Tensors are TOO intuitive

Описание: 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"   / what_exactly_is_a_tensor  
Monoidal monoidoidoids: https://grossack.site/2022/08/21/mono...
The math paper I referenced a couple times: http://www.tac.mta.ca/tac/volumes/199...
Poincaré quote: https://mathshistory.st-andrews.ac.uk...
XKCD 451: https://xkcd.com/451/

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Supporters:
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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?