What benefits might there be in describing the activity of mathematics in the following terms?
- Choosing axioms
- Choosing rules of inference
- Applying the chosen rules of inference to the chosen axioms
- Seeing what eventuates
1
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I have never seen such simplification but I am not qualified to judge it. I am intrigued to follow this thread if I am able.
Mcflewster
Level 8
June 26, 2021
1
My post comes from having started to learn mathematics at university, and following the field ever since. Gödel's incompleteness theorems fascinate me, and having seen
it has given me a slightly better understanding of how there are unproveable theorems, how Hilbert's dream of having a machine that could prove everything that was true in mathematics was dashed, and where the Halting Problem ( [en.wikipedia.org] ) comes into all this.
Isn't this addressed in Game Theory and Optimization Theory? Or am I misunderstanding your question?
t1nick
Level 8
June 26, 2021
1
It may well be addressed by Game Theory. In terms of optimisation theory, having worked for 18 years in optimisation, I do not see any connection with the question that I tried to pose.
It could well be that my question is poorly phrased or ill thought out (or both).
I will dig into Game Theory, and I thank you for that idea.
I think @Fernapple's observation is also worthwhile looking at.
I thought that that was the standard way.
Fernapple
Level 9
June 26, 2021
1
I started learning mathematics at university, and the students were all told about some axioms (the Axiom of Choice came later) and "all the rules of inference". Whether or not it is standard is something that is beyond me.
My thinking was guided by the discovery of non-Euclidean geometry, which is geometry from which the parallel postulate is excluded. The same approach makes me wonder what would happen if one or more of the currently accepted axioms, or one or more of the currently accepted rules of inference, were similarly excluded.
Yes, my thinking might at first glance seem outrageous, but I think it is worthwhile at least asking the question.
@anglophone Reminds me of the frequent religious thinking. Give that one of the axioms of god is that it exists, therefore this is how we prove it exists.
@Fernapple Ah, the logical fallacies of category error and of circular reasoning! 
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