Abstract
In the previous chapter we concerned ourselves with examples of “inward” Eddington inferences in science, that is, with inferences to claims about entities smaller than those we are capable of observing. But, of course, Eddington inferences can also take us “outwards”, to claims about states of affairs larger than those we can observe. To refer once again to the example of the fish trap of Chap. 5, if we blindly set the holes of the trap to exactly four inches and get fish, we may infer that there are probably fish less than four inches in the sea, but we may with equal justification infer that there are also fish longer than four inches. In this chapter we examine some inferences that take us “outward” to things not observable by us because they are too big or distant. Some of these are straightforwardly Eddington inferences, while others are not. But all the inferences to be considered have the same underlying logical structure as those earlier considered: they begin by noting that the location of our observations is blindly chosen. They then note that, given that the location of our observations is blindly chosen, it would be a highly improbable fluke if some assertion S were not true. The conclusion is drawn that (probably) S is true.
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Notes
- 1.
Analysis of the motions of some objects at the periphery of the observable universe has been claimed to show those objects are moving in a way that cannot be accounted for solely in terms of the influence of other objects within the observable universe and the expansion of the universe. These (alleged) motions have been called “dark flow”. Suggestions have been made that “dark flow”, if it is exists, might be best explained by postulating a concentration of mass outside the observable universe, and even that it might be due to influence from another universe in the multiverse. However, the very existence of “dark flow” has been claimed to be dubious. See P. A. R. Ade et al. “Planck intermediate results XIII Constraints on peculiar velocities” http://arxiv.org/abs/1303.5090
- 2.
The content of this section is elementary and introductory and many readers might wish to skip it.
- 3.
We can perhaps make this clearer by assuming that our observer is on a sphere, such as the Earth, and has traced out is a very large circle: in fact a “great circle” like the equator that divides the Earth in to two equal hemispheres. Then the distance from the central point to the circumference of the circle, as determined by pacing out that distance over the surface of the Earth, will be equal to one quarter of the circumference of the Earth. Say the distance from the centre of the Earth to the surface is 1. Then the circumference of the Earth is 2π. So, one quarter of this distance will be π/2, or about 1.57 units of length. That is, if we go from the central point (say, the north pole) to the equator by moving across the surface of the earth we cover 1.57 units of length.
Let us remind ourselves that – as noted two sentences back, the circumference of the Earth is 2π. The circle we arrive at (the equator) by traveling 1.57 units in a straight line from the pole, across the surface of the Earth, therefore has a circumference of 2π.
Now, let us assume that instead of being on the surface of the Earth, we are on a flat plain. We now imagine traveling out from our point of origin a distance of 1.57 units. But we are now traveling across a flat plain, not the curved surface of the Earth. Imagine a circle on this flat plain with a radius of 1.57. The circumference of this circle will, plainly, be 2π × 1.57. So, it will be a bigger circle than the one we form if we are on the curved surface of the Earth.
This is a simple geometrical fact. But this simple geometrical fact (or rather, the analogue of it in four dimensions) has a crucial role in enabling us to tell whether or not space is curved.
- 4.
A larger circle would be obtained if the plain were “saddle-shaped”.
- 5.
This is discussed in more detail in Sect. 3, below.
- 6.
See Andrew Liddle An Introduction to Modern Cosmology (John Wiley and Sons, 2003), p.1–2. Liddle, loc cit, describes the cosmological principle as the “cornerstone” of modern cosmology.
Taken strictly and literally, (CP) would seem to be obviously false. The way things are here and now as I write are very different from the way they are in Antarctica or at the bottom of the ocean, and they are even more different in the centre of the Sun or in the frozen wastes of space. Still, in some form or another, the cosmological principle has had, and continues to have a role, particularly in using General Relativity to derive a model of the universe. The principle was used extensively by Einstein.
On the face of it, this presents us with a puzzle. Why should the principle have been used, and continue to be used, when it is prima facie obviously wrong? It will be argued that a natural explanation of this follows from the position developed here. There is an argument for the cosmological principle that closely parallels the argument for induction given in Chapter Two. This is of course a probabilistic argument and so its conclusion is defeasible. If it is defeasible, it may be falsified. And, in fact, the cosmological principle CP – as stated above – plausibly has been falsified. But although CP (on a strict interpretation) may have been falsified, the logic of the probabilistic argument in favour of some form of the cosmological principle retains its force. And so, it may yet be rational to accept some other restricted or modified version of the cosmological principle. Moreover, we find that this is exactly what scientists have done.
First let us see how an argument for the cosmological principle CP can be constructed that parallels the argument given in Chapter Two for induction.
Suppose it were the case that volumes of space other than those observed contained significantly different properties from those that have been observed. If so, an improbable event would have occurred: the volume of space in which we, fortuitously, or by blind chance, find ourselves located, would have been an “island” with one set of properties in a sea of volumes with other sets of properties. Since this is a priori unlikely, we have reason to believe this is not the case, and therefore that other volumes probably exhibit more or less the same features as our own.
We will not here repeat the various arguments given in Chapter Two defending the thesis that “Other volumes are like our own” has a higher probability than any other specific claim about the nature of the other volumes.
However, as we have already noted, CP is false. The volume of space in this room as I write is very unlike an equivalent volume at the bottom of the ocean or the centre of the Sun. The inference to CP is not only defeasible but has in fact been defeated. However, scientists have not given up the cosmological principle altogether. Other, more restricted forms of the principle have been defended. While it seems to be clearly false that each room-sized volume of space is similar to each other room-sixed volume, perhaps larger volumes do resemble each other. After all, if we had a very large collection of marbles half of which were black and half white, it would not be hugely surprising if four marbles drawn at random were all white and another four all black. But, any million marbles drawn at random would be expected to closely resemble, in proportions of black and white, any other million drawn at random. Moreover, versions of the cosmological principle applying to larger volumes of space – more specifically, of the order of 250 million light years – seem to be as yet unfalsified and continue to be defended by astronomers. And this is in accord with the idea that there is a general, a priori, probabilistic but defeasible argument for some version of the cosmological principle. Even if certain versions of the principle prove to be false, this argument retains its force and so other versions of the principle are advanced and subjected to testing.
There is, moreover, a sense in which the version of the cosmological principle just mentioned – that is, that each sufficiently large volume of space is like every other sufficiently large volume – has been found to be false. It is not true for all volumes at all times through the history of the universe. According to the currently received view, there was once a time when all the matter in the universe was compressed down in to a very small volume: much smaller than a volume 250 light years across and in fact smaller than an atom. But still the principle is retained in a suitably qualified form: all sufficiently volumes of space at any one point in time contain (more or less) the same amount of matter as any other suitably large volume that exists at that point in time.
There are three more forms of the cosmological principle that are regarded as having stood up to testing. These are:
-
(i)
The nomological form of the principle: The same laws of nature operate at all points of space and time throughout the universe.
-
(ii)
The causal-historical form of the principle: All the entities and structures in the universe are the result of similar causal-historical processes. This is distinct from, and makes a more specific claim than, the nomological version of the principle. It asserts, for example, that even though galaxies in the distant past have very different properties from those around now, the ancient galaxies and the present ones are all evolving according the same causal-historical process of development. (This version of the principle might be regarded as the cosmological analogue of geological uniformitarianism.)
-
(iii)
The isotropy form of the principle: the universe is held to exhibit broadly the same features no matter from which direction it is observed.
In summary, there is an a priori but defeasible probabilistic argument for the cosmological principle. Strong forms of the principle have been falsified. But the general, probabilistic argument for the principle remains and scientists have advanced more qualified versions of the principle, many of which remain unfalsified. All of this comports well with the view scientific inference defended in this book.
-
(i)
- 7.
In cosmology, the value of k is given by the formula: −2 U/mc 2 x 3, where U is the total energy (kinetic energy and gravitational potential energy) of a mass m with respect to some other mass M, c is the speed of light, and x is the co-moving distance between m and the (centre of) the other mass M.
The co-moving distance between two objects can be explained as follows. It is now generally accepted that the universe is expanding, and this expansion is uniform, in the sense that the rate at which it is expanding is the same at all points within the universe. So, even if there are two objects (two stars, say) that are in all other respects at rest with respect to each other, the distance between them will be increasing merely in virtue of the expansion of the universe. Suppose, hypothetically, the two stars are initially one unit of distance apart. Then, in the course of some unit of time, the universe doubles in size. Then the two stars will now be two units of distance apart. Suppose after some longer span of time, the universe triples in size: the stars will now be three units of distance apart, and so on. So, the distance between the stars will, as a matter of physical fact, have increased. But, now let us imagine some grid or system of co-ordinates in space that itself expands at exactly the same rate as the universe expands. When the universe doubles in size, the spaces between the lines in the grid will also double in size, when the universe triples in size, so will the spaces between the lines of the grid. Clearly, relative to such an expanding grid, the distance between the stars will have remained the same. The co-moving distance between any two objects A and B can be thought of as the distance between A and B as measured by a grid or system of co-ordinates that expands (or contracts) exactly as the universe itself expands or contracts.
- 8.
This is something of an oversimplification. If the space of the observable universe is flat and Euclidean, then restricted Eddington inferences lead us to postulate indefinitely many other volumes of space beyond the observable universe that are also flat and Euclidean. However, it is possible for a space to be flat and non-Euclidean. For example, it is possible for a space to be flat and torus-shaped. Such a space would, plainly, be finite. However it seems to the present author that most cosmological discussions assume that the universe is not torus-shaped.
- 9.
See Matts Roos Introduction to Cosmology (John Wiley and Sons, 2015), p.218. A “gigaparsec” is 1 billion parsecs. A “parsec” is the distance from which one “astronomical unit” – the distance from the Earth to the sun, or about 93 million miles – would subtend an angle of 1/3600th of a degree.
- 10.
It is worth noting that the case, via an Eddington inference, for the existence of such a volume would seem to be pretty strong. The limit of the observable universe is about 14 gigaparsecs away. The future visibility limit is about 19 gigaparsecs away. So, a restricted Eddington inference does not need to be very restricted to get us to volumes of space beyond the future visibility limit: it only needs to postulate volumes of space about 1.35 times more distant than those we can see and therefore know exist.
A very high value for k, that is, a very strongly curving universe, could prevent there from being anything beyond the future visibility limit. However, later in the chapter it will be noted that the available empirical evidence mostly points to k having a much lower value than that which would be necessary to stop there being anything beyond the future visibility limit.
- 11.
See A. Stodolna et al. “Hydrogen Atoms under magnification: direct observation of the nodal structure of stark states” in Physical Review Letters 110, 213001, May 2013.
- 12.
See for example http://www.scienceline.ucsb.edu/getkey.php?key=530
- 13.
See for example Eng Wah Lim Longman Effective Guide to O level Chemistry (Pearson Education, 2007), p.56.
- 14.
This account comes largely from Liddle, op cit, pp.76–78.
- 15.
It is perhaps worth noting that this sphere is not a physical structure that would, for example, look like a sphere to any external observer. Rather, it is simply the set of all points that are a certain distance D from us such that any photons reaching us from that distance D will have been produced by the process of photon decoupling. Moreover, no photons from a greater distances – and therefore earlier times – could reach us since at those earlier times photons were unable to travel any great distance. So, the surface of last scattering is for us like a horizon when we look out at the universe. And just as different persons on different points on the Earth will have different horizons, so different observers in the universe (if there are any) will have different surfaces of last scattering. Also: just as the horizon on the earth for some observer on the surface of the Earth would not seem like a circular structure to an external observer, neither would our surface of last scattering.
- 16.
See, for example, http://www.star.le.ac.uk/nrt3/Cosmo/Cosmo11.pdf esp. p.12.
- 17.
See de Bernadis, P. et al. “A Flat Universe from High Resolution Maps of the Cosmic Microwave Background Radiation.” Nature, 404, (6781), (April, 2000), pp. 955–959. “BOOMERanG” is an acronym for “Balloon Observations Of Millimetric Extragalactic Radiation and Geophysics”.
- 18.
See “Detecting the anisotropies in the CMB” in http://www.cambridge.org/au/download_file/192050/
- 19.
See “Detecting the anisotropies in the CMB” loc cit.
- 20.
See “Applications of Bayesian model averaging to the curvature and size of the universe” by M. Vardanyan, R. Trotta and J. Silk, Monthly Notices of the Royal Astronomical Society, 413, L91-L95, (2011).
- 21.
See, for example, Liddle, op cit, p.50.
- 22.
See Ade, P. A. R. et al. “Planck 2013 Results Papers” Astronomy and Astrophysics, 571, A1, arXiv: 1303.5062.
- 23.
See Ade et al., op cit.
- 24.
See M. Gell-Man “Symmetries of Baryons and Mesons” in Physical Review Letters, 125, 1067, (1962).
- 25.
See V. E. Barnes et al., “Observation of a Hyperon With Strangeness Minus Three”., Physical Review Letters, 12, (8): 204. (1964).
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Wright, J. (2018). Eddington Inferences in Science – 2: The Size and Shape of the Universe. In: An Epistemic Foundation for Scientific Realism. Synthese Library, vol 402. Springer, Cham. https://doi.org/10.1007/978-3-030-02218-1_8
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