One For Sorrow: Gottlob Frege’s Julius Caesar Problem

by Jack Marley-Payne.

The founder of analytic philosophy, Gottlob Frege (1848-1925), was—like the founder of regular philosophy—a tragic hero. However, unlike Socrates’, his tragedy was fatal only to his life’s work. What brought him down was not an obsession with timeless ideals of truth and integrity, but, like the boyfriends of TikTok, with the Roman Empire. Frege’s particular quest was to prove that the number one was not identical to Julius Caesar. 

He was driven by the conviction that no branch of mathematics nor discipline dependent on it could be trusted if we did not have a comprehensive understanding of its foundational components—chief among them the number 1. Frege’s contemporaries, Dedekind and Peano, had shown that all numbers greater than 1 could be defined in terms of it using addition: 2 is defined as 1+1, 3 is defined as 2+1 etc.. They then showed that with these definitions, the truths of arithmetic—anything you might ask about the whole numbers regarding addition, multiplication and the like— could be proved logically, you didn’t need to take your elementary school teacher’s word for it.

So, our knowledge that 7 is prime is as secure as our knowledge that anyone who has committed patricide is a murderer—we just need to consult the definitions, we don’t need to gather any empirical evidence. 

As Frege noted, however, these conclusions still leave the question of the nature of 1 unanswered. In his influential book, The Foundations of Arithmetic (1884), he sought to address the matter definitively, and finally put mathematics on secure footing. He began by outlining his belief that the necessary foundation must be logical in nature, and that empirical or psychological investigation about how we acquire these basic concepts is irrelevant to their true nature. 

He had particular contempt for the idea, proposed by John Stuart Mill, that we could find the meaning of number terms in childhood learning experiences adding blocks and the like:

Often it is only through enormous intellectual work, which can last for hundreds of years, that knowledge of a concept in its purity is achieved, by peeling off the alien clothing that conceals it from the mind's eye. What are we then to say when someone, instead of carrying on this work where it still seems incomplete, ignores it entirely, and enters the nursery or takes himself back to the earliest conceivable stage of human development, in order there to discover, like John Stuart Mill, some gingerbread or pebble arithmetic! It remains only to ascribe to the flavour of the cake a special meaning for the concept of number.

—Foundations, p. vii.

He distilled this into a maxim: “Always separate sharply the psychological from the logical.” However such a rule is impossible to enforce: you cannot, by force of will, insulate yourself from the influence of your own psychology or childhood experiences. Believing otherwise is a kind of arrogance that tends to lead theorists to assume their context-specific conclusions are based on purely objective grounds and thus have universal applicability.

What he termed the “alien clothing” of a concept, which includes how we learned its meaning, its etymological history and psychological resonance, is everything human about language, while the “pure form” feels less real, since it is an ideal for using concepts that humans do not even begin to approximate. Indeed, Frege’s conception of meaning better applies to the Ariekei of China Mieville’s Embassytown (2011), an alien species who speak a language so closely tied to its subject matter that they cannot lie, or even use metaphor. The language’s usage is determined by what it refers to, not the whims of its users.

Despite such potential misgivings, Frege pursued his psychology-free method of analysis, with the over-precision characteristic to the field he initiated. He started with the question: How many number ones are there? He noted that “the word ‘one’ does not admit of a plural,” suggesting the answer is one (Foundations, p. 58). After all, the equation x=3-2 has a unique solution, unlike y²=4: There is a single number that is two less than three (i.e., the number 1), while there are two square roots of four (2 and -2).

But Frege articulates an essential problem with this definition by subjecting it to a commonsense interpretation of a very basic equation: “1+1=2”. Put into words, it might mean: “If you have one thing and another thing, then in total you have two things”. If Gottlob has a cat and Sigmund has a cat, then together they have two cats. The two cats are separate objects: The first ‘1’ in the equation and the second ‘1’ refer to different cats. 

But if you refer to one thing, then refer again to that exact same thing, you end up with only one thing. If Gottlob has a cat and Sigmund has a cat and they co-own the same cat, then together, then there’s only one cat. “1+1=1.” 

Their shared cat implies that multiple ones are needed to maintain the integrity of addition—an infinite number, even, since you need enough to add together to create every possible integer. This conclusion introduced a massive amount of ambiguity into the one part of language mathematicians hoped could remain pure of it. 

When we say ‘1,’ we could be referring to any number of objects, so how are we meant to evaluate a statement containing the expression with any security? Frege considered for us the opinion of a hapless mathematician who argued that, if we were speaking with true precision, we would disambiguate our ones, and write 1’+1’’=2, 1’+1’’+1’’’=3 etc.. But this creates all kinds of problems.

Consider 5-3, presumably interpreted as (1’+1’’+1’’’+1’’’’+1’’’’’)-(1’+1’’+1’’’). This equals 1’’’’+1’’’’’, which is not 2 by our definition above (2=1’+1’’). Trying to distinguish our ones creates more problems than it solves.

What if numbers are purely subjective? After all, we can describe the same entity as one football team, 11 guys, or approximately 80 billion molecules. Frege is almost as contemptuous of this line of reasoning as he is of Mill. Though number ascriptions can vary depending on the property we relate to them, they are not subjective. We cannot alter them at will. No subjective maneuvering will make 11 guys equal 5 guys plus 5 guys.

Frege at times in his Foundations seems deliberately obtuse. People know what they mean when they make numerical statements and his pedantry regarding their definitions doesn’t appear to increase clarity. 

Then, in the last instant of your patience, Frege reveals his hand with the deceptively simple claim that a number ascription applies not to an object but a concept. Though we may view the same object as one football team or 11 guys, the number of objects the concept is a player for Arsenal applies to is 11, whichever way you look at it, while the concept is a football team based in Holloway applies to one object.

This is one of the few truly satisfying payoffs in the history of philosophy. All the confusion and seemingly obstinate literalness Frege was putting us through was in fact laying the groundwork for him to say what we’d been trying to articulate all along, but explicitly and precisely, rather than just gesturing at the idea. Like Oedipus confronting the Sphinx, he solved the riddle and freed us all. 

Frege’s example suggests that perseverance and uncompromising pedantry can break a philosophical problem. The scales will fall from your eyes, and you will see the complete truth. Readers with tragedy on the brain may at this point worry about hubris.

From there, Frege made a series of brilliant logical moves that built on his central observation to define numerical concepts. What mattered to him was that he could construct these definitions using only the vocabulary of logic—concepts such as existence, negation and equality. For example, to say that the number of objects belonging to a concept, like childless victim of patricide, is zero is equivalent to saying there exists no object that the concept applies to—there does not exist such a victim. Similarly to say that one object belongs to a concept, like King of Thebes, is equivalent to saying there exists an object that belongs to the concept and for any object, if it belongs to the concept it is identical to that first object—there exists an object x that is the king of Thebes, and if we find something that is the king of Thebes that thing must in fact be x.

This was important to Frege because, in his view, the truths of logic and logic alone were self-evidently true and in need of no further justification. Only they could provide a suitable foundation for mathematics. Alternative approaches that rested on the vagaries of psychology and worldly experience, or entities in some non-physical realm, seemed to him to make mathematical theorems contingent and in danger of changing at a moment’s notice, suggesting for example that astronomers trying to make mathematical calculations about the past state of the universe “would hesitate to draw any conclusions about the distant past for fear of being charged with anachronism. . . . How could they progress to know that the proposition 2x2=4 already held good in that remote epoch?” (Foundations, p. vi).

This obsession with universal applicability was the tragic flaw in Frege’s project. Like Oedipus, he couldn’t stop asking a question he needed to let lie. According to Sophocles at least, Oedipus’ tragedy wasn’t that he killed his father and had sex with his mother—at the start of the story, he’d unwittingly done both and was living happily as king—but that he found out about it. His fatal mistake was insisting on discovering his origins despite being begged to leave the matter alone. 

For Frege, the matter he would not leave alone was the metaphysical nature of the number one. He had created a logical system for evaluating mathematical statements with complete rigor but refused to stop there—he needed to precisely situate numbers within the universe at large. 

Frege attached great significance to the idea that numbers are objects; that we can treat them as the subject of a sentence, like, “Two is a prime number,” or or ask whether a number exists that possesses a particular property, like, “Does there exist an object that is even and prime?”

One of Frege’s fundamental principles was that we must be vigilant about the distinction between objects and concepts. Viewing numbers as objects gives them a metaphysical heft that must be accounted for. In particular, we should be able to say whether a given number is identical to any other object. This made it a problem that, in his words, “we can never—to take a crude example—decide by means of our definitions whether any concept has the number JULIUS CAESAR belonging to it, or whether that same familiar conqueror of Gaul is a number or is not” (Foundations, p. 68).

Why demand a formal proof of a fact so manifestly obvious, not to mention crude?  Frege’s question was in keeping with his experiences in mathematics, where he assumed nothing except what he had logically deduced from the axioms and definitions. In his time, this was reasonable, as mathematicians were proving that various apparent absurdities were logically incontrovertible: Cantor proved that some infinities are bigger than others, while Banach and Tarksi showed that you can take a solid sphere and chop it into five pieces, then reassemble the pieces in such a way as to obtain two balls identical to the original. Demanding proofs of the seemingly obvious had proved incredibly fruitful in inspiring new branches of math.

Still, Frege was insisting on going further, applying mathematics’ principles to questions outside of the discipline. Just as Oedipus had found great success puzzling out the answer to riddles when dueling with sphinxes, but then assumed that the strategy was applicable to mysteries he encountered as king, Frege let his success in one arena spur him to overreach.

Frege thought he had a way to turn a concept into an object that could stand in for a number in a metaphysically appropriate way, so that he could then refer to it in numerical statements. Starting with a concept, he suggested one could create the set of all objects satisfying the concept. From the concept emperor, for example, you can obtain, as an object, the set of all emperors {Augustus, Genghis Khan, Napoleon…}. For each concept satisfied by just one object, you can obtain a set containing that object {O}, and going further, you can put them all together into the set of all one object sets { {O1}, {O2}, {O3}…}. This mega-set was what Frege identified with the number one and it had two great virtues: He could use it to prove all the necessary mathematical theorems the number one should satisfy and it was not Julius Caesar.

Unfortunately, the rule Frege relied on for creating objects was incoherent. He had proposed that for any property, there exists a set containing all objects that possess the property. As Bertrand Russell pointed out in a letter to Frege, one can apply this rule to the language of sets (1902) [this link will cause you to download a pdf of the letter—ed]. Being a set is a property, which means, by Frege’s rules, there exists a set containing all sets. Since the set of all sets is a set, it must belong to itself. From here Russell asked about the property of being a set that does not belong to itself—the set of emperors is not an emperor so it does not belong to itself, while the set of all sets does. Mercilessly applying Frege’s own rules once again, Russell defined a very special set: the set of sets that do not belong to themselves. The set of emperors belongs to this set, while the set of all sets does not. According to Frege’s rules, such a set must exist and crucially there must be an answer to the question of whether it belongs to itself—but both answers imply a contradiction, if it does belong to itself, then by definition we can conclude it does not do so, and vice versa.

This insignificant-seeming trick of language is a disaster to a logical system. According to the rules of logical inference, if the axioms of a system imply a contradiction, then those axioms imply everything is true: 1=2, 1=3, and 1=Julius Caesar. Frege’s system was rendered completely useless. While Socrates went to his death proud that he knew that he knew nothing, Frege was cursed with a philosophical system that knew everything. This was a great personal blow, which colored, in his estimation, his whole life’s work in failure. We might address him as the chorus addressed Oedipus: “Unhappy in your fate and in your mind which now knows all.”

As far as we know, Frege never spoke to the man who might have been his Tiresias; another disciplinary founder, born just a few years apart from him, on the opposite side of the Germanic world. Though the two never engaged directly, we can infer from Frege’s explicit rejection of psychology, his contempt for imprecision and ambiguity, as well as his late-in-life antisemitism, that he would have had little time for Freud, who would have noted that denying the existence of psychological forces does not eliminate them. So it was for Frege, who tried to eradicate the influence of psychology and ended up suffering the fate of Oedipus. Pre-empting Lacan by half a century, Frege repressed his Oedipal complex at the psychological level, only to have it re-emerge in the realm of the symbolic.

Like Frege, the Ariekei of Embassytown needed entities available for their words to stand for. Instead of relying on set theory, the aliens came up with a way to enrich their language by physically creating scenarios they could refer back to aphoristically. The story’s human protagonist was recruited to participate in a mysterious event in which she was “the girl in the dark who ate what was given to her”; since these events had occurred, the aliens could now use it as an expression, the way we are able to do with mere hypotheticals—just as Frege felt compelled to construct an object using set theory, that could stand in for the number one in numerical statements. At the end of Embassytown, the Ariekei’s language also came unstuck. Human influence caused their unmediated relationship with truth to drive them mad. Faced with an existential threat, however, they evolved, and forced themselves to lie in order to survive.