Slippery words, precise math
A challenge we face constantly in machine learning is that its words are slippery: the same term can point to different mathematical ideas depending on who is speaking. Before any equations, it pays to pin the vocabulary down.
Take the word “algorithm”. It is used in at least two distinct senses. In one sense, it means the system that makes predictions from input data — what we can more precisely call a predictor, or a model. In the other sense, it means the procedure that trains that system: the recipe that adapts a model's internal parameters using data.
When someone says “the algorithm learned”, they are mixing both senses in one sentence: a training algorithm (sense two) adjusted a predictor (sense one). Keeping the two apart makes every later concept clearer.