In mathematics, a paradox is really only puzzling to those that are uninformed. Unlike other areas, we either have a proof or the problem is unsolved. There can be arguments about the validity of a proof but once done, there is no doubt.

Hence, in order to develop paradoxes, it is necessary to interject other factors and it is the category of “Springer Briefs in Philosophy” that is used to develop the uncertainty. For example, there is the “Doomsday argument”, originally attributed to Brandon Carter and popularized by John Leslie. Among all people that have ever lived, any person alive right now would be ranked around 60 billion. If this is a low rank when the complete history of the human race is considered, then it will be a long time before humans vanish. However, if this birth order is of high rank, then the destruction of humans will occur very soon. This is of course more in the area of philosophy rather than mathematics, for this is not based on statistical reasoning using accurate data. If a rank is assumed, then the reasoning would be sound.

Another paradox considered in many forms is the two-letter problem, where you are handed two sealed envelopes and are told that one contains twice as much money as the other one. If you open your envelope and see the contents the question becomes, “do you swap envelopes or hold the one you have?”

There are seven paradoxes examined in this book and philosophical principles are interjected in order to create a paradox and overshadow the fundamental mathematics. While some mathematicians will find the interjections bordering on the absurd, many will find the speculations fun. For example I had never considered the possibility that I am only a mere simulation existing within a society containing solids.

Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing *The Journal of Recreational Mathematics*. In his spare time, he reads about these things and helps his daughter in her lawn care business.