 Field’s strategy only has a chance of working if Hilbert’s fear that in a very fundamental sense our best scientific theories may entail that there are only finitely many concrete entities, is ill-founded. If one sympathizes with Hilbert’s concern but does not believe in the existence of abstract entities, then one might bite the bullet and claim that there are only finitely many mathematical entities, thus contradicting the basic principles of elementary arithmetic. This leads to a position that has been called ultra-finitism (Essenin-Volpin 1961).