Any actual and last theory yet not being generalized can be considered effectively as Russell’s set of all sets Any meta-principle such as “Mach’s razor” is self-contradictorily to be incorporate within its proper principles because this supposes for its boundaries to be known, but they are in fact unknown This can be demonstrated by the case of Einstein’s “Mach’s principle” in general relativity

1. «Mach’s razor»
applied to itself
Sober, Einstein’s «Mach’s
principle», and the cosmological
constant

2. Vasil Penchev
Bulgarian Academy of Sciences:
Institute for the Study of Societies and Knowledge:
Dept. of Logical Systems and Models:
vasildinev@gmail.com
11:15 – 12:00, Friday, June 12th, Husova 4, Praha,
Department of Analytic Philosophy, Institute of
Philosophy, Czech Academy of Sciences
«Parsimony in Philosophy», June 12th
– 13th , 2015

3. Sober’s “Reconstructing the Past”
Sober (“1988: 59) interprets the principle of
parsimony relatively: “to a set of empirical
background assumptions”
This means that the background assumptions
determines a reference set, only according to which
can be define a certain subset by its properties or in
other words, a notion or theory by its intension
Then the principle of parsimony suggests equating
the extension and intension or the “principle of
abstraction”: any set can be equivalently defined both
by its properties and by its elements

4. Mach’s «Razor»
If that principle should be universal, a common
set of that kind should exist
That common set cannot help but the set of all
sets
Then the principle of parsimony would be really
universal for it would refer to any set
However the set of all sets is self-contradictory
as Russell’s paradox (1902) demonstrates

5. Mach’s «economy of thought» as
a «razor»
Mach’s “economy of thought” accepted as an
universal principle would be to be referred to
that most universal set of all sets
In fact Mach’s “economy of thought” is a
paraphrase of “Occam’s razor” and thus one can
speak of “Mach’s razor”
However both Occam’s and Mach’s “razor” turn
out to be self-contradictory each considered as
an absolute principle

6. The razor of Russell’s «barber»
According to the popular version of Russell’s paradox, a
barber has been employed in a village under the following
conditions:
To be an inhabitant of that village
To shave those villagers who do not shave themselves
Not to shave those villagers who shave themselves
Then the barber himself should begin to shave as he does
not shave himself, but immediately stop for he shave
himself, then again and again start and stop
The lesson is: Mach’s “razor” razors itself turning out to
be self-contradictory just as that of Russell’s “barber”

7. Einstein’s «Mach’s principle»
Einstein (1918) involved an additional principle
in his general relativity titled by himself “Mach’s
principle”
It “razors” any source of gravitational field
different than mass and energy
Furthermore Einstein’s application of it forced a
hypothetical constant called cosmological to
conserve the universe stationary as the simplest
conjecture

8. «Mach’s principle» as the “biggest
blunt”
The cosmological constant was called “the
biggest blunt” by Einstein (Gamov, “My World
line” 1970: 44) after Hubble’s discovery of the
expansion of the universe
Nevertheless, the “biggest blunt” of some
nonzero cosmological constant is very widely
utilized nowadays in different cosmological
models of the expanding universe and even
confirmed experimentally (“Supernova Search
Team” 1998; “The Supernova Cosmology
Project” 1999)

9. The history of the “biggest blunt”:
the cosmological constant
The history of Einstein’s “Mach’s principle” can
be considered as an example for the self-
contradictory of “Mach’s razor” as an absolute
principle
Its application forces for the “cosmological
constant” and other corollaries to be
successively accepted and refused just for their
refusing or accepting immediately before that
and just as a demonstration of Russell’s Barber
and his self-contradictory

10. Initially: no cosmological
constant
There was no cosmological constant in the
initial variant of general relativity (1915-1916)
However this infers two “ridiculous” corollaries:
The universe is not stationary
Gravitational field allows as its source some
entity, which is neither mass nor energy
Both did not confirm experimentally at that time
and should be “shaved” by some relevant
correction in the principles

11. Einstein’s revision of himself
Einstein introduced (1918) a revised variant of
his field equation containing an additional
member containing the cosmological constant
If the cosmological constant has any nonzero
value, both ridiculous corollaries turn out to be
“shaved”
The principle justifying that “shaving” Einstein
called “Mach’s principle” as a direct application
of Mach’s “economy of thought”

12. The intention of «Mach’s
principle»
He grounded it by “Mach’s principle” (only mass
and energy is the source of gravitational field) as
it implies that the cosmological constant is
nonzero
However Mach’s principle itself is an additional
assumption in relation to the initial corpus of
general relativity principles
Thus one can debate whether Mach’s razor
should not shave Einstein’s “Mach’s principle”

13. GeorgeGamow’s statement of the
“biggest blunt”
However Edwin Hubble observed experimentally
the expansion of all universe, which therefore turns
out not to be stationary
If the universe is not stationary, this rejects “Mach’s
principle” in general relativity
Once “Mach’s principle” is rejected, the other
conclusion about the cosmological constant should
be refused as well
According to Gamow (1970: 44), Einstein declared
the cosmological constant as his “biggest blunt”
after he had been invited by Hubble to observe the
evidences about the expansion of the universe

14. Now: The “biggest blunt” is ...
correct
However the cosmological constant could
conserve some nonzero value(s) on other ground,
different than that of “Mach’s principle”
Indeed the contemporary experiments are in favor
of a nonzero and even variable in time value of the
cosmological constant not less than in favor of
Einstein’s theory of general relativity
Then, the cosmological constant would still one
case in science for a correct conclusion from an
incorrect premise

15. Rejecting « Mach’s principle» ...
Anyway if Mach’s principle is rejected and
gravitational field can have some other source,
the gravitational field created by that unknown
source can be equivalently represented by the
same action of some hidden mass and energy
If that is the case, that unknown source of
gravitational field would seem as missing energy
and mass in the universe

16. “Dark” entities in physics:
“Dark matter” and “dark energy” are
experimentally absolutely confirmed nowadays
As the adjective “dark” shows, the
contemporary physical theory and the Standard
model first of all cannot even suggest any
possible source of them
However any source of gravitational field
different than mass and energy would explain
both facts above

17. The global average density
of mass and energy as well
Even more, the average density of mass and
energy in the universe globally seems to be zero
implying the cosmological constant to be zero
following Einstein’s introduction of it by Mach’s
principle
However if the cosmological constant is globally
zero (i.e. being nonzero only locally), it implies
some other source of gravitation field in turn

18. Sober’s cure
The self-contradiction of “Mach’s razor” can be
anyway removed and it can survive as
a principle if it is interpreted relatively, only to
background empirical assumptions as Sober
(1988) did
Just the change of those “background
assumptions” can explain that contradictory
history of Einstein’s “Mach’s principle”
When Mach’s principle had been introduced,
the empirical and experimental data had been
ones, then they changed and this should imply
its irrelevance to the new data

19. The cure in mathematics
Indeed mathematics does so, too: Sets of
obvious postulates ground axiomatic and
deductive method
Any theorem can be true only to some set of
axioms rather than at all
If one changes the background assumptions
whether for new experimental data or for newly
axioms, this implies corresponding changes in
the intension or the extension of the theory at
issue

20. The abstract setting of the
problem
An information approach to the problem is the
following:
One theory as a very extended notion can be defined
both by its extension, i.e. as the collection of data
(facts), and by its intension, i.e. as a certain subset of
a reference set
The two ways of definition are different in general
The quantity of information can serve as a measure
for the degree of mismatch between those two
definitions of one and the same theory

21. Theory as a very extended notion
Any theory can be considered as a relation
between
(A) principles, axioms, premises and any
statements, which are constant, and
(B) experiments, theorems, results and any fact,
which are variables
(A) is the intension of the notion, to which the
theory is supposedly equivalent
(B) is the corresponding extension
(A) and (B) turn out to be in information
equilibrium after a long enough period

22. Equating the extension and
intension
However both any new principle and
inconsistent fact (experiment) violate that
equilibrium generating unstable disturbance
converging to some new equilibrium and thus
requiring either new facts or new principles
Consequently, if one adds a new principle
(whether even of “Mach’s razor”), this one
generates disbalance and addresses implicitly
some new equilibrium supposing new facts

23. That equation
as information (entropy)
equilibrium
The coincidence of the extension and intension
of a theory implies the minimum of the function
of mutual entropy (information), i.e. the
information of intension to that of extension
That minimum is furthermore a state of
equilibrium, in which the theory would remain
arbitrarily long without an external action such
as new facts, experiments, principles or at least
interpretations

24. The self-contradictory of «Mach’s
razor»
“Mach’s razor” should remove those principles,
which are redundant to the available facts
However therefore it is a new principle, which
generates information inequilibrium if it makes
sense to be involved
That inequilibrium is unstable and converges to
some new balance of extension and intension by
adding new facts
In turn those new facts contradict to the “razor”
converging to its removing and so on just as in
the “fable about the barber”

25. «Mach’s razor» as that information
equilibrium
One can object that no theory is the set of all sets
However any theory can be effectively considered as
the set of all sets if it is not referred to any more
general theory
Just that is the case about any actual theory yet not
generalized by some new one
Consequently the self-contradictory of Russell’s
barber’s razor is quite relevant to it
In fact, any meta-principle involved in any valid and
actual theory generates the same contradictoriness
rather only than that of “Mach’s razor”

26. Conclusions:
Any actual and last theory yet not being
generalized can be considered effectively as
Russell’s set of all sets
Any meta-principle such as “Mach’s razor” is
self-contradictorily to be incorporate within its
proper principles because this supposes for its
boundaries to be known, but they are in fact
unknown
This can be demonstrated by the case of
Einstein’s “Mach’s principle” in general relativity

27. References:
Banks, Erik (2004) “The Philosophical Roots of Ernst Mach's Economy of Thought,” Synthese, 139
(1): 23–53.
Gamow, George (1970) My world line : an informal autobiography. New York : Viking Press, 1970.
Einstein, Albert (1918) “Prinzipielles zur allgemeinen Relativitätstheorie,” Annalen der Physik, 55
(4): 241-244.
Peebles, Phillip James Edwin and Ratra, Bharat., (2003) “The cosmological constant and dark
energy,”. Reviews of Modern Physics, 75 (2), 559–606.
Perlmutter, Saul. et al. (The Supernova Cosmology Project) (1999) “Measurements of Omega and
Lambda from 42 high redshift supernovae,” Astrophysical Journal, 517 (2): 565–86.
Riess, Adam et al. (Supernova Search Team) (1998) “Observational evidence from supernovae for
an accelerating universe and a cosmological constant,” Astronomical Journal, 116 (3): 1009–38.
Russell, Bertrand (1902) “Letter to Frege,” in Jean van Heijenoort (ed.), From Frege to Gödel,
Cambridge, Mass.: Harvard University Press, 1967, 124–125.
Russell, Bertrand (1986) The philosophy of logical atomism and other essays, 1914-19. London -
Boston: George Allen & Unwin.
Sober, Elliott (1988) Reconstructing the Past. Parsimony, Evolution, and Inference. MIT Press,
Cambridge (Mass), London.
The CSM Collaboration (2014) “Evidence for the direct decay of the 125 GeV Higgs boson to
fermions,” Nature Physics, 10, 557–560.
Trimble, Virginia. (1987) “Existence and nature of dark matter in the universe,” Annual Review of
Astronomy and Astrophysics, 25, 425–472.

28. Velice vám děkuji za vaši
laskavou pozornost
Těším se
na vaše dotazy a komentáře!