From: Joscha Bach Sent: Monday, December 31, 2018 4:54 PM To: Jeffrey Epstein Subject: Re: This is the solution of =he TSP for all towns of the US that have more than 500 =esidents: The fact that TSP is NP =ard does not mean that it cannot be approximated efficiently, and even =olved for most practical cases. Amoebas and many other biological =ystems can often implement a parallel graph traversal that gives a good =nough solution for a TSP like problem. However, approximate solutions =o TSP cannot usually mapped to solutions to other NP hard problems =i.e. crypto). l=suspect that we are not on the same page wrt the nature and scope of =omputation. The way I see it, mathematics is the domain of all languages. =11 modeling takes place in a language, so modeling (including thinking, =hich is a kind of modeling) is a subset of mathematics. "Classical =athematics" is a branch of math that has "time-less" semantics, i.e. =he mapping between the a statements and its semantics (for instance, =he axioms that a statement can be reduced to in a proof) can take an =rbitrary number of steps, including infinitely many. Godel could =how that if we allow self referential statements, we cannot always =uarantee that the proof does not lead to contradictions and the =emantics of the statement become undecidable. Turing generalized =0del's idea and demonstrated that the problem crops up once the =roof can have infinitely many steps, i.e. a machine that performs the =roof is not guaranteed to terminate its execution. These problems at the =oundations of classical mathematics disappear once we switch to =onstructive mathematics. We can recover the functionality of classical =athematics (like geometry, real numbers and continuous transitions) by =eplacing all entities that require infinite numbers of steps to =enerate by generator functions, and distinguish between values and =unctions. Evaluating a function comes at a cost that is being left to =he one that wants