’ presents the mountain peak as a peak that one reaches by following the route taken by the first explorer, while ‘Zhūmùlǎngmǎ’ presents that peak as the one reached by the route taken by the second explorer.

We can take Frege’s explanation to be based, at bottom, on a conception of the use of sentences to communicate information. Specifically, the communication of information has to make it possible for a speaker or writer to impart knowledge to their audience. Now, an ancient philosophical tradition has it that knowledge is tied to justification: to know something about a mountain peak called ‘Sagarmāthā’ requires not merely believing something about it, but also having good reasons for holding that belief.

So the question is, what is required for someone to have reasons for holding a belief about something? Frege’s idea is that she has to understand what counts as evidence for that belief. Suppose she believes that the mountain peak named ‘Sagarmāthā’ is 8,848 meters high. What counts as evidence for that belief? Evidence for that belief, from Frege’s perspective, depends on what counts as evidence for taking some object to be the mountain peak named ‘Sagarmāthā’, as opposed, for example, to one of the other peaks in its vicinity—Gaurisankar, Melungtse, Kangchenjunga, Lhotse, Makalu, …. Before the team of explorers discovered the identity of Sagarmāthā with Zhūmùlǎngmǎ, evidence for taking something to be the peak named ‘Sagarmāthā’ depends on the route that explorer A took to it. Thus, evidence for believing that Sagarmāthā is 8,848 meters high has to be evidence for the height of the mountain reached by this route. Otherwise, that evidence isn’t a good reason for believing that Sagarmāthā, as opposed to some other peak, is 8,848 m high. Similarly, evidence for believing that Sagarmāthā is Zhūmùlǎngmǎ is 8,848 meters high is evidence for the height of the mountain reached by explorer B’s route. The difference in conception of evidence is tied to the difference in the senses expressed by the names ‘Sagarmāthā’ and ‘Zhūmùlǎngmǎ’.

I should note that Frege is not claiming that the use of sentences to communicate knowledge is the only function of language. It is one of the many types of discourse that make up language, but it is one that is of some importance since, without it, we can’t build up a common store of knowledge.

This is not to say that Frege’s distinction is uncontroversial. In particular, starting in the 1960s with the work of philosophers such as Ruth Barcan Marcus, Hilary Putnam, David Kaplan, and, most influentially, Saul Kripke, the distinction has increasingly been questioned. Indeed, the current dogma in philosophy of language has become that names don’t have sense but only reference. But, even with this new dogma in place, there remains a philosophical problem of accounting for the way in which the difference in informational or communicative value of the two telegrams is clearly connected to how the names were introduced in language. Thus, Frege’s sense/reference remains relevant in contemporary philosophy of language.

CC: Although Frege is celebrated for his sense/reference distinction, he was not the first to draw it. John Stuart Mill was one of the earliest modern philosophers to examine these notions. He referred to them as the Connotation and Denotation of a name in his book System of Logic (1863). And even before Mill, one may find these notions explored by the Scholastics and even as far back as by the Stoics. What sets Frege’s distinction apart from the rest?

SS: What is unique about Frege’s sense/reference distinction can be brought out by contrasting it with another distinction drawn by Frege, the distinction between sense and tone or illumination (Beleuchtung). Consider the following sentence, which you might come across in a Victorian novel:

Marianne’s sister is poor, but she is honest.

Now, most English speakers would detect a difference in meaning between this sentence and

Marianne’s sister is poor, and she is honest

What is the difference? One way of spelling out the difference is that the second sentence is neutral, while the first is not; the first presumes that there is some sort of contrast between poverty and honesty, or that it is somehow surprising that the poor can be honest. So, the first has certain connotations that are missing from the second.

But these sentences do not differ with respect to sense. Evidence for the truth of both of these sentences has to consist of evidence for Marianne’s sister’s financial status and evidence for her moral character. So whatever justifies one of them justifies the other.

Now, contrast these sentences with

Sir John’s eldest niece once removed is poor, but she is honest

This sentence, on Frege’s theory, has a different sense from the other two, because what counts as evidence for this sentence depends on what counts as evidence for someone to be Sir John’s eldest niece, but not on what counts as evidence for someone to be Marianne’s sister.

Mill’s distinction between denotation and connotation is too coarse to give an account of the differences among sentences such as these three. So I would also say that although Mill’s denotation/connotation is a lot like Frege’s sense/reference distinction, Mill’s distinction is, at bottom, not Frege’s.

CC: Both Frege and Russell believed that only through an ideal language like the one mapped out in Russell’s Principia Mathematica could we solve many of the philosophical riddles that bedevil us. For instance, the notion of non-being was problematic. They both had solutions to this riddle that differed. What were they?

SS: Well, perhaps we’d better start with how non-being is problematic. Here’s one version of the puzzle, which really goes all the way back to Plato’s Sophist. Consider a statement like

Roger Federer doesn’t smoke

It’s clear what makes this statement true. The name ‘Roger Federer’ refers to a person, the phrase ‘smokes’ refers to a property of people, and what makes the statement true is that this person doesn’t have the property in question. But now consider

Santa Claus doesn’t exist

Isn’t this a true statement? What makes it true? Is it that ‘Santa Claus’ refers to a person, ‘exists’ refers to a property of people, and this statement is true because this person fails to have the property in question? This doesn’t seem to make any sense. What could it mean to say that ‘Santa Claus’ refers to a person except that there is something that this name refers to? But then this thing exists, no? So it doesn’t seem as if anything can make this statement true. Moreover, this line of reasoning seems to apply for any name that one can substitute for ‘Santa Claus’. So, it seems that it can never be right to deny existence.

Frege resolves this problem by holding that existence is not a property of things; it is, rather, a property of concepts. A concept has this property if there’s at least one object that falls under that concept; it fails to have this property if no object falls under it. So, if the statement ‘Santa Claus doesn’t exist’ is true, that is because the phrase ‘Santa Claus’ only seems to be a name but is, in fact, the expression of a concept under which no objects fall. The Platonic puzzle only arises because ‘Santa Claus’ seems to be a name rather than a concept expression. So, we can avoid getting ourselves entangled in such puzzles if we used a language in which the difference between names and concept expressions is clearly marked.

As for Russell, he gave different solutions to this puzzle at different points in his career. Early on, between roughly 1899 and 1905, Russell tended to hold that ‘existence’ is ambiguous. One meaning is occupying some part of space and lasting for some period. The other meaning is being the object of some belief; Russell prefers to use ‘being’ instead of ‘existence’ for this second meaning. Now, ‘Santa Claus doesn’t exist’ is true because ‘Santa Claus’ refers to a fictional character, and so does not occupy any portion of space. But the Platonic puzzle is right about the statement ‘Santa Claus doesn’t have being’; so long as a phrase isn’t, as Russell puts it, an empty sound, it picks out some entity about which we can have beliefs, and so it is false to deny it being.

At some point around 1905, Russell gave up this position, possibly because it led to a troubling conclusion. Consider the statement ‘The married spinster has being’. On Russell's earlier view, this statement is true, and it is about some entity to which the phrase ‘the married spinster’ refers. This entity is presumably both married and a spinster. But to be a spinster is not to be married. So we have to accept that this entity is both married and not married. That is to say, we have to accept a logical contradiction. The position that Russell then adopts is, in all essentials, Frege’s position. Existence or being are both properties of what Russell calls propositional functions rather than of individuals. A propositional function maps its arguments to propositions, the values of the propositional functions. A denial of being is true if the propositional function to which being is denied fails to have any values that are true propositions.

CC: Perhaps more than any other scholar, the late British philosopher Michael Dummett shone a brighter light on Frege by bringing his considerable critical skills to bear over a series of three books on his life and work. What new insights on Frege did Dummett’s scholarship produce?

SS: My views on most of the issues in Frege’s philosophy that we’ve discussed already are very much shaped by Sir Michael’s writings on Frege. In particular, the picture I sketched of Frege’s great advance in logic beyond traditional logic is, in essentials, taken from Sir Michael.

Another deeply influential part of Sir Michael’s interpretation of Frege is his view that Frege created a systematic theory of the functioning of language. Part of this theory is the sense/reference distinction, and the idea that this distinction rests on explaining how, through communication, we can impart knowledge to one another comes from Sir Michael. Another part of Frege’s theory of language is a theory for specifying the conditions under which sentences are true or false in terms of the referents of the expressions from which sentences are composed. The idea of such a theory, if not the details of Frege’s theory, plays a central role in the branch of contemporary linguistics called semantics.

Finally, Sir Michael took Frege to champion an approach to metaphysics that is based on Frege’s logic. Metaphysics, for philosophers, tends to be about the most general features of reality, and their relationship to our thought and knowledge. In particular, Sir Michael showed that Frege conceived of the denizens of reality to belong to metaphysical categories that stem, ultimately, from the multiplicity of logical structures of thoughts that are required to remedy the inadequacies of traditional logic.

CC: What do you feel is Frege’s lasting legacy?

SS: We can start with where I left off in answering your first question. What got me interested in Frege is his raising a question about the nature of numbers. The philosophical motivation for Frege’s logicism, I think, is one of his most important legacies. He brought out just how difficult it is to think coherently at all if one denies any arithmetical truth. Now, as we have seen, Frege’s attempt to give a rigorous demonstration of the continuity of arithmetic with logic failed because of Russell’s Paradox. But that, of course, doesn’t make his motivating question go away. If anything, it makes the question sharper: what are numbers, which are so entrenched in our world, and apparently so inextricable from reason itself? So, I take Frege’s most important legacy in philosophy to be his posing, clarifying, and drawing out the ramifications of this question about the nature of number.

There are two other far-reaching aspects of Frege’s philosophical legacy. First, the sense/reference distinction provides a philosophical conception of the nature of communicative discourse, which is not only a starting point from which much of contemporary philosophy of language and semantics developed, but is of continuing relevance to these disciplines. Second, Frege’s approach to metaphysics on the basis of logic will, in my view at least, continue to be a central and lasting, albeit also controversial, part of analytic philosophy.

Let me finish up what I take to be the two most important aspects of Frege’s legacy, which are only partly in philosophy. The first is his indisputably greatest achievement: the discovery of modern logic, which gives us a far clearer grasp of the heart of deductive rationality than had been available for 2,500 years. The second arose from one of the things that Frege saw as a requirement for his demonstration of logicism, the construction of a formal language together with a formal system specifying precisely whether a sentence of this language follows logically from another. This precise specification came to be understood as amounting to a set of precisely delimited steps of inference such that a proof can be carried out purely mechanically. In the hands of Turing, this view evolved into the conception of procedures that can be carried out by a machine. This conception, of course, is one of the early seeds of the idea of a computer program. So, for good or for evil, all the iPhones, tablets, and laptops that now rule our lives are, ultimately, the concrete living legacy of that little-known German mathematician and logician who is Gottlob Frege.

Recommended Reading