At what age did the idea that people could solve problems through science and thought become popular

The scientific revolution occurred between 1500 - 17000.
During this period, people began to discard the idea of assumption and began to embrace new idea that encouraged drawing conclusions from observations made
In the middle 1500, some scholar published papers that antagonized old way of thinking and making assumptions. The scientific revolution established a pattern of thinking about the natural world through observations. It also encouraged people to examine and challenge accepted beliefs.

Explanation:

This was part of Goring's speech when the Luftwaffe first started dropping bombs on London. Ironically it all started with an error which turned into a history turning moment.

A lost Luffwaffe bomber accidentally dropped its bombs on London. This had been prohibited by Hitler. The Brits responded by dropping bombs on Berlin. This infuriated Hitler so much he ordered London be bombed. As the German bombers flew toward London, Goring gave his speech which included the phrase you quoted.

Ironically the bombing of London, which was terrible for the city, gave the RAF airfields a break from being hit repeatedly so they regained strength to keep on fighting the Germans.