On Field’s Nominalization of Physical Theories

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232 FOUNDATIONS OF SCIENCE

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L S

formal system with quantification

over non−physical terms formal system without quantification over non−physical terms

observable phenomena

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γ α β

α β γ

γ α β

α, β

γ, δ α, β

γ, δ

α β γ δ γ δ

ε ζ α β ε

ζ

(L, S) L= (R²,Γ,Λ)

Γ(a, b, c) R²

Γ(a, b, c)⇐⇒

(a₁−b₁)²+ (a₂−b₂)²+

(c₁−b₁)²+ (c₂−b₂)²

=

(a₁−c₁)²+ (a₂−c₂)²

Λ(a, b, c, d) R²

Λ(a, b, c, d) ⇐⇒

(a₁−b₁)²+ (a₂−b₂)²=

(c₁−d₁)²+ (c₂−d₂)² S

R² α a= (a , a )∈R² β b= (b , b) ∈R²

Γ Λ

(L, S) (R²,Γ,Λ)

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(L, S) (L, S) (R²,Γ,Λ)

∀a∀b∀g∀d∀e∀z∃oΓ(a, d, b)∧Γ(b, e, g)∧Γ(g, z, a)

∧Λ(a, d, d, b)∧Λ(b, e, e, g)∧Λ(g, z, z, a)

→Γ(a, o, e)∧Γ(b, o, z)∧Γ(g, o, d)

b g

e z o

d a

α, β, γ, δ, ε ζ δ α β ε

β γ ζ γ α α, δ

δ, β β, ε ε, γ γ, ζ ζ, α

ω α ε

β ζ γ δ

β γ

ε ζ ω

α

δ

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(L, S)

Γ Λ

L S

L L

A, B, C, . . . Bet

Cong

∀A∀B Cong(A, B, B, A)

∀A∀B∀C Cong(A, B, C, C)→A=B

∀A∀B∀C∀D∀E∀F Cong(A, B, C, D)∧Cong(C, D, E, F)

→Cong(A, B, E, F)

∀A∀B Bet(A, B, A)→A=B

∀A∀B∀C∀D∀E Bet(A, D, C)∧Bet(B, E, C))

→ ∃F(Bet(D, F, B)∧Bet(E, F, A)

∃E∀A∀B φ(A)∧ψ(B)→Bet(E, A, B)

→ ∃F∀A∀B φ(A)∧ψ(B)→Bet(A, F, B)

φ ψ

E F A

ψ(B) B φ(A)

∃A∃B∃C ¬Bet(A, B, C)∧ ¬Bet(B, C, A)∧ ¬Bet(C, A, B)

∀A∀B∀C∀D∀E Cong(A, D, A, E)∧Cong(B, D, B, E)

∧Cong(C, D, C, E)∧ ¬D=E

→Bet(A, B, C)∨Bet(B, C, A)∨Bet(C, A, B)

∀A∀B∀C∀D∀E Bet(A, D, E)∧Bet(B, D, C)∧ ¬A=D

→ ∃F∃G Bet(A, B, F)∧Bet(A, C, G)∧Bet(G, E, F)

∀A∀B∀C∀D∀E∀F∀G∀H¬A=B∧Bet(A, B, C)∧Bet(E, F, G)

∧Cong(A, B, E, F)∧Cong(B, C, F, G)∧Cong(A, D, E, H)

∧Cong(B, D, F, H)→Cong(C, D, G, H)

∀A∀B∀C∀D∃E Bet(D, A, E)∧Cong(A, E, B, C)

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(L, S) L

∀A∀B∀G∀D∀E∀Z∃O Bet(A, D, B)

∧Bet(B, E, G)∧Bet(G, Z, A)∧Cong(A, D, D, B)

∧Cong(B, E, E, G)∧Cong(G, Z, Z, A)

→Bet(A, O, E)∧Bet(B, O, Z)∧Bet(G, O, D)

B G

E Z O

D A

α, β, γ, δ, ε ζ δ α β ε

β γ ζ γ α α, δ

δ, β β, ε ε, γ γ, ζ ζ, α

ω α ε

β ζ γ δ

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β γ

ε ζ ω

α

δ

1

R²

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∗

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