Constructive Formalism Quotes

“It has been contended, by Rudolf Carnap and others, that since we are unable to find in application an absolute standard by which the validity of a formal system may be tested we are free to choose what formalisation of mathematics we please, technical considerations alone leading us to prefer one system to another. If we accept this standpoint then the distinction between constructive and non- constructive systems is a distinction without a difference and the constructive system becomes little more than a poor relation of the non-constructive. I consider this view to be wholly mistaken. Even if we leave out of account the question of demonstrable freedom from contradiction, the Principia [Mathematica of Whitehead and Russell] and the Grundlagen [der Mathematik of Hilbert and Bernays] must be rejected as formalisations of mathematics for their failure to express adequately the concepts of universality and existence. Even though we do not discover a contradiction in a formal system by showing that the existential quantifier fails to express the notion of existence, for we have no right to pre-judge the meaning of the signs of the system—and to this extent Carnap is right—nonetheless when a mathematician seeks to establish the existence of a number with a certain property he will not, and should not, be satisfied to find that all he has proved is a formula in some formal system, which whatever it may affirm assuredly does not say that a number exists with the desired property.”
― Reuben Louis Goodstein, Constructive Formalism

“It has been contended, by Rudolf Carnap and others, that since we are unable to find in application an absolute standard by which the validity of a formal system may be tested we are free to choose what formalisation of mathematics we please, technical considerations alone leading us to prefer one system to another. If we accept this standpoint then the distinction between constructive and non- constructive systems is a distinction without a difference and the constructive system becomes little more than a poor relation of the non-constructive. I consider this view to be wholly mistaken. Even if we leave out of account the question of demonstrable freedom from contradiction, the Principia [Mathematica of Whitehead and Russell] and the Grundlagen[der Mathematik of Hilbert and Bernays] must be rejected as formalisations of mathematics for their failure to express adequately the concepts of universality and existence. Even though we do not discover a contradiction in a formal system by showing that the existential quantifier fails to express the notion of existence, for we have no right to pre-judge the meaning of the signs of the system—and to this extent Carnap is right—none-the-less when a mathematician seeks to establish the existence of a number with a certain property he will not, and should not, be satisfied to find that all he has proved is a formula in some formal system, which whatever it may affirm assuredly does not say that a number exists with the desired property.”
― Reuben Louis Goodstein, Constructive Formalism