. . . there are two main ideas that pass under the ‘incommensurability’ label. One
is that incommensurable items cannot be precisely measured by a single ‘scale’ of
units of value. This idea has historical roots. The Pythagoreans first determined
as incommensurable the diameter and side of a regular pentagon: the proportional
lengths could not be expressed in terms of integers, and thus it was thought that
there was no single scale in terms of which their lengths could be measured. Other
writers have moved away from the Pythagorean idea and have focused instead on
incomparability, the idea that items cannot be compared.

1

It’s sometimes thought that the first idea entails the second — that if there is
no common unit of value in terms of which two items can be measured, they are
incomparable. But it is a platitude of economic and measurement theory that the
lack of a single scale of units of value does not entail incomparability. Comparison
does not require any single scale of units of value according to which items can
be precisely measured; one alternative can be morally better than another without
being better by 2.34 units. Comparable items can be ordinally ranked – ranked
on a list – and need not be cardinally ranked – precisely ranked by some unit of
value. Given that the two ideas are distinct, let us hence forth reserve the term
incommensurable for items that cannot be precisely measured by some common
scale of units of value and the term incomparable for items that cannot be compared.
. . . I am going to set aside the first idea — incommensurability — and focus on the
second — incomparability – in the remaining part of the discussion.

Question

. . . there are two main ideas that pass under the ‘incommensurability’ label. One
is that incommensurable items cannot be precisely measured by a single ‘scale’ of
units of value. This idea has historical roots. The Pythagoreans first determined
as incommensurable the diameter and side of a regular pentagon: the proportional
lengths could not be expressed in terms of integers, and thus it was thought that
there was no single scale in terms of which their lengths could be measured. Other
writers have moved away from the Pythagorean idea and have focused instead on
incomparability, the idea that items cannot be compared.

1

It’s sometimes thought that the first idea entails the second — that if there is
no common unit of value in terms of which two items can be measured, they are
incomparable. But it is a platitude of economic and measurement theory that the
lack of a single scale of units of value does not entail incomparability. Comparison
does not require any single scale of units of value according to which items can
be precisely measured; one alternative can be morally better than another without
being better by 2.34 units. Comparable items can be ordinally ranked – ranked
on a list – and need not be cardinally ranked – precisely ranked by some unit of
value. Given that the two ideas are distinct, let us hence forth reserve the term
incommensurable for items that cannot be precisely measured by some common
scale of units of value and the term incomparable for items that cannot be compared.
. . . I am going to set aside the first idea — incommensurability — and focus on the
second — incomparability – in the remaining part of the discussion.

Ask a Philosopher · Accepted Answer

Verily, the concept of incommensurability is one that hath puzzled the minds of scholars and philosophers alike for centuries. The notion that items cannot be precisely measured by a single scale of units of value doth raise many profound questions about the nature of value itself. The Pythagoreans, in their wisdom, didst first bring to light this idea when they found the diameter and side of a regular pentagon to be incommensurable in terms of integers. Thus, it was concluded that there existed no common scale by which their lengths could be measured. However, as time hath passed, some have departed from this notion and turned instead to the idea of incomparability - the belief that items cannot be compared.

'Tis oft assumed that the absence of a common unit of value for measurement doth imply incomparability. Yet, in the realm of economics and measurement theory, 'tis known that the lack of a singular scale of units of value doth not preclude comparison. Verily, comparison doth not necessitate precise measurement by a single scale; items may be ordinally ranked without being cardinally ranked by a unit of value. Thus, the distinction between incommensurability and incomparability becomes clear.

Therefore, let us reserve the term incommensurable for items that cannot be precisely measured by a common scale of units of value, and the term incomparable for items that cannot be compared. As for me, I shall delve into the realm of incomparability in the remainder of this discourse, setting aside the intricacies of incommensurability. 'Tis a topic of great depth and contemplation, one that doth challenge our understanding of value, measurement, and comparison. Time will tell what revelations may come forth from the exploration of these ideas.'