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Understanding the Infinite
How can the infinite, a subject so remote from our finite experience, be an everyday tool for the working mathematician? Blending history, philosophy, mathematics, and logic, Shaughan Lavine answers this question with exceptional clarity. Making use of the mathematical work of Jan Mycielski, he demonstrates that knowledge of the infinite is possible, even according to strict standards that require some intuitive basis for knowledge.
- GenresPhilosophyMathematics
376 pages, Paperback
First published September 23, 1994
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September 1, 2018
A dense, technical book. The first half is a historical survey of the finer points of early set theory and the ZFC axioms. I learned quite a few things, but I suspect this may not be the best book on those topics. Lavine tries to do too many things in this book, and an author focused on the history alone is likely to be clearer and more accessible.
The second half presents a technique for recasting ZFC (the foundation of most modern mathematics) in finite form by using indefinitely large finite numbers ("a zillion") instead of infinity. It's an interesting gimmick, but doesn't really achieve anything. It's reminiscent of campaigns to switch from sets to mereology, or from predicate logic to more user-friendly logics closer to natural language, or (at a more mundane level) from natural language to a rational language like Esperanto. There's simply no reason to make a switch like this if the new system does exactly the same things the old system does. The massive retooling effort would only make sense if the new system yields new results/insights. The system of Mycielski/Lavine doesn't tell us anything mathematically new (i.e., it doesn't prove any new theorems), so mathematicians aren't interested in it, except as a curiosity.
Does this book give us any philosophical insight into infinity and how we come to know and understand it? In a word, no. Lavine assumes we can explain infinity by reducing the infinite to the finite. That doesn't work though because finite numbers can be grotesquely, frighteningly huge. The mystery is how small, feeble, limited creatures like humans can mentally interact with such grotesquely large entities, be they finite or infinite. The problem isn't how to reduce infinity to the finite; the problem is how to reduce the infinite & monstrously finite to human size. That's the crucial question, and Lavine doesn't address it. He offers a math book instead. (By that I mean: One wonders how the extremely finite, tiny biological processes occurring in a human brain can possibly grasp, manipulate and understand an infinite object—like the continuum of real numbers. Supplying pages full of theorems doesn't answer the question. Mathematics is an example of the thing to be explained, not an explanation.)
The second half presents a technique for recasting ZFC (the foundation of most modern mathematics) in finite form by using indefinitely large finite numbers ("a zillion") instead of infinity. It's an interesting gimmick, but doesn't really achieve anything. It's reminiscent of campaigns to switch from sets to mereology, or from predicate logic to more user-friendly logics closer to natural language, or (at a more mundane level) from natural language to a rational language like Esperanto. There's simply no reason to make a switch like this if the new system does exactly the same things the old system does. The massive retooling effort would only make sense if the new system yields new results/insights. The system of Mycielski/Lavine doesn't tell us anything mathematically new (i.e., it doesn't prove any new theorems), so mathematicians aren't interested in it, except as a curiosity.
Does this book give us any philosophical insight into infinity and how we come to know and understand it? In a word, no. Lavine assumes we can explain infinity by reducing the infinite to the finite. That doesn't work though because finite numbers can be grotesquely, frighteningly huge. The mystery is how small, feeble, limited creatures like humans can mentally interact with such grotesquely large entities, be they finite or infinite. The problem isn't how to reduce infinity to the finite; the problem is how to reduce the infinite & monstrously finite to human size. That's the crucial question, and Lavine doesn't address it. He offers a math book instead. (By that I mean: One wonders how the extremely finite, tiny biological processes occurring in a human brain can possibly grasp, manipulate and understand an infinite object—like the continuum of real numbers. Supplying pages full of theorems doesn't answer the question. Mathematics is an example of the thing to be explained, not an explanation.)
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