(These are excerpts from my book "Intelligence is not Artificial")
Footnote: Neural Networks as Vector Spaces
When a neural network is trained to "learn" some pattern, its neurons get organized in a rather geometric manner. A neural network basically constructs a highdimensional space in which the distance between two points mirrors the degree of relationship between two objects in the real world. For example, two words that tend to show up together in many sentences will be represented by two points very close to each other. These "points" in highdimensional spaces are "vectors". This was the discovery made by Tomas Mikolov's team at Google in 2013, using the "skipgram" method for constructing vector representations of words from analyzing large sets of text, a method now known as "word2vec". The same approach can be used to analyze images or speech, and, again, turn a large set into a highdimensional vector space. Now you can use an old mathematical tool called "vector arithmetic" and perform calculations on these vectors. Mikolov showed that, for example, one can perform this algebraic operation: "vec(king)  vec(man) + vec(woman)" and obtain "vec(queen)". Unfortunately (or luckily), this method ended up revealing embarrassing biases in the texts of our world. For example, in 2016 Tolga Bolukbasi at Boston University used a neural network to calculate "vec(father)  vec(doctor) + vec(mother)". You would expect the answer to be still "doctor" as there are many female doctors, but instead the answer (obvious once you see it) was "nurse": the line from male doctor to female doctor is not a straight one. As Einstein would put it, the vector space is warped!
For the record, vectors were introduced in the 19th century book "The Barycentric Calculus" (1827), published by August Moebius, the same German mathematician who in 1858 discovered the Moebius strip, and were indirectly popularized by James Maxwell's classic on electromagnetism, "Treatise on Electricity and Magnetism" (1873), and even more by Werner Heisenberg's version of Quantum Mechanics (1925) that used matrix and vector algebras instead of calculus (Erwin Schroedinger's version). It turns out that they seem to be ideal mathematical tools not only for electromagnetic waves and quantum systems but also for brains.
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