The first senctence: > Am I correct in understanding that it's not that intuition is invalid, but that it's insufficient for communicating a proof to another m

The first senctence: > Am I correct in understanding that it's not that intuition is invalid, but that it's insufficient for communicating a proof to another mathematician. is close to correct. It's a very useful tool, but it can sometimes lead you into incorrect assumptions. However, the rest of it incorrect. Math, and in particular, probability and statistics, are full of things that seem intuitive and easy at first glance, but are actually a bit more nuanced. Take for example, the Monty Hall Problem. It's probably familiar to most here, but essentially you have three doors. Behind one, is a car or some other desirable object, and behind the other two are goats or something undesirable. Select a door, and you get whatever is behind that door. What's the probability of getting the car in this scenario? Easy enough right? It's just 1/3. However, if Monty Hall opens one of the doors after you select a door, he always reveals a goat, and he offers to let you switch your door now, should you switch or keep your door? What's the probability that you get a car if you switch? Is it 1/2 or 1/3? What's the probability that you get a car if you stay? 1/3? Well, the somewhat surprising answer is that you have a 2/3 probability of winning the car if you switch, and a 1/3 probability of winning the car if you stay. Often things will break down in the limits as well, so if you try to apply your finite dimensional intuition to something that corresponds to an empty object or an infinite number of objects, you'll find that you're often wrong.