An ongoing reflection on impossibility

I have been working on the theme of impossibility in mathematics, especially in geometry, since 2006. Occasionally I venture into other domains, but my main obsession remains impossibility (what is it that we cannot do within certain theories or given certain instruments? How can we mathematically prove the limits of our mathematical methods? And so on).

Eventually, I wrote a thesis on the topic. Well, not on any impossibility theorem or proof ever formulated in maths, but on three old yet nice impossibility results regarding geometry. These are the impossibility of duplicating the cube, the impossibility of trisecting the angle and that of squaring the circle by ruler and compass. All these problems were proved to be insoluble in the 19th century. For a classical survey, see Felix Klein’s beautiful book.

However mathematicians had been discussing their unsolvability for a long time before the 19th century, often with interesting insights and remarkable results. This is at least what I think and what I tried to show in my dissertation. I make it available for anyone who may want to read it for her or his own sake, or use it in the classroom (prof. Jesper Lützen used it for his history of maths class in Fall 2014: I especially thank him and his students for having found out mistakes which had escaped me). For all other public uses, please  contact me. Few words to conclude:

I will not indulge in the conventional fatuity of remarking that [others] are not responsible for the errors this book may contain. Obviously, only I can be held responsible for these: but, if I could recognize the errors, I should have removed them, and since I cannot, I am not in a position to know whether any of them can be traced back to the opinions of those who have influenced me.

Michael A. Dummett, Frege; Philosophy of Language (New York, 1973), p. XII (quoted in S. Unguru (ed.), Witelonis perspectivae liber primus, preface, p. 11).