In a very interesting paper, Amanda Gefter, "The Infinity Illusion", New Scientist, August 17, 2013, pp.32-32, we are given a concise diagnosis of one of the many things, I believe, wrong with mathematical physics. Infinities abound in mathematics - in set theory, analysis, geometry and calculus - but infinities cause great problems in physics. As she says, "Trouble is, once unleashed these infinities are wild, unruly beasts. They blow up the equations with which physicists attempt to explain nature's fundamentals. They obstract a unified view of the forces that shape the cosmos. Worst of all, add infinities to the explosive mixture that made up the infant universe and they prevent us from making any scientific predictions at all".

Infinities arose first in the early study of the electron, but these problems were dealt with. Later further infinities arose in quantum electro-dynamics, the quantum mechanical theory of electromagnetic forces. This was in turn dealt with by "re-normalisation", a kind of way of dividing out the Infinities, which some logicians question as being, at best, less than rigorous; at worst, invalid if not delusional. Be that as it may, the infinities arising in the general theory of relativity are not so easily dealt with. Conditions inside a black hole seemed to involve an infinite density of matter and an infinite warping of space-time, whatever that means. If the matter inside a black hole is infinitely dense then it is reasonable to suppose that the black hole itself is infinitely dense and if this is literally so, the gravitational attraction between the rest of the universe and the black hole should be "infinite". Matter should then disappear into the black hole like a giant vacuum cleaner. That is my interpretation. As black holes, if they exist (I believe that there is a case against them), are not that powerful, it is unlikely that matter inside is really "infinitely" dense.

Gefter points out that in the present cosmological theory of cosmic inflation in the first instances after the Big Bang there was a rapid expansion of matter and space-time. This, however, leads to continuous universe creation, with the inflation of more space-time and the creation of multiple universes. Apparently in these multi-verses anything can happen. Some of these universities will have radically different laws of nature. All this arises from the assumption that space-time is like the real number line, a continuum infinitely divisible; without that assumption there will be no infinite explosion of universes.

Infinity in Mathematics

Since Aristotle almost all mathematicians and philosophers rejected the idea of an actual physical infinity and accepted instead the concept of a potential infinity, that one could add 1 to any number (to take the example of arithmetic) without reaching a last number. Many Christian philosophers and theologians reserved the notion of the infinite for God alone. But that changed with the 19th century mathematician Georg Cantor and his work in transfinite theory.

Galileo noted in 1638 that the set of all natural numbers {1, 2, 3, 4,…} could be put into a 1- 1 correspondence with one of its subsets {1, 3, 5, 7,…}, the set of odd natural numbers, even though the set of natural numbers has numbers that are not in that subset. Cantor did not see this as a paradox and said that the sets are the same size, both denumerably or countably infinite. However by an argument known as the diagonal argument, he allegedly showed that the set of natural numbers was a smaller infinite set than the set of real numbers because a parallel type of 1- 1 correspondence could not be set up. In a hypothetical list it is allegedly possible to always change a number in the diagonal of the set of reals, to create a real not in 1- 1 correspondence with the natural numbers. Hence the real numbers were of a higher cardianality than the natural numbers, non-denumerably or uncountably infinite, and that is just the start of the transfinite stairway to Platonic heaven.

Paradoxes: Logical

Not all mathematicians were pleased with the arrival of transfinite numbers. The great French mathematician Henri Poincoré regarded Cantor's transfinite set theory as a "disease from which one has recovered". Around the turn of the 20th century a series of logical paradoxes in set theory such as Russell's paradox of the set of all sets not members of themselves (a set which is a member of itself if and only if it is not a member of itself, and which is therefore contradictory), rocked the mathematical world. Infinity was thought to have a part to play in this, but was not the only problem: self-reference seemed to be problematic as well. Nevertheless said theory paradoxes continued to be uncovered, such as the paradox of the set of all truths, which conflicts with one of Cantor's central theorems, the power set theorem (see P.Grim, The Incomplete Universe, 1991, pp.92-93).

Perhaps what was worse was that it was proven that on the basis of the axioms of standard set theory it was not possible to prove or disprove various transfinite set theoretical statements such as whether or not there were other transfinite sets between the known ones, or the size of certain large transfinite sets (the continuum hypothesis). It is somewhat ironic to note that one of the mathematical establishment’s leading logicians and mathematicians, Abraham Robinson, who himself developed a theory of infinitesimals (known as non-standard analysis) said in his 1973 retiring presidential address at the Annual Meeting of the Association of Symbolic Logic ("Meta-mathematical Problems", Journal of Symbolic Logic, vol.38, 1973, pp.500-516) the following:

"While others are still trying to buttress the shaky edifice of set theory, the cracks that have opened up in it have strengthened my disbelief in the reality, categoricity or objectivity, not only of set theory but also of all other infinite mathematical structures, including arithmetic". (p.514) (I wonder if his skepticism extended to his own non-standard analysis?)

For example, infinity problems in probability theory raise havoc for the regularity or frequency view of probability. An example is: how probable is an infinite sequence of heads, in an imaginary infinite number of tosses? (See Analysis, vol.67, no.3, 2007, pp.173-180) One argument is that the probability of heads of such tosses is ½. ½. ½… which converges to zero. But by another argument the set {h, h, h,…} is one possible sample space and so is theoretically possible, so having a non-zero probability. In another example consider a circle with a non-denumerable infinity of points. A "magical" rotating pointer is assumed to be able to stop at one point. The probability of any point being stopped at is zero, as a denominator is "infinite", yet the pointer is assumed to be capable of stopping at a point.

Paradoxes: Physical

Paradoxes also arise when attempting to model infinity experiments in the physical world. One problem, given by William Lane Craig ("A Swift and Simple Refutation of the Kalam Cosmological Argument", Religious Studies, vol.35, 1999, pp.57-72) concerns someone who has an infinite number of marbles, and who wants to give to you an infinite number of marbles too. Thus all of the marbles M, could be given, so M- M=0. Or only the odd numbered marbles could be given so M-M=M. Hence dividing and subtracting equal amounts gives contradictory results. Quick as flashes the mathematicians will counter with Cantonian transfinite "arithmetic" is not like ordinary arithmetic permitting subtraction and division, and that is of course true. Nevertheless regardless of this Craig’s thought experiment is highly counter-intuitive and combined with other problems builds up evidential weight against Infinity. (See the useful Internet book by Kip Sewell, The Case Against Infinity (2010) which has been very helpful for me).

There are other "paradoxes of the infinite" (see British Journal for the Philosophy of Science,vol.42, 1991, pp.187-194). If one supposes that an infinite number of things could be done in a finite time (consider a lamp which is turned on in the first ½ + ¼ + ⅛… converges to 2, at the end of two minutes is the lamp on or off? It would appear to be both on and off, which is impossible. Along with this there are many other physics problems generated by infinity. (See British Journal for the Philosophy of Science ,vol.54, 2003, pp.591-599)

Finite Mathematics to the Rescue

There is a considerable amount of good quality logico-mathematical work attempting to construct a finite mathematics. I haven't digested it all yet so I will list some references for students who want to explore this area further:

on geometry:

(1) P.Forrest, "Is Space-Time Discrete or Continuous? An Empirical Question", Synthese, vol.103, 1995, pp.327-354.

(2) J.P.Van Bendegem, “Zeno’s Paradoxes and the Tile Argument", Philosophy of Science, vol.54, 1987, pp.295-302.

(3) J.P.Van Bendegem, “Alternative Mathematics: The Vague Way" Synthese, vol.125, 2000, pp.19-31.

(4) J.P.Van Bendegem, Finite, Empirical Mathematics: Outline of a Model (1987)

(5) D.H.Sanford, "Infinity and Vagueness", Philosophical Review, vol.84, 1975, pp.520-535 (Argues that due to the vagueness, time could be finite but unbounded).

There are also many fruitful attempts to construct a finite mathematics, not involving actual infinities and for some, not even the presupposition of the potentially infinite; here is what my internet search uncovered:

(6) P.T.Shepard, "A Finite Arithmetic", Journal of Symbolic Logic, vol.38, 1973, pp.232-248.

(7) J. Mycielski, "Analysis without Actual Infinity", Journal of Symbolic Logic ,vol.46, 1981, pp.625-633.

(8) S. Lavine, "Finite Mathematics", Synthese, vol.03, 1995, pp.389-420.

(9) S. Lavine, Understanding the Infinite, (Harvard University Press, Cambridge, Massachusetts, 1994).

(10) J.P.Van Bendegem, "Classical Arithmetic is Quite Unnatural", Logic and Logical Philosophy, vol.11, 2003, pp.231-249.

The basic idea is to operate with indefinitely large numbers instead of infinity. As well the axioms of Peano (standard) arithmetic need revision as there is a standard objection that there must be an infinity of numbers because if there was a last number Ω then add 1 to it to get another. But this is question begging because Ω +1 could be defined as non-denoting, that is, not being a number at all.

Even within conventional mathematics, if it makes sense to have an infinite decimal expansion, say in irrational numbers such as √2, then it also makes sense to consider numbers such as “1111…", “8888…" and “9999…" "infinite" numbers in the sense of having an infinite string of numbers. The number “9999…" could be defined as the last number because there is no way of representing a bigger number.

What about numbers such as √2? Such numbers disturbed the Pythagorean Greeks because it seems that a completed infinity can be represented. Thus any length of an object L is equal to itself. L can be moved through space, at one point defining a perfect right angle triangle and the length of the hypothenuse would be √2 taking L to be of unit length. So do infinities exist? In the real world there are no exact triangles so the problem does not arise. Irrational numbers can be regarded as "incomplete" and a "calculation process": T.Kalanov, "The Critical Analysis of the Pythagorean Theorem and the Problem of Irrational Numbers", Bulletin of Pure and Applied Sciences - Mathematics and Statistics, vol.370, no.1, 2013. Another way of thinking about √2 is "whatever integers are substituted for p and q in the expression p^{2}- 2q^{2} the result will not be 0". B.Rotman, Ad Infinitum: The Ghost in Turing's Machine, (Stanford University Press, Stanford, 1993), p.8.

The idea that √2 exists in some abstract realm as a "completed" infinity is in any case paradoxical, because the infinite expansion of decimals in the expansion of this irrational number must be assumed to be "completed" and yet by definition it is "uncompleted", which is contradictory.

Finally, what about Cantor's infinities? Cantor's diagonal argument was based on the idea that infinite sets can be placed in a 1- 1 correspondence. But P.O. Johnson, “Wholes, Parts and Infinite Collections", Philosophy, vol.67, 1992, pp.367-379 to my mind refuted this:

"We cannot match terms unless we know in advance that each series or set contains the same number, and there is no other justification for saying that they 'correspond'. This is, of course, exactly the opposite argument to Cantor's. Where Cantor argues that, where there is a one-to-one correspondence between two infinite classes, both must contain the same number of objects, I say that unless some classes can be shown to contain the same number of objects, their terms cannot be said to correspond". (p.372)

It is an assumption that the notion of a 1- 1 correspondence between infinite sets is meaningful.

In future essays for this site I will deal with questions of infinity and paradox in more detail. Rejection of the mathematical notion of infinity moves us back to a far saner period of time where infinity was the exclusive domain of God, and something to be regarded with awe, not a mere symbol to manipulate.