How does one “invent new maths”? Like Isaac Newton inventing Calculus, or John Napier logs. How does one answer a mathematical question that’s never been answered?
In: Mathematics
Here is a story from the 20th century. Hilbert, optimistic as usual, ask for people to find an algorithm to solve all logical problems. This is an interesting question that people want an answer to.
The issue is, that sounds impossible. Unfortunately, there is no ways to prove that it’s impossible, because the concept of algorithm have not been invented. That doesn’t mean people don’t know about algorithm at all; but more like, the ideas of algorithm is too vague. The same ways words in normal language are vague; if someone ask you “what is the precise definition of ‘porn’?” you might just answer “I don’t know, but I know it when I see it” (actual quote from a judge, btw, it’s a famous case).
For many other problems in the past, if people want an algorithm, you just give an algorithm, and everyone can check and agree that this is an algorithm. But to show that an algorithm doesn’t exist, you need to know exactly what is an algorithm. So Hilbert’s question is not a complete question, you have to fill in the definition of an algorithm to make it a complete question.
Turing and Church did this. They invented the concept of computability, to answer the question of what is an algorithm. What they invented isn’t a proof. They make concise and precise their intuitive idea of algorithm, and offer it to other people. Whether people agree or not, the key thing is, there is at least *a* definition. With this, Turing and Church had finally form a new question that nobody asked before. Then they show that the question have no answers, there are no algorithm. This is the start of computer science.
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