Philosophy Lexicon of Arguments

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Formalism: the thesis that statements acquire their meaning only from the rules for substituting, inserting, eliminating, forming, equality and inequality of symbols within a calculus or system. See also calculus, meaning, rules, content, correctness, systems, truth.

Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments.

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Waismann I 69
Intuitionism/Waismann: allows only proofs that can be constructed in a finite number of steps (that are constructive). All others are meaningless.
Formalism/Waismann: also allows non-constructive proofs. This dispute is, however, idle, if it is true that the word existence has no clear meaning from the outset. It is only obtained by the proof. And then a corresponding different one.
Waismann I 74
Formalism/FormalismVsPeano/Peano/Waismann: Formalism does not share Peano's assumption that we already know the meaning of the words "zero", "number", "successor". For the formalists the axioms are links of meaningless signs whose structure alone interests them. The symbols can then be interpreted in infinitely many ways, Russell e.g. suppose,
1. "0" should be 100 and "number" should be the numbers from 100 upwards. Then our principles are satisfied. Even the fourth applies: although 100 is the successor to 99, 99 is not a "number" in the newly defined sense. Can be done with any arbitrary number instead of 100.
2. "0" usual meaning: "number" is an "even number" "successor" should be a number that results from it by addition of 2.

: 0,2,4,6,8...

all 5 axioms by Peano are satisfied.
I 75
3. "0" should be the number 1, "number" should mean the series

1, 1/2, 1/4, 1/8, ...

and "successor" shall mean "half of". For the resulting series all 5 peanotic axioms apply.
They therefore do not characterize the notion of the series of numbers, but rather that of progession.
One could then understand under numerical terms some things that satisfy the axioms (Russell) WaismannVs: unsatisfying, e.g. we would then have no more possibility to distinguish the statement "There are 5 regular bodies" from the statement: "There are 105 regular bodies".
Were the axioms restricted by additions so as to give a complete characterization of the cardinal numbers?
> Löwenheim-Skolem has thwarted this hope.

Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.

Wa I
F. Waismann
Einführung in das mathematische Denken Darmstadt 1996

F. Waismann
Logik, Sprache, Philosophie Stuttgart 1976

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Ed. Martin Schulz, access date 2018-05-20