A Small Book in Jena
1879, Jena. Gottlob Frege was thirty, had been teaching mathematics at the university for five years, and that same year had just been promoted to associate professor. On his desk lay a manuscript of no great thickness, printed by a bookseller in Halle — a work of which today every student of logic has heard the name, and which in its own year almost no one opened: the Begriffsschrift (concept-script), a formal language of pure thought modeled on the language of arithmetic. The newly minted word in the title — concept-script — and the ambition he meant it to carry are written entire in the subtitle: a symbolic language designed, on the model of arithmetic, for pure thought.
The book's reception was cold to the point of farce. It garnered a single review, from his fellow mathematician Ernst Schröder, and that one critical throughout: clumsy notation, expensive printing, the wrong direction. Frege complained more than once, in later years, of the silence that had surrounded the work. More than twenty years would pass before it found the man destined to change its fate — and to end it with his own hands: in June 1902, in Cambridge, a letter from Bertrand Russell reached Jena. That letter opens the next chapter; what this chapter must first set out is what kind of building that belated letter collided with.
By the settled judgment of the field, the Begriffsschrift marks the birth of predicate logic, the logic of quantifiers. To weigh that sentence one must first weigh what stood before it: for more than two millennia, from the walking path of Athens to the colonnade, from the Arabic commentators to the doctors of the schools, the advance of logic had remained a refinement carried out within the two planes that Aristotle and the Stoics had demarcated. Leibniz in the seventeenth century had seen through this situation and left behind the famous vision: if the concepts of thought could be symbolized as numbers are, then every dispute could be settled by sitting down to calculate — when two philosophers fall out, there would be no need to argue, one need only say: let us calculate. He drafted sketches toward a universal characteristic, but never completed it. The one who truly set to work before Frege was George Boole: The Mathematical Analysis of Logic (1847) and An Investigation of the Laws of Thought (1854) mapped classes and propositions onto algebraic operations, proving that a part of inference could become calculation. But Boole's algebra could not handle the nesting of quantifiers or many-place relations, and a decisive step still lay between it and "all inference becomes calculation". That step was taken by the Begriffsschrift.
What the Symbols Did
Frege's operation was made of two cuts — one at grammar, one at existence.
The cut at grammar: he abolished subject–predicate analysis. Traditional logic carved the sentence into "subject–predicate": "Socrates is mortal". Frege replaced this with function and argument: "x is mortal" is a function; "Socrates" is the argument filled in. The substitution looks like a mere change of bookkeeping, but it released expressive power of a new order: functions can be many-place — "x knows y" — which subject–predicate structure cannot write; functions can be nested — "x knows the father of y" — which it still less can write. Relation, that entity which for two thousand years had held no registered residence in term logic, from that moment had a systematic notation.
The cut at existence: he invented the quantifier. "All men are mortal" is no longer a judgment about the subject "all men" but a structure with bound variables: for all x, if x is a man, then x is mortal. "Someone knows y" is: there exists an x such that x knows y. With quantifiers and variables working in concert, "everyone knows some general" and "some general is known by everyone" — a pair of sentences the Aristotelian system could not tell apart — at once fell each into its own place. Universal and particular were no longer the quantity of a proposition but became levels of binding — and existence can be defined outright as an abbreviation of universality plus negation: it is not the case that there is none .... With two primitive notions (if ... then; not) plus the quantifier, all the logical connectives are derived.
The wait along this road deserves one further question: from "let us calculate" to actually setting to work — why the gap of two hundred years? The answer lies not in any scarcity of ambition but in parts not yet ready. To turn inference into calculation, two prior inventions were needed: a symbolic boldness that dared to write thought as algebra, and a concept of function capacious enough to receive the quantifiers. The former was prepared by the algebra of the nineteenth century and by Boole — mathematicians had grown accustomed to manipulating unknowns without asking after their meaning; the latter was not completed until Frege grafted the mathematical concept of function onto grammar. Leibniz could see the terminus, but the road to it had first to pass through a discipline that matured only after his death. The order of convergence is decided not by the prophet's field of vision but by the order in which the parts mature — this sentence is, on this book's reading, the standing annotation to every phenomenon of being "ahead of its time".
With the two cuts together, the expressive power of logic went from plane to solid. Aristotle's term logic and the Stoics' propositional logic each became a special fragment within the new system: Barbara was no longer an axiom-grade starting point of the system but a pattern that could be derived afresh in the new language. Frege himself gave the demonstration in Part III of the Begriffsschrift — using only the new language, deriving from purely logical starting points theorems about sequences and heredity, which is precisely the kernel of mathematical induction. In other words, this little book that no one read did not merely announce a new grammar; on the spot, it grew a stretch of mathematics out of the new grammar.
The technical apparatus changed generation accordingly: Frege wrote inference as two-dimensional diagrams — premises and conclusion connected by lines, the dependencies evident at a glance; from the axioms onward, every step of transformation carried the explicit license of a rule. Chapter Three said that modern logic was born of the foundational crisis of mathematics, and that Frege constructed the concept-script precisely in order to lay a foundation for arithmetic: what he wanted was not a better instrument of argument but a language that could hold the whole of arithmetic and prove that it grew out of pure logic. This motive is about to be unfolded.
But first, on this book's reading, the apparatus must be classified. Chapter Three said that logic is like a chess player, deciding valid deductions within given rules; what Frege did was to press every move of the chess player into the gears of the rules. Symbolization is the compressing of the convergence paths of inference into mechanically executable transformation rules: which sign may be exchanged for which sign under which conditions — all written down and fixed in advance, no judgment permitted on the spot. Here, then, can be stated this chapter's positive thesis, the formulation Chapter Sixteen will quote directly: a formal system is the symbolization of an already-locked structure of convergence — axioms and rules are the explicitly locked achievements of convergence, and theorems are future deposits mechanically replayed.
An Artificial Accelerator of Consensus Reinforcement
The weight of this thesis can be read only under the axioms of RC.
RC discusses consensus reinforcement in its axiomatic system (Axiom A6): observations across subjects and levels verify one another, forming a positive reinforcement loop, converging into stable objective reality. Chapter One showed how expensive the daily running of A6 is: one persuasion requires rhetoric, authority, time, and luck; the checking of one argument depends on the acuity and the patience of the checker; the more complicated the account, the fewer the people able to audit it. Aristotle's metalanguage let inference be talked about, but the inference talked about still had to be understood by human beings — and understanding is the possible point of entry for divergence.
Symbolization cut the running cost of A6 to the floor. Once the rules are fixed, "anyone who follows the rules arrives at the same conclusion" turns from ideal into literal fact: whether a transformation is licensed no longer depends on the reader's intuition, only on the shapes of the signs. The sources of divergence are removed systematically — not by forbidding disagreement, but by leaving it nowhere to attach: let there be a dispute over whether step n is legal, and the first n steps are re-checked character by character, back to the character where the error lies. Chapter Two showed that maintaining the a priori feel requires dissipative feeding and closed-loop reinforcement; the formal system made this mechanism of maintenance self-driving: every character-by-character check is one re-emphasis; every new proof is one new deposit to the system of rules. Such is the verdict on this book's reading: a symbolic language is an artificial accelerator of consensus reinforcement — it does not change the mechanism of A6; it adjusts the gear clearances of the mechanism to industrial precision.
What the accelerator changed, and what it did not, must also be seen clearly. What changed is the threshold of checking: a lengthy proof can henceforth be taken apart and distributed among any number of checkers, each auditing a small stretch, and the auditing of no stretch requires a genius — this is the division of cognitive labor, one form of the same revolution as the factory's distribution of processes among ordinary workers. Chapter Three said that logic's community is all who reason, and mathematics' community all who can read a proof; formalization lowered the threshold of "being able to read" from insight down to patience, and the community's effective boundary expanded accordingly. What did not change is the community itself: the auditing still has to be done by human beings, and human beings have history, inertia, and partiality — the episode of the adoption of the notation (see below) will show this. As for the dream of dispensing with the checker as well — letting machines take over all the auditing — its exact boundary was not established until 1936, and that is another stretch of history, from after logic had walked off the main line of this book.
A ground on the side of language must also be entered. RC's General Outline, discussing the bidirectional representation of language in Section 2.2, says that language is the embryonic form of the convergence of possibility, and that any single expression is a projection from higher dimension to lower; the paper, in its epistemological part, points out that the abstractness of linguistic symbols necessarily entails a loss of experience, leaving an explanatory gap between the description of phenomena and the phenomena themselves — the ambiguity and context-dependence of everyday language are precisely the root of the symbols' inability to vouch for themselves. A formal language is a self-conscious execution of this principle: since every expression is a projection, then make the most thoroughgoing projection of all — project away meaning, tone, and intent, and keep the form only. All expressive power is sacrificed for one thing: absence of ambiguity. And the validity of inference happens to depend on the form only. The formal language is the product of language's two-sidedness pushed to its extreme: it is the poorest of languages, and the only language that can be checked without context.
The Ambition of Logicism
With such a language at hand, Frege's real goal enters: logicism.
The motive came from a crack. Nineteenth-century mathematics grew purer and purer while its foundations grew more and more suspect: the rigorization of analysis reduced the calculus to the real numbers, and the real numbers to the rationals and the natural numbers, and the inquiry finally came to a stop before the natural numbers — on what ground does arithmetic stand? Two ready-made answers lay to hand. Kant's traditional answer was intuition: counting depends on succession in time, an intuitive form; arithmetical judgments are synthetic a priori — they extend knowledge yet hold necessarily, by grace of the form of intuition. Mill's empiricist answer was induction: the facts of number come from repeated observation of the world, and are of the same lineage as "the sun rises". Frege dealt each of the two answers a heavy blow: if arithmetic rested on intuition, it should carry the vagueness and the privacy of intuition, yet arithmetic is precisely the most exact, the most public, the least exception-admitting of all knowledge — no one has a second intuition about whether 7 plus 5 equals 12; if arithmetic rested on induction, it should admit exceptions and revisions, yet "every natural number has a successor" accepts no counterexample from any tomorrow. So he gave the programmatic judgment: the truths of arithmetic are not synthetic but analytic — numbers and the laws of number can be reduced to purely logical concepts and laws, and the whole of their necessity comes from the contentlessness of logic itself. The Foundations of Arithmetic (1884) argued this direction in prose: number is something that grows out of concepts, and "the number that belongs to a concept" can be defined logically. Over the nearly ten years that followed, he built the program into a formal system: Basic Laws of Arithmetic, volume one published in 1893, volume two in press in 1903 — deriving arithmetic, character by character in the concept-script, from a few logical axioms.
The relay on this road came from Cambridge. Russell took up the baton with The Principles of Mathematics (1903), stating the program even more fully than Frege had: all mathematics is symbolic logic. At the International Congress of Philosophy in Paris in 1900 he met Peano, and saw at once that the other's system of notation was exactly the material this project needed — linear, concise, effortless for mathematicians to read. The three volumes of Principia Mathematica, written in collaboration with Whitehead, were issued in succession by Cambridge University Press in 1910, 1912, and 1913. This giant work of more than two thousand pages does one and the same thing, only at a terrifying scale: from very few logical axioms and the constraints of the theory of types, it derives number, order, and limits, all the way to the real numbers and cardinal arithmetic. Not until more than three hundred pages deep into volume one was the equation established by way of "the union of two disjoint unit classes is 2"; and 1+1=2 in the sense of arithmetical addition was not formally proved until ✳110.643 in volume two — beside which stands the famous dry remark: this proposition is occasionally useful.
This dry remark deserves an entry in this book's ledger, for it states, without intending to, the nature of the project. Three hundred pages for an equation every child understands: on the surface, waste; in truth, a conversion — the self-evidence of "one plus one equals two" as it stands in everyday understanding was exchanged for a formal guarantee of more than three hundred pages checked character by character; every fraction of self-evidence was converted into explicit license by rule. Chapter One said that self-evidence is an account grown so thick that people have forgotten it was ever opened; what Principia Mathematica did was to call out the whole account for a fresh audit, attaching a receipt to the provenance of every deposit. The ambition of logicism, on this book's reading, can be transcribed without the change of a word: to converge all of mathematics into one explicitly locked path — leaving no deposit of untraceable provenance, and letting no step depend on intuition that has not been registered.
The division of labor of Chapter Three shows itself here once more. Logic and mathematics are two labors: the chess player decides within given rules; the architect lays her own foundation and closes the building upward. Logicism was a crossing of the boundary, a general contract taken in over the other trade: the chess player declared he could lay the architect's foundation for her — the foundation of mathematics, arithmetic, would be proved never to have been a foundation at all, but an interior floor of the greater building of logic. The success or failure of that declaration hung upon one unresolved premise: the greater building itself must be free of cracks. And the crack was already on its way.
The Fate of a Notation
This chapter has one episode left to tell, and it concerns the notation of that small book itself.
Frege's two-dimensional diagrammatic notation, unrivaled in expressive power in its day, met a fate all but total extinction. Russell, after the letter of 1902, became Frege's most important reader, yet what Principia Mathematica adopted was the Peano-style linear notation; and the textbooks of the century that followed carried on this Peano–Russell lineage. Frege himself never made peace with it — to his last years he held that his two-dimensional notation was the superior, and it does have the merit of showing the dependencies at a glance. But the community had voted with its feet.
In the standard historical narrative this is mere anecdote; on this book's reading it is key testimony. If the merits of notations were decided by truth, then the better notation should have prevailed; what decided in fact was the path dependence of convergence: the mathematical community of the late nineteenth century had already built up, in the practice of algebra and of Peano's axiomatization of mathematics, a thick account of linear symbolism, and the switching cost of adopting two-dimensional notation outweighed the long-term gain in expressive power. Every community receives new observations from out of the snapshot it has already locked — the interest of the old account weighs more heavily than the potential gain of the new proposal. RC's General Outline, discussing ideas and consensus in Section 2.4, says that when a particular interpretive framework shows predictive power and practical value, individuals and groups will form a consensus of ideas through comparison, and the established consensus will in turn act back upon the subjects' perception, closing the loop. The rise and fall of notations is a specimen of this principle at its smallest scale: it was not the best notation that won, but the notation most absorbable by the account of the time.
This verdict can be run once more in reverse, as a calibration of this book's own method: to say that Frege's notation was "superior" is itself an assessment after the fact — the criteria of superiority (expressive power, transparency of dependence) are likewise judgments made within some framework of convergence. What matters is not to rehabilitate history, but to remember the shape of this fact: even a symbolic language born for the elimination of disagreement could not exempt its own history of adoption from path dependence. The purest instrument of reason rode an impure history of convergence through its dissemination. And there is a closer implication still: that symbolic systems became in time the standard form of logic and mathematics is not because they are the only possible form, but because they are the form that one historical community absorbed with the thickest account. This note will take effect again in the comparisons of Part Five — the boundaries of the three forms of reason are, in part, the sediment of their respective histories of adoption, not an inventory of pure natures.
The Architect's Foundation
Back to the main line. By 1910, when volume one of Principia Mathematica appeared, the enterprise of logicism looked like this: an unambiguous symbolic language; an edifice of arithmetic derived character by character from very few axioms; a plan to take the whole of mathematics into the bag. The chess player had very nearly contracted for the architect's foundation.
But at the margin of the ledger there had stood all along an annotation that was not quite right. It belonged to Frege's system: Basic Law V permitted the construction, out of any concept whatever, of a corresponding "extension" — a set. This law is everywhere in the derivations of Basic Laws of Arithmetic, one of the load-bearing members at the very bottom of the edifice. It looked harmless to the point of self-evidence: concepts have extensions, and extensions can be gathered up — every science uses its words in this way; who would object? Self-evidence, Chapter One said, is an account grown so thick that people have forgotten it was ever opened; and the thickest deposit in an account is often precisely the one no one remembers making.
On June 16, 1902, Russell asked him a question in a letter. The question was only a few lines long. It was these few lines that made Frege reply acknowledging himself "thunderstruck", and that drew from the appendix to volume two of Basic Laws of Arithmetic the sentence quoted for a hundred years — at the moment the edifice was completed, the foundation collapsed. One intriguing footnote: the questioner was not an adversary come to demolish, but the program's most faithful fellow traveler — it is precisely the one who would lock all of mathematics into one explicit path who most needs to walk that path to its darkest end, and who therefore first touches, in the dark, the crack. The next chapter begins with this letter: what happens, when a system has gained a sufficient capacity for self-observation, and the observer falls into his own range of observation.