If every object and every measuring instrument were shifted together, what physical difference would remain? Leibniz uses questions of this kind to challenge the idea of space and time as independent containers.
An order of relations
In his Third Paper to Clarke, sections 4–6, Leibniz describes space through coexistence and time through succession. His argument invokes sufficient reason: an otherwise unchanged world placed elsewhere in a uniform absolute space would lack a reason for that placement. He similarly questions shifting the entire creation to an earlier date while preserving all successions.
The argument is metaphysical, not merely a report that an instrument failed to detect a shift. It asks whether the supposed alternatives are different worlds at all. Its premises and its conclusion must therefore be assessed together.
A modern mathematical example can clarify one part of the issue without claiming to prove Leibniz's complete position. We can specify exactly which transformations preserve a chosen set of comparisons.
What a triangle retains
Place three labelled points at A = (−1,0), B = (1,0) and C = (0,1.5). Their pairwise distances are 2, √3.25 and √3.25; the enclosed area is 1.5. All lengths and areas here use dimensionless model units.
Rotate every point by the same angle and translate every point by the same vector. Their coordinates change, but subtracting the two translated positions cancels the translation. A rotation preserves the norm of that difference. Thus the three distances remain unchanged.
The demonstration also offers a vertical stretch before the rigid transformation. Multiplying the height by s changes AC and BC to √(1 + 2.25s²) and the area to 1.5s. That operation changes the relational configuration rather than just its coordinate placement.
Change placement, then change shape
Rotate and translate the whole triangle. Then change its height before the rigid transformation. Lengths are dimensionless; both axes use the same scale.
Solid green: transformed triangleDashed tan: original reference
Read transformed coordinates
Which tests define sameness?
The comparison assumes that the three labelled points and their mutual distances are the objects of interest. If an unshifted wall is added and distances to it can be measured, moving the triangle relative to the wall becomes detectable. A transformation of a whole system and a motion of one part within a larger system are different operations.
Even three distances do not retain every conceivable property. In a larger setting, handedness, orientation relative to another object, velocities or field values might matter. An equivalence claim must identify its admitted tests and the structures those tests use.
This is why coordinate freedom does not automatically establish that all motion is physically indistinguishable. The present picture concerns rigid transformations of one static configuration. Time-dependent rotations would introduce questions about dynamics that the picture does not answer.
The temporal comparison
Suppose three events have labels 2, 5 and 9. Adding one hundred to every label gives 102, 105 and 109. The successive differences remain 3 and 4. If the zero point is a convention, the two lists describe the same elapsed intervals.
By contrast, replacing the labels with 2, 8 and 16 preserves order but changes those intervals to 6 and 8. The change could represent a different unit if all relevant quantities are transformed consistently. If the unit is held fixed, it represents different durations. Specifying the equivalence relation prevents these cases from being confused.
A relational account therefore needs more than the word relation. It must say which relations exist, which are measurable and how they evolve. Order, metric distance, duration and causal accessibility are different candidates.
From relations to completed states
The SMM/DTT book defines operational equivalence through future tests. Histories may be identified only when all admitted later observations agree, and the update must preserve that identification. This is stronger than matching a present picture.
Its account of distance from admissible comparisons gives a further mathematical task: derive the comparison rule and its calibration from the proposed generative structure. The triangle assumes Euclidean geometry in two dimensions; it does not derive space, its dimension or a physical interaction from one source.
Sources and further reading
- G. W. Leibniz, Third Paper to Clarke, sections 4–6, in the correspondence published in 1717; normalized text, Newton Project.
- Mohamed Haj Yousef, When are two histories the same? and Distance from admissible comparisons. The triangle is an original teaching construction.