What proofs are (split from amusing test answers)
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What proofs are (split from amusing test answers)
I split this off the other thread because, as was pointed out, it had kind of diverged from the original intent. I don't know if the discussion will continue now that oxoiron seems to have given up, but figured I'd make the new topic.  gm
As a kid, I barely scraped by with a passing grade in geometry because I refused to do proofs. The conversation leading up to this refusal went something like this:
Teacher: Prove [some really obvious theorem] is true.
Me: That's silly. Anybody can see that is true. Why prove something when I already know it is true?
Teacher: Not everybody can see it as easily as you.
Me: O.K. How do I prove it?
Teacher: Prove it using postulates, like I showed you.
Me: How do I know that the postulates are true?
Teacher: We just accept that they are.
Me: That's stupid. Why do we just accept them?
Teacher: Because they are obviously true.
Me: So is [some really obvious theorem]. Why should I prove something that is obviously true using something that I just have to accept as being true? Why I can't I just accept that [some really obvious theorem] is true, too? What's the difference?
Teacher: Postulates are true because [blah, blah, I don't really know, I never thought about it before]. That is why we use them to prove theorems.
Me: The theorem is just as obviously true as the postulate and I refuse to use something that is unproven to prove something else. If you expect me to just accept one as being true because it is obvious, I see no reason why I shouldn't accept the other as being true when it is just as obvious.
Teacher: If you don't learn how to do this, it will haunt you later when you do higher math.
That never happened.
As a kid, I barely scraped by with a passing grade in geometry because I refused to do proofs. The conversation leading up to this refusal went something like this:
Teacher: Prove [some really obvious theorem] is true.
Me: That's silly. Anybody can see that is true. Why prove something when I already know it is true?
Teacher: Not everybody can see it as easily as you.
Me: O.K. How do I prove it?
Teacher: Prove it using postulates, like I showed you.
Me: How do I know that the postulates are true?
Teacher: We just accept that they are.
Me: That's stupid. Why do we just accept them?
Teacher: Because they are obviously true.
Me: So is [some really obvious theorem]. Why should I prove something that is obviously true using something that I just have to accept as being true? Why I can't I just accept that [some really obvious theorem] is true, too? What's the difference?
Teacher: Postulates are true because [blah, blah, I don't really know, I never thought about it before]. That is why we use them to prove theorems.
Me: The theorem is just as obviously true as the postulate and I refuse to use something that is unproven to prove something else. If you expect me to just accept one as being true because it is obvious, I see no reason why I shouldn't accept the other as being true when it is just as obvious.
Teacher: If you don't learn how to do this, it will haunt you later when you do higher math.
That never happened.
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Re: Amusing answers to tests
oxoiron wrote:The conversation leading up to this refusal went something like this:
Looking back, though, you understand that your teacher was correct, right?
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Re: Amusing answers to tests
Buttons wrote:oxoiron wrote:The conversation leading up to this refusal went something like this:
Looking back, though, you understand that your teacher was correct, right?
I hope so. (This exact conversation will take place right before I murder a philosopher, mark my words...)
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Re: Amusing answers to tests
Using Postulates are ifthen operators.
If you can show that the Postulates are true, then anything you can prove from the Postulates are true.
You don't have to accept the Postulates as true in order to use Postulates this way. You only have to build the ifthen chain.
And the important part is not always what you produce, but what you cannot produce. If X then Y means that if you detect ~Y, you know X isn't true! And if you cannot prove Z from X, then you have some evidence that Z might be independent of the postulates X.
If you can show that the Postulates are true, then anything you can prove from the Postulates are true.
You don't have to accept the Postulates as true in order to use Postulates this way. You only have to build the ifthen chain.
And the important part is not always what you produce, but what you cannot produce. If X then Y means that if you detect ~Y, you know X isn't true! And if you cannot prove Z from X, then you have some evidence that Z might be independent of the postulates X.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
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Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Re: Amusing answers to tests
To prove A = E, I have to assume that A + B = 180 deg. and that B + C = 180 deg. and so on.
That A + B = 180 degrees is obvious. That B + C = 180 degrees is also obvious. That A = E is just as obvious, but one is expected to "prove" that by going through steps showing the relationship between angles until you have worked your way from A to E.
I say that proving A = E by basing an argument on A + B = 180 degrees is foolish, since we haven't proven that A + B = 180 degrees; we just accept that postulate to be true, because it is obvious within our chosen framework. I submit that A = E is just as obvious and if we accept one premise as true, we ought to accept the other. If we have to prove one premise, we should also have to prove the other.
(If my terminology is driving you math people nuts, I apologize. I gather from the shocked responses to my last post that I have insulted the great field of mathematics, but I hope you are able to figure out what I'm saying without picking at semantic issues.)
That A + B = 180 degrees is obvious. That B + C = 180 degrees is also obvious. That A = E is just as obvious, but one is expected to "prove" that by going through steps showing the relationship between angles until you have worked your way from A to E.
I say that proving A = E by basing an argument on A + B = 180 degrees is foolish, since we haven't proven that A + B = 180 degrees; we just accept that postulate to be true, because it is obvious within our chosen framework. I submit that A = E is just as obvious and if we accept one premise as true, we ought to accept the other. If we have to prove one premise, we should also have to prove the other.
(If my terminology is driving you math people nuts, I apologize. I gather from the shocked responses to my last post that I have insulted the great field of mathematics, but I hope you are able to figure out what I'm saying without picking at semantic issues.)
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Re: Amusing answers to tests
oxoiron wrote:That A + B = 180 degrees is obvious. That B + C = 180 degrees is also obvious. That A = E is just as obvious, but one is expected to "prove" that by going through steps showing the relationship between angles until you have worked your way from A to E.
Yeah, clearly you still don't get the point. Which is to prove, from the fact (assumption?) that the sums of those angles are 180 degrees, that A=E. The point is to see how those things all logically relate to each other. How, even if it weren't "obvious" that A=E, being told that the sums of those other angles was 180 still tells you that A=E.
I admit that geometry proofs at that basic a level do seem rather silly. But that's why you were doing them in such a basic class.
Proofs are not about what is true, full stop. They're about what is true if something else is true, namely the axioms we start with. In higher math, you begin to see the axioms not as these "selfevident" things (and I'll grant that your teacher was skirting the issue by calling them such), but rather as a list of properties defining whatever object you're working with. If you have a set of elements obeying one set of axioms, it's a group. If an equalsized set merely obeys a bit more, it suddenly becomes a ring or a field.
Re: Amusing answers to tests
gmalivuk wrote:Proofs are not about what is true, full stop. They're about what is true if something else is true, namely the axioms we start with.
The part that irks me is that the whole argument is predicated on that "if". I certainly see how I can prove something after I accept the original premise, but I hate that my proof depends on something merely posited, not proven.
On a separate note, are you English, Austalian or some other English speaking nonAmerican? I ask because you say "full stop", not "period". During a presentation in my research group (which is about 75% nonAmericans) somebody pointed out that a comma needed to be replaced with a full stop. We Americans were very confused until an English friend of mine pointed out that a full stop is the dot at the end of the sentence, and a period is "when a woman bleeds."
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Yep. Now take your postulates, and presume that the lines are drawn on a sphere.
Now does angle A equal angle E?
If you had a proof of A=E from postulates, and you knew which of those postulates held under the "lines drawn on a sphere" case, you might be able to see the step where things get questionable (because you relied on a postulate that wasn't true when you draw lines on a sphere), or be able to easily see "it still holds!" free of charge because all of your proof steps rely only on postulates that still hold in spherical geometry!
What is more, saying "A=E is obvious" is not convincing to someone who doesn't believe you. Meanwhile, saying "A=E if the two angles split by a line sum to 180 degrees (or however you want to put it formally", together with the proof of that statement, is selfevident. Note that the "X if Y" construct is key to make it self evident: the "X if Y" can be selfevident even if Y isn't selfevident.
This is the lesson and the beauty of proofs. What you assume is explicit, and if you can demonstrate that which you assume is true, you can know that your result holds in this case as well.
The A=E statement would be a bitch to express formally, and determining if it held in a strange geometry would be harder than checking if A+B=180 degrees.
And that is why you deserved to fail geometry.
All proofs are "Ifthen" structures. That is the point of the proof, not to find ultimate cosmic truth.
Now does angle A equal angle E?
If you had a proof of A=E from postulates, and you knew which of those postulates held under the "lines drawn on a sphere" case, you might be able to see the step where things get questionable (because you relied on a postulate that wasn't true when you draw lines on a sphere), or be able to easily see "it still holds!" free of charge because all of your proof steps rely only on postulates that still hold in spherical geometry!
What is more, saying "A=E is obvious" is not convincing to someone who doesn't believe you. Meanwhile, saying "A=E if the two angles split by a line sum to 180 degrees (or however you want to put it formally", together with the proof of that statement, is selfevident. Note that the "X if Y" construct is key to make it self evident: the "X if Y" can be selfevident even if Y isn't selfevident.
This is the lesson and the beauty of proofs. What you assume is explicit, and if you can demonstrate that which you assume is true, you can know that your result holds in this case as well.
The A=E statement would be a bitch to express formally, and determining if it held in a strange geometry would be harder than checking if A+B=180 degrees.
And that is why you deserved to fail geometry.
All proofs are "Ifthen" structures. That is the point of the proof, not to find ultimate cosmic truth.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
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Re: Amusing answers to tests
oxoiron wrote:To prove A = E, I have to assume that A + B = 180 deg. and that B + C = 180 deg. and so on.
That A + B = 180 degrees is obvious. That B + C = 180 degrees is also obvious. That A = E is just as obvious, but one is expected to "prove" that by going through steps showing the relationship between angles until you have worked your way from A to E.
Obvious is in the eye of the beholder. The assumption that B + C = 180 degrees may seem obvious to you, and seemed obvious to every mathematician until the 1800's. Then they were shocked by the discovery of geometries in which that wasn't true. Here is one of the simpler ones.
Draw a unit circle. Let's declare "straight line" to mean any circle or line which intersects that unit circle at right angles. It turns out that this satisfies the first four of Euclid's axioms. It does not satisfy the last. In particular given a line and a point not on that line, there are an infinite number of parallel lines passing through that point. And consequently B+C can vary from being 180 degrees.
This geometry is known as hyperbolic space. It is not hard to define area and distance for it. And it is studied for a number of reasons. Including artistic ones. Escher's famous circle limit drawings explore some of the interesting tilings there are for this geometry.
oxoiron wrote:I say that proving A = E by basing an argument on A + B = 180 degrees is foolish, since we haven't proven that A + B = 180 degrees; we just accept that postulate to be true, because it is obvious within our chosen framework. I submit that A = E is just as obvious and if we accept one premise as true, we ought to accept the other. If we have to prove one premise, we should also have to prove the other.
What you're saying is trivial and wrong.
It is trivial in that there are many different sets of axioms that you can choose. And from any set of them you can derive the others. When you set about choosing a set of axioms you therefore have many choices to make between equally obvious statements. There is really no reason a priori to choose one set instead of another.
It is wrong in that the entire point of axiom systems is that there are many mathematical systems that are similar in a great many respects but are different in some critical way. (For instance the system I gave above and traditional Euclidean geometry.) The important skill isn't picking which set of obvious statements about a mathematical system you will use as axioms. The important skill is that once you've chosen a set of axioms, to be able to show what does and does not follow from it.
oxoiron wrote:(If my terminology is driving you math people nuts, I apologize. I gather from the shocked responses to my last post that I have insulted the great field of mathematics, but I hope you are able to figure out what I'm saying without picking at semantic issues.)
You have not insulted mathematics. You've just completely missed a critical point of great importance to mathematicians.
For more background on why this point is so important to mathematics I'd suggest reading either The Mathematical Experience or else Hilbert! by Constance Reid. The former is a collection of short essays on all sorts of mathematical topics, and touches on a lot of interest, including the importance of axioms. The articles vary widely in difficulty. The second is a biography of David Hilbert, who was a key player in the axiomatization of mathematics. Better than any other book that I know of, it explains the philosophical issues mathematicians were grasping at without requiring any technical knowledge of mathematics.
Yakk wrote:Yep. Now take your postulates, and presume that the lines are drawn on a sphere.
Now does angle A equal angle E?
Bad example because you can't draw the diagram on a sphere. Because every pair of lines intersect, and therefore you don't have parallel lines.
That's why I'm using hyperbolic geometry as my example. Because there you have parallel lines  parallel just doesn't mean what you'd expect it to mean.
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Re: Amusing answers to tests
If I recall correctly, I think in junior high we developed almost everything from Euclid's axioms (although we didn't call them that). So the answer to "how do I know the postulate true?" is "because we already proved it".
As for the obviousness of proofs, you need to remember two things. First, as mentioned, it's a tool to teach you how to prove more difficult, not so obvious ideas. Second, try to prove the theorem to someone who's blind. You can't just say something's obvious in that case because the blind man can't see it (and let's assume he can't feel a model etc.). So you need to do things logically and in a structured manner.
As for what people said, regarding how the original premise doesn't have to be true, they're right. If you go back enough to the origins of geometry you'll reach a basic set axioms, things we assume are true and we have no way of proving. On the basis of these axioms you can build your postulates and theorems.
However, you could also take a different set of axioms (which don't contradict each other) and create new postulates and theorems. So you see that there's not always a "right" thing to base yourself on. You can prove different things in different systems of axioms (for example on a flat plane the sum of angles in a triangle is 180 degrees. on a sphere it doesn't have to be so).
As for the obviousness of proofs, you need to remember two things. First, as mentioned, it's a tool to teach you how to prove more difficult, not so obvious ideas. Second, try to prove the theorem to someone who's blind. You can't just say something's obvious in that case because the blind man can't see it (and let's assume he can't feel a model etc.). So you need to do things logically and in a structured manner.
As for what people said, regarding how the original premise doesn't have to be true, they're right. If you go back enough to the origins of geometry you'll reach a basic set axioms, things we assume are true and we have no way of proving. On the basis of these axioms you can build your postulates and theorems.
However, you could also take a different set of axioms (which don't contradict each other) and create new postulates and theorems. So you see that there's not always a "right" thing to base yourself on. You can prove different things in different systems of axioms (for example on a flat plane the sum of angles in a triangle is 180 degrees. on a sphere it doesn't have to be so).
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Re: Amusing answers to tests
Zohar wrote:If I recall correctly, I think in junior high we developed almost everything from Euclid's axioms (although we didn't call them that). So the answer to "how do I know the postulate true?" is "because we already proved it".
if you want to be evil in a class like that start a discussion like this http://www.ditext.com/carroll/tortoise.html with the teacher
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Re: Amusing answers to tests
Zohar wrote:As for what people said, regarding how the original premise doesn't have to be true, they're right. If you go back enough to the origins of geometry you'll reach a basic set axioms, things we assume are true and we have no way of proving. On the basis of these axioms you can build your postulates and theorems.
However, you could also take a different set of axioms (which don't contradict each other) and create new postulates and theorems. So you see that there's not always a "right" thing to base yourself on. You can prove different things in different systems of axioms (for example on a flat plane the sum of angles in a triangle is 180 degrees. on a sphere it doesn't have to be so).
I'll restate this only one more time. I understand the purpose of a proof is to show that something is true based upon our suppositions about other somethings. However, as several of you have pointed out, we have no proof that those suppositions are correct, we just agree they are correct within our defined framework. If (and that is the first key word) our original postulates are true, then (the other key word) we can prove things using these postulates.
The point you people all seem to miss is that we have no way of proving the postulate, so we don't know in an absolute sense whether our theorem is true; we only know that it is true if (there's that word again) our postulate is true, which we all agree cannot be proven.
Ironically, considering the discussion we are having, I use 3D group theory on a regular basis to spectroscopically "prove" things about molecules. However, I leave open the vanishingly small possibility that at some point in the future, someone will find an exception to the rules regarding orbital mechanics and group theory.
"Dubito, ergo cogito, ergo sum" is the only thing I know for absolute truth, which doesn't fly in a math forum.
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What proofs are (split from amusing test answers)
oxoiron wrote:I understand the purpose of a proof...
No, you still don't.
so we don't know in an absolute sense whether our theorem is true; we only know that it is true if (there's that word again) our postulate is true, which we all agree cannot be proven.
No, you got it wrong again. Our theorem does not merely say "conclusion". It says "If [postulate] then [conclusion]". So the theorem as a whole is still absolutely true.
You seem to be understanding theorems as saying something like "Bob is unmarried", from the postulate, "Bob is a bachelor". But that's not anything like a theorem. Its truth is contingent on whether the postulate is true. A real theorem, however, would state, "If Bob is a bachelor, then Bob is unmarried." This is absolutely true, regardless of Bob's actual marital status in the real world. It is a necessary truth, not a contingent one. This is how proofs work.
You do not have to accept any premise, other than definitions and the basic rules of logic, to accept a theorem as true.
Re: Amusing answers to tests
The tail end of this topic fails badly in the <amusing> department. Please repair.
Re: Amusing answers to tests
gmalivuk wrote:"If [postulate] then [conclusion]". So the theorem as a whole is still absolutely true.
You seem to be understanding theorems as saying something like "Bob is unmarried", from the postulate, "Bob is a bachelor". But that's not anything like a theorem. Its truth is contingent on whether the postulate is true. A real theorem, however, would state, "If Bob is a bachelor, then Bob is unmarried." This is absolutely true, regardless of Bob's actual marital status in the real world. It is a necessary truth, not a contingent one. This is how proofs work.
You do not have to accept any premise, other than definitions and the basic rules of logic, to accept a theorem as true.
Precisely my point. I do have to accept that the definiton of bachelor is "unmarried". The proof is contingent upon the truth of that definition. We have no way of knowing the definition is universally correct, we just know it is correct in our system, which makes the proof correct within our system. My entire argument is that I don't have to accept the postulate.
I certainly understand that if I accept the postulate, then I can prove the theorem. Return to the parallel line example. If I accept that the sum of the angles formed on one side of a line has to be 180 degrees and if I accept the definiton of parallel lines, then I can prove that A = E. Once again, this is predicated upon my acceptance of the postulate. If I don't accept even one of those postulates, the proof goes out the window.
As I said before about group theory, I use it to prove things about molecules. However, this proof stems from my acceptance of the postulates put forth as symmetry operations. It's a nice system as long as I accept it, which I do.
I can't make it more simple than this: If A, then B. A must be true to prove B, but I can't prove A, so my proof only works when I accept A.
I may choose not to accept A, which I did on principle as a snotnosed geometry student. I have no argument with proofs as long as the postulates are acceptable. I thought the geometric postulates were acceptable, but I just wanted to be a pain in the ass and I didn't care about my grade. And I certainly deserved the D.
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Re: Amusing answers to tests
gmalivuk wrote:oxoiron wrote:I understand the purpose of a proof...
No, you still don't.so we don't know in an absolute sense whether our theorem is true; we only know that it is true if (there's that word again) our postulate is true, which we all agree cannot be proven.
No, you got it wrong again. Our theorem does not merely say "conclusion". It says "If [postulate] then [conclusion]". So the theorem as a whole is still absolutely true.
You seem to be understanding theorems as saying something like "Bob is unmarried", from the postulate, "Bob is a bachelor". But that's not anything like a theorem. Its truth is contingent on whether the postulate is true. A real theorem, however, would state, "If Bob is a bachelor, then Bob is unmarried." This is absolutely true, regardless of Bob's actual marital status in the real world. It is a necessary truth, not a contingent one. This is how proofs work.
Question for you: If Bob got married then divorced, can you say that he is really unmarried? After all he has gotten married?
(In highschool we had a 2 week debate on this exact issue. It gets really interesting if, say, they had a Catholic wedding and a civil divorce. Then Bob may be a bachelor, but he's still married according to some people.
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Re: Amusing answers to tests
oxoiron wrote:Precisely my point. I do have to accept that the definiton of bachelor is "unmarried". The proof is contingent upon the truth of that definition. We have no way of knowing the definition is universally correct, we just know it is correct in our system, which makes the proof correct within our system. My entire argument is that I don't have to accept the postulate.
No, the proof exists regardless. The proof, if you include the axioms, is roughly "if Bachelor means Unmarried, and Bob is a Bachelor, then Bob is Unmarried". Now you can go up the "but what if 'if' doesn't work like that"?
The proof is correct regardless of system, but it may or may not apply within a given system. The theorem does not state what is true: it simply states "if X, then Y"  the correctness of the "if X then Y" statement does not depend on the truth of the statement X.
Group theory, as an example, lists the properties that a group has to have. It then proves things from those properties. These proofs "work" regardless of the existence of an actual group.
When you apply group theory, if the axioms hold in your problem space, then the results from group theory also hold in your problem space. (This is a metamathematical theorem).
I certainly understand that if I accept the postulate, then I can prove the theorem.
You don't have to accept the postulates of group theory to prove a result from group theory. And that is the beauty of mathematics.
Return to the parallel line example. If I accept that the sum of the angles formed on one side of a line has to be 180 degrees and if I accept the definiton of parallel lines, then I can prove that A = E. Once again, this is predicated upon my acceptance of the postulate. If I don't accept even one of those postulates, the proof goes out the window.
No, you don't have to accept that this is true. You can just prove the relationship between your postulates and your conclusion independently of any of them actually being true.
You can then try to determine if, in a particular situation, your postulates hold. If you find a situation in which your postulates hold, you get "free of charge" the conclusions you proved from those postulates.
However, if you never ever find a situation in which the postulates hold, then your proof is still valid and "true". And if you discover that a domain over which you thought your postulates held, your postulates don't hold, your theorem is still true  you just cannot use the theorem in the situation involved.
As I said before about group theory, I use it to prove things about molecules. However, this proof stems from my acceptance of the postulates put forth as symmetry operations. It's a nice system as long as I accept it, which I do.
Yes, taking group theory and saying that the postulates hold in the symmetry of molecules is a step you have to take before you can state that the theorems of group theory produce results about the symmetry of molecules. But the theorems of group theory don't depend on the fact that molecule symmetry lines up with group theory.
I can't make it more simple than this: If A, then B. A must be true to prove B, but I can't prove A, so my proof only works when I accept A.
No, A doesn't have to be true to prove B. A can be false, and the theorem that proves B is implied by A will still hold.
You can even use this in practical situations. Suppose you don't know if your postulates hold in a particular domain.
So you check some of the theorems. They say "if the postulates hold, then X is true". You happen to easily be able to check X  and it turns out that X isn't true. And as ~X is true, you now know that at least one of the postulates are not true!
Viola  I've just used a theorem without assuming any of the truths of the postulates: the theorem remained true, even if the conclusion of the theorem wasn't! And, as a bonus, I got a useful, concrete result from it.
Mathematics is, in a way, a web of "if then" clauses. It tells you how X must be if Y is true, and how X can vary with the variation in Y. These "if then" clauses are, of course, dependent on the rules of "if then", etc: but that is another spiral, and there the goal is similarly to find some really simple postulates that you can accept.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Re: Amusing answers to tests
gmalivuk wrote:Stuff
Yakk wrote:Stuff
btilly wrote:Stuff
Good enough. I'm done.
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Re: Amusing answers to tests
McHell wrote:The tail end of this topic fails badly in the <amusing> department. Please repair.
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Re: Amusing answers to tests
oxoiron wrote:Precisely my point. I do have to accept that the definiton of bachelor is "unmarried".
But not in the same way as you thought you had to accept the postulates as true. The definition, "bachelor means unmarried" is fundamentally different from the proposition "Bob is unmarried".
And if that still bothers you for some reason, then go with what Yakk proposed: "If we (locally) define 'bachelor' to mean a male over 25 who has never married, and if Bob is a bachelor, then Bob is not married."
You don't have to believe anything about what the word "bachelor" normally means outside of the theorem. You don't have to believe anything about Bob's marital status. You don't have to believe anything about whether or not the word "bachelor" applies to Bob. The above statement is still true unless you deny the most basic rules of logical inference.
Re: Amusing answers to tests
gmalivuk wrote: The above statement is still true unless you deny the most basic rules of logical inference.
Is there any compelling reason to accept these rules?
 jestingrabbit
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Re: Amusing answers to tests
taemyr wrote:gmalivuk wrote: The above statement is still true unless you deny the most basic rules of logical inference.
Is there any compelling reason to accept these rules?
If you want to keep talking to us, you have to accept these rules.
is that a compelling reason? how about
If you have a problem with the rules, tell us about it, and we can talk, but a lot of people find these rules intuitively satisfying/correct, so you'll have to do a lot of convincing to get another collection of rules accepted.
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Re: Amusing answers to tests
jestingrabbit wrote:If you have a problem with the rules, tell us about it, and we can talk, but a lot of people find these rules intuitively satisfying/correct, so you'll have to do a lot of convincing to get another collection of rules accepted.
I don't know if I accept the law of the excluded middle for infinite sets.
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Re: What proofs are (split from amusing test answers)
And that is the trick. You reduce the actual "proof" down to a meaningless moving around of symbols by rules. These rules don't have to mean anything for the game of moving symbols around to work.
Then, you assign meaning to the game of moving symbols around. "This one means the law of excluded middle, where every statement is either true or not true", etc.
So now the attachment of your proofgamerules to the rules of logic is about as open to interpretation as "what is an angle", or "what is a line". You can prove things in different sets of logic, using logics that disagree with each other, etc.
Now the mechanical moving around of symbols and checking that the manipulation is legal still needs a rather basic level of "yes, that's a legal move in our mechanical game", but those rules are really simple, and don't involve any infinities or other weird things. If you disagree with the "infinite excluded middle", then that happens at a level above the mechanical proof steps. If you have ever done the "tedious" type of proof, where you cite an axiom or derivation rule for every single line that you write, and each line only uses one derivation rule  that is closer to the "raw mechanical symbol manipulation" stuff I'm talking about, but still overly abstract.
A proof is all about what entails what. Even the rules of what "implication" means can get shoved into the postulates of your proof. It isn't an attempt to say "X is true", it is an attempt to say "if Y is true, then X is true"  that Y being true entails X being true.
Then, you assign meaning to the game of moving symbols around. "This one means the law of excluded middle, where every statement is either true or not true", etc.
So now the attachment of your proofgamerules to the rules of logic is about as open to interpretation as "what is an angle", or "what is a line". You can prove things in different sets of logic, using logics that disagree with each other, etc.
Now the mechanical moving around of symbols and checking that the manipulation is legal still needs a rather basic level of "yes, that's a legal move in our mechanical game", but those rules are really simple, and don't involve any infinities or other weird things. If you disagree with the "infinite excluded middle", then that happens at a level above the mechanical proof steps. If you have ever done the "tedious" type of proof, where you cite an axiom or derivation rule for every single line that you write, and each line only uses one derivation rule  that is closer to the "raw mechanical symbol manipulation" stuff I'm talking about, but still overly abstract.
A proof is all about what entails what. Even the rules of what "implication" means can get shoved into the postulates of your proof. It isn't an attempt to say "X is true", it is an attempt to say "if Y is true, then X is true"  that Y being true entails X being true.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Re: What proofs are (split from amusing test answers)
@gm: I'll stick around just to see how many ways people can restate their arguments.
EDIT: Oops! I can see how that could be interpreted as a jab at gm, but it is merely regarding the comment he made upon splitting the thread. I wasn't being a smartass and I wasn't implying that he was the only one restating arguments. We all kept restating our arguments over and over, but it's like we never understood what the other person was saying. I've given up on getting others to see it from my perspective.
EDIT: Oops! I can see how that could be interpreted as a jab at gm, but it is merely regarding the comment he made upon splitting the thread. I wasn't being a smartass and I wasn't implying that he was the only one restating arguments. We all kept restating our arguments over and over, but it's like we never understood what the other person was saying. I've given up on getting others to see it from my perspective.
Last edited by oxoiron on Wed Nov 28, 2007 6:17 pm UTC, edited 1 time in total.
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Re: What proofs are (split from amusing test answers)
oxoiron wrote:@gm: I'll stick around just to see how many ways people can restate their arguments.
Is this an attempt to be insulting while pretending to be polite?
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
 gmalivuk
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Re: What proofs are (split from amusing test answers)
oxoiron wrote:@gm: I'll stick around just to see how many ways people can restate their arguments while I still fail to understand the point they're trying to make.
Fix'd for truth value. Unless you now do understand the point, in which case, please say so in order to save people the trouble of continuing to find ever simpler ways of explaining it.
Re: What proofs are (split from amusing test answers)
Summary: both sides are correct in what they are trying to say (although some are unclear on the proper vocabulary used to say it) and both sides are arguing vehemently about completely different issues (one philosophical, one mathematical).
Re: What proofs are (split from amusing test answers)
Yakk wrote:oxoiron wrote:@gm: I'll stick around just to see how many ways people can restate their arguments.
Is this an attempt to be insulting while pretending to be polite?
No. I think you all made excellent points and I understand what you are saying; I just don't seem to be able to make you see it from a different perspective. I genuinely feel that is a failure on my part.
For the most part, I've given up on insulting people, except in a joking fashion.
quintopia wrote:Summary: both sides are correct in what they are trying to say (although some are unclear on the proper vocabulary used to say it) and both sides are arguing vehemently about completely different issues (one philosophical, one mathematical).
I believe you've hit the nail on the head.
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Re: What proofs are (split from amusing test answers)
..."if Y is true, then X is true" is always true...
Doesn't the tortoise and Achilles story point out this is wrong. You can't use "if X then Y" statements without modus ponens. And you can't apply modus ponens to it self so you have to except at least something a priory.
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Re: What proofs are (split from amusing test answers)
zmljuk wrote:..."if Y is true, then X is true" is always true...
Doesn't the tortoise and Achilles story point out this is wrong. You can't use "if X then Y" statements without modus ponens. And you can't apply modus ponens to it self so you have to except at least something a priory.
That might be a good point... But you are quoting yourself, because I can't find anyone in this thread who made the statement you are saying is wrong?
I've seen people make statements that are slightly different than the statement you are quoting. But not once did anyone say the words "always true" as far as I can find.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Re: What proofs are (split from amusing test answers)
I've seen people make statements that are slightly different than the statement you are quoting. But not once did anyone say the words "always true" as far as I can find.
OK I made up the quote, and it's actually very very wrong.
But you still can't apply modus ponens to it self, you have to accept it.
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Re: What proofs are (split from amusing test answers)
So, you invented a quote, then claimed it was wrong?
I disagree. But only mildly.
zmljuk wrote:I am a fool! A silly bunny! A frou frou chicken!
I disagree. But only mildly.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
 jestingrabbit
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Re: Amusing answers to tests
btilly wrote:jestingrabbit wrote:If you have a problem with the rules, tell us about it, and we can talk, but a lot of people find these rules intuitively satisfying/correct, so you'll have to do a lot of convincing to get another collection of rules accepted.
I don't know if I accept the law of the excluded middle for infinite sets.
Constructivist Corner is thatta way *points towards a well lit but largely empty part of the mathematical world*
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Re: What proofs are (split from amusing test answers)
I think the question is not "Are the logical rules of inference true?".
Nor is it "Are these postulates true?".
But rather "Do these rules model how we think well?" and "Are these postulates good models for our intuitive idea of the object at hand?" I don't think that truth really enters in at any point. As for the answer to the first question, I would say that the answer is a resounding yes. We created the rules, so the rules came from our thought process. Our thought process might be corrupt (to an outsider or an alien) but that doesn't even matter because we are not studying math from the perspective of an alien. Math to me is sort of a study of our own ideas about the world we perceive, and that world includes ourself and our perception of ourself.
As for the answer to the second question, I would say that if you don't have a good enough idea of what it is you are trying to define, then any definition is just as good as another. Once you want to make distinctions, you can branch off from there (as in the examples given of euclidean geometry and noneuclidean geometry).
This is why I consider Math to be a science, in a sense. I think the main differences between Math and other sciences is that there is a lot less focus on observation, and a lot more focus on implication. Although, I guess that depends on your point of view. Because in a sense, applying logical implications is observation.
I hope this is not just a rehashing of what's already been said.
Nor is it "Are these postulates true?".
But rather "Do these rules model how we think well?" and "Are these postulates good models for our intuitive idea of the object at hand?" I don't think that truth really enters in at any point. As for the answer to the first question, I would say that the answer is a resounding yes. We created the rules, so the rules came from our thought process. Our thought process might be corrupt (to an outsider or an alien) but that doesn't even matter because we are not studying math from the perspective of an alien. Math to me is sort of a study of our own ideas about the world we perceive, and that world includes ourself and our perception of ourself.
As for the answer to the second question, I would say that if you don't have a good enough idea of what it is you are trying to define, then any definition is just as good as another. Once you want to make distinctions, you can branch off from there (as in the examples given of euclidean geometry and noneuclidean geometry).
This is why I consider Math to be a science, in a sense. I think the main differences between Math and other sciences is that there is a lot less focus on observation, and a lot more focus on implication. Although, I guess that depends on your point of view. Because in a sense, applying logical implications is observation.
I hope this is not just a rehashing of what's already been said.
Re: What proofs are (split from amusing test answers)
One of the best definitions of "proof" that I've heard:
"A proof is an argument that will convince a mathematician"
If I was in the teacher's position in the original post, I would say that accepting an axiom as true is sort of a dirty thing, philosophically, so we try to use as few as possible. If something can be proven in terms of other axioms, it is redundant to call it an axiom and it makes the situation one step closer to elegance to chuck it.
The bonus in doing this is that when we reduce a mathematical tool like group theory down to the most basic axioms (modded out by pedantry), it turns out that a LOT of things can be looked at as a group, simply because they fit the axioms. Then everything we've proven about groups holds for this other object, and all of a sudden, we know a whole lot more about this other object.
"A proof is an argument that will convince a mathematician"
If I was in the teacher's position in the original post, I would say that accepting an axiom as true is sort of a dirty thing, philosophically, so we try to use as few as possible. If something can be proven in terms of other axioms, it is redundant to call it an axiom and it makes the situation one step closer to elegance to chuck it.
The bonus in doing this is that when we reduce a mathematical tool like group theory down to the most basic axioms (modded out by pedantry), it turns out that a LOT of things can be looked at as a group, simply because they fit the axioms. Then everything we've proven about groups holds for this other object, and all of a sudden, we know a whole lot more about this other object.
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Re: What proofs are (split from amusing test answers)
Hmm... Lemme see if I can tackle this.
Postulates, despite what your teacher says, are not things that are "obviously true." They *are* basic statements that are unprovable with statements more basic than themselves.
Rather than saying "these postulates are obviously true", we should say "we will use these postulates to build a formal system of geometry." You're not required to use those postulates, but making observations on the system you build with them will tell you that they happen to model the real world fairly well.
Put another way, we don't *assume* the postulates to be true, we *define* them to be true and build a system from them.
Postulates, despite what your teacher says, are not things that are "obviously true." They *are* basic statements that are unprovable with statements more basic than themselves.
Rather than saying "these postulates are obviously true", we should say "we will use these postulates to build a formal system of geometry." You're not required to use those postulates, but making observations on the system you build with them will tell you that they happen to model the real world fairly well.
Put another way, we don't *assume* the postulates to be true, we *define* them to be true and build a system from them.
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Re: Amusing answers to tests
jestingrabbit wrote:btilly wrote:jestingrabbit wrote:If you have a problem with the rules, tell us about it, and we can talk, but a lot of people find these rules intuitively satisfying/correct, so you'll have to do a lot of convincing to get another collection of rules accepted.
I don't know if I accept the law of the excluded middle for infinite sets.
Constructivist Corner is thatta way *points towards a well lit but largely empty part of the mathematical world*
pssst... Yakk's a constructivist!
Also, let me take a stab (poor word choice, perhaps!) at explaining away the controversy.
What is an example of this strange beast that mathematicians call a theorem?
Is "a^{2} + b^{2} = c^{2}" a mathematical theorem? No. I don't even know what a, b, and c are! It's certainly not true in most cases. Convention fails us: there exist plenty of 3tuples of reals that don't satisfy this property.
How about, "For a right triangle, with legs of length a and b, and hypotenuse of length c, a^{2} + b^{2} = c^{2}"?
Not really. What's a right triangle? We haven't defined it formally in the statement. Hell, we haven't defined it at all. Also, we would do well to define 'length,' '+' (which presumably signifies addition), and '^{2}'.
At this point I will defer to Euclid and his five postulates; to Peano arithmetic; and to ZFC set theory, which I understand is the most common beginning for these large monsters.
I will not say categorically that Euclid's postulates (or any of that other nonsense) are true. That way lies madness.
I will not ask you to assume they are, either.
I will not even state that these statements correspond to anything in the real world! That's for other people.
I will ask you if the statement
Code: Select all
"These definitions (of number, length, and geometry) imply a[sup]2[/sup] + b[sup]2[/sup] = c[sup]2[/sup]"
is true.
If it is, the theorem is true.
The truth of a theorem is a very limited statement. It tells you nothing, in itself, about the real world.
It tells you flawlessly how the real world would behave if only the conditions that we need for a^{2} + b^{2} = c^{2} were true.
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