Der formalistische Standpunkt
Ein Gegenentwurf zum Platonismus ist die formalistische Haltung, die ihre Sicht der Mathematik aus der Möglichkeit eines finiten formalen Rahmens für die mathematische Tätigkeit gewinnt. Seine radikalste Form ist: Es gibt die mathematischen Objekte nicht wirklich, es gibt nur Zeichenketten auf dem Papier, und diese Zeichenketten werden nach festen Regeln manipuliert. Aus Axiomen − bestimmten ausgezeichneten Zeichenketten − gewinnt man durch diese Manipulationen mathematische Sätze.
Der Formalismus hat den Vorteil großer Klarheit. Er scheut die metaphysischen Grundannahmen des Platonismus wie der Teufel das Weihwasser, und gibt eine präzise Anwort auf die Frage, was Mathematik ist.
Die in diesem Buch eingenommene Haltung tendiert stark zum platonischen Entwurf. Die formale Welt ist uns ein willkommener, das Gebäude tragender Keller, der insbesondere metamathematische Aussagen ermöglicht.
Zur Illustration des formalistischen Standpunkts schließen wir dieses Kapitel mit einem neueren Kommentar zum Formalismus.
H. Dales über Formalismus
„ … mathematicians are ambivalent between realism [Platonismus] and formalism. For example, I quote from Davis and Hersh (1981, p. 320):
… the typical working mathematician is a [realist] on weekdays and a formalist on Sundays. That is, when he is doing mathematics he is convinced that he is dealing with an objective reality whose properties he is attempting to determine. But then, when challenged to give a philosophical account of this reality, he finds it easiest to pretend that he does not believe in it after all.
… It seems to me that most mathematicians are formalists for all the days of the week. It is of course very useful when seeking proofs within the formal system to have a ‘realistic picture’ in one’s mind, and so it is temporarily convenient … to be a realist …
I think that the success of the major mathematicians in resolving problems and advancing the subject owes much to their ability to formulate in their mind an appropriate image of the abstract problem: it must be sufficiently subtle and complicated to capture the essential features of the question at issue, yet remain sufficiently simple to allow our limited minds absolutely and fully to explore, in quiet contemplation, all aspects of this image until we understand it sufficiently to begin the attempt to transfer this understanding to a written account of the general, abstract situation …
Thus my view is that we are genuine, believing formalists who temporarily act as realists for reasons of expediency in solving problems.
… The first remark is that formalists practically never use a truly formal language in their writings (and may not know to do this, even under pressure); they formulate their theorems in the naive language of set theory developed in the XIXth century by Dedekind and Cantor. But they are confident that, if their results had to be formalized, this could be done; and doubtless they are correct in this.“
(H. Dales, „The mathematician as a formalist“. In: Dales, Olivieri 1998, „Truth in Mathematics“ )