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This whole argument reminds me a bit of the book, Godel-Escher-Back, which is (partially) about the interplay of formal and informal systems. Both are necessary

This whole argument reminds me a bit of the book, Godel-Escher-Back, which is (partially) about the interplay of formal and informal systems. Both are necessary, and one without the other tend to be useless or sterile. Sometimes, when I've asked for an reason/explanation about some physics problem, I get a response like: "because of this equation". And in my brain there is a mismatch, were it doesn't feel at all like an actual explanation, because abstract symbol manipulation feels like, well, symbol manipulation. But I think that's the wrong interpretation, because someone who has been really deep in mathematics for a long time feels the math at a deeper level than I do. For example, it's like explaining something that's true because F=MA. Before I had any physics knowledge someone answering a question with "It's because F=MA" would have felt off, in the same way. After doing a bunch of problems then F=MA is a perfectly fine answer. They are just a lot better at getting meaning from the symbols and following the manipulations.

rramadass 2y ago
The essay is not about formal vs. informal systems. In fact your example illustrates the very point that Arnold is making. In "F=ma" the symbols actually map-to something in the real world. The "domain of discourse/interpretation" for the symbols is well known and there is no confusion on what the equation stands for. This article, Why “F = ma” is the most important equation in physics explains the nuances of the above equation - https://bigthink.com/starts-with-a-bang/most-important-equation-physics/ https://bigthink.com/starts-with-a-bang/most-important-equat... As the article points out it might be tempting to think of the above as a special case of the equation of a straight line (i.e. y=mx+c) but that would not be correct. Symbolic logic is concerned only with the "forms" of the argument and inference (hence the name "Formal" system) and not their meaning/context. It is the latter which brings mathematics "alive" else as the GEB book shows we are just playing games.

rramadass 2y ago

The essay is not about formal vs. informal systems. In fact your example illustrates the very point that Arnold is making. In "F=ma" the symbols actually map-to something in the real world. The "domain of discourse/interpretation" for the symbols is well known and there is no confusion on what the equation stands for. This article, Why “F = ma” is the most important equation in physics explains the nuances of the above equation - https://bigthink.com/starts-with-a-bang/most-important-equation-physics/ https://bigthink.com/starts-with-a-bang/most-important-equat... As the article points out it might be tempting to think of the above as a special case of the equation of a straight line (i.e. y=mx+c) but that would not be correct. Symbolic logic is concerned only with the "forms" of the argument and inference (hence the name "Formal" system) and not their meaning/context. It is the latter which brings mathematics "alive" else as the GEB book shows we are just playing games.