# Economics Dictionary of Arguments

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Mathematics: Mathematics is the study of patterns and relationships. It encompasses arithmetic, algebra, calculus, geometry, trigonometry, topology, and others. Mathematics is used in science, engineering, business, and finance. See also Physics, Logic.
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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.

Author Concept Summary/Quotes Sources

Hartry Field on Mathematics - Dictionary of Arguments

I 80
Existence/Field: Existence should not be part of the logic - therefore, mathematics cannot be reduced to logic - otherwise too many properties would have to be assumed.
>Mathematical entities
.
I 80f
Mathematics/Knowledge/Field: nevertheless, mathematical knowledge is simply logical knowledge because of deflationism.
>Deflationism.
E.g. Knowing a lot/little about maths: two kinds of knowledge: mathematical knowledge: non-logical knowledge: e.g. what other mathematicians accept.
I 112
This knowledge is empirical.
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III 9
Pure mathematics/application/Field: e.g. number theory: is not applicable at all to the world. - E.g. set theory: must allow the use of elementary elements.
>Set theory.
Solution: "impure mathematics": functions that map physical objects to numbers. - Then the comprehension axioms must also contain non-mathematical vocabulary. - E.g. instances of the separation axiom.
III 13
Mathematics/Field: Mathematics can prove to be inconsistent. - Even if it is extremely improbable - then it would also be non-conservative. - So mathematics is not a priori true.
>Conservativity.

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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. Translations: Dictionary of Arguments
The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.

Field I
H. Field
Realism, Mathematics and Modality Oxford New York 1989

Field II
H. Field
Truth and the Absence of Fact Oxford New York 2001

Field III
H. Field
Science without numbers Princeton New Jersey 1980

Field IV
Hartry Field
"Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67
In
Theories of Truth, Paul Horwich, Aldershot 1994

> Counter arguments against Field
> Counter arguments in relation to Mathematics