🛡️AI Hallucinations

The famous universal approximation theorem (Cybenko 1989; Hornik, Stinchcombe, and White 1989) says a single hidden layer with enough neurons can approximate any continuous function. Read the fine print the authors themselves wrote: the theorem says such a network exists. It does not say how many neurons, and it does not promise gradient descent will find it. It explains why networks are expressive, never why training works. Quoting it correctly is a reliable signal you have read past the headline.

Confabulation is the default, not a glitch

A generative model always produces its best-scoring output, whether or not the world backs it up. When an LLM states a false citation fluently, it is not lying, it is doing exactly what it was trained to do: produce plausible next tokens. Grounding (retrieving real documents), training models to decline, and verification tooling reduce the failure; nothing eliminates it. Calibrate accordingly, especially for facts that matter.

The brain question, resolved honestly: it isn't

Does the brain do backpropagation? Crick's 1989 objection stands: cortex has no evident mechanism for shipping exact error signals backward through synapses. The 2020 reply from Lillicrap, Santoro, Marris, Akerman, and Hinton argues feedback connections might locally approximate those signals. The debate is open, and the safe summary is the one this tutorial has used throughout: artificial networks are mathematics that works, inspired by, not evidence about, your cortex.

Where to go from here

You now hold the full basic picture: what AI is, how machines learn, and what the ingredient under everything actually does. The expert path from here is well marked and entirely free: Karpathy's Zero to Hero to build everything from scratch, Nielsen's book for depth on this chapter, Goodfellow for the mathematics, CS231n for the engineering craft, and this site's 3D guides whenever a mechanism needs to be seen instead of read.