What is Number? (a Peano arithmetic primer)

The need to formalise arithmetic was not well appreciated until the 1860s. Then, Hermann Grassmann showed that many facts in arithmetic could be derived from more basic ideas. This video gives a simple introduction to the system introduced by Giusseppe Peano. Peano's axioms have had an influence to this day, for example in modern notation ∈, for membership of a set. This video is an updated version of one of the earliest uploads on this channel. 0:00 Intro: where do numbers come from? 0:31 Peano, Grassman and Dedekind at the end of the 19th century 0:51 A simplfied version of two of Peano's axioms 1:10 Examples of the "successor" function 1:45 Peano's definition is recursive and allows you to build a number of any size 2:05 What can be known? Outro "Nine" is often listed as the number of Peano's axioms. This video uses "five" as the original number. History is complicated. A nice (technical) summary is found in the Sec 9.7 Notes of the Lectures on the Curry-Howard Isomorphism, by Morten Heine Sørensen, Pawel Urzyczyin https://www.sciencedirect.com/topics/mathematics/peano-axiom Peano photo: https://commons.wikimedia.org/wiki/File:Giuseppe_Peano.jpg (Copyright term expired: public domain) Herman Grassman photo: https://mathshistory.st-andrews.ac.uk/Biographies/Grassmann/Grassmann.jpeg (Public domain) Dedekind photo https://www.sunsigns.org/famousbirthdays/d/profile/richard-dedekind/ All other visuals through VideoScribe, under licencse. Soundtrack: Night Chase Song by Raz Burg under license from artlist.io https://artlist.io/song/41496/night-chase ► Support the channel Like: Give a 👍 to this video (thanks!) Subscribe: https://www.youtube.com/channel/UCNbNqtl8DpH_LJf54wBxIlg?sub_confirmation=1