Paradoxes also arise when attempting to model infinity experiments in the physical world, especially involving so-called “super-tasks.” 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 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 amount gives contradictory results. Quick as a flash, the mathematician will counter that Cantorian transfinite “arithmetic” is not like ordinary arithmetic, permitting subtraction and division, and of course, they will be right. Nevertheless, this thought experiment does not appear to be logically contradictory (it begs the question to assume the correctness of the Cantorian position, when it is that position which is open to critique), and combined with other problems, begins to build up a balance of reason against Cantorianism. See Kip Sewell, “The Case Against Infinity,” (2010), at http://philpapers.org/archive/SEWTCA

Another related area where problems of infinity directly come into play with physical reality are Zeno’s paradoxes, posed by Zeno of Elea (born C. 490 BC). A standard introduction is “Zeno’s Paradoxes,” “Stanford Encyclopedia of Philosophy,” at http://plato.stanford.edu/entries/paradox-zeno/

Zeno believed that motion did not exist because he was a cosmic monist who followed his teacher Parmenides in holding that all that existed was an undivided “one.” He produced a number of ingenious paradoxes dealing with space, time, motion, change and infinity, that are still being refuted today, over and over again. All on a different basis.

The Dichotomy Paradox: for a body to reach some given point, it must first travel half of that distance. But, before it can complete that it must travel half of the previous distance, and so on ad infinitum, so that it can never get started, having to pass through an infinite number of such divisions.

Achilles and the tortoise: there is a hypothetical race between Achilles and a tortoise, with the tortoise being given a lead. Achilles must first travel the distance from the starting point to the tortoise, but then the tortoise has advanced further, and that distance must then be covered and so on, so Achilles never catches the tortoise.

The Arrow Paradox: the orthodox account of motion (or velocity, if directed), has it that motion is a relational state. Motion/velocity is the occupation of different places at different times, so there is no instantaneous state of motion. There is no real difference between a body in motion and a body at rest at an instant, because both are at that instant located at a single point in space. However, this leads directly to Zeno’s Arrow Paradox. If one considers an arrow, the point of the arrow at instant t0 is not advancing, being at a point A in space. But, to move between points A and B, the arrow needs to progress through all of the points between A and B, and if it does not progress on its journey, at any of these points, then it cannot traverse the line segment AB. Hence, it seems that the arrow cannot move – but I bet that Zeno wouldn’t have stood down range from one!

Another related problem, which was not given by Zeno, is the instant of change problem. Before a time t0 a system was in state S0, but after t0 the system is in state S1. So what state was the system in at t0? The possibilities are: (1) in one of S0 or S1; (2) in neither S0 nor S1 or (3) in both S0 and S1. Applied to motion the states are “at rest” and “in motion” and none of the options seem to be justified. See J. R. McKie, “Transition and Contradiction,” “Philosophica,” vol. 50, 1992, pp. 19-32. So what the fuck is going on?

Now, many think that the differential calculus does give us an instantaneous velocity, that is a velocity at an instant. But, according to the limit definition, a limit of velocities is considered at successively shorter periods of time, converging to a given instant. That, however, assumes that motion is already occurring, and so begs the question against Zeno. See J. Lear, “A Note on Zeno’s Arrow,” “Phronesis,” vol. 26, 1981, pp. 91-104.

Another paper says that in the Arrow Paradox Zeno assumed that the velocity v=0, but what we have is an indeterminate form of the type zero over zero i.e. v=0/0, which could be any real number. See M. Zangari, “Australasian Journal of Philosophy,” vol. 72, 1994, pp. 187-204. To my decayed mind, this still does not say what v actually is, so Zeno’s difficulties remain.

W. McLaughlin, “Resolving Zeno’s Paradoxes,” “Scientific American,” no.5, 1994, argued that the solution lied in supposing that motion occurs in infinitesimals, non-standard numbers, greater than zero, but less than any real number. Between real instants, there are allegedly infinitesimals, and objects move through them. But, that seems to make matters even worse, because now one also has to explain how objects move through infinitesimals to real points.

Some very good papers outlining the difficulties with the mathematical physics approach to Zeno’s paradoxes are: A. Papa-Grimaldi, “Why Mathematical Solutions to Zeno’s Paradoxes Miss the Point: Zeno’s One and Many Relation and Parmenides’ Prohibition,” “Review of Metaphysics,” vol. 50, 1996, pp. 299-314; A. Papa-Grimaldi, “The Presumption of Movement,” “Axiomathes,” vol. 17, 2007, pp. 137-154; T. Glazebrook, “Zeno Against Mathematical Physics,” “Journal of the History of Ideas,” vol. 62, 2001, pp. 193-210.

The answer lies in the idea that space and time are discrete, not continuous, that is, that there are both quanta of space and time. This idea raises some problems of working out a finite geometry. But the McKie paper cited earlier discusses this and makes some useful suggestions, as do these two papers; J. P. van Bendegem, “Zeno’s Paradoxes and the Tile Argument,” “Philosophy of Science,” vol. 54, 1987, pp. 295-302 and P. Forrest, “Is Space-Time Discrete or Continuous? – An Empirical Question,” “Synthese,” vol. 103, 1995, pp. 327-354.

Discussions of Zeno’s paradoxes have also considered whether an infinite number of tasks can be completed in a finite time (a “supertask”), and whether or not infinite processes can be completed in reality. J. Benardete, “Infinity,” (Oxford University Press, 1964), p. 237, asked us to consider a book which is in one-inch-thick, with the first page of the book being ½ inches thick, the second page ¼ inches thick, and so on ad infinitum. While the whole book is one-inch-thick (as the series converges), if one looks at the book from the back cover there is nothing to see as there is no last page in the book. As a form of Zeno’s paradox, any book, or object could be modelled in this way.

John Earman and John Norton in their paper, “Infinite Pains: The Trouble with Supertasks,” in “Benacerraf and His Critics” (1996), see no problem with an infinite series of tasks being completed in a finite time. Their example is an idealized perfectly elastic ball bouncing on a hard surface. With a single release, with each bounce the speed on rebound is reduced by a fraction of its speed prior to the bounce. Assuming that each bounce takes no time at all, it can allegedly be shown that the time between successive bounces forms a converging series, so an infinite number of bounces can be completed in a finite time. But, I doubt it. Even for an ideal physical system, the assumption that a bounce takes no time at all is incoherent, because a “bounce” is a physical event that by definition takes time, however small. Hence, the thought experiment is flawed. And, where is the proof that the ball actually stops? It may look like it has, but, who knows?

The “Thomson lamp” was devised as a thought experiment to show the incoherence of the idea of supertask. The hypothetical lamp, which doesn’t burn out etc., has an on/off switch, turned on in the 1st minute, off in the next half minute, on in the next ¼ minute, and so on. At the end of two minutes is the Thomson lamp on or off? The series 1-1+1-1… diverges, so no meaningful answer is possible, unless we say that the lamp is both on and off, which paraconsistent logicians may go for. The generally accepted rebuttal is by P. Benacerraf, that being on or off only applies to times prior to the supertask being completed, and that in itself shows nothing about the lamp’s state at 2 minutes. He should have been called out on this. If that was so, then there is nothing at all to justify any conclusion about the state of the lamp at 2 minutes. That seems to be even more problematic.

Earman and Norton say that “informally we assume that if a lamp is left unswitched, it persists in its current state. Therefore, the state of the lamp at a time when it is not switched is automatically fixed by the prior history of switching.” (p.238) They claim that if the “persistence assumption” is rejected and some other way of fixing the state of the lamp at 2 minutes is presented, then there is no logical contradiction. But, they don’t give any such method, and it is irrelevant any way because the state of the hypothetical lamp, by hypothesis is fixed by its prior states. That is just how the problem is defined. See C. Ray, “Time, Space and Philosophy”(1991).

One way out of all of this is to move to a finite mathematics. There is a considerable amount of work in this genre and some papers include: J. P. van Bendegem, “Alternative Mathematics: The Vague Way,” “Synthese,” vol. 125, 2000, pp. 19-31; J. P. van Bendegem, “Finite, Empirical Mathematics: Outline of a Model,” (1987); D. H. Sanford, “Infinity and Vagueness,” “Philosophical Review, vol. 84, 1975, pp. 520-535; P. T. Shepard, “A Finite Arithmetic,” “Journal of Symbolic Logic,” vol. 38, 1973, pp. 232-248; J. Mycielski, “Analysis Without Actual Infinity,” “Journal of Symbolic Logic,” vol. 46, 1981, pp. 625-633; S. Lavine, “Finite Mathematics,” “Synthese,” vol. 103, 1995, pp. 389-420; S. Lavine, “Understanding the Infinite,” (1994); J. P. van Bendegem, “Classical Arithmetic is Quite Unnatural,” “Logic and Logical Philosophy,” vol. 11, 2003, pp. 231-249.

Hence, there are finitist alternatives, but this does not in itself refute infinitism, a task which still needs to be undertaken. Let us first discuss some aspects of physics where infinity seems to have made a home, then move to mathematics.

[This bullshit will continue in part II]

The "problem," insofar as any exists, lies in our words and concepts not in reality. The arrow strikes its target; Achilles catches the tortoise at 111.11 yards; and our spacecraft land on the moon. In a contest between man and reality bet against man.