Hilbert's Infinite Hotel in 60 seconds
There are 5 other installments in the wonderful series '60-second Adventures in Thought' by the Open University: Zeno's paradox, the grandfather paradox, the Chinese room, the twin paradox, and Schrödinger's cat. Go check them all out!
The clear answer to the Hilbert's Hotel paradox is that mathematics doesn't inherently involve content. Whenever content is involved, one must adopt criteria of the content.
ReplyDeleteThus, mathematically speaking, if there is no way to count trans-finite numbers, then the odd-number proportion becomes mathematically unsound. It is easy to imagine how infinity might have a definition even if it is not a quantitative one. We can't just throw out these definitions because we have adopted a quantitative model. For example, what if every infinity were a circle, but circles imposed some physical restraint on the number of residents?
This is what I call the 'exceptional exception,' 'exception for exceptions,' or 'exceptionalist exception. But another way to phrase it is 'radical technicalism'.
In some sense, we cannot say that it is just the rules we adopt which are the only rules. However, the major concern is that mathematics doesn't necessarily involve a standard of content. Content is always more than technical. For example, technicalism might have a similar, although more general form of the mathematical paradox.
To solve the problem, one must have 'areas' with 'functions'. Locating a number in a lattice is not sufficient. The simple way to look at the Hilbert Hotel paradox is that we must treat infinity as a number if that is all it represents (e.g. in the same way that numbers in a lattice must occupy space).
Does anyone agree with me?
Are you joking, trolling, or insane?
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