Gottlob Frege

This article records tradition as it has been passed down and reported. Its sources are not yet part of the atlas's verified catalogue.

On June 16, 1902, a letter arrived from Bertrand Russell informing Gottlob Frege that the foundation of his life's work contained a contradiction. Frege had spent decades building a formal system meant to show that arithmetic was nothing more than logic in disguise, a project called logicism, and the keystone of that system was a principle he called Basic Law V, which let him treat any well defined concept as having an extension, a collection of everything falling under it. Russell showed that this innocuous seeming rule allowed the formation of a concept, extension which is not an element of itself, and that concept generates a straightforward contradiction, since it is an element of itself if and only if it is not. The Stanford Encyclopedia's account of Frege's career records the moment starkly, noting that he never fully recovered from the flaw discovered in the foundations of his two volume Grundgesetze der Arithmetik. That is one honest way to tell the story, a system builder undone by the very rigor he introduced.

But the more interesting scholarly argument concerns how much of Frege actually goes down with that ship. Basic Law V is inconsistent, and no repair of it has ever satisfied logicians since. Crispin Wright showed in 1983, as a modern reconstruction, that the Dedekind or Peano axioms for arithmetic could be derived from a much narrower principle called Hume's Principle, which says roughly that two concepts have the same number just when their instances can be paired off one to one. It was R. Heck who showed, in 1993, something sharper still: that Frege himself, in his own original text, had already validly derived those same axioms from Hume's Principle, without needing Basic Law V in its full strength. Hume's Principle is not obviously safe either, but it is far more defensible than unrestricted Basic Law V, and the derivation Heck uncovered inside Frege's own work now goes by the name Frege's Theorem. The Stanford entry is candid that philosophers appreciated the importance of this work only relatively recently, which means that for most of a century Frege was read as the architect of a failure, when in fact a working piece of logicism had been sitting inside the wreckage the whole time, unnoticed because nobody had separated it from the part that actually broke.

That recovery changes what the paradox means. It is no longer simply the story of a genius refuted by a younger rival, though it is also that. It is instead a case study in how much of a formal system's real content can survive the collapse of its official foundation, and how long it can take a field to notice. The same entry traces a second, quieter dispute running through Frege's work on meaning, over whether his commitment to compositionality, the idea that a sentence's meaning is built from the meanings of its parts, can be reconciled with his equally firm insistence, in the Context Principle, that a word only means something inside the context of a full sentence. Scholars still argue over whether these two commitments can be made to sit together in one coherent picture, or whether Frege simply held both without fully reconciling them. Between the paradox he did not see coming and the theorem he did not know he had proved, Frege's legacy is less a monument than an argument still being adjudicated.

This article records tradition as it has been passed down and reported. Its sources are not yet part of the atlas's verified catalogue.

Gottlob Frege's 1892 essay "On Sense and Reference" opened a problem that occupies philosophy of language to this day, namely how to explain why two ways of referring to the same thing can carry such different cognitive weight. The puzzle begins with identity statements. The morning star is the morning star is knowable just by inspecting the sentence, true by logical form alone. The morning star is the evening star, by contrast, states something that took centuries of astronomical observation to establish, namely that Venus appears both at dawn and at dusk. Yet both sentences, on the usual view of meaning current before Frege, say the same thing, since the morning star and the evening star pick out the same planet. If meaning were exhausted by what a term denotes, the two identity claims would be equally trivial. They plainly are not, and Frege took that asymmetry as proof that reference cannot be the whole story.

His solution was to split the meaning of an expression into two components. Its reference, or Bedeutung, is the object it denotes, Venus in this case. Its sense, or Sinn, is the particular way that object is presented, the route of thought by which a speaker arrives at it. Morning star presents Venus as the last bright point visible before sunrise, evening star presents the same planet as the first visible after sunset. Same Bedeutung, different Sinn, and it is the difference in sense, not any difference in reference, that makes the identity statement informative rather than empty. Frege generalized the point beyond astronomy. Four and eight divided by two denote the same number through different arithmetical routes, and ordinary proper names work the same way, so that a sentence identifying two names for one historical figure can carry real news even though, referentially, it says an object is identical to itself.

Frege pressed the distinction further than most successors have needed to. He held that sense attaches not just to names but to whole sentences. A sentence's reference, on his account, is startlingly austere, nothing more than a truth value, the True or the False, since substituting any expression for another with the same reference inside a sentence leaves the sentence's truth value unchanged. What a sentence expresses beyond its truth value, the actual content asserted, is what Frege called a thought, a Gedanke, and he identified the thought as the sense of the sentence. Thoughts are not psychological events. Frege insisted, against the psychologism common among his contemporaries, that a thought is not a mental image or an act of judging but something a mind can grasp without owning it, the way many people can grasp one and the same theorem. He placed thoughts in what he called a third realm, distinct both from the physical world of Venus and sunlight and from the private inner world of sensations and ideas, precisely because a thought had to be available to be shared, checked, and disputed by different thinkers at different times without becoming a different thought each time it was grasped. Two mathematicians proving the same theorem grasp one identical thought, not two resembling private copies of it, and this mind independence is what let Frege treat logic as a science of objective truths rather than a description of how human beings happen to reason.

The distinction also solved a puzzle about belief. John believes Mark Twain wrote Huckleberry Finn can be true while John believes Samuel Clemens wrote Huckleberry Finn is false, even though Twain and Clemens are one person, because inside a belief context a name no longer picks out its ordinary object but instead denotes its own customary sense. Since the two names carry different senses, they behave, inside belief reports, like names of different things, and substitution fails exactly where a theory of meaning built on reference alone could not explain why it should. The sense and reference distinction thereby did far more than resolve a puzzle about Venus, it supplied the machinery later philosophy of language still uses to explain how thought and language can be both about a shared, mind independent world and, at the same time, informative, mistaken, and disputable.