giovanelli, marco (2012) the forgotten tradition. how the logical empiricists missed the...
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Abstract: The paper attempts to show how the Logical Empiricists’ interpretation of the relation between geometry and reality emerges from a “collision” of mathematical traditions. Considering Riemann’s work as the initiator of a 19th century geometrical tradition, whose main protagonists were Helmholtz and Poincaré, the Logical Empiricists neglected the fact that Riemann’s revolutionary insight flourished rather in a non-geometrical tradition dominated by the works of Christoffel and Ricci-Curbastro roughly in the same years. I will argue that in the attempt to draw the line Riemann-Helmholtz-Poincaré-Einstein Logical Empiricists were led to argue that General Relativity raised mainly a problem of mathematical under-determination, i.e. the discovery that there are physical differences that cannot be expressed in the relevant mathematical structure of the theory. However, a historical reconstruction of the alternative line of development Riemann-Chritoffel-Ricci-Einstein shows on the contrary that the main philosophical issue raised by Einstein’s theory was rather that of mathematical over-determination, i.e. the recognition of the presence of redundant mathematical differences that do not have any correspondence in physical realityTRANSCRIPT
7/18/2019 Giovanelli, Marco (2012) the Forgotten Tradition. How the Logical Empiricists Missed the Philosophical Significance …
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7/18/2019 Giovanelli, Marco (2012) the Forgotten Tradition. How the Logical Empiricists Missed the Philosophical Significance …
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7/18/2019 Giovanelli, Marco (2012) the Forgotten Tradition. How the Logical Empiricists Missed the Philosophical Significance …
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n
n x1, x2, x3, . . . , xn
ds
dx
x
n
n
x1, . . . xn
ds2
dx2
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ai,idxidxi
ai,i
dx2
i
xi i = 1, 2, 3)
(ιι
, ι
ι
)
ai,i xi
(ιι, ιι)= 0
ai,i
ι,ι bι,ι dsι dsι
n
bι,ι dsι dsι
ι
d2ι
(s1, s2, . . . , sn)
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F =
ωikϑxiϑxk F =
ω
ikϑx
iϑx
k
ωik
∂ωgk
∂xh
ghk
ilr
=
E rkE
ghk
(ghki)
ωik
(ghki) (ghki)
n
ϕ =
rsarsdxrdxs ars
x1, x2, . . ,xn
x1, x2, . . ,xn
u1, u2, . . ,un
ϕ = ϕ
ErkE
ωik
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ars
(a pq)
a
(rs)
a
( pq)
ars a(rs)
ars,i
ahi,jk
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ds2 =
gµν dxµdxν
gµν
δ ´
ds
= 0
gµν
∂ 2ϕ∂x2 + ∂ 2ϕ
∂y2 + ∂ 2ϕ∂z2 = 4πkρ
gµν = gνµ ϕ
ρ Θµν
Γµν
gµν
κ ·Θµν = Γµν κ
(ik,lm)
∆φ
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d2xτ
ds2 =µν
µν
τ
dxµ
ds
dxν
ds
gµν T µν
Rµν
T µν
Gµν = Rµν − 12Rgµν R = gµν Rµν
Gµν
Gµν = −κT µν
gµν
gµν T µν
gµν(x) gµν (x)
x
x
gµν(x) gµν(x) x
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Γτ µν = −
µv
τ
gµν Γτ µν = 0
gµν
Γτ µν = 0
Γτ µν Rρ
µντ
Rρµντ
gµν
gµν
gµν
ds2
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d2xτ
ds2
= Γ
τ
µν
dxµ
ds
dxν
ds
gµν Γτ µν
gµν Γτ µν
gik Γihl
Γihl gik
Γihl
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Γisr = Γi
rs
Γikl
Γikl
gik
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ds2 =4
Σ1
dx 2ν
ds2 =4
Σ1
gµν dxν dxν
gµν
gµν
gµν
gµν
gµν
gµν
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gµν
gµν
gµν
gµν
gµν
gµν
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P
G + P
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X
X
X
X
X = 0
X
gµν
ds2 =
dx2ν
gµν
gµν
ds2
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gµν
gµν
gµν
∂gµν∂xσ
∂ 2gµν∂xσ∂xρ
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F = 0
G
θ G
G F
G + F
F = 0
G
G
F
gµν
gµν
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gµν
gµν
gµν
ds2
g
µν
gµν
gµν
gµν
g
µν
x
µ
gµν
g
µν
ds2
= gµνdxµdxν = g
στ dx
σdx
τ
ds2
gµν
(1, 1)
√ 2, π/4
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gµν
gµν
F.
F
g
µν
G gµν G F µν F
g
µν + F µν = gµν
g
µν
F µν
g
µν
gµν
F
g
µν
F µν
gµν ds2 = gµν dxµdxν = (g
µν +F µν )dx
µdx
ν gµν
g
µν
ds2 = gµν dxµdxν = g
µν dx
µdx
ν
gµν
g
µν
ds2
gµν
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gµν
gστ
gµν
gµν
gµν
gµν g
µν ds2 = ds2
gµν
gµν
∂gµν∂xσ
∂ 2gµν∂xτ ∂xσ
Rµνστ
Rµνστ
Rµνστ
Rµνστ = 0
Rµνστ
Rµνστ gµν .
gµν
gµν
Γτ µν
gµν
gµν Γτ µν
gµν
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K
g
στ
K
gµν
gµν
K
K
K
g
στ gµν
gµν
Γτ µν
gµν
gµν
gµν
gµν
gµν
Γτ µν
∂gµν∂xτ
Γτ µν
gµν + γ µν
gµν
gµν
gµν
γ µν
Γτ µν γ µν
gµν
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Γτ µν
gµν
∂gµν∂xτ
g44
gµν
g
στ
Rµνστ
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gµν
gµν
gµν
Γτ µν
Rµνστ
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Γτ µν (xν
gµν (xν )
Γτ µν gµν
dAτ = −Γτ µν A
µdxν
l2 = gµν AµAν
Γτ µν
gµν
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Γlik Riklm
gik
Γlik
P
P δAν= −Γ
νστ A
σdxτ
gik
Γ
Riklm
Rkl
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gµν
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