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It always saddens me how mathematicians seem to look down on “intuition”. Maybe higher math as a full-time job is just hard work and stubborn precision but for

by nothis 9y ago

It always saddens me how mathematicians seem to look down on “intuition”. Maybe higher math as a full-time job is just hard work and stubborn precision but for me, an intuitive, visual look is probably getting me closer to understanding than any cold-hard-facts book does. Not to mention how much easier it is to appreciate the beauty of it.

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.
curuinor 9y ago
what the hell kind of mathematicians have you been talking to, if they look down on intuition?
eigenman 9y ago
Mathematians (such as myself) don’t look down on intuition. It is the only way to construct a map through a complex proof. However what you may have picked up on is we disstain people who insist they have a brilliant idea for which someone should prove they are right. That’s like saying a car engine consists of cyclinders and piston; someone else just needs to do the work to put the pieces together.
gracenotes 9y ago
You can take it from one of the most productive living mathematicians: https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/theres-more-to-... I recommend reading the entire post - it's not too long, and links to other great materials - but this is a good relevant quote: > The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition.
dvt 9y ago
> It always saddens me how mathematicians seem to look down on “intuition”. They do this for good reason. I had to take, as part of my philosophy concentration, a bunch of mathematical logic classes. First Order and Second Order logic, as you might imagine, are pretty simple. The rules make sense in a very intuitive way. I'd get 100 on exams just because I'm good at programming. But when I had to study metalogic, model theory, Henkin Proofs, and Godel's Theorems, that intuition quite literally flies out the window. I guess my point is that most interesting stuff is rarely intuitive.
- goldenkey 9y ago
  That is ludicrous. The intuition is just harder to find, harder to grasp. But it is still there because these theorems are logical consequences of the base axioms. And the base axioms are logical consequences of our desire for sensible assumptions.
  - lostpathfinder 9y ago
    Working mathematician does derive his ideas from axioms, it is ridiculous perception of mathematical process. And axioms are not logical consequences of sensible assumptions, it is try and fail process, long historical process. Intuition in mathematics has nothing to do with common sense intuition. Respectfully.
    - goldenkey 9y ago
      In abstract mathematics, one can choose any axioms one wants. To derive all kinds of consequences/theorems. But respectfully, ZFC came about because of logical paradoxes that couldn't be accepted as consistent. You can create a new number system and derive all kinds of consequences but the truth is, most mathematicians care more entirely about prime number theory on the naturals that are entirely based on counting. Most of modern math is based on the natural numbers. You can't remove all intuition. The thread that holds our love for math is also the same one that tells us we are exploring consequences that tell us a dearth about our universe.
  - 9y ago
gowld 9y ago
> It always saddens me how mathematicians seem to look down on “intuition”. citation?
adyavanapalli 9y ago
You'll find that no such thing is true. Intuition is something you develop over time in mathematics, and mathematicians will often use this intuition to have certain "feelings" about problems e.g. a feeling that a certain conjecture must be true.

by nothis 9y ago

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.

curuinor 9y ago

what the hell kind of mathematicians have you been talking to, if they look down on intuition?

eigenman 9y ago

Mathematians (such as myself) don’t look down on intuition. It is the only way to construct a map through a complex proof. However what you may have picked up on is we disstain people who insist they have a brilliant idea for which someone should prove they are right. That’s like saying a car engine consists of cyclinders and piston; someone else just needs to do the work to put the pieces together.

gracenotes 9y ago

You can take it from one of the most productive living mathematicians: https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/theres-more-to-... I recommend reading the entire post - it's not too long, and links to other great materials - but this is a good relevant quote: > The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition.

dvt 9y ago

> It always saddens me how mathematicians seem to look down on “intuition”. They do this for good reason. I had to take, as part of my philosophy concentration, a bunch of mathematical logic classes. First Order and Second Order logic, as you might imagine, are pretty simple. The rules make sense in a very intuitive way. I'd get 100 on exams just because I'm good at programming. But when I had to study metalogic, model theory, Henkin Proofs, and Godel's Theorems, that intuition quite literally flies out the window. I guess my point is that most interesting stuff is rarely intuitive.

goldenkey 9y ago
That is ludicrous. The intuition is just harder to find, harder to grasp. But it is still there because these theorems are logical consequences of the base axioms. And the base axioms are logical consequences of our desire for sensible assumptions.
- lostpathfinder 9y ago
  Working mathematician does derive his ideas from axioms, it is ridiculous perception of mathematical process. And axioms are not logical consequences of sensible assumptions, it is try and fail process, long historical process. Intuition in mathematics has nothing to do with common sense intuition. Respectfully.
  - goldenkey 9y ago
    In abstract mathematics, one can choose any axioms one wants. To derive all kinds of consequences/theorems. But respectfully, ZFC came about because of logical paradoxes that couldn't be accepted as consistent. You can create a new number system and derive all kinds of consequences but the truth is, most mathematicians care more entirely about prime number theory on the naturals that are entirely based on counting. Most of modern math is based on the natural numbers. You can't remove all intuition. The thread that holds our love for math is also the same one that tells us we are exploring consequences that tell us a dearth about our universe.
- 9y ago

gowld 9y ago

> It always saddens me how mathematicians seem to look down on “intuition”. citation?

adyavanapalli 9y ago

You'll find that no such thing is true. Intuition is something you develop over time in mathematics, and mathematicians will often use this intuition to have certain "feelings" about problems e.g. a feeling that a certain conjecture must be true.