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What do you mean something other than mathematics? It's just a language problem nothing else. Just because the English language lacks the descriptive power it does not mean those objects are "mathematical", just as a round ball is not "English" in it's nature just because you can describe it's properties and behavior using that language. A quantum field is just that, a field, it exists just as much as a magnetic field exists and that you can observe directly. It is an area of space where a specific "force" you might say has impact on objects that enter that area of space. It's properties are measurable and they produce consequences in the world and it has a precise description using the language of mathematics

What do you mean exist? All money is just a promise basically. A promise to get some good at a future date and exchange it for that illusory number, be it paper or electronically. By that description yes all that money existed. They were promises made in contractual and legal forms. And behind those promises lie pensions, payrolls and a big part of the economy. Without those promises holding and some basic level of trust being maintained economic activity becomes very difficult

Fair enough, maybe compression is a too specific term to apply here but I does not matter if it's compression or not to violate copyright. Compression was a good example to mention because it is already familiar to laypeople and established law. The main point is that it stores some sample of the original data - and if it's more it is derived from the original data (your strokes example) and applying some algorithm to reconstruct it to some approximation that we humans might find indistinguishable

Compression (especially a lossy one) means storing a smaller sample of the original data in whatever form you desire and then using some algorithm to reconstruct the original data up to some acceptable approximation. I would argue that in the situation we are discussing the network does just that and it is obvious to everyone involved.

It is exactly remembering the pixels. Just not all of them and it obviously fills in gaps (more hair as mentioned in a another post). You can consider the way it stores those pixels as a lossy compression format. If I copy a music sample but I store a compressed version of it (mp3 for example) you will not find the original bits in my database at all. I am still violating copyright.