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Full Idea
Modern logic requires that logical truths be true in all models, including ones devoid of any mathematical objects. It follows immediately that the existence of mathematical objects can never be a matter of logic alone.
Gist of Idea
Logical truth is true in all models, so mathematical objects can't be purely logical
Source
Øystein Linnebo (Philosophy of Mathematics [2017], 2)
Book Ref
Linnebo,Øystein: 'Philosophy of Mathematics' [Princeton 2017], p.21
A Reaction
Hm. Could there not be a complete set of models for a theory which all included mathematical objects? (I can't answer that).