I'm not sure why I'm writing this down now, but: I don't agree that "Balaam's error" has anything to do with money. Based on his actions, monetary reward seems to be a small concern for him. His error comes from this: He is afraid to contradict the Moabites. He is too polite, too unwilling to offend.
Sometimes things need to be said which are offensive-- causing offense is rarely good in its own right, but offensive things are important. Balaam was too afraid (either socially, or for his life) to tell the Moabites something offense, "God says 'no!'"
We should learn from this.
Tuesday, January 19, 2010
Saturday, December 12, 2009
Balaam
I wrote this sometime last year, but apparently forgot about it.
***
I tread on with the Moabites, seeking their praise, silver and jewels. "I cannot contradict the ineffable," but I march on, afraid to contradict these messengers.
But still my ass, wiser than I, stubbornly refuses to move. Twice she's flogged, and twice she stands up and walks. Again she's flogged, but finally she speaks her mind:
"Why do you treat me so? Have I not carried you from your home?"
And thus I lie: "No."
***
I tread on with the Moabites, seeking their praise, silver and jewels. "I cannot contradict the ineffable," but I march on, afraid to contradict these messengers.
But still my ass, wiser than I, stubbornly refuses to move. Twice she's flogged, and twice she stands up and walks. Again she's flogged, but finally she speaks her mind:
"Why do you treat me so? Have I not carried you from your home?"
And thus I lie: "No."
Friday, December 4, 2009
Friday, November 20, 2009
Mathematical insight...
This is from a reply I posted here to a question about gaining mathematical insight.
* Nothing is "obvious".
Try to be extremely formal with all of your proofs. Make sure your steps all follow immediately from previous steps, definitions or theorems. Spend some time proving the "really basic" properties that follow immediately from applying the definition. Also, ask yourself what sort of objects satisfy certain properties, and which don't. Eg. For complete metric spaces, come up with a "canonical" example of a complete metric space, a "canonical" incomplete metric space, and a degenerate example of each. For example, the discrete metric is complete (if you know about metric spaces, you may want to prove this), but it really doesn't match our intuition for what a complete metric space "should be."
On that note, try to understand what the intuition for a property or object is-- what does it "mean" for a set to be a group under an operation? Also, try to keep track of where intuition departs from math-- For example, we like to think of topological spaces geometrically, but there are some very non-geometric topological spaces.
* Rewrite the same thing as many different ways as you can.
For example, if the problem asks a question about a normal subgroup, you should be thinking of all the characterizations of normality-- It's the kernel of a homomorphism, it's invariant under conjugatian (which really is the same as its left and right cosets are the same), if a and b are in the same coset of N, then a-b is in N.
* When working on a proof, pay attention to everywhere you use your assumptions.
* After writing a proof, make sure the result seems to make sense.
Does it match up with intuition? If not, figure out why. If the problem is with your intuition, try to figure out what you are assuming to be true, and make a note of it.
Are any basic examples of the structure a counter-example to your "theorem"? Does each step follow from the last? Are you sure?
(I have a friend who has written 3 or 4 wrong proofs this semester, and every time, he realized it was wrong based on these checks, although normally I had to pick out the false step for him :D )
* Learn to look for counter-examples.
If you're asked to prove something wrong, look at some basic examples of the structure you're looking at. Does the statement hold for them? If so, can you see what properties make it work? If so, try to come up with an example where that property doesn't hold. Does the statement fail now? Rinse and repeat.
* Rewrite your assumptions. Rewrite them in different words. Rewrite them with the definitions of any terms you are uncomfortable with.
* Look for connections.
* Rewrite any objects you're looking at in terms of other objects. The complement of an open set is closed. The complement of a closed set is open. A connected space has proper (non-empty) clopen sets. g is in the Center of G means gh=hg for any h.
* State the obvious. Often. And then state it again.
* Ask stupid questions. Then answer them.
Is R complete? Why is a polynomial continuous? Is Z abelian? Finitely generated? What about Z^n? What does Abelian mean anyway?
* Don't be afraid to ask someone else stupid questions.
* Don't be discouraged when you sit for hours without understanding what to do; let the gears keep grinding.
Put on some music and rock out while you think. Rewrite the assumptions. Try to do something. When you get stuck, try to figure out why that doesn't work. Does it get you anywhere at all?
* Don't be afraid to go do something else for an hour or 2 and then come back to work on a problem.
This is when some of the best insights happen-- go make some tea, read a book, watch a movie, get coffee with a friend, do something. Then come back and start again. Sometimes it'll be hard to get back in the zone-- redo some easier problems: Try to reword your argument or try to find a cleaner argument.
* Work on a simpler problem.
Need to separate two compact sets? Don't! separate a compact set from a point. Can you use this same argument again? Will a similar argument work for two sets?
* Work on a more general problem.
Don't show that n is divisible by 3, show that all numbers of a certain form are divisible by 3. Then show that n has that form.
Hope these give you something useful to think about.
* Nothing is "obvious".
Try to be extremely formal with all of your proofs. Make sure your steps all follow immediately from previous steps, definitions or theorems. Spend some time proving the "really basic" properties that follow immediately from applying the definition. Also, ask yourself what sort of objects satisfy certain properties, and which don't. Eg. For complete metric spaces, come up with a "canonical" example of a complete metric space, a "canonical" incomplete metric space, and a degenerate example of each. For example, the discrete metric is complete (if you know about metric spaces, you may want to prove this), but it really doesn't match our intuition for what a complete metric space "should be."
On that note, try to understand what the intuition for a property or object is-- what does it "mean" for a set to be a group under an operation? Also, try to keep track of where intuition departs from math-- For example, we like to think of topological spaces geometrically, but there are some very non-geometric topological spaces.
* Rewrite the same thing as many different ways as you can.
For example, if the problem asks a question about a normal subgroup, you should be thinking of all the characterizations of normality-- It's the kernel of a homomorphism, it's invariant under conjugatian (which really is the same as its left and right cosets are the same), if a and b are in the same coset of N, then a-b is in N.
* When working on a proof, pay attention to everywhere you use your assumptions.
* After writing a proof, make sure the result seems to make sense.
Does it match up with intuition? If not, figure out why. If the problem is with your intuition, try to figure out what you are assuming to be true, and make a note of it.
Are any basic examples of the structure a counter-example to your "theorem"? Does each step follow from the last? Are you sure?
(I have a friend who has written 3 or 4 wrong proofs this semester, and every time, he realized it was wrong based on these checks, although normally I had to pick out the false step for him :D )
* Learn to look for counter-examples.
If you're asked to prove something wrong, look at some basic examples of the structure you're looking at. Does the statement hold for them? If so, can you see what properties make it work? If so, try to come up with an example where that property doesn't hold. Does the statement fail now? Rinse and repeat.
* Rewrite your assumptions. Rewrite them in different words. Rewrite them with the definitions of any terms you are uncomfortable with.
* Look for connections.
* Rewrite any objects you're looking at in terms of other objects. The complement of an open set is closed. The complement of a closed set is open. A connected space has proper (non-empty) clopen sets. g is in the Center of G means gh=hg for any h.
* State the obvious. Often. And then state it again.
* Ask stupid questions. Then answer them.
Is R complete? Why is a polynomial continuous? Is Z abelian? Finitely generated? What about Z^n? What does Abelian mean anyway?
* Don't be afraid to ask someone else stupid questions.
* Don't be discouraged when you sit for hours without understanding what to do; let the gears keep grinding.
Put on some music and rock out while you think. Rewrite the assumptions. Try to do something. When you get stuck, try to figure out why that doesn't work. Does it get you anywhere at all?
* Don't be afraid to go do something else for an hour or 2 and then come back to work on a problem.
This is when some of the best insights happen-- go make some tea, read a book, watch a movie, get coffee with a friend, do something. Then come back and start again. Sometimes it'll be hard to get back in the zone-- redo some easier problems: Try to reword your argument or try to find a cleaner argument.
* Work on a simpler problem.
Need to separate two compact sets? Don't! separate a compact set from a point. Can you use this same argument again? Will a similar argument work for two sets?
* Work on a more general problem.
Don't show that n is divisible by 3, show that all numbers of a certain form are divisible by 3. Then show that n has that form.
Hope these give you something useful to think about.
Wednesday, October 21, 2009
Wednesday, October 14, 2009
Cruel Irony.
Worth watching. I won't say you should never buy Monster again, or anything... actually I will, but mostly because Monster is awful-- if the sugar is over-saturated, there is too much.
Also in the Cease and Desist letter, the following quote' "VERMONSTER in connection with beer will undoubtedly create a likelihood and/or dilute the distinctive quality of Hansen's MONSTER marks." Self-fulfilling prophecy, much?
Also in the Cease and Desist letter, the following quote' "VERMONSTER in connection with beer will undoubtedly create a likelihood and/or dilute the distinctive quality of Hansen's MONSTER marks." Self-fulfilling prophecy, much?
Sunday, October 11, 2009
Art
I've finally found a definition of art that I think I agree with... Came to me as I woke up this morning.
****
Art (as a verb) is a creative or transformative process undertaken primarily as an appeal to some aesthetic, in order to induce a sense of "aesthetic euphoria" in those who experience the resulting object.
An object created (or transformed) in this way (this is, with this aesthetic goal as a primary objective) is a work of art.
The broad category of all artistic processes is art-- Everything which is done primarily as an appeal to an aesthetic. Any category of process which is primarily undertaken for aesthetic appeal is an artistic discipline.
Anything which has an aesthetic appeal, but was not designed with the aesthetic appeal as the primary objective is craft.
****
This definition is pretty loose (yet mathematically precise; I won't apologize for who I am), but it seems to explicitly exclude "useful" objects from the category of art... This isn't entirely true. An object which is useful, but was designed with its aesthetic appeal as a primary objective is still art: Something can be both craft and art.
I also am not trying to be derogatory towards craft: many great artists are primarily craftsmen, and a lot of craft is more aesthetically appealing than a lot of art. Further, what separates a good craftsman from a great craftsman, is that a great craftsman elevates the artistic value of his creation to an equal footing with it's utility-- without sacrificing function for form.
It's also, I assume, a very modernist definition... So my poetry and my artistic ideals are 70+ years behind the times; C'est la vie.
The one thing I'm struggling with is how kitsch fits into this. I would like to say kitsch is not art, but I don't think this definition excludes it.
On the other hand, I tend to refer to kitsch as "the unart" in the same way that zombies are undead. So it makes sense that kitsch will fit the definition of art; now how does in fit the definition of non-art?
****
Art (as a verb) is a creative or transformative process undertaken primarily as an appeal to some aesthetic, in order to induce a sense of "aesthetic euphoria" in those who experience the resulting object.
An object created (or transformed) in this way (this is, with this aesthetic goal as a primary objective) is a work of art.
The broad category of all artistic processes is art-- Everything which is done primarily as an appeal to an aesthetic. Any category of process which is primarily undertaken for aesthetic appeal is an artistic discipline.
Anything which has an aesthetic appeal, but was not designed with the aesthetic appeal as the primary objective is craft.
****
This definition is pretty loose (yet mathematically precise; I won't apologize for who I am), but it seems to explicitly exclude "useful" objects from the category of art... This isn't entirely true. An object which is useful, but was designed with its aesthetic appeal as a primary objective is still art: Something can be both craft and art.
I also am not trying to be derogatory towards craft: many great artists are primarily craftsmen, and a lot of craft is more aesthetically appealing than a lot of art. Further, what separates a good craftsman from a great craftsman, is that a great craftsman elevates the artistic value of his creation to an equal footing with it's utility-- without sacrificing function for form.
It's also, I assume, a very modernist definition... So my poetry and my artistic ideals are 70+ years behind the times; C'est la vie.
The one thing I'm struggling with is how kitsch fits into this. I would like to say kitsch is not art, but I don't think this definition excludes it.
On the other hand, I tend to refer to kitsch as "the unart" in the same way that zombies are undead. So it makes sense that kitsch will fit the definition of art; now how does in fit the definition of non-art?
Subscribe to:
Posts (Atom)
Cory Knapp.