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There's nothing wrong with intuition, but intuition does not write proofs. It can help you write proofs, but you need more than that. To pick an example: Intu

by pflats 9y ago

There's nothing wrong with intuition, but intuition does not write proofs. It can help you write proofs, but you need more than that. To pick an example: Intuition is great for understanding the intermediate value theorem. If the temperature was 40 degrees at 8 am, and is 60 degrees at noon, it must have been 50 degrees somewhere in between. Why, though? How can you guarantee that there will always have to be a time where it was 50 degrees[1]? [1] provided that temperature is real-valued and temperature change is continuous, the debate of which I'll leave to the science people

tkzzbneig 9y ago
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. What I believe to be true is that a solid intuitive proof will be sound, but not necessarily transcribable until formalized. Formalization of the proof is essentially the matter of making a proof communicable. Putting it one more way: purely-intuitive proofs would be just fine if one could share one's mental state with another. What do you think?
- JHonaker 9y ago
  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.
- mkl 9y ago
  Not who you're asking, but I think that is incorrect. Intuitive explanations are how mathematicians usually communicate proofs to each other in person, when discussing mathematics. The receiving mathematician is convinced of the proof's validity if their own intuition and experience tells them that the nitty-gritty details can be formalised correctly.
- gowld 9y ago
  Intuition is not proof. We all have many intuitive ideas that are not actually true!
- gizmo686 9y ago
  >purely-intuitive proofs would be just fine if one could share one's mental state with another. No, I have had purely intuitive proofs myself that turned out to be wrong.

by pflats 9y ago

tkzzbneig 9y ago

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. What I believe to be true is that a solid intuitive proof will be sound, but not necessarily transcribable until formalized. Formalization of the proof is essentially the matter of making a proof communicable. Putting it one more way: purely-intuitive proofs would be just fine if one could share one's mental state with another. What do you think?

JHonaker 9y ago
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.
mkl 9y ago
Not who you're asking, but I think that is incorrect. Intuitive explanations are how mathematicians usually communicate proofs to each other in person, when discussing mathematics. The receiving mathematician is convinced of the proof's validity if their own intuition and experience tells them that the nitty-gritty details can be formalised correctly.
gowld 9y ago
Intuition is not proof. We all have many intuitive ideas that are not actually true!
gizmo686 9y ago
>purely-intuitive proofs would be just fine if one could share one's mental state with another. No, I have had purely intuitive proofs myself that turned out to be wrong.