Is there an ultimate goal to mathematics progress?

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SUMMARY

The discussion centers on the philosophical and practical implications of mathematical progress, drawing parallels with physics, particularly the search for the Higgs boson. It asserts that while fundamental questions in mathematics were addressed 80 years ago, particularly through Hilbert's and Godel's work, modern mathematicians often overlook these historical milestones. The conversation emphasizes that despite the theoretical significance of Godel's incompleteness theorems, they hold little practical value for the majority of mathematicians today, similar to the Higgs boson in physics. Ultimately, the dialogue reflects on the nature of scientific inquiry and the ongoing pursuit of knowledge.

PREREQUISITES
  • Understanding of Hilbert's axiomatization and Godel's incompleteness theorems
  • Familiarity with the basics of mathematical logic
  • Knowledge of the relationship between mathematics and computer science
  • Awareness of the significance of the Higgs boson in physics
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  • Research the implications of Godel's incompleteness theorems on modern mathematics
  • Explore the role of algorithms in mathematical problem-solving
  • Investigate the historical context of Hilbert's 1900 problems
  • Examine the impact of theoretical physics discoveries on practical applications in technology
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Mathematicians, philosophers of mathematics, physicists, and anyone interested in the intersection of theoretical concepts and practical applications in science.

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Just as physicists are trying to answer their most fundamental questions with the LHC, are mathematicians in a similar bind? Is there a foreseeable time when they will no longer be faced with daunting mysteries?
 
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Newai said:
Just as physicists are trying to answer their most fundamental questions with the LHC, are mathematicians in a similar bind? Is there a foreseeable time when they will no longer be faced with daunting mysteries?

The world isn't a mystery to solve. The search for the Higgs boson makes a pretty headline, but it won't be the end of science. Think of it like chess or football or whatever game you like. The rules are simple. The game is not.

The most fundamental questions of mathematics were resolved 80 years ago. In 1900, Hilbert wanted all of mathematical logic completely axiomatized in a consistent way. In 1931, we found it wasn't possible thanks to Godel. With the invention of computer science, we discovered that all of mathematics can be boiled down to a search algorithm and "solved" essentially the same way Big Blue plays chess. The caveat: to do anything as useful as that sentence sounds would take more resources than a billion universes would be able to provide.

The funny thing is that most modern mathematicians don't care or are even totally unaware of Hilbert and Godel. Godel's work isn't considered significant in the realm of mathematics (it is more important in logic and philosophy), and the reason is that it isn't really applicable to what everyone does.

That sounds a lot like the Higgs boson to me. Yeah, it's the truth. It's fundamental. But it's also totally useless to 99.99% of people in physics. Food chemists will still churn out new products to turn kids' mouths green. Bioengineers will work on new ways to use bacteria to process goods. And all of the material scientists and electrical engineers will continue to pretend that nothing is smaller than the electron.
 
Wow. That's a lot for me to think about, given the perspective I've had for many years. Thanks!
 

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