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 reality
Transcript of Giovanelli, Marco (2012) the Forgotten Tradition. How the Logical Empiricists Missed the...
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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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
7/18/2019 Giovanelli, Marco (2012) the Forgotten Tradition. How the Logical Empiricists Missed the Philosophical Significance …
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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)
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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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
7/18/2019 Giovanelli, Marco (2012) the Forgotten Tradition. How the Logical Empiricists Missed the Philosophical Significance …
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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
7/18/2019 Giovanelli, Marco (2012) the Forgotten Tradition. How the Logical Empiricists Missed the Philosophical Significance …
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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
7/18/2019 Giovanelli, Marco (2012) the Forgotten Tradition. How the Logical Empiricists Missed the Philosophical Significance …
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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
7/18/2019 Giovanelli, Marco (2012) the Forgotten Tradition. How the Logical Empiricists Missed the Philosophical Significance …
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d2xτ
ds2
= Γ
τ
µν
dxµ
ds
dxν
ds
gµν Γτ µν
gµν Γτ µν
gik Γihl
Γihl gik
Γihl
7/18/2019 Giovanelli, Marco (2012) the Forgotten Tradition. How the Logical Empiricists Missed the Philosophical Significance …
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Γisr = Γi
rs
Γikl
Γikl
gik
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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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ds2 =4
Σ1
dx 2ν
ds2 =4
Σ1
gµν dxν dxν
gµν
gµν
gµν
gµν
gµν
gµν
7/18/2019 Giovanelli, Marco (2012) the Forgotten Tradition. How the Logical Empiricists Missed the Philosophical Significance …
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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
7/18/2019 Giovanelli, Marco (2012) the Forgotten Tradition. How the Logical Empiricists Missed the Philosophical Significance …
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F = 0
G
θ G
G F
G + F
F = 0
G
G
F
gµν
gµν
7/18/2019 Giovanelli, Marco (2012) the Forgotten Tradition. How the Logical Empiricists Missed the Philosophical Significance …
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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
7/18/2019 Giovanelli, Marco (2012) the Forgotten Tradition. How the Logical Empiricists Missed the Philosophical Significance …
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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
7/18/2019 Giovanelli, Marco (2012) the Forgotten Tradition. How the Logical Empiricists Missed the Philosophical Significance …
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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
7/18/2019 Giovanelli, Marco (2012) the Forgotten Tradition. How the Logical Empiricists Missed the Philosophical Significance …
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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
7/18/2019 Giovanelli, Marco (2012) the Forgotten Tradition. How the Logical Empiricists Missed the Philosophical Significance …
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Γτ µν
gµν
∂gµν∂xτ
g44
gµν
g
στ
Rµνστ
7/18/2019 Giovanelli, Marco (2012) the Forgotten Tradition. How the Logical Empiricists Missed the Philosophical Significance …
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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
7/18/2019 Giovanelli, Marco (2012) the Forgotten Tradition. How the Logical Empiricists Missed the Philosophical Significance …
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Γlik Riklm
gik
Γlik
P
P δAν= −Γ
νστ A
σdxτ
gik
Γ
Riklm
Rkl
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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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gµν
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