Abstract
This paper aims to investigate certain aspects of Weyl’s account of implicit definitions. The paper takes under consideration Weyl’s approach to a certain kind of implicit definitions i.e. abstraction principles introduced by Frege. Abstraction principles are bi-conditionals that transform certain equivalence relations into identity statements, defining thereby mathematical terms in an implicit way. The paper compares the analytic reading of implicit definitions offered by the Neo-Fregean program with Weyl’s account which has phenomenological leanings. The paper suggests that Weyl’s account should be construed as putting emphasis on intentionality of human mind towards certain invariant features of the elements of initial domains of discourse that are involved in equivalence relations. Definition of terms like direction, shape, number etc. is achieved by a kind of transformation of those invariants into ideal objects that is involved in intuition. Then the paper argues that at the period of 1926 Weyl’s writings on implicit definitions, he is inclined to endorse symbolic construction as a way to explicate the objectivity of certain processes as those that are carried out in case of implicit definitions.
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Notes
Citations of (1926) in the text follow the 2009 publication.
The Neo-Fregean program (Neo-Logicism, Neo-Fregeanism) offered a systematic analytic defense of Frege’s two basic claims according to which arithmetic is reduced to logic and natural numbers are abstract objects.
For this formulation in ordinary language cf. Grundlagen §64.
The name is due to the fact that when Frege introduced the principle in question, he was reminded of Hume’s considerations about numerical identity.
For Frege’s explicit definitions cf. Grundlagen §68: “the direction of the line a is the extension of the concept ‘parallel to the line a’” and “the number of the concept F is the extension of the concept ‘equinumerous to the concept F’”.
A Carnap conditional has the form “\( \exists {\text{x}}\left( {\# {\text{x}}} \right) \to \# {\text{f}} \)” (: if something satisfies such and such conditions then f satisfies those conditions). It is taken to be an implicit definition of a theoretical scientific term f, e.g. ‘electron’ (cf. Hale and Wright (2001); Psillos and Christopoulou (2009).
Non-arrogance is a property of an implicit definition. An implicit definition should be non-arrogant, i.e. it should not need any a posteriori work for its affirmation. For example, according to Hale and Wright (2001a, b, 142–150), HP avoids arrogance because of its (double) conditional form. Besides, an implicit definition is conservative if and only if it does not imply new consequences that were not already implied by the initial language before the very stipulation takes place. In fact, HP does not imply new consequences that were not already implied before the extension of the initial vocabulary takes place by means of the introduction of the operator N. Some abstraction principles are consistent, conservative, non arrogant and work successfully as implicit definitions. The discussion about the virtues of abstraction principles that work as good implicit definitions is broad as well as rich (For example, cf. discussion about the so called “bad company” i.e. a class of abstraction principles that lack certain virtues of good implicit definitions (Linnebo 2009; Eklund 2009, etc.).
Weyl cites Frege, Die Grundlagen der Arithmetik, Breslau (1884), §§ 63–68 as well as Helmholtz, 1887, Zählen und Messen, 1887, Wissenschaftliche Abhandlungen, III, 377.
It is easy to see that this relation is reflexive, symmetrical and transitive.
Equivalently, x and y are congruent mod 5 if and only if, divided by 5, they leave the same remainder.
Addition: \( {\text{T}}_{\text{m}} \left( {\text{x}} \right) + {\text{T}}_{\text{m}} \left( {\text{y}} \right) = {\text{T}}_{\text{m}} \left( {{\text{x}} + {\text{y}}} \right) \), multiplication: \( {\text{T}}_{\text{m}} \left( {\text{x}} \right)*{\text{T}}_{\text{m}} \left( {\text{y}} \right) = {\text{T}}_{\text{m}} \left( {{\text{x}}*{\text{y}}} \right),\quad {\text{x}},{\text{ y}},{\text{ m integers}},\quad {\text{m}} > 1 \). It can be proved that the set {C0, C1, … Cm−1} is an abelian group with regard to addition and a reversing semi-group with an identity element with regard to multiplication. Besides, multiplication is distributive to addition.
His Investigations VI, §§ 1–29 offer an account of the issue of how an intention is fulfilled.
The intuitionist leanings of Weyl (which involve the importance of intuition too) are not adequate enough to explain Weyl’s persistence on the transformation of the invariant features into ideal objects. The latter is a phenomenological notion. So it is the phenomenological leanings of Weyl that justify his claim about the transformation in question.
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Acknowledgments
This research has been co-financed by the European Union (European Social Fund – ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) – Research Funding Program: THALIS-UOA-Aspects and Prospects of Realism in the Philosophy of Science and Mathematics (APRePoSMa). I thank three anonymous reviewers too for their comments that were very helpful for improvement of this paper.
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Christopoulou, D. Weyl on Fregean Implicit Definitions: Between Phenomenology and Symbolic Construction. J Gen Philos Sci 45, 35–47 (2014). https://doi.org/10.1007/s10838-014-9239-7
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DOI: https://doi.org/10.1007/s10838-014-9239-7