Overview: From Picture to Practice
Late Wittgenstein (1949–1951) is a radical departure from his early Tractatus Logico-Philosophicus (1921). There, he thought language pictures facts, and philosophy could find the logical foundations of the world.
In his later work, language is not a picture but a toolbox of language-games — rule-governed practices embedded in forms of life. There is no single crystalline logic underneath.
- Epistemology (On Certainty): We don’t rest knowledge on self-evident foundations. We rest on hinges — certainties that make doubting possible.
- Mathematics (RFM): We don’t rest math on logic or Platonic objects. We rest on techniques of counting, proving, and acting.
Both works were published posthumously from notebooks. They are not systems, but 1,000+ remarks that invite you to look at how we actually use words like "know," "doubt," "prove," and "true."
On Certainty (OC) — Doubt Needs Ground
Context: 1949–51, 676 remarks Written in response to G.E. Moore's famous "proof" of an external world — Moore held up his hand and said "Here is one hand, therefore I know an external world exists."
Core Thesis: Doubt Presupposes Certainty
Doubting is not a free-floating mental act. It is a move inside a language-game. To doubt that this is a hand, you must already be certain of what "hand," "here," and "doubt" mean — and of a hundred background facts.
Knowledge vs. Certainty
Moore says "I know I have hands." Wittgenstein replies: that's a misuse. "I know" is used where we can give grounds or be mistaken ("I know the train leaves at 3"). But for hinges, doubt is senseless. We are certain, not knowledgeable.
Acting Lies at the Bottom
We do not learn hinges as propositions first. A child learns to reach for things, to trust hands, to count. Certainty shows in what we do, not what we say.
Hinges & the River-Bed
Hinge propositions are not true or false in the ordinary sense. They are the pivot on which true/false turns.
Examples Wittgenstein gives:
- "The earth existed long before my birth." (OC §84)
- "I have two hands." (OC §1)
- "12 × 12 = 144." (OC §447)
- "My name is L.W."
- "Every human has parents."
The River-Bed Metaphor (OC §96–99)
Our world-picture is like a river-bed of thoughts. Some parts are hard rock — rarely move. Some are sand — shift gradually. What counts as a hinge can change over time, but not by argument alone; by changes in practice.
Hinge Tester
Type a proposition. Is it a hinge, an empirical claim, or nonsense outside a game?
Moore's Hands Simulator
Click to raise radical doubt. Watch what happens to the game.
River-Bed Visualization
Remarks on the Foundations of Mathematics (RFM)
Context: 1937–44 Notes critiquing three schools: Logicism (Frege/Russell: math reduces to logic), Formalism (Hilbert: math is symbol manipulation), Intuitionism (Brouwer: math is mental construction).
Mathematics as Practice, Not Discovery
Mathematics does not describe a Platonic realm. It is a motley of techniques for counting, measuring, inferring, building. A mathematical proposition gets its life from its use in our practices.
Proof as Concept-Formation
A proof does not discover a pre-existing truth. It creates a new norm — a new rule for using signs.
Before we have a proof that 12×12=144, we can calculate and get 144. After the proof is accepted and taught, "12×12=144" becomes a hinge — to deny it is to show you don't know how to multiply, not to make an empirical error.
Following a Rule
Connected to Philosophical Investigations §201: no rule contains its own interpretation. What counts as "going on the same way" is decided by training, practice, and community agreement — not by a private mental act.
Proof, Surveyability & Gödel
The Critique of Foundations
Finitism & Surveyability
Wittgenstein insists a proof must be something we can take in — write down, check, teach. Appeals to infinite totalities we cannot survey are not proofs but pictures that can mislead us.
On Gödel's Incompleteness
Wittgenstein's notorious remarks (RFM I Appendix) do not deny Gödel's technical result. He asks a philosophical question: what do we do with the sentence P = "P is not provable in S"?
His point: "true" in mathematics normally means "provable according to the rules." If P is not provable in S, calling it "true" smuggles in an ordinary, extra-systemic notion of truth. If we add P to S, we've changed the game — created S', where P is now provable. Truth is system-relative, not a glimpse of Platonist heaven.
This is not a mathematical refutation of Gödel, but a reminder not to be bewitched by the word "true."
Interactive Lab: Mathematics as Technique
1. Rule-Following Game
Sequence: 2, 4, 6, 8, ...
PI §185: a pupil continues the series differently. No fact in your mind determines "the" continuation.
2. Proof as Grammar: 12 × 12
Click squares to count. Then lock it as a rule.
3. Gödel Lab: System-Relativity of "True"
P = "P is not provable in System S"
Connections: Two Sides of One Coin
- Against ultimate grounds: In epistemology, hinges replace foundations; in math, practice replaces Platonist realms. Certainty is not derived, it is shown in action.
- Certainty is logical, not psychological: "I am certain" describes my role in the game, not my feeling of conviction. Mathematical certainty is hinge-like: to deny 2+2=4 is not to be wrong, but to have left arithmetic. (OC §655)
- Meaning as use: Both OC and RFM apply "meaning is use." Doubting, knowing, proving, following a rule — all are techniques learned in forms of life.
- Legacy: Hinge epistemology (Annette Baier, Daniele Moyal-Sharrock, Crispin Wright), social practice theories of mathematics (Kreisel, Wittgensteinian finitism), and therapeutic philosophy.