" What is it that lets them put their faith in the model given that the model is something purely mathematical and not instinctual? "

The counter question should be asked. Your question implies that instinct is a better way to develop trust in knowledge rather than logic and mathematics? Why do you believe so? Can you appraise your own "knowledge base" and determine how much of what you believe you know, you developed through actual observation and consideration? How much of what you think you know is "borrowed" from someone else's experience and knowledge? Why do you trust your knowledge in this circumstance? How do you make this evaluation?

One example, most people "know" that 1+1 = 2. I am confident that most people don't know how to prove this. Yet they believe this to be true? Why? It cannot be instinct, surely?