Now, citing axioms and theorems to justify a step in a proof is not a mere social convention to make mathematicians happy. It is a useful constraint on your cognition, allowing you to make only inferences that are actually valid.
When you are trying to build up a new argument, temporarily accepting steps of uncertain correctness can be helpful (if mentally tagged as such). This strategy can move you out of local optima by prompting you to think about what further assumptions would be required to make the steps correct.
Techniques based on this kind of reasoning are used in the simulation of physical systems and in machine inference more generally (tempering). Instead of exploring the state space of a system using the temperature you are actually interested in, which permits only very particular moves between states (“provably correct reasoning steps”), you explore using a higher temperature that makes it easier to move between different states (“arguments”). Afterwards, you check how probable the state is that you moved to when evaluated using the original temperature.
When you are trying to build up a new argument, temporarily accepting steps of uncertain correctness can be helpful (if mentally tagged as such).
Agreed. You just have to remember once you’ve figured out all those steps leading you your conclusion, you have an outline, not a completed proof. Being able to produce such outlines that most of the time can be successfully turned into proofs, or at least interesting reasons why the proof failed, is an important skill.
When you are trying to build up a new argument, temporarily accepting steps of uncertain correctness can be helpful (if mentally tagged as such). This strategy can move you out of local optima by prompting you to think about what further assumptions would be required to make the steps correct.
Techniques based on this kind of reasoning are used in the simulation of physical systems and in machine inference more generally (tempering). Instead of exploring the state space of a system using the temperature you are actually interested in, which permits only very particular moves between states (“provably correct reasoning steps”), you explore using a higher temperature that makes it easier to move between different states (“arguments”). Afterwards, you check how probable the state is that you moved to when evaluated using the original temperature.
Agreed. You just have to remember once you’ve figured out all those steps leading you your conclusion, you have an outline, not a completed proof. Being able to produce such outlines that most of the time can be successfully turned into proofs, or at least interesting reasons why the proof failed, is an important skill.