This extract is taken from the sub-section "The Standard Solution to the Paradoxes" of the article "Zeno’s Paradoxes"
In the 5th century B.C.E., Zeno of Elea offered arguments that led to conclusions contradicting what we all know from our physical experience–that arrows fly, that runners run, and that there are many different things in the world. The arguments were paradoxes for the ancient Greek philosophers. Because the arguments turn crucially on the notion that space and time are infinitely divisible, Zeno can be credited with being the first person in history to show that the concept of infinity is problematical.
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In the 5th century B.C.E., Zeno of Elea offered arguments that led to conclusions contradicting what we all know from our physical experience–that arrows fly, that runners run, and that there are many different things in the world. The arguments were paradoxes for the ancient Greek philosophers. Because the arguments turn crucially on the notion that space and time are infinitely divisible, Zeno can be credited with being the first person in history to show that the concept of infinity is problematical.
As usual kindly put down the summary in the Comments Section.
The Standard Solution to the Paradoxes
A paradox is an argument that reaches a contradiction or other absurd conclusion by apparently legitimate steps from apparently reasonable assumptions, while the experts at the time can not agree on the way out of the paradox, that is, agree on its resolution. It is this latter point about disagreement among the experts that distinguishes a paradox from a mere puzzle in the ordinary sense of that term. Zeno’s paradoxes are now generally considered to be puzzles because of the wide agreement among today’s experts that there is at least one acceptable resolution of the paradoxes. This Standard Solution presupposes calculus, the rest of classical real analysis, and classical mechanics. It implies that motions, durations, distances and line segments are all linear continua composed of points, then employs these ideas to challenge various steps made by Zeno. A key background assumption of the Standard Solution is that this resolution is not simply employing some concepts that will undermine Zeno’s reasoning–Aristotle does that, too, at least for many of the paradoxes–but that it is employing concepts which not only do that but also are needed in the development of a coherent and fruitful system of mathematics and physical science.
What are continua? Intuitively, a continuum is a continuous entity; it is a whole thing that is smooth, having no gaps. Two abstract examples are the path of a runner’s center of mass and the time during this motion. Two concrete examples of continua are oceans and metal rods because treating them as continua is very useful for many calculations in physics even though we know the objects are lumpy with atoms, and thus discontinuous at the microscopic level. The ocean has the continuous property of degree of salinity, and the rod has the continuous property of temperature, so these properties are assigned real numbers as values rather than merely fractions or integers.
The distinction between “a” continuum and “the” continuum is that “the” continuum is the paradigm of “a” continuum. The continuum is the mathematical line, which has the same structure as the real numbers in their natural order. Real numbers are assigned one-to-one to its points; there are not enough rational numbers for this assignment. For Zeno’s paradoxes, the most important features of any continuum are that (a) it is undivided yet infinitely divisible, (b) it is composed of points, (c) the measure (such as length) of a continuum is not a matter of adding up the measures of its points nor adding up the number of its points, (d) any connected part of a continuum is also a continuum, (e) it is so dense that no point has any point next to it since the distance between distinct points is always positive and finite, and (f) the total distance traveled when crossing a convergent series of point places is defined by an infinite sum.
Knowing a continuous object is infinitely divisible does not tell you how many elements or points or ultimate parts it has, other than that there are an infinite number. The Standard Solution says there are in fact an aleph-one number of elements between any two elements in a continuum.
Physical space is not a linear continuum because it is a three-dimensional continuum. But it has one-dimensional subspaces such as paths of runners and orbits of planets; and these are linear continua if we use the line created by only one point on the runner and only one point on the planet. Regarding time, each (point) instant is assigned a real number as its time, and the duration of an instant is zero. Well-defined events that are not instantaneous are assigned an interval of real numbers rather than a single real number. For example, the time taken by Achilles to catch the tortoise is an interval, a linear continuum of instants, according to the Standard Solution (but not according to Zeno or Aristotle).
What are continua? Intuitively, a continuum is a continuous entity; it is a whole thing that is smooth, having no gaps. Two abstract examples are the path of a runner’s center of mass and the time during this motion. Two concrete examples of continua are oceans and metal rods because treating them as continua is very useful for many calculations in physics even though we know the objects are lumpy with atoms, and thus discontinuous at the microscopic level. The ocean has the continuous property of degree of salinity, and the rod has the continuous property of temperature, so these properties are assigned real numbers as values rather than merely fractions or integers.
The distinction between “a” continuum and “the” continuum is that “the” continuum is the paradigm of “a” continuum. The continuum is the mathematical line, which has the same structure as the real numbers in their natural order. Real numbers are assigned one-to-one to its points; there are not enough rational numbers for this assignment. For Zeno’s paradoxes, the most important features of any continuum are that (a) it is undivided yet infinitely divisible, (b) it is composed of points, (c) the measure (such as length) of a continuum is not a matter of adding up the measures of its points nor adding up the number of its points, (d) any connected part of a continuum is also a continuum, (e) it is so dense that no point has any point next to it since the distance between distinct points is always positive and finite, and (f) the total distance traveled when crossing a convergent series of point places is defined by an infinite sum.
Knowing a continuous object is infinitely divisible does not tell you how many elements or points or ultimate parts it has, other than that there are an infinite number. The Standard Solution says there are in fact an aleph-one number of elements between any two elements in a continuum.
Physical space is not a linear continuum because it is a three-dimensional continuum. But it has one-dimensional subspaces such as paths of runners and orbits of planets; and these are linear continua if we use the line created by only one point on the runner and only one point on the planet. Regarding time, each (point) instant is assigned a real number as its time, and the duration of an instant is zero. Well-defined events that are not instantaneous are assigned an interval of real numbers rather than a single real number. For example, the time taken by Achilles to catch the tortoise is an interval, a linear continuum of instants, according to the Standard Solution (but not according to Zeno or Aristotle).
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