Imagine if we had a machine that let us change some earlier moment in history and see what developed. We wouldn't actually change history--that would lead to paradoxes and other disasters, but we could see what would develop. Suppose Archduke Ferdinand was never assassinated. How would that have changed 20th century history and beyond?

The same could be said for an academic field. Science flows like a story, with building results from other results, "standing on the shoulders of giants" so to say. But not all theorems are dependent on earlier ones and suppose that things happened in a different order. How would that have changed our field?

Points to ponder:

Suppose Gauss was alive today instead of two centuries ago. Would he still be as famous? Would there be a big hole in mathematics that would have been left for the current Gauss to solve?

Suppose Fermat's last theorem was still a conjecture. Would there be more budding mathematicians inspired by the wonders of that famous open problem like my generation was? Would the Clay Mathematics Institute still have produced a list of those millennial problems, including P v NP?

Suppose we knew Primes in P back in 1975. Would randomized algorithms and subsequent derandomization techniques have happened without its prime example? Same for Undirected connectivity in log space. These both are small cheats as the AKS and Reingold proofs are at their core derandomization arguments and may not have happened if we didn't think about randomized algorithms.

What if someone settled P vs NP right after Cook? Would it have stopped most of the research in computational complexity theory? Would it depend on whether P = NP was answered positively or negatively?

What if you were never born? Ultimately that would be the only true measure of your influence in the world. What if your research somehow prevented other great theorems from happening? Would you even want to see the results of that experiment?

I also like to think about this in the following form: Who are those scientists, who proved something that wouldn't have been proved for decades without them?

ReplyDeleteps. Who the hell is Cardinal Ferdinand? Do you mean the Habsburg heir? I doubt anything would have changed except for the casus belli.

I mixed my Cardinals and Archdukes. Fixed the post.

DeleteThis kind of reasoning is called Counterfactual analysis, BTW.

ReplyDeleteAs far as science is concerned I don't believe that the great minds are needed anymore. They only speed things up that would later have been developed by an industry of a great mass of brilliant but more mediocre reseachers. The best examples are Godel's and Turing's work which would have been done neccessarily by the next generation of logicians or theoretcal computer scientists. Regarding your own contributions it is fair to say that "the future doesn't need you." Anything important would also be created sooner or later by others, only some idiosyncratic work of minor importance could be left undone forever.

ReplyDeleteTaking a slightly different tack: if some sort of reasonable computers had existed in Gauss's day would he done a lot of his mathematics -- a large motivation in all of the work that he did in solving equations over the reals was motivated by his "day job" at the observatory. He had to do all of the computation by hand, so there was great motivation to make it as efficient as possible.

ReplyDelete