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Down the Rabbit Hole

By Daniel Hubbard | October 16, 2011

By all accounts, Kurt was a strange man, insane eventually. He was paranoid and his life came to an end when he starved himself, fearing that he would be poisoned.

Kurt was also brilliant. Not many people took their evening stroll from work at The Institute of Advanced Study with Albert Einstein.

Kurt was a mathematician and produced what may be the most intriguing piece of mathematics in the last hundred years.

Self-reference, a very odd thing

There is a fascinating process of self-reference in complex research. Last week I mentioned using evidence against itself. Based on what we know, we find new information. That new information “sheds new light” on the old information, perhaps even proving some of that old information to be wrong. But wait, didn’t we use the old information that we apparently didn’t understand to find the new information that disproved the old? Yes, and that is where things can get a little strange if you stop to think about it. They become strange because of self-reference.

When doing research we need to go back over it—make sure that the old stands up to the new. Some of the old may need to be adjusted, reinterpreted or discarded. I like to ask if, with my new understanding of the old information, would I perform the search differently? If I would, then is it possible that something more might be found with that somewhat different search? Should I look at the new evidence from a different angle?

People are usually fascinated by self-reference. Children love the parallel mirrors at a barbershop. Each reflection containing a reflection of itself. I remember the first time I sang “Row, Row, Row Your Boat” with my children as a round. There was an almost stunned look on their faces. They probably only realized that it sounded good when it really shouldn’t. They couldn’t have known that what they were hearing was a melody that contains its own harmony within itself.

Perhaps the most famous example of self-reference is the liar’s paradox. There are many ways to formulate it. Here is one “This sentence is a lie.” Clearly, if the statement is true then it says it is a lie, which means it is, by definition, false making it a lie and therefore true… and down the rabbit hole we go.

Research often needs to loop back upon itself many times, narrowing in on correct answers or reaching a contradiction that says, “time to rethink this.” The data allows us to form a hypothesis. A hypothesis helps us to understand the data and gather new data. The new data may change the hypothesis and alter our understanding of the old data. Often getting to the truth in research is a process of using new results to analyze their own origins. Genealogist don’t normally run the risk of falling down some rabbit hole of paradoxes but we do need to know how research works. It can be a simple linear process if the records are almost perfect and the digging doesn’t go too deep. If the records are poor or the digging very deep and detailed, then there is a need to go back over what we think we know. Does an old conclusion need changing? Would an old search done with new knowledge have yielded different results? Where do those results lead and what do they have to say about our old information and conclusions? I think one of the joys of complicated research is to constantly discover, reevaluate, discover more and go on, aware that there will be limits to what one can know yet having the privileged of pushing those limits.

Back to Kurt

The liar’s paradox actually brings us back to Kurt Gödel. In the early twentieth century some mathematicians thought that they were on the verge of being able to derive the theory of numbers from a few basic statements (axioms) and formal logic. The feeling had been that mathematics could do many amazing things but that its foundations were weak and contradictions and inadequacies were being found. Think of the Taj Mahal balanced on toothpicks. That is how they saw things at the time. Slowly but surely, some mathematicians began the work of establishing the formal foundations of mathematics.

Kurt Gödel proved that any mathematical system that was rich enough to be used for carrying out arithmetic could be turned back against itself using something like the liar’s paradox. He found a way to show that you could always form the mathematical equivalent of the statement “This statement is not provable.” If it is false then it is provable, and a provable statement that is false is a contradiction, a very bad thing in mathematics. So, it must be true, except that then it is, according to itself, not provable, so we have a true statement that cannot be proven—exactly the problem that people had hoped to avoid. Self-reference, at least in number theory, won the day. Mathematics didn’t fall down the rabbit hole but the hope had been that there was no rabbit hole at all.

A Migratory Aside

There is a much more concrete reason that Kurt Gödel is genealogically interesting. He was born in what was then Austria. He lived out much of his life in Princeton. He and his wife arrived in the U.S. via California, an odd place to arrive for a couple leaving Austria for New Jersey. In this case history is the key to a very odd migration route.

By 1940, Nazi Germany had annexed Austria, conquered Poland and was at war with England and France. The North Atlantic was a very dangerous place. However, Germany and the Soviet Union had cooperated on the partition of Poland and there was still peace and safety in the east. As an academic living in Vienna, Kurt naturally had many Jewish associates and that was enough for life to become very uncomfortable.  He and his wife, Adele, reached Moscow, took the Transsiberian Railroad across Russia and eventually boarded a boat in Yokohama, Japan. There was still peace in the Pacific and the couple arrived safely in California. By the time they reached New Jersey, they had traveled three-fourths of the way around the world. Knowing the history makes the bizarre quite reasonable.

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