Tensors are TOO intuitive

A tensor is a multidimensional array that transforms like a tensor A tensor is an element of the span of a basis given by a product basis A tensor is a multilinear map A tensor is a scalar-valued multilinear map from some fixed number of vectors and covectors A tensor is an element of a tensor product The tensor product is the most general way to multiply two vectors The tensor product is a space that satisfies a particular universal property The tensor product is a particular quotient of the free space over the Cartesian product of the original spaces A tensor product is an associative bifunctor from the definition of a symmetric monoidal category A tensor is an element of a tensor power The tensor power is the tensor product of a vector space with itself several times A tensor is an element of a tensor algebra The tensor algebra is the most general way to multiply and add an arbitrary number of vectors The tensor algebra is an algebra that satisfies a particular universal property The tensor algebra is the direct sum of all tensor powers The tensor algebra is a particular quotient of the free space of the free monoid of the original space Sources: "I'm not going to tell you what a tensor is" https://www.reddit.com/r/AskPhysics/comments/1qotlk8/what_exactly_is_a_tensor/ Monoidal monoidoidoids: https://grossack.site/2022/08/21/monoidal-monoidoidoids.html The math paper I referenced a couple times: http://www.tac.mta.ca/tac/volumes/1999/n5/n5.pdf Poincaré quote: https://mathshistory.st-andrews.ac.uk/Extras/Poincare_Intuition/ XKCD 451: https://xkcd.com/451/ Discord: https://discord.gg/3Zj59zA2Rg Patreon: https://www.patreon.com/sudgylacmoe Supporters: David Johnston Jason Killian jerrud Richard Penner trb Sections: 00:00 Your doom 02:03 Introduction 02:45 The role of intuition 03:00 Issues with intuition 03:56 The solution: formality 05:12 The recent emphasis on intuition 05:34 The issues with the recent emphasis on intuition 07:59 Tensors are TOO intuitive 08:55 How DO we actually learn about tensors?