The individuation of mathematical objects

Area of Science:

• Philosophy of Mathematics
• Metaphysics
• Epistemology

Background:

• Mathematical platonism posits the existence of abstract mathematical objects.
• A key objection is that mathematical objects are mysterious and their individuation is unexplained.
• Nominalism challenges the nature and existence of abstract mathematical entities.

Purpose of the Study:

• To evaluate the nominalist objection regarding the individuation of mathematical objects.
• To explore and analyze different modes of mathematical object individuation.
• To determine if the individuation of mathematical objects poses a genuine problem for mathematical platonism.

Main Methods:

• Analysis of three proposed modes of individuation for mathematical objects: intrinsic properties, relations, and underlying substance.
• Metaphysical examination of the implications of each individuation mode for platonism.
• Argumentative evaluation of the nominalist objection's validity based on the analysis.

Main Results:

• Individuation by intrinsic properties and relations does not pose significant metaphysical challenges for platonism.
• Individuation by underlying 'substance' is the only mode that raises metaphysical problems potentially supporting the mystery objection.
• Platonism is not obligated to adopt the problematic 'substance' individuation, thus retaining explanatory power.

Conclusions:

• The nominalist objection, as elaborated through the individuation problem, does not undermine mathematical platonism.
• Platonists can account for the individuation of mathematical objects through non-problematic means.
• The mystery objection concerning individuation lacks persuasive force against mathematical platonism.