Tractatus Logico-Philosophicus · Chapter
5.154 explained
Wittgenstein uses the example of drawing balls from an urn to distinguish between mathematical facts and empirical probabilities.
What happens
Wittgenstein uses the example of drawing balls from an urn to distinguish between mathematical facts and empirical probabilities. He argues that the statement of equal probability (1/2) is not a mathematical truth but a reflection of our ignorance of the specific circumstances, which the experiment itself can verify as independence from unknown factors.
Themes in this chapter
Logic and Tautology
Distinguishes empirical probability from mathematical or logical necessity.
Limits of Language and Thought
Highlights how probability statements express the limits of our knowledge rather than objective facts.
Characters to notice
- Mr. Wittgenstein
Presents the urn example to illustrate the nature of probability statements.
Key passages
“In an urn there are equal numbers of white and black balls (and no others). I draw one ball after another and put them back in the urn. Then I can determine by the experiment that the numbers of the black and white balls which are drawn approximate as the drawing continues. So this is not a mathematical fact.”
Repeated draws from an urn with equal black and white balls show that the ratio of colors drawn approaches equality, but this convergence is an empirical observation, not a mathematical certainty.
Wittgenstein emphasizes the empirical nature of probability, contrasting it with a priori mathematical truths.
“If then, I say, It is equally probable that I should draw a white and a black ball, this means, All the circumstances known to me (including the natural laws hypothetically assumed) give to the occurrence of the one event no more probability than to the occurrence of the other.”
Saying the probability is 1/2 means that, given everything I know (including assumed natural laws), there is no reason to expect one outcome over the other.
Probability is tied to our state of knowledge, not to an inherent property of the world.