Introduction

The position of "Fregean Platonism" advocated by Crispin Wright et al. Was a

bold attempt to defend the existence of abstract objects such as numbers, but it
was also a source of much critical controversy. One of the most important of the
various criticisms is the disputation by Dammett [1991]. There was also an
exchange of discussions between Wright and Dammet over this issue, right
[1998a], Dummett [1998], Wright [1998b]). In this essay, I would like to
examine the controversy between them over the pros and cons of "Fregean
Platonism."

1. Light's Fregean Platonism

In Frege's idea of number as an object (Wright [1983]), Wright attempts to
rescue Frege's logicism and Platonist ideas from Frege's own failure. The part of
Frege's conception that is being modified and rescued by Wright is especially
called "Fregean Platonism". Therefore, in order to understand Wright's position,
we will take a brief look at what Frege himself is supposed to have done with
respect to Platonism's claim (introduction of abstract objects). In The
Foundations of Arithmetic, Frege seeks to introduce an abstract object (number)
by appealing to his "contextual principle". According to the contextual principle,
words have meaning only in the context of sentences, so in order to give
meaning to numbers, it is necessary to clarify the meaning of sentences that
include numbers. In this way, what is submitted as a contextual definition of a
numeral (in this case, the radix operator) is the following formula called Hume's
principle;

Hume’s Principle:
The number of Fs is identical to the number of Gs if and only if F and G are
equinumerous.
Using our notation ‘#F’ to abbreviate ‘the number of Fs’, we may formalize as
follows:
#F=#G≡F≈G

However, Frege dismisses this N = because he cannot avoid the Caesar problem
(the problem of not being able to determine if the number of planets is the
same as Julius Caesar). Instead, the cardinality operator is given an explicit
definition using the extension of the concept. Specifically, #F is defined as an
extension of the concept of "equal to concept F". However, in fact, this explicit
definition does not solve the Caesar problem. In order for this definition to be a
solution to the problem, we must know about the extension of the concept (we
must be able to determine if the extension of a given concept is the same as
Caesar). In this way, the problem is carried over to the "basic law of arithmetic".
In "The Basic Law of Arithmetic" Frege introduces what is called the "course of
values", which is a generalization of the extension of the concept. (Therefore,
we must understand what course-of-values are.) And it is that axiom (V) that
Frege establishes as the axiom that governs the course of values. (The axiom
(V) has the form "ε (ε) = αΨ (α) ⇔ ∀x ((x) = Ψ (x))".) However, this axiom (V) is
inconsistent. (Russell's paradox) is guided, where Frege's logicism and the idea
of Platonism fail.
Since Frege's own attempts are believed to have failed, Wright, who defends
Platonism, makes a partial modification of Frege's conception. According to
Wright's diagnosis, Frege's mistake is that he abandoned the contextual
definition of N = and gave the cardinal operator an explicit definition. (The
extension of the concept was brought up to give an explicit definition, and this
extension of the concept is the source of the contradiction.) Wright also argues
that it is possible to solve the Caesar problem by adding some clauses to N =,
and Frege's vision should remain within a contextual approach.
Basic Law V:
,ϵf(ϵ)=,αg(α)≡∀x[f(x)=g(x)]ϵ,f(ϵ)=α,g(α)≡∀x[f(x)=g(x)]
This principle asserts: the course-of-values of the function ff is identical to the
course-of-values of the function gg if and only if ff and g map every object to the
same value.

Thus, Wright's position on Frege's Platonism seeks to introduce numbers, which

are abstract objects, in a manner that relies entirely on the contextual definition
of N =.

2. Criticism from Dammet

At the heart of Dummett's book Frege: Philosophy of Mathematics (Dummett
[1991]) is a discussion of criticisms of Frege's Platonism. In this section, we will
look at some of Dammet's discussions that will be problematic in later
discussions.
First, Dummett points out that Wright cannot completely ignore the problem of
conceptual extension (even if the Caesar problem can be solved in the way
Wright says) (Dummett [ 1991] p.187). As mentioned earlier, Frege introduces
the course of values in "The Foundations of Arithmetic" to deal with the problem
left in "The Foundations of Arithmetic", which is the extension of the concept.
And it is confirmed that there is a clear formal similarity between Frege's axiom
(V), which is the axiom governing the course of values, and N =, which is used
to introduce numbers contextually. Here, Dammet argues that: If Wright's
advocacy of introducing cardinality operators by contextual means is sound,
then abstract operators introduced in a similar way must also be valid
(Dummett [1991] p. 188). However, the introduction of abstract operators (that
is, the introduction of "course of values") leads to contradictions and is far from
valid. Dammet's criticism is that Wright should have some accountability in this
regard. (This criticism is called Bad Company Objection.)

The next question that Dammet is heading to is whether the contextual

definition method explains the existence of the referent of name.

First, three different perspectives on the contextual definition are introduced.

There are three views: austere view, robst view, and intermediate view
(Dummett [1991] p.191). In an austere view, the contextual definition is nothing
like a referent for the name in question. The contextual definition gives a
statement that contains a name an equivalence statement that does not, which
is taken as an indication that the name in question is just an apparent name. ..
The robust view, on the other hand, is that the contextual definition gives the
meaning of the sentence containing the name in question, which qualifies the
name as a true referent with a referent (this is). Wright's view). And it is an
intermediate view that lies between the two, and Dammet adopts this position.

Dummett's intermediate view is explained based on the distinction between

Frege in "The Foundations of Arithmetic" and Frege in "The Foundations of
Arithmetic" (Dummett [1991] pp.192-193). .. The difference in these is that a
formal system not given in the former is given in the latter, and Dammet states
his intermediate view within the framework of the "Basic Law".
The significance of the Singular term is usually the way the referent is given.
Also, understanding a sentence (the idea it represents) is to understand what
makes the sentence true and what makes it false. Therefore, understanding a
sentence containing a singular term leads to the referent through the meaning
of the term, and considers whether the referent has appropriate properties (true
if it has, false if it does not). Will have a procedure. However, such a picture
does not apply at all if the singular term is introduced by a contextual definition.
Our understanding of a sentence containing such a singular term relies on the
understanding of an equivalence sentence not containing such a singular term.
In other words, here, the singular term's function of "refer" is not working at all
and is useless.
That is, the attribution of instructions to the singular term introduced by the
contextual definition is semitically-idle.

The above is the content of the intermediate view, but Dammet says that this
view cannot be applied to the "basic" framework. According to Dammet, in the
"foundation" there is no distinction between a target language without a formal
system and a metalanguage (which gives its semantics), and the contextual
principle there is about the instructions of the singular term. It is like retrieving
a question into an in-language problem and denying the possibility of semantics
from outside the language (Dummettl 1991] p.192 and p.195). Therefore, there
is no question in "basic" such as "does the name" a "have an instruction?" (Such
a question is a question in a metalanguage), and within the framework of
"basic". In the first place, it is not possible to give an intermediate view.

However, Dammet's argument, which takes an intermediate view of the

contextual definition, is not a criticism of Wright, who is trying to stay within the
"foundation" framework.

No. Rather, Dammet's argument is this. As I just mentioned, there is no

metalanguage for semantic theory in the "foundation", so there is no
substantive concept of instruction (Dammet says that there is only a thin
concept of instruction).

However, the instructions that Wright is trying to attribute to the singular term
are instructions with substance (rather than thin), and such a substance of
instruction is within the framework of the target language and its semantic
theory.

That is, as long as it stays in the "foundation", the concept of the substantive
instruction that Wright wants cannot be obtained, and in order to assign a
substantive instruction to the singular term, Wright must stand in the
framework of the "basic law" of the target language and its semantic theory.
And in doing so, Dummett’s intermediate view of the contextual definition
should be introduced.

Finally, Dammet goes to criticism that he considers decisive. The suspicion

expressed in the Bad Company Objection section above (the suspicion that the
procedure itself for introducing an abstract object by contextual means is
actually unhealthy) is a reality. I'm trying to show you.
This is done by a detailed analysis of how the course of values introduced in the
same procedure as the introduction of the cardinality operator by contextual
definition causes a contradiction in Frege's system.いいかえ

Dummett tries to depict Frege's contradiction by showing the process in which

Frege's proof of consistency fails.

The proof of consistency here is, in fact, a proof that any legally formed singular
term has a definite indication, which is done by induction on the complexity of
the expression.

First, as an induction Basis, it must be shown that any primitive expression has
an instruction. In doing so, Frege stipulates the following conditions for the
expression to be directive.

(1) A function representation is directive when the resulting representation

always has an indication when the directive term is placed in that term location,

(2) A singular term is directive when it is placed in the term location of the
directive function expression and the resulting expression always has an
instruction. In light of this condition, it is proved that the primitive function
expressions such as "horizontal line function", "negative function", "conditional
method", "identity", and "universal quantifier" are directive.

What remains is proof that the domain name "ε (ε)" has an indication. Here
Frege takes only primitive function representations (horizontal function,
negation, conditional mood, identity) that have already been established to be
directive, and puts the domain name in place of that term.

When a domain name is placed in a horizon, negation, or conditional mood, it

can be reduced to the case of identity, so it is only necessary to consider the
case of identity as a primitive function expression.

As for the expression in which both sides of the identity symbol are the domain
name, the axiom (V) can be said to indicate the directive, so the problem is how
to think about the value when the right side of the identity is the domain name
and the left side is the singular term. be. (Caesar problem carried over to "Basic
Law"!)

And this problem is solved by the promise that the only objects given so far are
the truth value and the course of value, and that true and false are each equal
to a particular course of values.
In other words, it is only necessary to consider the case where the identity is
sandwiched between the truth value name and the course of values, and in this
case as well, it can be reduced to the case where the axiom (V) can be used.

In this way, it is shown that the course of values has an instruction, and it can
be seen that the primitive expression has an instruction. Since all complex
expressions are made from these, it can be concluded that any singular term
has a definite indication.

The central part of Frege's proof is the place where the indication of the domain
name is finalized. Since his language contains only range names as
mononominal names, this is considered to be a task that gives the area of
interest in his theory, so to speak. Also, since the axiom (V) should hold in that
area, this gives a model to his theory, and should result in consistency.
However, Frege's system was inconsistent, so there must have been a failure
somewhere in the proof. In the domain's diagnosis, the failure of the proof lies in
the indication of the (exactly this) domain name.
As is clear so far, the procedure of introducing an abstract object based on the
contextual principle involves the above cycle and is unhealthy. This is the
criticism that Dammet considers decisive.