A field is an abstract system of numbers where one can add and subtract, multiply and divide. (You study them a fair bit in Algebra II, but that’s not required to enjoy this talk). The real numbers, complex numbers, and rational numbers are examples. But there are also fields that have a finite number of elements (you get very familiar with these guys if you do Galois Theory). An important fact is that the smallest field has exactly 2 elements: it’s the integers mod 2. So why on earth would some people talk about a mythical “field with one element” that clearly can’t exist? In this talk I’ll give a tour of some of the strange and imaginative ways in which mathematicians have tried to make sense of this idea. Along the way I’ll try to tell some entertaining stories about famous mathematicians of the 20th century.