Tractatus Logico-Philosophicus · Chapter
6.36111 explained
Wittgenstein discusses the Kantian problem of incongruent counterparts, such as right and left hands, arguing that their inability to coincide is not a matter of intrinsic difference but of spatial dimensionality.
What happens
Wittgenstein discusses the Kantian problem of incongruent counterparts, such as right and left hands, arguing that their inability to coincide is not a matter of intrinsic difference but of spatial dimensionality. He illustrates that in one-dimensional space, congruent figures cannot be made to overlap without leaving that space, and similarly, a right-hand glove could fit a left hand if rotated through four-dimensional space.
Themes in this chapter
Limits of Language and Thought
The example shows how spatial intuition and language can mislead about the nature of congruence, highlighting the limits of what can be said versus shown.
Logic and Tautology
The logical structure of spatial relations is revealed through the possibility of higher-dimensional rotation, akin to tautological transformations.
Characters to notice
- Mr. Wittgenstein
Author of the Tractatus, presenting a logical analysis of the Kantian problem of incongruent counterparts.
Key passages
“The Kantian problem of the right and left hand which cannot be made to cover one another already exists in the plane, and even in one-dimensional space; where the two congruent figures a and b cannot be made to cover one another without moving them out of this space.”
The puzzle of why a right hand and left hand don't match up is not unique to three dimensions; it also occurs in two dimensions and even one dimension, where two identical shapes cannot be superimposed unless you lift them into a higher dimension.
Wittgenstein generalizes Kant's problem to show it is a logical issue about dimensionality, not a special feature of three-dimensional space.
“A right-hand glove could be put on a left hand if it could be turned round in four-dimensional space.”
If we could rotate a right-handed glove through a fourth spatial dimension, it would fit a left hand perfectly.
This illustrates that the apparent difference between left and right is not absolute but depends on the dimensionality of the space considered.