3 ms·

My point is, the need for semantics is a human construct, it's not required to solve practical problems. Practical value of MVT is that we can apply it in the p

by js8 3y ago

My point is, the need for semantics is a human construct, it's not required to solve practical problems. Practical value of MVT is that we can apply it in the process of describing a solution, i.e. an algorithm. But if you have a SAT algorithm that can somehow inherently apply a finite instance of MVT, you no longer need to know what MVT is. And this is at the heart of P vs NP question, how to solve SAT, and once we understand that, all the math that is about solving equations (i.e. that of practical value) will become very boring. Kinda like when a game becomes solved with a computer. (This reminds me of Chomsky and Norvig debate on the machine learning and nature of understanding. I wouldn't think I would take Norvig's position, yet here we are.) Addendum: I am also not arguing that computers writing proofs are a necessity for this, but rather efficiently (as possible) solving SAT instances. You don't need to state a hypothesis (and thus prove it) for an infinite number of cases to solve practical problems. I am also not judging value of math - yeah, it is fun. All I am saying it inherently has a human dimension in the sense we don't need it (assuming we have an efficient SAT solver) to solve practical problems.

seanhunter 3y ago
Most equations that can be solved can already be solved symbolically by things like mathematica or numerically by a grab-bag of numerical techniques that every maths, engineering and physics student learns at some point. You don't need to have found the solution to P vs NP to do any of that. I think most mathematicians would say that most of maths is not in solving equations because once you get the right equations, solving them is a simple mechanical process for the most part. Most of maths is figuring out what the right equations are in the first place and proving that what you have done is legitimate.
- js8 3y ago
  I disagree that converting from an equation to a solution algorithm is a straightforward process. It is very much where mathematics helps, for example just to ensure convergence of the method. Also, the way you model reality (choose the equation) is often done with the limitations of solvability in mind. I also disagree that maths can help you pick the right equations, just like there is no formal method that can verify whether a program specification is appropriate to the problem. Equation (or a model) is just an infinite family of SAT problems, typically parametrized on our understanding what a "number" is. If we had a really good SAT solver, then getting a solution would indeed be a really straightforward process (just pick the number representation) of solving the instance. The same is also true for checking "legitimacy" (which I interpret as consistency) of the model - SAT solver would tell you that the instance is unsatisfiable and why. But you can never formally determine that the chosen SAT instance is appropriate model for the task at hand.
constantcrying 3y ago
None of this is true and your understanding of the subject is obviously very poor.
- js8 3y ago
  Before there was AlphaGo, many top Go players believed that understanding Go is somehow special in helping us to understand the universe. The AlphaGo was a shock, because it turned Go into "just another game", which can be beaten by enough calculations on a powerful computer. I think the mathematics is the same. Lot of its beauty we cherish, because we don't properly understand it on the "raw" level of SAT (I would write logic, but I really mean the simplest metalogic you need), which is evidenced by our lack of understanding of P vs NP. Once it is understood, the process of "doing math" will be understood algorithmically, and it will become "boring" or "dull". But as I said, I don't think lack of universal meaning (or lack of mystique) has to detract from the beauty of mathematics.
  - constantcrying 3y ago
    Some facts to consider: - Polynomial problems can still require enormous computing power to the point they are unsolvable. Even if tomorrow there is an algorithm transforming every NP problem to a P problem, not all problems will be feasibly solvable. A O(n^100) Problem is still P. - P vs. NP is about algorithmic complexity, it has no implications about mathematics being an algorithm or not. - The most likely outcome is that P and NP are different, in which case there are zero implications on computational efficiency. - Considering mathematical statements as algorithms has happened for a long time. Any outcome of P vs. NP has limited implications on the philosophy behind it. It just might make it somewhat better. You just don't understand what P vs. NP is about, your posts are bizarre in the actual context of the problem statement. The implications on mathematics just aren't what you think they are.
    - js8 3y ago
      I think what's more fundamental than just answering P vs NP, is what exactly makes boolean functions hard to (pseudo)inverse? And it's the latter where answering (unless the answer comes from an oracle) the former will help. And this question is crucial, because everything we can ask computers (or restated, math with real-world applications) can be rephrased in terms of finding a solution to a SAT instance. It can be a very large instance but if there is a practical way of approaching the problem, it shouldn't matter. I think you're assuming that future computer scientists and mathematicians will continue to run (many different) algorithms, and I don't. I think there might be a single algorithm, a SAT solver, which will subsume all others.

by js8 3y ago

seanhunter 3y ago

Most equations that can be solved can already be solved symbolically by things like mathematica or numerically by a grab-bag of numerical techniques that every maths, engineering and physics student learns at some point. You don't need to have found the solution to P vs NP to do any of that. I think most mathematicians would say that most of maths is not in solving equations because once you get the right equations, solving them is a simple mechanical process for the most part. Most of maths is figuring out what the right equations are in the first place and proving that what you have done is legitimate.

js8 3y ago
I disagree that converting from an equation to a solution algorithm is a straightforward process. It is very much where mathematics helps, for example just to ensure convergence of the method. Also, the way you model reality (choose the equation) is often done with the limitations of solvability in mind. I also disagree that maths can help you pick the right equations, just like there is no formal method that can verify whether a program specification is appropriate to the problem. Equation (or a model) is just an infinite family of SAT problems, typically parametrized on our understanding what a "number" is. If we had a really good SAT solver, then getting a solution would indeed be a really straightforward process (just pick the number representation) of solving the instance. The same is also true for checking "legitimacy" (which I interpret as consistency) of the model - SAT solver would tell you that the instance is unsatisfiable and why. But you can never formally determine that the chosen SAT instance is appropriate model for the task at hand.

constantcrying 3y ago

None of this is true and your understanding of the subject is obviously very poor.

js8 3y ago
Before there was AlphaGo, many top Go players believed that understanding Go is somehow special in helping us to understand the universe. The AlphaGo was a shock, because it turned Go into "just another game", which can be beaten by enough calculations on a powerful computer. I think the mathematics is the same. Lot of its beauty we cherish, because we don't properly understand it on the "raw" level of SAT (I would write logic, but I really mean the simplest metalogic you need), which is evidenced by our lack of understanding of P vs NP. Once it is understood, the process of "doing math" will be understood algorithmically, and it will become "boring" or "dull". But as I said, I don't think lack of universal meaning (or lack of mystique) has to detract from the beauty of mathematics.
- constantcrying 3y ago
  Some facts to consider: - Polynomial problems can still require enormous computing power to the point they are unsolvable. Even if tomorrow there is an algorithm transforming every NP problem to a P problem, not all problems will be feasibly solvable. A O(n^100) Problem is still P. - P vs. NP is about algorithmic complexity, it has no implications about mathematics being an algorithm or not. - The most likely outcome is that P and NP are different, in which case there are zero implications on computational efficiency. - Considering mathematical statements as algorithms has happened for a long time. Any outcome of P vs. NP has limited implications on the philosophy behind it. It just might make it somewhat better. You just don't understand what P vs. NP is about, your posts are bizarre in the actual context of the problem statement. The implications on mathematics just aren't what you think they are.
  - js8 3y ago
    I think what's more fundamental than just answering P vs NP, is what exactly makes boolean functions hard to (pseudo)inverse? And it's the latter where answering (unless the answer comes from an oracle) the former will help. And this question is crucial, because everything we can ask computers (or restated, math with real-world applications) can be rephrased in terms of finding a solution to a SAT instance. It can be a very large instance but if there is a practical way of approaching the problem, it shouldn't matter. I think you're assuming that future computer scientists and mathematicians will continue to run (many different) algorithms, and I don't. I think there might be a single algorithm, a SAT solver, which will subsume all others.