Another Semester, Another Sh*t Show
I don't know why you feel you have to defend proofs, they're pointless.
you are dismissing them, possibly out of frustration, too easily. I hate proofs too, but I can't ignore their value at times. And as useful as they are, I can live on without having to do any more of them in relative bliss.
you are dismissing them, possibly out of frustration, too easily. I hate proofs too, but I can't ignore their value at times. And as useful as they are, I can live on without having to do any more of them in relative bliss.
| They were believed because the people who read them didn't fully understand them |
This is pretty dodgy. This is true of every claim ever made. This would make a proof no more convincing than any old claim that seems to have reasoning behind it. To me, the point of a proof is to PROVE something. If a proof seemingly proves something but then turns out it didn't, then the proof has lost all meaning.
| The problem is "how do I prove P?" The solution is the proof of P. You can check the solution by following each step in the reasonining of the proof looking for fallacies, but you don't need to prove P yourself to check the proof. |
If you don't understand P, then how can you know if the proof accurately relates to P? You can check the whole proof and see that it's logical, but then someone who actually understands P will realize that the proof doesn't relate to P or doesn't fully prove P by nature.
| I think this is a valid proof |
Thanks, I did too. Apparently the TA didn't feel the same.
| I'm sorry you don't see their value, but they're certainly not pointless. Even for non-mathematicians, they're good training to express ideas clearly and to exercise mathematical rigor. |
Maybe I exaggerated my claim a tad. I USE proofs. I'll sometimes in a math test find a pattern, and then I'll try to prove whether or not this pattern always exists, to test whether or not I can use the pattern in my favor and it wasn't a coincidental pattern. In an argument, I may have to prove something to myself to make sure it's a rule and not a one time fluke. I use proofs to MY OWN STANDARDS. Whatever I'm doing in this class is meaningless. The proofs themselves usually carry no meaning to anyone who'd read it, and the proofs simply seem to have no value at all.
For example, the proof I did with the GCD and P/A, I wrote it in a way that follows a logical progression, it makes sense. Anyone who reads it will understand. Not all proofs have to be this way (or CAN be this way). It makes them pointless most of the time. I can think of a few rare scenarios where they may be useful, but most (if not all the time) proving something to my own standards is more than enough.
| I can live on without having to do any more of them in relative bliss. |
Hopefully that'll be me when I'm taken out in the next school shooting.
| This is pretty dodgy. This is true of every claim ever made. This would make a proof no more convincing than any old claim that seems to have reasoning behind it. |
| To me, the point of a proof is to PROVE something. |
This confusion was one of the critiques to the last proof of the four color theorem. "How can we know that the computer didn't encounter an error in its execution of the proof? Even if you run it a thousand times, that doesn't assure us that no errors happened." The implicit assumption is that human brains are infallible, which is grossly incorrect.
| If you don't understand P, then how can you know if the proof accurately relates to P? |
| Hopefully that'll be me when I'm taken out in the next school shooting. |
| I said it right at the beginning. A proof is just an argument in favor of a statement. |
The thing about proofs is that it has its own standards as if those mean something and make a proof indestructible. However, a proof isn't anymore valid than a proof by other standards that don't require the same pretentiousness. This is my issue with them, that these proofs aren't good for all situations and seem only helpful in rare cases.
| But what exactly would that entail? When you write a correct proof the paper doesn't glow or anything, you know? |
Yes, so calling them "proofs" seems pretentious. The more we dig, the more pretentious it gets.
| If you agree this is the only way to know if a proof is correct, you need to accept that some proofs will be incorrectly believed to be true, and some incorrectly believed to be false. |
The standards for when to use these mathematical proofs shouldn't be "whenever!" Why would you need to ever prove that the GCD between a prime number and another number is 1 only if the prime doesn't divide the other number? When exactly will proving such an obvious statement do anything for anyone?
The issue I find is that if a proof is correct, then likely there were many other standards that could have been used to prove it. A proof, again, seems only useful in a handful of rare cases. It's just so pretentious it pisses me off. I hate using the word since the word itself is kind of pretentious, but I don't know how else to describe it.
| "How can we know that the computer didn't encounter an error in its execution of the proof? Even if you run it a thousand times, that doesn't assure us that no errors happened." The implicit assumption is that human brains are infallible, which is grossly incorrect. |
I'll be sure to have this one handy when the professor says my proof is wrong.
| Some proofs involve clever and unintuitive transformations that maybe even the person who wrote it doesn't know how they came up with them. But you don't need to know that to know if the proof is valid. As long as the reasoning used is valid, the proof should hold up. |
I understand what you mean, I don't understand how someone can validate a proof is they can't verify how someone went from one step to the next. For example, one the proofs in class was easy to follow, then suddenly... BAM! Where the hell did that step come from!? Unintuitive and crazy transformation. But if he doesn't explain he did that, how can anyone validate whether that's a correct next step?
| Jeez, this has taken a turn... |
I remember high school, someone came with a gun and was loading it in the bathroom when he was caught. My area gets a bit freaked out by these things since its happened here quiet a few times.
| The thing about proofs is that it has its own standards as if those mean something and make a proof indestructible. A proof has its own axioms and they do make the proof indestructible. That's the whole outcome of a proof. |
Two points arise from your misunderstanding.
1. The rules of (predicate) logic may or may not be immutable, but so far there has been nothing found by human endeavour that the rules are not universal and not immutable. I suppose that means good luck to you if you ever find otherwise, in which case you will be in the history books as a founder of a whole new direction for human thought and civilisation.
2. Change the axioms as much as you want - it's been done plenty of times before - just ask Euclid about his geometry. Trouble is though, given someone else's proposition to prove make sure you mutually agree/accept the axioms.
| However, a proof isn't anymore valid than a proof by other standards that don't require the same pretentiousness. |
| This is my issue with them, that these proofs aren't good for all situations and seem only helpful in rare cases. |
| The thing about proofs is that it has its own standards as if those mean something and make a proof indestructible. However, a proof isn't anymore valid than a proof by other standards that don't require the same pretentiousness. |
| I'll be sure to have this one handy when the professor says my proof is wrong. |
| I understand what you mean, I don't understand how someone can validate a proof is they can't verify how someone went from one step to the next. |
I love cplusplus.com
The only place you can find a flamewar over the usefulness of proofs in a thread topic about how useless college is.
Proofs are like many mathematical studies: useful for expressing an idea in a firm manner. Useful to practice and train the mind to be more detail oriented in all aspects. Useful if you want to win a fight against an equally asinine anally-retentative opponent. A tool to be used in the appropriate circumstances.
Mostly useless and unused by most people outside of its one-semester scope in the classroom.
edit:
That. That made me laugh, thank you.
The only place you can find a flamewar over the usefulness of proofs in a thread topic about how useless college is.
Proofs are like many mathematical studies: useful for expressing an idea in a firm manner. Useful to practice and train the mind to be more detail oriented in all aspects. Useful if you want to win a fight against an equally asinine anally-retentative opponent. A tool to be used in the appropriate circumstances.
Mostly useless and unused by most people outside of its one-semester scope in the classroom.
edit:
| However, a proof isn't anymore valid than a proof by other standards that don't require the same pretentiousness. |
| Given you've had a lot of trouble clearly saying what you mean, I think the only course now is to prove your assertion. |
That. That made me laugh, thank you.
Last edited on
| Given you've had a lot of trouble clearly saying what you mean, I think the only course now is to prove your assertion. |
I've stated myself clearly. If you want me to prove that they're useless, well, by what standards? My standards would say this: If a proof doesn't give you new information or proves something that was already known - then it was meaningless. Since this is the vast majority of proofs, most proofs are pointless - proof over ;)
| A proof of that would be good here too. A rare case indeed. |
Rare in this case would refer to the small minority of cases where it may be useful. For example, the proof of the hand multiplication thing that Helios uploaded. That proof might be useful in truly understanding why it works. Most proofs don't do that, and they only seem useful when a more intuitive explanation is harder to find.
| The point of rigor and formality is to make the proof indisputable. When you skip steps or are less strict you run the risk of someone (such as a teacher ;-)) not accepting the proof. |
I didn't get an F on my test because my proofs were inadequate, I got an F because he wanted certain guidelines followed which he only gave afterwards. I got full points for many proofs - doesn't mean that the standards of a mathematical proof aren't any less pretentious.
| They'll say that your brain is no less fallible, but theirs has the advantage of experience over yours. |
I could easily say that they've aged, and the reliability of their brains has diminished while mine is still fresh. Since we're going by useless arguments, I may as well tell them that their brains exist within a simulation I created.
| That's the thing. You're confusing "verify" and "replicate". You can verify. All you need to do is look at the transformation and decide whether it's valid. It's okay while reading a proof to not understand why the author decided to perform a step, all that matters is whether the step is fallacious. |
Again, this makes no sense. If you verify that the proof has no contradictions and every step can logically come from the one before it, you still have no idea if they've tried all the needed test cases for the proof or if they've even correctly chosen how to prove it. You'd only be able to say that the proof is valid, but not whether or not it's correct.
| Mostly useless and unused by most people outside of its one-semester scope in the classroom. |
Two semesters for my degree. Double the uselessness.
| you still have no idea if they've tried all the needed test cases for the proof |
| You'd only be able to say that the proof is valid, but not whether or not it's correct. |
| "Test cases"? It's a proof, not software testing. Only proofs by exhaustion test the entire (usually small) problem space. For example, "prove that the first 2^32 naturals have P property". |
It was only an example, but test cases are used for some of our in class-proofs.
| If the proof is valid and the premises are true, then the proof is correct; that is, the statement being proven is actually true. I don't understand how you can be proving things and not know this. |
I think you're completely misunderstanding my point. Let's say someone writes a proof for X. My argument from the start is that you can read someone's proof and not at all know not only how they got to it, but if it's valid. To go from one step to the next, you're not required to show how. Someone can look at the proof and not know if what they did was valid because they don't understand how they got there. Moreover, if you don't understand what's required to prove X, how can you know if the proof adequately does so? Maybe a variable was missed? A test case was missed? How would you know? The point is that you can't, and some proofs are accepted, as you said, before it's found out later that something WAS actually missed or wrong.
| To go from one step to the next, you're not required to show how |
The idea there is that the writer of the proof should have the proof for that step in their arsenal, ready to be displayed if asked how they made the jump. It's kind of like how libraries work in coding. You call a function because you know how to use it and your audience generally understands its use. You know that given the 5 arguments input in the call are correct, you will get a usable output. You might have to reference other people's libraries and have to trust that they have done the work properly, but if someone asks why/how you used a specific function call then you can send them to the documentation.
You can't call a function that doesn't have a definition. You can create your own functions so long as they are built using the correct building blocks, otherwise they are gibberish.
Just saying 5X = X + X + X + X + X because multiplication is basically expanded addition did not work because you called a function that didn't have a definition. A test is like when the compiler tries to put everything together and the teacher is spitting out errors about missing references.
Unfortunately the teacher has the ability to say something like "yeah, that step is true, but I don't think this student actually built it based on the tools currently available, so the answer is invalid." So a show-your-work approach is recommended whenever it is within reason.
This is the mindset I had to take in order to get through the class.
Last edited on
| if someone asks why/how you used a specific function call then you can send them to the documentation. |
Here's my documentation for 5X = X + X + X + X + X :
https://www.youtube.com/watch?v=h0RF0N5TOPE
| This is the mindset I had to take in order to get through the class. |
It doesn't seem like a mindset to use when trying to do anything useful. The point of taking math classes that may never actually help you is to learn how to think critically. This was true at least until High School. At this point, not only are we not thinking critically for most of these math classes, but the only reason people pass is due to intense curves. So again, what's the point of a proof? To teach it to people who never use it? To use it yourself and feel special? Again, not useless 100% of the time, but close enough to the point where I don't see why anyone would defend it's importance outside of a few rare cases.
Your argument is so absolutist it's self-defeating. You're saying "if we can be wrong about the validity of an argument then making an argument is pointless because we'll never know for sure if we've understood it correctly". Basically you're saying that conveying rational thoughts is impossible because it can't be done with perfect accuracy 100% of the time. Well, your argument is itself a rational thought, so if you really believe that, why did you bother trying to convey it?
We're always going to be imprisoned inside our own faulty brains looking at the world through our faulty senses.
A true solipsist is unable to do science because he can never know if his eyes are deceiving him. Science starts from the assumption that the scientist is perceiving the world at least approximately as it is. If there is some malevolent entity deceiving you, you have no way to help anyway.
Likewise, to do mathematics you have to start from the assumption that you haven't completely misunderstood an argument and that your brain is capable of understanding logic. Again, if your brain makes a mistake or if there's some external force controlling your mind, what can you possibly do about it?
If you look at a proof and you don't understand a leap in logic, you're free to reject the proof or ask for clarification on a step. A wide enough valid leap in logic is indistinguishable from a non sequitur.
However, I wasn't talking about skipping steps, but about non-obvious transformations. A transformation that usually trips up students in propositional logic proofs is
x = y
x = y OR (foo AND NOT foo)
Stuff like that is usually done to make the algebra simpler, but students often ask where the extra condition came from, even though the logic is obviously valid.
Maybe I don't understand the question. In deductive logic there are only two types of true statements: axioms and theorems. Axioms are statements that are defined to be true, and theorems are statements that are reducible to tautological logical expressions that are some set of axioms combined in some way with logical operators (conjunction, disjunction, negation, etc.). For example, "the greatest common divisor of a prime and a number that's indivisible by the prime is 1" is eventually reducible to some expression where the only variables are Peano's axioms.
You seem to be seriously confused about how syllogisms work. I'd recommend grabbing a book on propositional logic. I've said and I'll say it again: if a proof starts from true propositions and doesn't contain any fallacies, then necessarily whatever it concludes is true.
We're always going to be imprisoned inside our own faulty brains looking at the world through our faulty senses.
A true solipsist is unable to do science because he can never know if his eyes are deceiving him. Science starts from the assumption that the scientist is perceiving the world at least approximately as it is. If there is some malevolent entity deceiving you, you have no way to help anyway.
Likewise, to do mathematics you have to start from the assumption that you haven't completely misunderstood an argument and that your brain is capable of understanding logic. Again, if your brain makes a mistake or if there's some external force controlling your mind, what can you possibly do about it?
| Someone can look at the proof and not know if what they did was valid because they don't understand how they got there. |
However, I wasn't talking about skipping steps, but about non-obvious transformations. A transformation that usually trips up students in propositional logic proofs is
x = y
x = y OR (foo AND NOT foo)
Stuff like that is usually done to make the algebra simpler, but students often ask where the extra condition came from, even though the logic is obviously valid.
| Moreover, if you don't understand what's required to prove X, how can you know if the proof adequately does so? |
You seem to be seriously confused about how syllogisms work. I'd recommend grabbing a book on propositional logic. I've said and I'll say it again: if a proof starts from true propositions and doesn't contain any fallacies, then necessarily whatever it concludes is true.
| You're saying "if we can be wrong about the validity of an argument then making an argument is pointless because we'll never know for sure if we've understood it correctly". |
Again, I don't think my point is getting across. If all proofs are subject to the fallible human brain and the fact that we can't possibly have all the knowledge, then what's the point of proving obvious/easy to prove things using the pretentious standards of mathematical proofs? There's no point when proving it by much more logical standards would be the same if not better.
| We're always going to be imprisoned inside our own faulty brains looking at the world through our faulty senses. A true solipsist is unable to do science because he can never know if his eyes are deceiving him. |
Yes, but this isn't what my argument was. It was about how a proof holds things to a standard as if it makes the proof perfect, when clearly it's bound to the same issues with a proof by, let's say, MY standards. So what's the point of this extra pretentiousness? Again, only seems useful in rare cases, not worth more than a few lectures if that, definitely not worth a semester or two.
| Maybe I don't understand the question |
I think so. You're attacking a position I never held. The breakdown you provided with axioms and theorems aren't the only tools for a proof. There are many ways to go about a proof. The kind of proofs your talking about here with axioms and logical operators would be with symbols. If X, then Y would mean X -> Y. But that's not a proof, this would be something TO prove. How do you prove this? You can't without knowing what X/Y are (the typical proofs I do in this class), or by being given premises to work with (the proofs I was doing in a class before this one which was about symbolic logic).
| I've said and I'll say it again: if a proof starts from true propositions and doesn't contain any fallacies, then necessarily whatever it concludes is true. |
No, this statement has faults. The proof must start with ALL true propositions if you want to reach a conclusion that is true. Meaning if you want to prove, for example, something that would need 3 cases, you can't give 2 of them and then conclude with a true proof just because proof contained only true propositions. This is why some proofs have been believed until they were provided a counterexample - they were missing a factor which wasn't known when the proof was constructed.
| If all proofs are subject to the fallible human brain and the fact that we can't possibly have all the knowledge, then what's the point of proving obvious/easy to prove things using the pretentious standards of mathematical proofs? |
That question holds the answer inside itself. It's a way of identifying the building blocks of the mathematical language and using them to create a stable foundation that is undeniable and can be expressed within a usable syntax. Take that video that you posted. It's an explanation of how multiplication can be interpreted as addition, but it relies on intuition to teach this. How might he show that a near infinite number times a number is also addition of beachballs? Complex proofs cannot rely on intuition as their low level base, because it leaves open too many factors. The point of a proof is to nail a concept onto the table and dissect it; no matter how many tentacles the creepy thing has.
Wow. I'm defending proofs. What have I become?
edit:
Your teacher expects you to put aside your assumptions and only use the tools that are given by the book and in class.
Show that m*n is even if m is odd and n is even.
An even number is defined as 2k where k is some integer.
An odd number is defined as 2j+1 where j is some integer.
m*n can be expressed as (2j + 1) * (2k)
By the distributive property (2j + 1) * (2k) = 4jk + 2k
4jk + 2k = 2(2jk + k)
Since the multiplication of integers is an integer,
and the addition of integers is an integer
2(2jk+k) = 2(h)
where h is some integer, which meets the definition of an even number, QED
(I possibly would get points off for doing two steps in one go, my teacher allowed it but warned it was "bad form") No tennis balls or images of dolls required in this proof.
| Proofs are like many mathematical studies: useful for expressing an idea in a firm manner. Useful to practice and train the mind to be more detail oriented in all aspects. Useful if you want to win a fight against an equally asinine anally-retentive opponent. A tool to be used in the appropriate circumstances. Mostly useless and unused by most people outside of its one-semester scope in the classroom. |
Yep. I shall now hit the power button an go stare at the sun for a bit.
Last edited on
| If all proofs are subject to the fallible human brain and the fact that we can't possibly have all the knowledge, then what's the point of proving obvious/easy to prove things using the pretentious standards of mathematical proofs? |
But that aside, the point of the class is, I would assume, to introduce you to writing proofs. Obviously they're going to have to start you from simpler statements, but for the class to make sense they'll require you to have the same level of rigor as if you were proving something more complex.
| But that's not a proof, this would be something TO prove. |
| No, this statement has faults. The proof must start with ALL true propositions if you want to reach a conclusion that is true. Meaning if you want to prove, for example, something that would need 3 cases, you can't give 2 of them and then conclude with a true proof just because proof contained only true propositions. |
For example, consider:
1. I want to prove P(x) for all x integers in [0; 2^16)
2. When I run a program Q that checks the truth of the statement, it reports that the property is true for all numbers sampled.
3. Therefore the statement is true.
If Q actually has a bug and misses 42, then conclusion #3 is fallacious.
It's really a basic principle in deductive reasoning. Honestly I can't believe you're being asked to write proofs when you haven't taken an introductory course in logic.
https://en.wikipedia.org/wiki/Deductive_reasoning
| If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true. |
A proof that turns out later to be invalid is not a proof. It's that simple.
Of course now we go down the track of the law of the excluded middle which no doubt @zapshe will say is the whole/original point.
Sadly, constructs like those of @zapshe are external to the world as most of us know it. We can desribe this external world as a @zapshe-godel state of utter incompleteness, one where the axioms will never be enough how many they are because new one's will be introduced the moment any resolution draws near. BTW axioms aren't science.
The Veil of Ignorance has draped itself over the Brain in a Vat and bingo, we have a zapshe who won't play the game.
Of course now we go down the track of the law of the excluded middle which no doubt @zapshe will say is the whole/original point.
Sadly, constructs like those of @zapshe are external to the world as most of us know it. We can desribe this external world as a @zapshe-godel state of utter incompleteness, one where the axioms will never be enough how many they are because new one's will be introduced the moment any resolution draws near. BTW axioms aren't science.
The Veil of Ignorance has draped itself over the Brain in a Vat and bingo, we have a zapshe who won't play the game.
| Complex proofs cannot rely on intuition as their low level base, because it leaves open too many factors. |
Obviously complex proof wouldn't rely on intuition. Everything above algebra likely isn't very intuitive for most people. At that point, you can still prove many things without the hideous standards of mathematical proofs.
For example, you want to prove that F(x) = 5x^2 - 5 is negative iff -1 < x < 1. Easy, right? You just look at it and realize that only at x = 0 will 5x^2 every be smaller than 0, so this is the only case in which it comes out negative. How would you go about proving this? Yea, for a simple one like this it's not too difficult, but why even bother with such an obvious proof?
| Wow. I'm defending proofs. What have I become? |
Your father
| (I possibly would get points off for doing two steps in one go, my teacher allowed it but warned it was "bad form") No tennis balls or images of dolls required in this proof. |
Now I REALLY don't understand why you defend proofs.
| By whose standard is a statement "obvious"? |
We can go by 100IQ standard for the average person. If that doesn't suit your fancy, then how can a proof, which may require explanation from the one who proved it, be worth anything to anyone who wouldn't find something like 5X always being even if X is even obvious?
| I challenge you to find a proof that's not reducible to a single tautological logical expression. |
I'm sure they can be. I'm saying that you can't use those tautological expressions from start to finish to solve a proof. Not that you might not be able to reduce every step to a tautological form, but using those alone obviously wouldn't get you anywhere.
| If proving P definitely requires using propositions A, B, and C, and a proof for P uses only A and B, then the proof contains a fallacy of some kind. |
Depending on the proof, it can be valid yet incorrect as you've stated before. For example, let's say:
If you have an umbrella, then it's raining outside.
You have an umbrella.
Therefore it must be raining outside.
Within the confines of this proof, it's valid - because the premises are all true and the logic is followed. However, it's incorrect, because what if they also use an umbrella when it's sunny?
| It's really a basic principle in deductive reasoning. Honestly I can't believe you're being asked to write proofs when you haven't taken an introductory course in logic. |
I did take a logic class. I can't believe you're defending proofs against basic logic that I'm putting forward. We might not be on the same page of what a mathematical proof entails.
| If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true. |
If going by your standards, what would be "clear"? And this doesn't even state that you needed all premises, only that the ones you have are true.
| A proof that turns out later to be invalid is not a proof. It's that simple. |
It's an incorrect proof. It doesn't just stop being a proof.
| Sadly, constructs like those of @zapshe are external to the world as most of us know it. We can desribe this external world as a @zapshe-godel state of utter incompleteness, one where the axioms will never be enough how many they are because new one's will be introduced the moment any resolution draws near. BTW axioms aren't science. |
Uh, definitely. I agree. Because the mighty againtry read 3 sentences and concluded this, it must be true. Your words don't even need a proof, they're simply true by nature.
Last edited on
@zapshe
A couple of words of wisdom, pearls before a swine if you like, better still the start of true learning and discipline experience for you.
1. Denial and reaction to something new, especially for callow youth, is normal and vital. It is akin to skepticism as a foundational part of the scientific method.
2. Denial to the extent of coming up with a new paradigm is commendable. Again foundational for the whole of civilization - worth it or not.
3. Never accept words as absolutely true without questioning them.
4. Denial in the face of really trying hard and understanding, with or without acceptance, is simply arrogance, ignorance, stubbornness. That's the history of many disgruntled losers.
5. Running away is intellectual cowardice but admitting defeat is honorable.
6. Fault can be assigned to many externalities but that rarely applies to intellectual defeat. It is your problem to solve not ours. You'll be better for it.
7.Asking questions is how it works but asking questions that are easily confused with callow youth in denial is not rewarding.
8. Selfishly, I get nothing out of you kow-towing to the greatness you unilaterally bestow on me.
A couple of words of wisdom, pearls before a swine if you like, better still the start of true learning and discipline experience for you.
1. Denial and reaction to something new, especially for callow youth, is normal and vital. It is akin to skepticism as a foundational part of the scientific method.
2. Denial to the extent of coming up with a new paradigm is commendable. Again foundational for the whole of civilization - worth it or not.
3. Never accept words as absolutely true without questioning them.
4. Denial in the face of really trying hard and understanding, with or without acceptance, is simply arrogance, ignorance, stubbornness. That's the history of many disgruntled losers.
5. Running away is intellectual cowardice but admitting defeat is honorable.
6. Fault can be assigned to many externalities but that rarely applies to intellectual defeat. It is your problem to solve not ours. You'll be better for it.
7.Asking questions is how it works but asking questions that are easily confused with callow youth in denial is not rewarding.
8. Selfishly, I get nothing out of you kow-towing to the greatness you unilaterally bestow on me.