Infinity and Eternity of the Universe

A change of speculative opinions does not imply an increase of the data upon which those opinions rest, but a change of the habits of thought and mind which they reflect... Reasoning which in one age would make no impression whatever, in the next age is received with enthusiastic applause.

                     -- W. E. H. Lecky,  History of the rise and influence of the spirit of rationalism in Europe (1879)

The first step towards modern cosmology

The idea of the universe held five hundred years ago, at the end of the middle ages, was essentially the one described by Aristotle in the 4th century BC, retrofitted for consistency with Christian teaching by St. Thomas Aquinas.  In almost every respect, this was quite different from our modern scientific view, which almost had to be established from scratch. That this was so is something of a historical accident: two thousand years ago, among the competing philosophies of the Roman Empire were ideas very much closer to our present ones. Ironically, Aristotle prevailed precisely because he was so much more scientific than the alternatives: his system was built on close observation of nature and generally gave a better account of the facts. But his underlying assumptions about the way the world worked led him in the wrong direction.

The first and most famous break with Aristotle was the revolution initiated by Copernicus, that placed the Sun and not the Earth in the centre of the universe. The Copernican revolution has been worked over intensively by historians of science, but from the perspective of modern cosmology it was a side-show: neither the Earth nor the Sun remain as significant features in cosmology, which now works on vastly larger scales. The true origin of modern cosmology occurred in parallel with the Copernican revolution, but started and finished a little later: the replacement of Aristotle's idea of a finite material universe, like a closed shell, with the idea of infinite space. [1]

What is particularly striking about this change is that it did not occur in response to any new discoveries (at least, not directly); really, it was not based on facts at all, despite its crucial role as a starting point for (allegedly empirical) modern science.  It was essentially a radical change of philosophical fashion, so that what seemed self-evident to Aquinas eventually came to seem absurd, and vice versa.  The clearest example of this is the way our place in the Universe changed from obviously being at the centre to obviously being at no particularly special place. But there were deeper and more fundamental changes, particularly in the concepts of space and time, and God.

Aristotle's finite universe

The question of whether the Universe is finite or infinite was a traditional one in natural philosophy, and in the ancient world was one of the main bones of contention between Aristotelians and the "atomist" philosophies of Democritus and Epicurus. As a result, the Aristotelians had sophisticated replies to the obvious objections to a finite universe. For instance, the atomists asked, what would happen if you made your way to the edge of the universe and tried to throw a spear outward? The Aristotelian answer has points in common with  the explanation in General Relativity of why light can't escape a black hole. They said that in the outer universe, the region above the moon, the natural motion of all things is circular around the centre of the universe (the Earth). Nearer to the Earth, the motions are actually quite complicated, being composed of several circular motions combined (this was the famous theory of epicyclic motion of the planets), but as one approaches the outer limit of the Universe, the "sphere of fixed stars", the motions become more and more perfectly circular: thus the fixed stars themselves seem to describe almost perfect circles every day around the Earth (of course nowadays we see that this is actually caused by the daily rotation of the Earth on its axis). Thus as the spear thrower ascended higher and higher, his motion becomes more and more circular, so that eventually at the limit the only possible motion for the spear is to circle around with the stars and not to move any further outwards. [2]

Aristotle also insisted that there was no contradiction in a finite volume of space. To describe this anachronistically, in his universe only the set of points within some finite distance of the centre of the Earth existed; we have to agree that there is nothing in principle contradictory in this; his idea of space is just not the same as the modern idea of a 3-D continuum. In fact Aristotle went further and denied the possibility of any space not occupied by matter ("nature abhors a vacuum"), again in marked contrast to the atomists, for whom the world consisted entirely of small material particles moving in the void. From Aristotle's point of view a vacuum is literally nothing and therefore does not exist. Thus a universe containing a finite amount of matter is necessarily finite in size.

Whereas Aristotle's universe was finite in space, he taught that it was eternal in time. One of his main motives was to avoid the need for a Creation by the gods, since traditional Greek myths of the Creation seemed ridiculous to him. This posed a major problem for Aquinas since the Creation is one of the basics of Christian belief.  Aquinas' solution was twofold. While accepting that the world had an origin in time as described in the Bible, he argued most subtly that Aristotle was not in error in principle, in that the Christian God could, if he had chosen, have created an eternal universe of the kind described by Aristotle.

Downfall of the Crystal Spheres

In one, perhaps crucial, way Aristotle's universe turned out to be vulnerable to observation. The space between the edge of the Earth's atmosphere and the edge of the universe was believed to be filled by the crystal spheres which carried the planets in their complex motions. The stars, as mentioned above, are carried by one of the outermost of these (not quite the last, in the most detailed models). One consequence of the Copernican revolution was that the existence of the crystal spheres was disproved,  making the concept of a spatially finite universe much harder to grasp imaginatively.

Copernicus himself did not of course reject the crystal spheres; his great work was entitled  On the revolutions of the celestial spheres, and by this he meant the crystal spheres, not the planets themselves. All Copernicus did was to re-arrange the spheres to make what he thought of as a more elegant system; but this had dire consequences for the sphere of fixed stars, and hence for the idea of an outer limit to the universe.

In the first place, as the Earth moves around in its orbit, it in turn approaches and recedes from any given star.  This would lead to significant parallax, that is, change in the angular separations of the stars, unless the radius of the sphere of stars was vastly bigger than the radius of Earth's orbit.[3]  Thus Copernicus' universe had to be much bigger than Aristotle's, and in particular there was a huge empty space between the last planetary sphere, carrying Saturn, and the fixed stars.  By giving up the notion that the spheres were nestled tightly one inside the other, Copernicus abandoned one of the features of Aristotle's system that made it seem a physical picture, rather than just a mathematical description.

In the second place, and even more importantly, once the Earth was seen as rotating on its axis, the fixed stars appeared truly fixed, not only relative to each other, but absolutely.  In fact, in one of his more physical arguments, Copernicus said that it was irrational to imagine the huge sphere of fixed stars rotating and the relatively tiny Earth stationary, when the same appearances would result if the motions were the other way round. [4]  But one of the main reasons for imagining that there was a sphere of stars at all was that it provided a straightforward explanation for the common motion of all the stars: with that motion gone the sphere itself was not needed.  Copernicus knew this and commented that there was no observational difference between an infinite universe and one with the stars in a distant spherical shell; he claimed that the question was therefore not an empirical one, but in fact he clearly believed in his explicitly finite universe.

These are not themselves observational results of course. Copernicus gave us a new mathematical model, not new observational facts, and his model did not fit the data much better than the old one. But later, when evidence began to accumulate in favour of Copernicus,  it made the finite universe seem less plausible for the reasons just given.

A generation after Copernicus died, the leading astronomer in the world was the Danish nobleman Tycho Brahe. [5]  Tycho remained convinced by Aristotelian physics (indeed, there was no other available at the time), and so could not accept that the Earth moved.  In  the year of the Spanish Armada, 1588, Tycho published his own model for the solar system, in which the stationary Earth was orbited by the Moon and Sun, but all the other planets orbited the Sun.  To a modern eye, Tycho's system might seem effectively identical to that of Copernicus: certainly the relative motion of all the planets is identical.  But there is a crucial difference: in Copernicus' system orbits of planets do not intersect, and so it is consistent with solid crystal spheres. But in Tycho's system the orbit of Mars around the Sun crosses the orbit of the Sun around the Earth, which could not work if the planets were carried by real spheres. Tycho wrote:

"Now it is quite clear to me that there are no solid spheres in the heavens, and those that have been devised by authors to save the appearances, exist only in their imagination, for the purpose of permitting the mind to conceive the motion which the heavenly bodies trace in their courses."
Of course Tycho's model of the solar system  was only evidence against the spheres if you believed  it (in fact it did become quite popular for a while). But  Tycho had stronger evidence against the spheres: in 1577 he had made detailed observations of a great comet, and shown that it was not only beyond the Moon but moved unhindered through the crystal spheres, which therefore did not exist.  Earlier still, his observations of the new star of of 1572 (now called Tycho's supernova) showed that it moved only with the fixed stars, and hence was part of the same system: thus the fixed stars were not unchangeable, as Aristotle had taught.

With the planetary spheres gone, a solid sphere of fixed stars now made no sense at all. But Tycho  still believed that the stars were all more or less at a fixed distance, even if not attached to a physical sphere.  He had a good reason: his careful timing of stars passing his sight marks had shown that nearly all stars have roughly the same angular size; about two minutes of arc (or a fifteenth of the size of the Moon).  This only made sense if the stars were similar sized objects all at roughly a common distance.

Although  the professional astronomers would not abandon the sphere of  stars, not all their readers were so conservative.  In 1576 one Thomas Digges issued a free translation of the first part of Copernicus' book,  which describes the heliocentric system; but in Digges' version the stars are spread out to infinity.  But Digges' cosmology is also far from ours; he followed traditional Christian belief in placing Heaven beyond the material universe, and to Digges our view of the stars is nothing less than a direct view of infinite Heaven.  Digges' pamphlet must surely have been seen by the heretic monk Giordano Bruno, who visited England  a few years later, in 1584,  and made the acquaintance of many in the intellectual circle surrounding the court of Elizabeth I.  It was Bruno who famously first claimed a truly infinite material universe, in which the stars are other suns, orbited by other planets; this claim was made in several works that he wrote in England at this time.

After Tycho died in 1601, the observational case against Bruno and his followers was made by Tycho's assistant and successor Johannes Kepler. Kepler relied on Tycho's argument from  the near-constant apparent size of the stars: if the stars were spread randomly through a large volume, it would mean that the more distant the star, the larger its physical size, which contradicted the whole idea of the Sun as a typical star at no special place in an infinite universe.

This argument had to be revised after Galileo Galilei turned his telescope to the skies in 1610. He found that, unlike the Moon and planets, the stars seemed  brighter but no larger than before, indicating that their apparent size to the naked eye was some kind of optical illusion.  He also found many more stars than could be seen with the naked eye. Kepler remained true to Tycho, merely revising down slightly the size of the stars. But as telescopes improved over the following decades, and the apparent sizes of stars continued to decrease, astronomers realized that the size is indeed entirely an illusion; and stars are effectively just points of light (or in other words too far away for any true size to be measurable).  In modern terms the size measured by Tycho was just the resolution of the human eye.

This is also the time when observational evidence began to decisively favour the Copernican arrangement for the Solar system. Galileo's discovery of Jupiter's satellites showed that Copernicus was not "cheating" by making the Moon orbit the Earth when the other planets orbit the Sun. His observations of the phases of Venus clearly showed that Venus, at least, orbited the Sun.  Finally, in 1627 Kepler completed his life's work by using his famous laws of planetary motion and Tycho's data to produce tables to predict planetary positions.  He died in 1630, a year before the accuracy of his tables was first demonstrated when Pierre Gassendi observed a predicted transit of Mercury across the Sun's disk.  This gave very strong support for Kepler's laws, which only make sense for a Sun-centred system.

This was the end of empirical input into the debate. The large number of faint stars was broadly consistent with the infinite universe of Bruno. But equally, it was consistent with the sort of shell of stars imagined by Tycho and Kepler, which would have still fitted within a finite Aristotelian universe.  Only actual measurements of the distances to stars could resolve this dilemma experimentally, but by the time that was done, in the late nineteenth century, scientists had long made up their minds to side with Bruno.

God, Space and Vacuum

Throughout this period, and indeed until late in the eighteenth century, it was obvious to all involved that any discussion of the Universe necessarily involved a discussion of God. This was not because of the political control exercised by the Church (or rather churches, as we are dealing with the Reformation when the Protestant churches emerged and eventually dominated the countries of northern Europe), but because Christian belief was very deep in Europe; the bloody religious conflicts of the time are testament to that.  Although some were accused of atheism, all involved in this debate seem to have had deep religious feelings, and indeed it was probably those feelings which drew them towards the potentially dangerous task of discussing the nature of the Universe.

A major problem with the idea of infinite space was that infinity was held to be a defining characteristic of God.  Indeed for Bruno everything in the universe contained an element of the divine. But his form of pantheism does not claim that his infinite universe was identical with God: the incorporeal infinity of God is different in kind to, and more perfect than, the infinity of the world.  It does not make his writings easier to understand that he subscribes to the doctrine of the medieval philosopher Nicholas of Cusa, who claimed that the infinitely large and the infinitely small are identical, like two ends of a circle.  Bruno's other religious views were even more radical: he got into trouble successively with the Catholics, the Calvinists and the Lutherans,  and eventually was burned at the stake by the inquisition at Rome in 1600, after refusing to recant.  Bruno is often seen as a martyr for science,  but his downfall probably had more to do with his scathing attacks on the Catholic hierarchy (appreciated and encouraged by the Protestants) combined with a tendency to feuding and heretical views [6] which left him with no sanctuary in either Catholic or Protestant Europe.  Nevertheless, Bruno's fate had a large impact on subsequent discussion: feuding philosophers were quick to accuse each other of heresy, and careful to insist that their views remained orthodox.

One factor which probably made the idea of actual infinity seem more reasonable was the connection forged by René Descartes between algebra and geometry, published in 1637.  Through the use of co-ordinates, Cartesian geometry focussed attention on the infinite space in which geometry operated, rather than on geometric figures such as triangles and circles.  The effect of applying Cartesian geometry to the Universe was that it became even harder to imagine that the Universe might have some sort of spatial limit.

The importance of Cartesian geometry was amplified because it came as part of a package, a comprehensive system of philosophy which Descartes explicitly intended to replace Aristotelianism in its entirety.  Rather astonishingly, he largely succeeded, partly because he made himself the chief spokesman for the new "mechanical philosophy" which in various forms was already gaining ground in Europe, and perhaps partly because of the prestige of France, which after the destruction of the Holy Roman Empire in the Thirty Years War emerged as the most powerful state in Europe.  At any rate, not only did Descartes deeply influence all the major philosophers and scientists of his own and the following generation,  but by the end of the seventeenth century Cartesianism had become the basis for courses in natural philosophy in universities throughout Europe.

It was in Descartes' time that Blaise Pascal, in his Pensées, imagined a "man without god" who finds himself

"engulfed in the infinite immensity of spaces of which I am ignorant and which know me not...The eternal silence of these infinite spaces frightens me."
The universe alluded to in the Pensées at first seems the traditional Aristotelian one, but this turns out to be only an atom in an infinitely larger universe; moreover Pascal imagines within the particles that make up the everyday world a hierarchy of universes extending down to nothingness.  The Pensées is intended to put the fear of  God into the reader, rather than to outline a consistent cosmology, but Pascal was writing to convince his worldly friends and so his constant harping on the infinity of the Universe surely reflected contemporary ideas.

In 1647, before he turned to religion,  Pascal had conducted experiments in barometry. By showing that  pressure decreased with height, his work supported Torricelli's new theory that a vacuum lay beyond Earth's atmosphere.  Descartes was not impressed; after a visit to Pascal he commented that "he has too much vacuum in his head". Like Aristotle,  Descartes denied the possibility of a vacuum, and actually identified matter and space (which, as Newton later pointed out, made it impossible to explain differences in density).  Although Descartes believed that the stars were other suns with their own solar systems, he claimed that the Universe was only indefinitely extended, rather than truly infinite.  This distinction was intended to avoid the problem Bruno had run into,  especially because Descartes wanted to make an absolute distinction between the material and spiritual universe (one aspect of this was the mind-body dualism for which he was so much criticized). But many, if not most, of Descartes readers regarded the distinction between indefinite and infinite as a false one, included only to satisfy the Church.

An alternative approach was suggested by Henry More, leader of a group of Platonist philosophers at Cambridge.  More had a way out of the argument for the non-existence of the vacuum: as a Platonist, material things were to him far from the only things in the world. In addition, he believed in "spirits", which included such items as ghosts, magnetic fields (as we would now say), and, of course, God.  Clearly, spirits could interpenetrate ordinary matter, and equally clearly matter could be removed to leave a vacuum without removing all the spirits.  In particular,  God was always present.  From More's point of view, the infinity of God, which all agreed, practically guaranteed the infinity of space (and time). Side-stepping Bruno's pantheism, he identified space and time as aspects of, but not identical with, God. This viewpoint, in truth, rather trivializes God, as Descartes pointed out in an exchange of letters with More.  To theologians at least since St. Augustine in the 5th Century, God was transcendent, that is, beyond space and time: not merely lasting forever and existing everywhere, but perceiving the whole history of creation as a unity.

Nevertheless, More's interpretation of space and time proved extremely fruitful for science: it was taken over by Isaac Newton, and underlies the famous "absolute, true, and mathematical" time and space of the Principia, near the end of which we read:

He is eternal and infinite, omnipotent and omniscient; that is, his duration reaches from eternity to eternity; his presence from infinity to infinity; he governs all things, and knows all things that are or can be done. He is not eternity or infinity, but eternal and infinite; he is not duration or space, but he endures and is present. He endures for ever, and is every where present; and by existing always and every where, he constitutes duration and space....He is omnipresent not virtually only, but also substantially; for virtue cannot subsist without substance. In him are all things contained and moved; yet neither affects the other: God suffers nothing from the motion of bodies; bodies find no resistance from the omnipresence of God.
Although this passage clearly shows that Newton, following More, believed space and time infinite and eternal, it does not say that he thought them populated by an infinite number of stars. Certainly, More in later life thought that the material content of the universe was finite.  In 1692, Newton himself corresponded with Richard Bentley about the possibility of an infinite system of matter.  Newton does not clearly favour one or the other, pointing out that in both cases God must intervene from time to time to prevent the system's collapse. Of course Newton, like all Christian philosophers, believed the material world had a beginning in time, as described in Genesis.

The concept of God as literally omnipresent got Newton over a serious difficulty with his theory of gravity.  As as good mechanical philosopher,  he was not prepared to contemplate action at a distance; as he told Bentley:

It is inconceivable that inanimate brute matter should, without mediation of something else which is not material, operate upon and affect other matter without mutual contact.
So Newton had tried to find a contact-force explanation for gravity, presumably involving emanations of invisible particles as favoured by Descartes; but he convinced himself this could never work.  More's ideas allowed Newton to believe in a local but immaterial "something": in fact God Himself.  Perhaps wisely, Newton only hinted at this in his public writings, but he emphasised that the Principia provides only a mathematical description of gravity.

After the publication of the Principia Newton got involved in a long-running controversy with Gottfried Wilhelm Leibniz over the invention of calculus.  As the arguments grew ever more vitriolic, Leibniz broadened his attack to the whole of Newton's system, leading to a famous exchange of letters with Newton's follower, Samuel Clarke.  Several aspects of the debate are relevant here.

Firstly, although both sides were fully convinced of the infinity of space, they had different reasons; for Leibniz this was intimately tied up with his belief in an infinite material universe. Leibniz' most famous argument for this echoed Bruno: what possible reason could induce God to make only a finite universe when he might have made an infinite one?  In the twentieth century, this argument was labelled the Principle of Plenitude.  In effect, Leibniz said that to claim a finite universe is to belittle the Creator.  At the time this argument had considerable force, and later in the eighteenth century it became the standard wisdom.  Leibniz, following Aristotle and Descartes, denied the vacuum, and following Aquinas and Descartes denied the identification of space with God; thus to him the infinity of space was just a re-statement of the infinity of matter.  Clarke of course followed Newton's line in clearly distinguishing space and matter, and argued that God could have made a finite material world if he chose.

Secondly, Leibniz held that space and motion were only relative, whereas Newton claimed the existence of absolute space and therefore absolute motion relative to it.  Here modern physicists often feel that Leibniz was correct, but in fact neither side represented the modern view of a set of equivalent inertial reference frames.  To Newton, absolute space, guaranteed by the omnipresence of God, was needed to define absolute acceleration; [7] while to Leibniz, space and motion were not relative to some inertial frame, as in modern physics, but relative to matter and ultimately to the totality of matter in the universe: essentially his view is what is now known as Mach's principle.  On this point the authoritative statements in the Principia, coupled with the success of Newtonian physics, made Newton's view the generally received opinion, although it was always criticized by a few.

Thirdly, Leibniz held that Newton  re-introduced "occult" forces into physics, which the mechanical philosophy was dedicated to explaining away. For God to cause gravity directly seemed to make it a continual miracle.  Clarke's reply was that miracles were by definition something out of the ordinary, so gravity was no miracle; after all, it was only by God's will that the world continued to exist at all, and the existence of the world was not normally considered a miracle.  Such a position was effectively a complete rejection of the mechanical philosophy, since it allowed any regular behaviour whatsoever to be attributed directly to God without the need to imagine mechanical causes. Newton had already signalled his disenchantment with Descartes' approach in his famous claim  "I feign no hypotheses".  By "feign" he meant to fancifully make up, in a way not justified by experiment or observation; and by "hypothesis" he meant the kind of mechanical explanation, based on the properties of unobservably small material particles, which fill the pages of Descartes' works on physics.  Both Leibniz and Newton's views on this point quickly gave way to the idea that there was nothing unnatural about action at a distance, provided it obeyed mathematical laws.

Finally, Leibniz latched on to the imperfections in Newton's universe, which require God's active intervention to prevent the collapse of the system. [8] Leibniz ridicules this:

According to their Doctrine, God Almighty wants to wind up his Watch from Time to Time: Otherwise it would cease to move. He had not, it seems, sufficient Foresight to make it a perpetual Motion.  Nay, the Machine of God's making is so imperfect, according to these Gentlemen, that he is obliged to clean it now and then...and even to mend it...
Now to Newton, this was not a defect in his system, but essential, because it guaranteed that God was actively overseeing his creation.  Clarke in his turn pointed out that Leibniz' idea of a perfect creation left God with nothing to do; it the Notion of Materialism and Fate, and tends (under pretence of making God a Supra-Mundane Intelligence) to exclude Providence and God's Government in reality out of the World.
What neither Newton nor Leibniz foresaw was that, as Newton's laws were applied in more and more detail, to better and better astronomical data,  the flaws in the system gradually disappeared.  As the eighteenth century progressed, faith in conventional Christianity waned.  The mathematisation of Nature was one of the main factors behind this, but also important was increasing sophistication in Biblical scholarship, which undermine its absolute authority, a skepticism caused by the large number of disagreeing protestant sects, and increased contacts with the rest of the world, forcefully bringing home to Europeans that Christianity itself was just one of many religions, and not obviously morally superior to the others.  These factors underlay the increasing fashionablility of the Enlightenment idea of a very abstract, detached deity (deism), and even of atheism.  With the obstacle of Genesis out of the way, the same Principle of Plenitude that implied spatial infinity also implied that the material universe was eternal. Further, to the atheists, as to Aristotle, an eternal universe seemed necessary to avoid a divine Creation for the universe.  At the end of the eighteenth century, Pierre-Simon Laplace was widely agreed to have brought Newtonian astronomy to perfection.  When he presented his masterpiece to Napoleon,  Napoleon said: "M. Laplace, they tell me you have written this large book on the system of the universe, and have never even mentioned its Creator".  "Sire," replied Laplace, "I had no need of that hypothesis".

Occam's Razor and the Edge of the Universe

As far as I know, no-one involved in the debate on the infinity of the universe explicitly invoked Occam's razor, the principle of reasoning that "plurality should not be posited without necessity".  Perhaps this was only because nothing was less fashionable to early modern philosophers than the work of late medieval scholastics.  Nevertheless, the debate sheds interesting light on the way this principle can be applied.

On the face of it, nothing could transgress Occam's razor more than the postulate of an infinite and eternal space, full of stars and planets.  But there is an alternate point of view which makes Occam strongly support the infinite universe.  As we have seen, a standard argument against a finite universe was that the properties of the edge of the universe would be absurd, if not contradictory.  Now in fact there are numerous ways of incorporating an edge to the universe without contradictions. When we perform simulations of some physical situation on a computer, we of course only include a finite region of space (computers only have finite memory!), and so we have a lot of experience in setting up suitable boundary conditions.  But anyone who has written a program to do this sort of thing knows that most of the work is indeed in setting up the boundaries correctly; in other words, incorporating a boundary involves a whole new set of "physical laws" which apply only at the boundary.  Now if we apply Occam's razor to our theories, rather than to the world that they produce, it tells us that we should not include all this extra "physics" unless absolutely necessary.  In this sense, it is perhaps correct to claim that a boundary to the universe is an absurd concept.

This is not an isolated example: one generally finds that simplifying our theories leads to a more complicated picture of the world, and vice-versa. For instance, Copernicus simplified the model of the solar system by replacing a whole set of epicycles with one motion, that of the Earth in its orbit. But this immensely complicated the physical picture of the universe, because by making the Earth a planet Copernicus promoted the other planets from being mere lights in the sky to being worlds in themselves.  Similarly Maxwell's successful unification of electric and magnetic forces greatly simplifies the equations, but predicted  the existence of wholely unexpected phenomena in nature, in particular the spectrum of electromagnetic radiation. The ideology of modern physics is to continue this trend to simpler and simpler theories, and at each step the world becomes more complicated: each new symmetry in the theory predicts new particles.  Historically, this has been a staggeringly successful strategy, giving strong support to the notion that Occam's razor should always be applied to the theory and not to the world that it produces.

Paradoxes of infinite space and time

It was not instantly realised, although it could have been, that the new infinite and eternal universe was subject to two serious inconsistencies, which did not affect the old finite version.

The first of these is the paradox of gravity.  In an infinite universe filled with stars, the gravitational force becomes undefined because there is an infinite force in each direction.  The net force on the Earth, for instance, is then not dominated by the Sun but by any asymmetry in the distribution of distant stars. In his correspondence with Newton, Richard Bentley had put his finger on the problem, although Newton had fobbed him off by claiming that the infinite forces from the distant stars exactly cancelled; essentially this was an appeal to his theorem that a uniform shell of matter had no gravitational force on objects within it.  In fact this theorem cannot be extended to an infinite shell, and in any case it was obvious that the stars are not uniform in space.  Newton's manuscripts show that he was worried enough by Bentley's question to set to work to compare the numbers of stars of different magnitudes with a model of stars uniformly distributed in space (this came to nothing because there was no quantitative definition of magnitudes). [9]  Finding no obvious solution, Newton chose not to publicise the problem (perhaps it underlay his unwillingness to commit himself to an infinite universe), and it was only re-discovered by Hugo Seeliger in 1895.

In Newton's work on the distribution of stars, the second paradox was staring him in the face, but he didn't see it.  This is the paradox of the dark night sky, often called Olbers' paradox although Olbers has no particular rights to it. A universe with a quasi-uniform distribution of stars should have either an infinite brightness, or at least a brightness equivalent to the surface of a star, which follows from the inverse-square law for light, Euclidean geometry, and the assumption that the universe is infinitely old. It has been claimed that Kepler knew of this problem, and perhaps he should have done, as he was the discoverer of the inverse-square law and also argued strongly against an infinite star system.  But in fact Kepler did not use this particular argument; he always focussed on the size rather than the brightness of the stars.  The first published account of the problem was in a pair of presentations to the Royal Society made by Edmond Halley in 1721: "Another argument I have heard urged, that if the number of Fixt Stars were more than finite, the whole superfices of their apparent Sphere would be luminous." Halley dismissed this through spurious arguments; it also now seems likely that he was not aware of the full impact of the problem, and by "luminous" he only meant the faint luminosity of the Milky Way. Newton, who was in the chair at the time, did not object; but Newton, then nearly eighty, had been known to snooze through meetings of the Society. [10]  The paradox was later discussed in its modern form by Philippe Loys de Chéseaux in 1744, who concluded that either the system of stars must be spatially finite, or that there was some obscuring material in space so that light from distant stars diminished faster than the inverse square.  Wilhelm Olbers, who had read  de Chéseaux's book around 1790, published essentially the same arguments in 1823, without mentioning de Chéseaux. This would be a clear case of plagiarism, except that Olbers had no obvious motive and a reputation for honesty; more likely he had long forgotten where the idea came from.  In fact the idea of  obscuring material does not solve the problem, as shown by John Herschel in 1848. The first satisfactory solution (that the universe had existed for only a finite time) had to wait for the work of  Johann Mädler in 1861.


The idea of an infinite universe reached by the end of the eighteenth century is familiar to all, since it is essentially the one we learn in school.  It was the starting-point from  which modern cosmology set out, but modern cosmology has undermined many of its certainties.  The argument between action at a distance and local forces swung back the other way with the introduction of field theories by Faraday, Maxwell and ultimately Einstein (with his general relativity); but the same battle is being fought again in relation to the notorious measurement problem in quantum mechanics.  The paradoxes of gravity and light that Newton swept under the carpet are solved in the new universe of Einstein.  Quantum mechanics has re-introduced a host of problems concerning the vacuum: very clearly the vacuum of modern physics is not nothing, but its nature remains hidden pending a theory of everything.

We have learned again that the Universe may be finite in space and time, without contradictions.  In all likelyhood this is not an issue that can be settled experimentally: we can never prove that the universe is infinite, and even if finite, the universe is almost certainly too big to prove it so.  Many people, including some cosmologists, imagine that our present time is like the age of discovery that once and for all pinned down the size and shape of the world.  In a few years time, the argument goes, the size, age and geometry of  the universe will be settled, leaving our sucessors only the task of improving the accuracy in the nth decimal place.  To me such closure seems unlikely.  It is based on the assumption that general relativity provides a fundementally correct description of the large-scale structure of space-time, for which there is not much evidence. It also assumes that we have a decent understanding of the physics of the matter in the universe, which is hopeful, to say the least, when most of it is "dark", undetected except through gravity.  More than likely, future theories and observations will undermine these assumptions, as they have done repeatedly in the past.  Cosmology may still come to an end,  but only because it becomes too expensive for the world to afford, or perhaps because the habits of mind of some future generation see no hope of, or no point  in, answering the questions it poses.

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[1] This transition is the subject of Alexander Koyré's From the Closed World to the Infinite Universe (1957, The Johns Hopkins Press, Baltimore) which traces the change largely through quotations from original sources; starting with Nicolas of Cusa in 1440, and culminating in the long controversy between Leibniz and the Newtonians at the start of the Eighteenth Century.

[2] This is a considerable over-simplification, as the first line of defense would have been to deny that anyone from the imperfect world below the Moon could physically enter the region of the heavenly spheres at all. The spheres were believed to be composed of a different and more perfect fifth element (quintessence) from the corruptible earth, water, air and fire that made up our sub-lunar world.

[3] In the ancient world the Aristotelian astronomer Ptolemy had already used the analogous argument to show that the sphere of stars was vastly bigger than the Earth itself; but this still left a universe small enough that the crystal spheres carrying the planets (which in those days meant the Sun and Moon as well) could be tightly packed to fill all space out to the fixed stars.

[4] This argument had been used previously by Heracleides of Pontus (387-212 BC)  and by Nicholas Oresme, Bishop of Lisieux (c. 1325-1382).

[5] With a significant fraction of the resources of the Danish state behind him, Tycho built on his private island of Hven not one but two observatories featuring instruments of unprecedented precision, and employed and trained some of Europe's best astronomers as his assistants.  Tycho's observations pushed astronomy as far as it could go before the invention of the telescope; not only were his instruments very accurate (mainly because they were very large and solid) but Tycho ensured that observations were made routinely night after night, and different instruments were checked against each other. This gave him two crucial advantages over previous astronomers. First, he had an accurate idea of the uncertainties in the measurements; and second he could test astronomical models in detail against the observed positions of each planet, rather than just against a few special points in the planet's motion, as had been done in the past.

[6] Descriptions of Bruno's heresies are various and inconsistent, but he certainly doubted the doctrine of the Trinity, and claimed that Jesus was a magician rather than the Son of God in the Christian sense.  The actual records of his trial in Rome have been lost (unlike the case of Galileo).

[7] At one level Newton knew that his laws of motion made it impossible to single out a particular frame of reference as absolute space.  In the Principia, the one claim explicitly labelled as a hypothesis is "That the centre of the system of the world is immovable". He comments "This is acknowledged by all...".  But his concept of God implicitly singled out a particular frame (i.e. the rest frame of God) as Absolute, in the fullest sense of the word!

[8] One of these is the gravitational instability of an infinite universe. In his correspondence with Bentley, Newton had claimed (correctly) that if matter was distributed evenly throughout infinite space, it would be highly unstable:

And much harder it is to suppose that all ye particles in an infinite space should be so accurately poised one among another as to stand still in a perfect equilibrium. For I reccon this as hard as to make not one needle only but an infinite number of them (so many as there are particles in an infinite space) stand accurately poised upon their points.
Newton concluded from this that the particles would collapse into "an infinite number of great masses scattered at great distances from one to another... And thus might ye Sun and Fixt stars be formed...".  Bentley took the argument a step further and argued that the stars themselves would collapse into each other without God's active intervention.

[9] In fact, by working backwards, Newton showed that the magnitude system was logarithmic in brightness. Newton's unpublished work on the fixed stars was discovered by  M. A. Hoskins, 1977. J. History of Astronomy, 8,  77.

[10] A nice piece of detective work suggests that the person who "urged" this argument was William Stukeley, now more famous as a founder of British archeology and the originator of the myth that the druids had something to do with Stonehenge.  See M. A. Hoskins, 1985. J. History of Astronomy, 16, 77. (Who is also the origin of the suggestion that Newton was asleep during Halley's talk).

Last modified: 2000 June 27
J. P. Leahy

University of Manchester
Nuffield Radio Astronomy Laboratories
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Cheshire SK11 9DL
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