Can someone explain the cognitive logic of the math here?

[cdux]

Can someone explain the "cognitive" logic of the math here?

I find it impossible to concentrate or learn math without understanding the underlying meaning as much 'physically' as possible (I wonder if that's a problem or a virtue by the way).

I had to prove that 1/P(A) + 1/(P(A') >= 4 given that 0<P(A)<1

The solution goes on to prove it by concluding that (2P(A)-1)^2 >= 0 which is true after transformations on the initial formula (P(A') = 1 - P(A) if you're unaware of it).

Now, the thing is, how is that explained cognitively? What is the logical position of that last squaring there in the end? How does that "start" the logic of it (or ends it)?

I mean, I see the math doing it, but not the cognitive logic behind it. If someone does not know of the solution, is there a cognitive meaning that would derive it early?

Did someone just come up with the solution and did calculations to reach the initial to-be-proven formula? Does that mean it's just 'normal' to be 'unpredictable'?

 

[tiny-tim]

hi cdux!

…What is the logical position of that last squaring there in the end? How does that "start" the logic of it (or ends it)?

I mean, I see the math doing it, but not the cognitive logic behind it. If someone does not know of the solution, is there a cognitive meaning that would derive it early?

Did someone just come up with the solution and did calculations to reach the initial to-be-proven formula? Does that mean it's just 'normal' to be 'unpredictable'?

they simplified the original equation by getting rid of the fractions by multiplying by P(A)P(A')

that produced a quadratic in P(A), so they then completed the square

 

[HallsofIvy]

I find it impossible to concentrate or learn math without understanding the underlying meaning as much 'physically' as possible (I wonder if that's a problem or a virtue by the way).

Since mathematics, while it can be used in specific applications, does NOT necessarily have a "physical meaning", that is at least restricting concept. I certainly would not call it a "virtue".

I had to prove that 1/P(A) + 1/(P(A') >= 4 given that 0<P(A)<1

The solution goes on to prove it by concluding that (2P(A)-1)^2 >= 0 which is true after transformations on the initial formula (P(A') = 1 - P(A) if you're unaware of it).

Now, the thing is, how is that explained cognitively? What is the logical position of that last squaring there in the end? How does that "start" the logic of it (or ends it)?

I mean, I see the math doing it, but not the cognitive logic behind it. If someone does not know of the solution, is there a cognitive meaning that would derive it early?

Did someone just come up with the solution and did calculations to reach the initial to-be-proven formula? Does that mean it's just 'normal' to be 'unpredictable'?

 

[V0ODO0CH1LD]

Mathematics is a game. It is very complex, full of rules and tricks, but just a game. The fact that it applies to real life is irrelevant to mathematics itself. Don't confuse math with reality. Mathematics is a useful "language" for modeling real problems, because you can use algorithms and logic to get the answers and then translate whatever you get back to reality. It's like if you could turn a real life problem into an unsolved rubiks cube in a certain configuration. Then you solve that rubiks cube using the various tricks and algorithms and you interpret the solved rubiks cube as your solution somehow.

A lot of problems in mathematics are just visual puzzles. You don't check your errors in equations by rationalizing weather every step makes "physical" sense. You check by looking at it and going: what is this 2 doing here? where did it come from? It's almost as if you were looking for the errors in the aesthetics of your equation. Don't try to rationalize every thing you do in mathematics. Like complex numbers.. They're just part of the game. They have rules and properties that make them fit nicely in the whole of it. But in the end is just a complex and very fun game.

 

[Stephen Tashi]

I find it impossible to concentrate or learn math without understanding the underlying meaning as much 'physically' as possible (I wonder if that's a problem or a virtue by the way).

It's a problem if you are attempting to learn math "step by step" - i.e. understanding one thing "completely" before progressing to the next topic. For example, one way to understand this question is "simply" to apply calculus to investigate the minimum value of \frac{1}{x} + \frac{1}{1-x} on the interval [0,1]. You can take that point of view without having an intuitive grasp of everything that is involved in calculus. (So you can understand something using a mix of things you previously understand and things you don't yet understand.)

A certain amount of intuition is necessary (for me, at least) in order to adequately remember the various theorems and definitions involved in mathematics. But intuition is essentially a private matter and different people have different intuitions about the same mathematical facts.

I think some authors admire proofs that make us say "That's darn clever! Who would have ever thought of that?" and such proofs may give you useful "ammunition" for tackling other problems.

 

[Mandelbroth]

Now, the thing is, how is that explained cognitively? What is the logical position of that last squaring there in the end? How does that "start" the logic of it (or ends it)?

Math is the cognitive logic behind it.

Since mathematics, while it can be used in specific applications, does NOT necessarily have a "physical meaning"...

Well, of course it has physical meaning! I mean, I can cut a ball into 5 pieces and make 2 balls of the same volume as the original from those pieces. I don't know about the rest of you.

Mathematics is a game. It is very complex, full of rules and tricks, but just a game. The fact that it applies to real life is irrelevant to mathematics itself.

This says it all. Very poetic, too.

 

[cdux]

OK let's take math as the set of its rules and as a game. Do you see a way to predict the solution of that problem without trying to transform the to-be-proven inequality first? Or is the only way to do it to start the transformations and then when you reach a quadratic equation to say "hey! That's a quadratic and it happens to be >= 0".

I know people have different intuitions (I remember Feynman saying he was first surprised when he realized everyone thinks of simplistic concepts completely different than others) but I wonder how you'd do it practically in this case. ..Since we've concluded 'just play with it'.

 

[micromass]

OK let's take math as the set of its rules and as a game. Do you see a way to predict the solution of that problem without trying to transform the to-be-proven inequality first? Or is the only way to do it to start the transformations and then when you reach a quadratic equation to say "hey! That's a quadratic and it happens to be >= 0".

Depends on the situation. Very often, you already know what you want to prove before you actually do the manipulations. You may know this because of intuition or a physical problem you're trying to solve.

However, when you actually do the proof, it may indeed happen that you do some mindless computations and then end up with "hey, this gives a nice result!"

 

[cdux]

OK, and how do you think the inventor of the problem first set it up? Is it most likely that he started from the quadratic / the solution and reached 'backwards' the to-be-proven inequality?

 

[micromass]

OK, and how do you think the inventor of the problem first set it up? Is it most likely that he started from the quadratic / the solution and reached 'backwards' the to-be-proven inequality?

That is very plausible. An other way is that he "experimentally" saw that the inequality was true, and then proved it in the way you mentioned in the OP. There's no real way to know here. That's a bit a problem with mathematics that it's never mentioned how people came up with things.

 

[cdux]

That is very plausible. An other way is that he "experimentally" saw that the inequality was true, and then proved it in the way you mentioned in the OP. There's no real way to know here. That's a bit a problem with mathematics that it's never mentioned how people came up with things.

Hah, that's interesting. I assumed you learn it at a course.

 

[micromass]

Hah, that's interesting. I assumed you learn it at a course.

If the course is good, then you might learn things like that. But not always. And sometimes, there's no way to know.

 

[HallsofIvy]

Math is the cognitive logic behind it.


Well, of course it has physical meaning! I mean, I can cut a ball into 5 pieces and make 2 balls of the same volume as the original from those pieces. I don't know about the rest of you.

I'd like to see you do that! Cutting those non-measurable sets requires some really tricky scissor work!

 

[johnqwertyful]

If the course is good, then you might learn things like that. But not always. And sometimes, there's no way to know.

On a completely unrelated note, I see nothing to solve in your signature. There is no question.

 

[micromass]

On a completely unrelated note, I see nothing to solve in your signature. There is no question.

Better now?