1

The question of what is a mathematical object was a deep source of debate in the last part of the eighteenth and first part of nineteenth century. There has only been at best a partial resolution, caused in part by Godel's work in logic and in part by exhaustion.

What's the meaning of "by exhaustion"?

  • Despair people feel despair in solving the question
  • Tired people feel tired to solve the question
  • The question was exhausted
2

Exhaustive research means that you have thoroughly examined every possible aspect that could be researched (you have exhausted all possibilities; there are no possibilities left). One definition:

ex·haus·tive adj.

  1. Treating all parts or aspects without omission; thorough: an exhaustive study.

So what is being said in the sentence you quote is that the resolution that has occured is partly due to Godel's work and partly due to the fact that all other possibilities have been exhausted, so what is left must be true. So the theory has been proven by the process of elimination; if nothing else is true, this must be. This is aptly described by this quote from Sherlock Holmes:

"When you have eliminated the impossible, whatever remains, however improbable, must be the truth."

1

I think here "by exhaustion" means "by a thorough study of the subject." See the third explanation.

It is unthinkable to interpret the phrase as meaning that mathematicians are despair or tired of the question; their job is to answer the question, after all.

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