Good point, I should have clarified this more. I’m not saying that people shouldn’t know how to calculate the area and circumference of a circle as people may actually use that. It’s more to do with all the things to do with tangents and chords and shapes inscribed in circles.
1. Passing tests—in a geometry class, taking the ACT, (I don’t know, maybe it’s a part of getting a GED).
2. Your interest in geometry is not merely theoretical, but practical. Maybe you construct things, perhaps out of wood using power tools. (You may find it useful to design/implement a coordinate system on a piece of wood to assist with getting the dimensions of things right, as you cut them out with a saw. Someone may have already invented this.) If you are trying to find the area under a curve, you may find it useful to buy very fine, high quality paper, graph the shape of the curve, and weight it, and use the average wight of the paper per inch or centimeter (squared) to find the answer. (This relies of the material being consistent through out, and weighing about the same everywhere.)
3. Despite your claims that you would never use math, or this part of math, someday you find yourself* designing a dome, or even a half sphere, perhaps as a place to live. The floor plan is a circle.
4. You enjoy math. You enjoy learning this/using this knowledge on puzzles to challenge your wits. (See 6)
5. You end up as a teacher, assistant, or tutor. The subject is math. (Perhaps you realize that not every geometry student that will one day teach geometry is aware of this fact.) Whether or not you learned all the fancy stuff the first time, if you didn’t retain it you have to learn it again—well enough to teach it to someone that doesn’t like the subject as much as you—and you hated geometry (class). (It was required.)
6. You learn visual calculus. Other mathematicians may compose long, elaborate arguments that they publish in papers that may take days to decipher (that seem to push the world ever closer to proofs people can’t read, but computers have apparently checked—or been used to produce). Perhaps your proofs employ no words, but consist of a picture instead, that employs esoteric knowledge (such as that of tangents and chords) to solves problems beautifully (and quickly). Perhaps this profound knowledge makes you both a better mathematician, and a better teacher of math.
In summary:
It’s arbitrary, but it’s part of the curriculum that’s tested on. (However students or teachers feel about it.)
You use the knowledge—specifically*, or generally**.
You enjoy learning, or a challenge.
Teaching.
“Higher math.” (You can see more up here, but the air is thinner. You feel kind of dizzy.)
*Your life has turned into a geometry problem. Is this hell?
**The square-cube law is general and useful. I think the second property is a result of the first.
That doesn’t mean your view can’t be correct. It’s as true as you are claiming to be. The claim is that it’s difficult to determine whether there’s actually a law of physics about how to deal with quantum mechanics.
If there wasn’t, then you would be far wrong. If there were, then either you and I would have different opinions. But what I would be proposing is a way for our disagreement about what ‘true’ means: that we should not be too confident or too skeptical about other people’s points on the theory, which could give us an overly harsh criticism, or make us look like the kind of fool who hasn’t yet accepted them yet.
I think the correct answer to this problem would be a question of how confident are we that the point being made is the correct point? It seems obvious to me that we have no idea about the nature of the dispute. If I disagree, then I think I’ll go first.
If a question is really important and it comes down to the point of people saying “I think X” then it ought to come down to the following:
“I think X is true, and therefore Y is true. If we disagree, then I don’t think X is true, and therefore Y is true.”
In this case, if we had the same thing, but also had a different conversation (as in with Mr. Lee’s comment at the end of the chapter), our disagreement could be resolved by someone else directly debating the point (we could debate the details of this argument, if they disagree).
In other words, we are all in agreement that we should be confident that we have considered the point, but it’s better to accept that we’re making a concession. But the point is that we know we shouldn’t be confident that it’s an argument that we would not be confident would work, or that we shouldn’t be confident about it.
In all cases, this is the point that it often seems to be getting.
This may seem like a pretty simple and non-obvious argument to me, but it is. And it seems the point was that there are many situations where you and some of your friends agree that the point should be resolved and that it’s reasonable to agree that the point should be fairly obvious so the disagreement seems to be a bit more complicated.
I read somewhere that there’s a norm in academia that it should never be controversial for a student to
Good point, I should have clarified this more. I’m not saying that people shouldn’t know how to calculate the area and circumference of a circle as people may actually use that. It’s more to do with all the things to do with tangents and chords and shapes inscribed in circles.
Possible uses:
1. Passing tests—in a geometry class, taking the ACT, (I don’t know, maybe it’s a part of getting a GED).
2. Your interest in geometry is not merely theoretical, but practical. Maybe you construct things, perhaps out of wood using power tools. (You may find it useful to design/implement a coordinate system on a piece of wood to assist with getting the dimensions of things right, as you cut them out with a saw. Someone may have already invented this.) If you are trying to find the area under a curve, you may find it useful to buy very fine, high quality paper, graph the shape of the curve, and weight it, and use the average wight of the paper per inch or centimeter (squared) to find the answer. (This relies of the material being consistent through out, and weighing about the same everywhere.)
3. Despite your claims that you would never use math, or this part of math, someday you find yourself* designing a dome, or even a half sphere, perhaps as a place to live. The floor plan is a circle.
4. You enjoy math. You enjoy learning this/using this knowledge on puzzles to challenge your wits. (See 6)
5. You end up as a teacher, assistant, or tutor. The subject is math. (Perhaps you realize that not every geometry student that will one day teach geometry is aware of this fact.) Whether or not you learned all the fancy stuff the first time, if you didn’t retain it you have to learn it again—well enough to teach it to someone that doesn’t like the subject as much as you—and you hated geometry (class). (It was required.)
6. You learn visual calculus. Other mathematicians may compose long, elaborate arguments that they publish in papers that may take days to decipher (that seem to push the world ever closer to proofs people can’t read, but computers have apparently checked—or been used to produce). Perhaps your proofs employ no words, but consist of a picture instead, that employs esoteric knowledge (such as that of tangents and chords) to solves problems beautifully (and quickly). Perhaps this profound knowledge makes you both a better mathematician, and a better teacher of math.
In summary:
It’s arbitrary, but it’s part of the curriculum that’s tested on. (However students or teachers feel about it.)
You use the knowledge—specifically*, or generally**.
You enjoy learning, or a challenge.
Teaching.
“Higher math.” (You can see more up here, but the air is thinner. You feel kind of dizzy.)
*Your life has turned into a geometry problem. Is this hell?
**The square-cube law is general and useful. I think the second property is a result of the first.
That doesn’t mean your view can’t be correct. It’s as true as you are claiming to be. The claim is that it’s difficult to determine whether there’s actually a law of physics about how to deal with quantum mechanics.
If there wasn’t, then you would be far wrong. If there were, then either you and I would have different opinions. But what I would be proposing is a way for our disagreement about what ‘true’ means: that we should not be too confident or too skeptical about other people’s points on the theory, which could give us an overly harsh criticism, or make us look like the kind of fool who hasn’t yet accepted them yet.
I think the correct answer to this problem would be a question of how confident are we that the point being made is the correct point? It seems obvious to me that we have no idea about the nature of the dispute. If I disagree, then I think I’ll go first.
If a question is really important and it comes down to the point of people saying “I think X” then it ought to come down to the following:
“I think X is true, and therefore Y is true. If we disagree, then I don’t think X is true, and therefore Y is true.”
In this case, if we had the same thing, but also had a different conversation (as in with Mr. Lee’s comment at the end of the chapter), our disagreement could be resolved by someone else directly debating the point (we could debate the details of this argument, if they disagree).
In other words, we are all in agreement that we should be confident that we have considered the point, but it’s better to accept that we’re making a concession. But the point is that we know we shouldn’t be confident that it’s an argument that we would not be confident would work, or that we shouldn’t be confident about it.
In all cases, this is the point that it often seems to be getting.
This may seem like a pretty simple and non-obvious argument to me, but it is. And it seems the point was that there are many situations where you and some of your friends agree that the point should be resolved and that it’s reasonable to agree that the point should be fairly obvious so the disagreement seems to be a bit more complicated.
I read somewhere that there’s a norm in academia that it should never be controversial for a student to