We Can’t Prove Most Theorems with Known Physics | Summary and Q&A
Summary
In this video, the speaker discusses Godel's theorem and Turing's proof of computability, highlighting the concept that the majority of theorems in mathematics are impossible to prove. The speaker emphasizes that what is computable depends on the physical limitations of the computers we can create in our universe, and that mathematicians cannot escape the laws of physics. The video also discusses the concept of inherently uninteresting theorems that have no bearing on our physical universe.
Questions & Answers
Q: What is Godel's theorem?
Godel's theorem is a concept that arises from Turing's proof of computability. It states that the overwhelming majority of theorems in mathematics are impossible to prove.
Q: How does Turing's proof of computability relate to Godel's theorem?
Turing's proof of computability helps to solidify Godel's theorem by showcasing that the number of things that are not computable is significantly greater than the number of things that are computable. This highlights the limitations of what can be proven mathematically.
Q: What determines what is computable?
What is computable depends entirely upon the computers we can create in our physical universe. If we could make computers that operate outside the boundaries of our known laws of physics, then we could potentially prove different theorems. However, we are bound by the limitations of our universe.
Q: Can mathematicians "get outside" the laws of physics?
No, mathematicians cannot escape the laws of physics. The human brain itself is a physical computer and must obey the laws of physics. If mathematicians existed in a universe with different laws of physics, they would be able to prove different theorems. However, in our universe, we are restricted by the laws of physics.
Q: What limits mathematicians in their understanding of certain theorems?
One limitation that mathematicians face is the finite speed of light. This means that there may be certain theorems or concepts in abstract space that we would have a more complete understanding of if we could surpass the limitations imposed by the laws of physics. However, in our current reality, we are bound by these restrictions.
Q: Are the theorems that cannot be proven inherently interesting?
No, the theorems that cannot be proven at the moment are considered inherently uninteresting. These theorems have no bearing in our physical universe and are not relevant to our understanding of reality. They exist in abstract space and do not contribute to our knowledge of the physical world.
Q: Why are theorems that cannot be proven considered inherently uninteresting?
Theorems that cannot be proven are inherently uninteresting because they have no connection to our physical universe. These theorems do not pertain to our understanding of reality and have nothing to do with the laws of physics. Therefore, they lack relevance and significance in the context of our world.
Takeaways
The majority of theorems in mathematics are impossible to prove, as demonstrated by Godel's theorem and Turing's proof of computability. What can be proven mathematically depends on the physical limitations of the computers we can create, which are bound by the laws of physics in our universe. Mathematicians cannot escape these restrictions, as their brains are also physical computers. Furthermore, there are inherently uninteresting theorems that have no bearing on our physical universe and do not contribute to our knowledge of reality.
