Return to Answer

See more in SEP's Nonexistent Objects and Negative Existential Beliefs.

See more in SEP's Nonexistent Objects, and Negative Existential Beliefs.

Plato's beard has an interesting application in mathematics. In proofs by contradiction negation of the intended conclusion is treated as an additional premise, and an auxiliary unsound argument is given using it. The contradiction in the conclusion of the auxiliary argument is then interpreted as entailing the intended conclusion. But at the onset of the auxiliary argument we do invoke inconsistent objects, implicitly or explicitly. For instance, Euclid's proof of the irrationality of square root of 2 ostensibly involves defining a rational number with square 2, and then reasoning about it. This non-existent number can be interpreted in Meinongian or in Russellian manner.

Plato's beard has an interesting application in mathematics. In proofs by contradiction negation of the intended conclusion is treated as an additional premise, and an auxiliary valid, but unsound argument is given using it. The contradiction in the conclusion of the auxiliary argument is then interpreted as entailing the intended conclusion. But at the onset of the auxiliary argument we do invoke inconsistent objects, implicitly or explicitly. For instance, Euclid's proof of the irrationality of square root of 2 ostensibly involves defining a rational number with square 2, and then reasoning about it. This non-existent number can be interpreted in Meinongian or in Russellian manner.

There is a related fallacy of "defining into existence" when implicitly defined entities are illicitly declared existent, versions of the ontological argument are often accused of defining God into existence. Kant clearly expressed the issue in his thesis that "existence is not a predicate". Even for ideal objects in mathematics it must be proved from axioms that objects fulfilling the defining conditions exists, the object is then said to be "well-defined". E.g. Euclid defines equilateral triangle as a triangle with equal sides, but he gives a straightedge and compass construction of it before using it in demonstrations (in modern texts the two steps are often combined into a single "theorem-definition").

There is a related fallacy of "defining into existence" when implicitly defined entities are illicitly declared existent, versions of the ontological argument are often accused of defining God into existence. Kant clearly expressed the issue in his thesis that "existence is not a predicate". Even for ideal objects in mathematics it must be proved from axioms that objects fulfilling the defining conditions exists, the object is then said to be "well-defined". For example, Euclid defines equilateral triangle as a triangle with equal sides, but he gives a straightedge and compass construction of it before using it in demonstrations (in modern texts the two steps are often combined into a single "theorem-definition").