It is, in my opinion, important to distinguish between the formal system X (which is timeless, unchanging, and Platonically real), and our experiences of applying X.
Why is it important? I have actually difficulties seeing the boundary. What other information we have about a formal system except experiences of its application?
This reply seems a bit too brief. When speaking about e.g. physics, it’s clear what’s the map (our models and theories) and what’s the territory (nature around us). There are two principal ways of obtaining knowledge: exploring the territory by conducting experiments and exploring the map by calculating implications of our theories. In reality experimental results are viewed indirectly through prism of the theory, but the distinction between experimentally established facts and theoretically derived propositions is still pretty clear.
I don’t see similar distinction for formal system. All we know about a formal system we know because we apply its rules and derive theorems. We can in fact analyse the system on the meta level, but you seem to be objecting to that. So, what practical importance lies in the map-territory distinction here? What actual mistakes I could commit because I forget to make the distinction?
Why is it important? I have actually difficulties seeing the boundary. What other information we have about a formal system except experiences of its application?
Map-territory. You’re not distinguishing formal system X from our knowledge of formal system X.
This reply seems a bit too brief. When speaking about e.g. physics, it’s clear what’s the map (our models and theories) and what’s the territory (nature around us). There are two principal ways of obtaining knowledge: exploring the territory by conducting experiments and exploring the map by calculating implications of our theories. In reality experimental results are viewed indirectly through prism of the theory, but the distinction between experimentally established facts and theoretically derived propositions is still pretty clear.
I don’t see similar distinction for formal system. All we know about a formal system we know because we apply its rules and derive theorems. We can in fact analyse the system on the meta level, but you seem to be objecting to that. So, what practical importance lies in the map-territory distinction here? What actual mistakes I could commit because I forget to make the distinction?