Sets are multitudes which are also unities. It is surprising that the fact that multitudes are also unities leads to no contradictions: this is the main fact of mathematics.Kurt Gödel (Hao Wang,A Logical Journey: From Gödel to Philosophy)In what sense can something be at the same time one and many? The problem is familiar since Plato (for example,Republic524e). In recent times, Whitehead and Russell, inPrincipia Mathematica,have been struck by the difficulty of the problem: ‘If there is such an object as (...) a class, it must be in some senseoneobject, yet it is only of classes that many can be predicated. Hence, if we admit classes as objects, we must suppose that the same object can be both one and many, which seems impossible.' It is, however, in Frege's great work,The Foundations of Arithmetic(henceforth,Grundlagen),that many see the final resolution of the old question: how can something be at the same time one and many? (shrink)

One of the objections against the thesis of underdetermination of theories by observations is that it is unintelligible. Any two empirically equivalent theories — so the argument goes—are in principle intertranslatable, hence cannot count as rivals in any non-trivial sense. Against that objection, this paper shows that empirically equivalent theories may contain theoretical sentences that are not intertranslatable. Examples are drawn from a related discussion about incommensurability that shows that theoretical non-intertranslatability is possible.

Some authors claim that ‘I’ and ‘now’ are essential indexicals, in the sense that they cannot be eliminated in favor of other indexicals or nonindexical expressions. This article argues that three indexicals of direction—one for each spatial dimension (e.g., ‘up’, ‘front’, and ‘left’)—must also be regarded essential, insofar as they are used as pure indexicals and not as demonstratives.