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Philosophical Perspectives on Infinity

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This book is the other half of Graham Oppy's "Arguing About Gods". They were originally supposed to be a mega-book titled "God and Infinity". It is good that Dr. Oppy separated them because both are heavy duty and may have been too much and too expensive if joined.

This book is a review of many of the puzzles surrounding infinity through the ages (for some reason, the other reviewer can not appreciate professor Oppy's goal of making this book definitive). While it covers some classical problems, it mostly focuses on those that have made their way into philosophical discussions involving mathematics and religion. The author's style is meticulous and concise. He carefully lays out his preliminary questions about different areas of mathematics, then dives right in. As someone who loves mathematics, physics, and philosophy, I really enjoyed this book.

There is no question that the concept of infinity is a difficult one. It is compounded by the fact that there are different levels of infinity, broadly split into the categories of countable and uncountable. It is also possible for two different infinite sets to have the same number of elements and yet one is completely contained in the other. For example, it is possible to match up the set of counting numbers {1, 2, 3, ...} with the set of even numbers {2, 4, 6, ...} by matching each number in the first with its double in the second. Clearly, the second set is completely contained in the first.
Mathematicians work constantly with the concept of infinity. The first year of undergraduate mathematics involves limits, summing infinite series and infinitesimals. Getting to this point has not been an easy journey, the finest mathematicians struggled with summing some series and it took years to successfully resolve the problems.
Oppy covers most of this ground and it is unfortunate that he doesn't seem to grasp some of the basic ways in which mathematicians deal with infinity. For example, on page 100 there is the statement: "Similarly, in topology, we have it that a one-dimensional line can be composed of nothing but zero-dimensional points. Yet how can one put together things that are all zero-dimensional and end up with something that is one-dimensional?" This is a supposed paradox that was settled by the mathematical community long ago.
Oppy also brings up the classic paradox of Zeno involving Achilles and the tortoise. This is a classic example of the sum of an infinite series being finite and even advanced precalculus students can determine the precise point where Achilles will pass the tortoise. He also brings up the famous "Olbers' paradox", which was raised by Heinrich Olbers in 1823. Olbers argued that the universe could not be statically infinite because otherwise the combined starlight would prevent the sky from ever getting dark. While there are still some arguments that take place on the edges, the overall expansion of the universe and the corresponding redshift easily explains why it is possible for the universe to be infinite and the sky dark at night.
Oppy spends a great deal of time on this problem, concluding that "It is not true that we can explain the darkness of the night sky by adverting to any of these versions of the claim that, even though there are stars in any direction we choose to look, there is a reason why we fail to see some of those stars." Unfortunately, while Oppy uses the term cosmology on a regular basis, he does not seem to be up on the current model of the universe.
Mathematicians long ago reached a comfort level when dealing with infinities. So much so that even first year college math students are expected to work with them without difficulty. While he certainly mentions a lot of mathematics and how it is used, Oppy seems not to have achieved this basic level.