# Zeta function (Cambridge Computing)

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Summer 1937, Cambridge England. Wittgenstein and Turing have made plans to go see a flick (Secret Agent).

Turing arrives in the hallway outside Wittgenstein's rooms just as the door opens and a flock of philosophy students depart from their session with Wittgenstein. Turing stands aside and listens to the very loud voice of Wittgenstein from inside, "You can not doubt just this and that. It is the number of things that you **can not** doubt that you can count on one hand."

The voice of the student is hard to hear, something like, "Well, I doubt that."

Wittgenstein laughs and looks out the door, sees Turing. "Here's Turing! He's a mathematician, he believes an infinite number of things ... he can not even write a proof that does not include infinity."

Turing nods to the student and does an little dance as they both try to go through the door at the same time. Wittgenstein is striding around the room pushing furniture around and standing folding chairs in corners. Turing knows from experience that Wittgenstein will be in no mood to talk until he quiets the bubbling thoughts that arose during the session with the students. Finally, all the furniture has been pushed multiple times and Turing realizes that Wittgenstein is looking for something. Turing picks up a piece of paper from Wittgenstein's writing table. On it is a sketch.

Wittgenstein notices what Turing is holding and he says, "What is the most wonderful thing for a mathematician?"

Turing replies, "Finding a simple solution to a problem that has been thought to be difficult."

"And how can anything first appear difficult and then be found to be easy?"

"You just find the right way to look at."

Turing lets the image of the duckrabbit flip between being a duck and a rabbit in his mind a few times then puts the paper on the desk. Wittgenstein is still looking around the room. "If searching your room fails to reveal your coat, what's another way of looking at THAT problem?"

"Turing, good man, I was just waiting for you to confirm the EXISTENCE of my coat. Now we can attack the problem of how to find it." Wittgenstein still can do nothing but look around the room.

Turing goes to the door and holds it open for Wittgenstein. "It's almost worth testing you to discover if you would stand there an infinite amount of time waiting for the coat to rematerialize, but I saw that your coat is downstairs in the common room when I came in."

They leave Wittgenstein's rooms and collect his coat then head out into the warm evening. Wittgenstein asks, "So have you agreed to dump your ridiculous Ph.D. project?"

A week before, Turing had described his meeting at Princeton with Gödel and Gödel's ideas for how to (or not) formalize the process of finding an infinite set of axioms that would be an infinite basis for mathematics. Wittgenstein had been outraged and had sputtered, "A complete waste of effort, Turing! An infinite set of axioms is oxymoronic. Let Gödel find some other lackey to do his dirty work. If you want to waste your time you should waste it on something that you want to do."

Turing had decided to formulate a proposal for a new Ph.D. topic. "What I need is a mathematics topic that can be solved by a computing machine."

Wittgenstein guffawed. "Applied mathematics at Princeton? Is that allowed?"

They were walking along River Cam towards the Botanic Garden and would end up at the newly opened Regal theater for the flick. "The Ph.D. project would be demonstrating an improved approximation method for the zeta function. Of course, my real goal is to do the actual calculations by machine."

They left the shore of the Cam and turned up towards the Gardens. Wittgenstein asked, "Is there really a different approach that you would take to the zeta function if you were going to have one of your machines do the calculations?"

Turing replied. "I would be happy to use an existing formula for zeta, but producing electronic circuits to perform the calculations and hold all of the calculated values during the calculation is a serious technical problem. An available trick is to use a Fourier transform and to put the problem into a form in which approximations could be made by analog computing components....sets of interconnected wheels. This would greatly simplify the amount of electronic components to be constructed."

They walked through the Garden in silence until Wittgenstein said, "I'm surprised that nobody has ever solved the Fourier analysis of the zeta function."

Turing shrugged, "It is not widely known that you can explore the distribution of prime numbers in terms of harmonic frequencies. The zeros of the zeta function constrain those frequencies."

Wittgenstein turned to Turing and put his fists on his hips. "Turing, you are becoming a scientist! Mathematicians can not prove the Riemann hypothesis so you have to build a piece of equipment and collect experimental data? Do the first 100 zeta zeros fit your prediction? Do the first 1000? No matter how long your calculator runs, you will never know ALL of the zeros. Its not even science. Its something between science and mathematics."

They started hurrying out of the Garden so as not to be late for the flick. Wittgenstein said, "What I want to know is if your computing machines will ever be able to perform a mental transformation. If you made a machine that could recognize a duckrabbit as a duck, would it ever be able to also see it as a rabbit? What changes in your mind when the image remains the same but your perception shifts? If your machines can not see both aspects, they will never do mathematics, they will just be calculating slaves. What is the formal system that defines creativity?"

Turing had been asking himself such questions with increasing frequency. He could only dimly conceptualize a 50 year process in the future during which computing machines would be built and tested, revealing the details of their mechanics and testing if they could replicate all of human thought.

They approached the theater and Wittgenstein elbowed Turing, "Creativity must be a secret agent. A secret agent in the brain."

Continue with Turing at Princeton university.

List of Pages in the Cambridge Computing alternative history