By Paul Lorenzen
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Talk of concepts is not a problem, however, when concepts are introduced as I have done here. We have just indicated what we mean when we say that two predicates represent the same concept. We must now determine how to talk intelligibly about concepts themselves. To do that we must indicate how we advance to predicates that concern concepts. We make this advance by first introducing predicates concerning predicates, that is, predicate predicates. We can, for example, exemplarily introduce a distinction between long predicates and short predicates.
What is going on here is that people are proposing axiomatic theories that use signs like x e M as formula symbols, where e is uninterpreted. Because we can propose as many axiomatic theories as we like, the only objection to this is that the significance and purpose of this activity remain completely obscure. On the other hand, if in the future someone were able to construct levels of sets for which the axioms are true, then an axiomatic foundation for set theory would be superfluous. What are we to make of the fact that axiomatic set theories are recognized in mathematics as the only acceptable theories, despite the obvious unintelligibility of such theories?
It is said that these theories are aesthetically pleasing or that they are empirically confirmed. These are the more harmless answers to the question. From more aggressive quarters comes the counterquestion: Do you doubt the validity of modern physics, which as everyone knows is built upon the foundation of the mathematics of infinitesimals? It is certainly true that physics is built upon the foundation of modern mathematics, but no one has yet demonstrated that physics would not flourish even better with an intelligible (constructive) mathematics of infinitesimals than it has so far with an unintelligible (axiomatic) mathematics.
Constructive Philosophy by Paul Lorenzen