Lexicon of Arguments

Philosophical and Scientific Issues in Dispute
 
[german]


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Versus
Sc. Camps
Theses I
Theses II

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I 20
Formalism/Thiel: Carries out, so to speak, the "linguistic turn" in mathematics. It is now asked what the object of the mathematician's work is. Rules for actions. Symbols are replaced by others. The formalist does not ask for the "meaning". Mathematics: Theory of formalisms or formal systems. >Formalism.
In addition to this "calculus-theoretical variant" of formalism, there is the "structure-theoretical variant". (>Hilbert). Different formal systems can be interpreted as valid from exactly the same mathematical object domains. We can call this their "description" by the formal systems.
>Mathematical entities.
I 279
Formalism/Geometry/Hilbert/Thiel: In 1899 Hilbert had used terms such as point, straight line, plane, "between", etc. in his foundations of geometry, but had understood their meaning in a previously unfamiliar way. It should not only enable the derivation of the usual sentences, but in its entirety should also determine the meaning of the terms used in them.
I 280
Later this was called "definition by postulates", "implicit definition".
>Definitions, >Definability,
The terms point, straight line, etc. should at most be a convenient aid for mathematical understanding.
FregeVsHilbert: clarifies in his correspondence that his axioms are not statements but forms of statements.
>Statement form.
He contested the fact that their combination gave meaning to the terms appearing in them. Rather a (in Frege's terminology) "second level term" is defined, today one would also say a "structure".
HilbertVsFrege: N.B.: Hilbert's approach is precisely that the meaning of "point", "straight line" etc. is left open.
Frege and Hilbert could have agreed on it, but did not.
Axioms/Frege/Thiel: an axiom should be a simple statement at the beginning of a system.
Axioms/Hilbert: forms of statement that together define a discipline. This has developed into the "sloppy" way of speaking, e.g. "straight line" in sphere geometry is a great circle.
Thiel I 342
Intuitionism and formalism are often presented as alternatives to logicism. The three differ so strongly that a comparison is even difficult.
I 343
Formalism/Thiel: 1. "older" formalism: second half 19th century creators Hankel, Heine, Thomae, Stolz. "formal arithmetic," "formal algebra". "The subject of arithmetic are the signs on the paper itself, so that the existence of these numbers is not in question" (naively).
Def "principle of permanence": it had become customary to introduce new signs for additional numbers and then to postulate that the rules valid for the numbers of the initial range should also be valid for the extended range.
Vs: this should be considered illegitimate as long as the consistency is not shown. Otherwise a new figure could be introduced, and
one could simply postulate e.g. § + 1 = 2 and § + 2 = 1. This contradiction would show that the "new numbers" do not really exist. This explains Heine's formulation that the "existence is not at all in question".
I 343/'344
Thomae treated the problem as "rules of the game" in a more differentiated way.
FregeVsThomae: he did not even specify the basic rules of his game, namely the correspondences to the rules, figures, and positions.
This criticism of Frege was already a forerunner of Hilbert's theory of proof, in which mere series of signs are also considered with disregard for their possible content on their creation and transformation according to given rules.
I 345
HilbertVsVs: Critics of Hilbert often overlook the fact that, at least for Hilbert himself, the "finite core" should remain interpreted in terms of content and only the "ideal" parts that cannot be interpreted in a finite way have no content that can be directly displayed. This note is methodical, not philosophical. For Hilbert's program, "formalism" is also the most frequently used term.
Beyond that, the concept of formalism has a third sense: namely, the concept of mathematics and logic as a system of schemes of action for dealing with figures free of any content.
HilbertVsFrege and Dedekind: the objects of number theory are the signs themselves. Motto: "In the beginning was the sign."
I 346
The term formalism did not originate from Hilbert or his school. Brouwer had stylized the contrasts between his intuitionism and the formalism of the Hilbert School into a fundamental decision.
Brouwer: his revision of the classical set and function concept brings another "Species of Mathematics".
Instead of the function as assignment of function values to arguments of the function, sequences of election actions of a fictitious "ideal mathematician" who chooses a natural number at every point of the infinitely conceived process take place, whereby this number may be limited by the most different determinations for the election action, although in the individual case the election action is not predictable.

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