Logical check-up of the anthropic, multiversal and fine-tuning arguments in cosmology
Jesus Mosterin (Oxford, 2013)
Models
A model is a simplified mathematical representation of a complex real system.
The logical structure of the model is a certain mathematical structure.
The pragmatic function of the model is to answer questions about the modeled real system. A model of something is a machine for answering questions about it.
A mathematical structure or model need not represent anything. There is no representation without presentation.
Models are proposals we make to reality, and which reality can accept or reject. Many of the models considered in today’s cosmology and theoretical physics are mere proposals, not yet accepted or rejected by nature.
Standard model and cloud of speculative proposals
The theoretic situation of an area of physics at a certain time is characterized by the interaction between the standard model of the discipline at that time and the cloud of speculative proposals surrounding it. The standard model consists of a series of well established, empirically supported and mutually consistent theories, together with certain formal tools and values of parameters. It accounts for many of the known facts, predicts some new ones, and is not contradicted by any known data. The standard model underlies teaching, research and technological applications. Outside of the standard model, creative and audacious scientists propose new and speculative theories lacking empirical support and technological applications, and frequently incompatible with each other. They try to develop the new untested proposals up to the point where they make new predictions confirmed by experiment. If and when that happens, they are admitted into the standard model, that has to be rearranged. The big bang model
The standard big bang model is the simplest model that is in approximate agreement with all observed phenomena.
One version is the cold dark matter model with dark energy (ΛCDM). It accounts for: the accelerating expansion of the universe the existence and spectrum of the cosmic microwave background radiation the large scale structure of galaxy clusters the distribution of hydrogen, helium, lithium and oxygen
1 8G R g R g T ab 2Theab ab logicalc 4 ab structure of the big bang model (1)
The underlying mathematical structure of the big bang model (and of general relativity) is that of a 4-dimensional differentiable manifold. The manifold is assumed to be Hausdorff, connected, paracompact and without boundary. The manifold is provided with a locally Lorentz metric g of signature 1 8G (‒+++). R g R g T ab 2 ab ab c 4 ab Einstein's equivalence principle. It implies that the effects of gravity must be equivalent to the effects of living in a curved spacetime. Einstein's field equations, describing the dependency of spacetime curvature on mass-energy distribution. Time-orientability and stable causality condition, which imply that it is possible to introduce a cosmic time, defined at every point (or event) of spacetime in such a way that this time is strictly increasing along any time-like geodesic.
The logical structure of the big bang model (2)
Robertson-Walker metric, corresponding to a perfectly homogeneous and isotropic universe. Friedmann solutions (with zero pressure and cosmological constant). Hawking-Penrose singularity theorems, which imply the big bang proper. Parameters to be filled by hand, like the Hubble constant and the average density ρ. Thermal history of the universe from 10−10 s after the big bang till now, including the passage from a radiation- to a matter- dominated age. Chemical history of the universe, including the account of primordial nucleosynthesis. Cold dark matter. The cosmological constant Λ for dark energy.
Empirical support for the big bang model
Rich empirical support for General Relativity: . Discovery of gravitational lenses. . Exact computation of the minute changes (due to the emission of gravitational waves) in the period of the binary pulsar discovered by Hulse and Taylor in 1974.
The big bang model deals adequately with 3 large empirical facts:
. the expansion of the universe
. the abundance of chemical elements
. the cosmic microwave background radiation Sensible questions with no answer
What is dark matter?
What is dark energy?
Is there extraterrestrial life?
Are there extraterrestrial civilizations?
When will I die?
Speculation, from mild to wild
Inflationary scenario
Eternal inflation
Superstring theory
An infinity of unconnected universes
The Anthropic Principle
Empirical and speculative science
(Up to a point) reliable empirical science:
Astronomy General theory of relativity Standard big bang model in cosmology Quantum field theory Standard model of particle physics
Unreliable speculative science:
Inflationary cosmology Superstring (and M) theory The multiverse
The radio sky, tuned at 408 MHz The sky in microwaves
Gamma ray all sky map
The Milky Way in visible light Andromeda (M31) in visible light Andrei Linde Edward Witten
Karl Popper
Criterio falsacionista de demarcación:. Solamente los enunciados o teorías falsables (refutables) son científicos.
John Earman y J. Mosterín Lee Smolin
Large Hadron Collider (CERN) A SIMPLE idea underpins science: “trust, but verify”. Results should always be subject to challenge from experiment. That simple but powerful idea has generated a vast body of knowledge. Since its birth in the 17th century, modern science has changed the world beyond recognition, and overwhelmingly for the better.
But success can breed complacency. Modern scientists are doing too much trusting and not enough verifying—to the detriment of the whole of science, and of humanity. Nobel Prizes in Physics (1)
14 of the 15 Physics Nobel Prize winners in cosmology have been experimentalists, with the only exception of Chandrasekhar.
2011: Saul Perlmutter, Brian P. Schmidt and Adam G. Riess, "for the discovery of the accelerating expansion of the Universe through observations of distant supernovae".
2006: John C. Mather and George F. Smoot, "for their discovery of the blackbody form and anisotropy of the cosmic microwave background radiation".
2002: Raymond Davis Jr. and Masatoshi Koshiba, "for pioneering contributions to astrophysics, in particular for the detection of cosmic neutrinos“. And Riccardo Giacconi, "for pioneering contributions to astrophysics, which have led to the discovery of cosmic X-ray sources".
Nobel Prizes in Physics (2)
1993: Russell A. Hulse and Joseph H. Taylor Jr., "for the discovery of a new type of pulsar, a discovery that has opened up new possibilities for the study of gravitation".
1983: Subramanyan Chandrasekhar, "for his theoretical studies of the physical processes of importance to the structure and evolution of the stars".
1978: Arno Allan Penzias and Robert Woodrow Wilson, "for their discovery of cosmic microwave background radiation".
1974: Sir Martin Ryle and Antony Hewish, "for their pioneering research in radio astrophysics: Ryle for his observations and inventions, in particular of the aperture synthesis technique, and Hewish for his decisive role in the discovery of pulsars".
Robert Wilson Allan Penzias The Cosmic Microwave Background temperature fluctuations from the 7-year Wilkinson Microwave Anisotropy Probe data over the full sky. The average temperature is 2.725 Kelvin. Saul Perlmutter Classical Logic Contradiction
In classical logic, a contradiction implies anything. The semantic definition of the truth of a negation is: ¬φ is true (in interpretation ℑ) if and only if φ is not true (in interpretation ℑ). It follows that under no interpretation can both φ and ¬φ be true. No interpretation can make (φ ∧ ¬φ) true. It follows that for any formula ψ and any interpretation ℑ: if ℑ makes (φ ∧ ¬φ) true, then it also makes ψ true. So, for any formula ψ: (φ ∧ ¬φ) ⊨ ψ
Syntactically, 1) φ ∧ ¬φ 2) φ 1) 3) ¬φ 1) 4) φ ∨ ψ 2) ψ 4) and 3) So, for any formula ψ: (φ ∧ ¬φ) ⊢ ψ
Inconsistent theories
A theory is a set of sentences closed under the relation of consequence.
Let be a set of sentences of language ℒ: ℒ. is a theory iff for every sentence ℒ: if ⊨ , then iff = { ℒ : ⊨ }
The sentences of a theory are its theorems.
A theory T is inconsistent iff T is not consistent iff for some φ ℒ, φ T and ¬φ T iff T is identical to its language: T = ℒ(T)
Two theories T and are incompatible iff TΣ is an inconsistent theory.
The bane of inconsistency
An inconsistent formal theory is identical to its language. It affirms everything and it denies everything, and so it is utterly useless.
To be inconsistent is the worst bane that can afflict a theory, much worse than being false. Small wonder that scientists tend to drop a theory as soon as they discover contradictions in it.
The notion of consistency is a regulatory idea (in the Kantian sense) in the progress of science.
The discovery of a contradiction always produces a deep crisis and pushes us to make strenuous efforts to find or invent a new and more satisfactory theory.
Consistency and progress in science
It was the perceived inconsistency between Newtonian mechanics and Maxwell’s electromagnetism that led Einstein to the creation of the special theory of relativity.
It was the contradiction between Newtonian gravity and special relativity that led Einstein (and Hilbert) to search for a new theory of gravity: the general theory of relativity.
Schrödinger equation of quantum mechanics was not Lorentz invariant. It was the perceived inconsistency between quantum mechanics and the special theory of relativity that led Dirac and others to the invention of quantum field theory.
Primordial nucleosynthesis
William Fowler and Fred From helium to carbon Hoyle (1967) The universe
Etymology
Greek: το ὃλον (the whole)
Latin: universum (poured into one) uni-versum, from unum and vertere (to pour)
¿How to represent the whole from the inside?
The “multiverse”
Max Tegmark:
. “The key question is not whether the multiverse exists but rather how many levels it has.”
. “Why was only one of the many mathematical structures singled out to describe the universe?”
. “As a way out of this conundrum, I have suggested that complete mathematical symmetry holds; that all mathematical structures exist physically as well. Every mathematical structure corresponds to a parallel universe. […] This hypothesis can be viewed as a form of radical Platonism, asserting that the mathematical structures in Plato’s realm of ideas … exist in a physical sense.”
Infinity does not imply that any arrangement is present
Consider an infinite set of binary sequences sn with the i-th member xi = 1 if i n,
and xi = 0 if i = n:
s1 = 0111111111111111111...
s2 = 1011111111111111111...
s3 = 1101111111111111111...
s4 = 1110111111111111111...
s5 = 1111011111111111111... And so on. As n ranges over all the natural numbers, we get an infinity of different binary sequences that are almost everywhere = 1, but differ in the place where they are = 0. This set of binary sequences is infinite, but most binary sequences you can think of (for example, any containing two or more 0's, such as 1010101010...) are not in it. And no sequence is repeated. Infinity does not imply that any arrangement is present
Notion that an infinity of objects characterized by certain numbers or properties implies the existence among them of objects with any combination of those numbers or characteristics. This suggestion is mistaken. An infinity does not imply that any arrangement is present or repeated.
Counterexamples. The infinite set of the even numbers does not contain any of the odd numbers. The infinite set of the numbers greater than a trillion does not contain any of the numbers up to one trillion. In general, all infinite sets contain proper infinite subsets. This property was famously used by Dedekind to define infinity. The same happens with uncountable domains, like the n-dimensional Euclidean spaces. Any interval of the real line is an infinite set of real numbers, but does not contain all real numbers; most of them remain outside. Any straight line in the plane is an infinite set of points of the plane, but does not include all points of the plane.
ALMA