Einstein said he went to his office at the Institute for Advanced Study “just to have the privilege of walking home with Kurt Gödel.”
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“So, who was Gödel?” Kate asked.

“All right, would you like me to tell you fun stories about Gödel? He was arguably the most celebrated logician of all time,” Father said.

“No, thanks. Better not,” George said.

“He can probably be considered as the most well-known logician of all time,” Father repeated, disregarding his son’s reply. “He was a member of the Vienna Circle. Do you know what the Vienna Circle was about?” Father asked.

“No,” Alex and George replied together.

“The Vienna Circle was a circle of intellectuals and young scientists of the time,” Father began. “Vienna, the capital of modern-day Austria, was at one point the capital of the Austro-Hungarian Empire and a major centre of culture and science. The Vienna Circle brought together young scientists who shared interest in philosophy, logic, mathematics and science in general. However, in the 1930s, Nazis became active in Vienna who certainly clamped down on clever, independent thinkers. The founder of the Vienna Circle Moritz Schlick was killed, and Gödel was afraid that he would share the same fate, and so he decided to flee to America. However, no passenger planes operated at the time, and he didn’t want to travel via the Atlantic Ocean.

“Why not?” Kate interrupted.

“The war was coming and he thought that it was not very safe,” Father replied. “So, he chose a very long journey. He travelled through enormous territory by a Trans-Siberian Railway, basically all of Europe and Asia. Then he sailed from Japan to San Francisco. That’s how it was,” Father picked up a globe from the shelf and showed them Gödel’s route with a finger.

“I’m actually surprised he was allowed to enter the Soviet Union, but never mind,” Father muttered. “He worked at the Institute of Advanced Studies in Princeton. Princeton is one of the leading scientific centres in the US. Do you know who his closest friend was at Princeton? Albert Einstein – a very famous physicist who created the theory of relativity.”

“Yeah, I’ve heard about Einstein,” George said.

“Apparently, as he was about to become a US citizen, Gödel said during the interview that he’d found a logical contradiction in the US constitution. It was good Einstein was close by and didn’t let him expand on this. Also, the judge who was to decide on Gödel’s citizenship, was apparently, as you’d say, Einstein’s fan. So, for Einstein’s sake, he didn’t pick on Gödel’s words. Otherwise, who knows, perhaps Gödel would have been left without citizenship. Do you know who else was Gödel’s friend?”

“Who?” the boys asked in unison.

“Oskar Morgenstern. Apparently, he too attended Gödel’s citizenship interview. Einstein and Morgenstern were Gödel’s companions during the interview.”

“Who was Morgenstern?” Kate asked.

“Morgenstern was an economist, the co-author and creator of the so-called game theory, together with John Von Neumann.”

“Did you say games?” Kate asked again.

“Yes, but not quite your kind of games. This is a mathematical theory which studies decision taking strategies under certain economic, political and military circumstances,” Father replied.

“That sounds rather complicated,” Alex said. “Anyway, Gödel seems to have had good friends. I wouldn’t mind having friends like that myself.”

“Trust me, he was just as good as them,” Father said. “Do you know what Einstein said in his old age? That he only went to work in order to be able to take walks with Gödel. Apparently, they often went out for walks for hours, conversing. Many people at Princeton apparently wondered what it was that that Einstein and Gödel talked about between themselves during their walks,” Father added.

“Einstein was awarded a Nobel prize, wasn’t he?” Alex asked.

“Yes,” Father replied.

“Did Gödel also get his Nobel?” George asked.

“No, he didn’t. By the way, mathematicians aren’t given Nobel prizes.”

“Why not? What have they done wrong?” Kate asked.

“There’s a legend that Alfred Nobel, the founder of the Nobel Prize, was in love with a woman who left him for a famous mathematician, so the angry Nobel decided that no mathematicians would be awarded his prize. However, this hasn’t been proven and is more like a legend than the truth. The real reason is probably that Nobel was less interested in mathematics.”

“Poor mathematicians,” Kate said softly.

“Mathematicians have their own prize called the Fields Medal which Gödel didn’t get either by the way. This prize is only given to mathematicians under 40 years of age, and some believe that when Gödel was that age, many people hadn’t yet realised the interesting and important discovery he had made.”

“Why is it given to under 40-year-olds?” Alex asked.

“To encourage talented young scientists. Otherwise, elderly mathematicians would get many such medals,” Father replied.

“Does that mean you’ll never get that medal then?” Alex continued.

“Yeah, that’s right,” Father murmured with a smile.

“And what did Gödel do?” George asked.

“Gödel arrived at lots of interesting results in logic, including you know what? The fact that even if you develop the strongest of computers in the world, there will still be facts the computer won’t be able to determine: so, we can’t learn everything about mathematics with the use of a computer,” Father said, moving onto a new topic.

“Not even with ChatGPT?” Alex asked. He’d only found out about ChatGPT a short while ago and really wanted to show off his knowledge.

“Not even that. In logic and mathematics, ChatGPT isn’t proving to be too competent but even if it was, it wouldn’t have fared much. For example, let’s take the natural numbers 0, 1, 2, 3, 4 and so forth. How many such numbers are there?

“Oh, I know that,” George said. “Infinite. Because even if it stops somewhere, you add one and it will increase. So, it can’t possibly be finite.”

“That’s absolutely right. There will always be some true statement about these numbers which none of the computers, even the most powerful ones, will be able to deduce,” Father continued.

“Yeah, but why don’t we teach the computer that statement. Then it would know it and that would be the end of it,” Alex said, probably still under the influence of ChatGPT.

“It’s the right thinking but even if you teach it this statement, Gödel’s ingenious technique will find another new statement that the computer doesn’t know,” Father said, looking pleased.

“Is that good or bad?” Kate asked.

“Depends on how you look at it. It’s good in a sense that computers can’t do everything. We also need ourselves. It’s bad in that we can’t delegate everything onto a computer. But I suppose that is good too. In general, ChatGPT is already an example of how much computers and artificial intelligence can do. By the way, many scientists believe that artificial intelligence needs to learn much more logic. It should be able to get better at logical reasoning for creating high quality output. Then it can get better at solving mathematical problems,” Father ended with a long answer.

He probably did that on purpose so that he could take a little break from the children’s questions. But in the end, he just moved on to a slightly different topic.