Ponderings on Belief and Understanding
Dec. 30th, 2008 05:10 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
tcepsaI'm currently enjoying The Little Schemer as a way to attempt to learn more of the Lisp/Scheme programming paradigm. It's been a useful book for me so far, in that I'm 30 pages in and haven't yet had to stop and go, "Wait, WTF?!"*
The book is arranged in an almost-entirely question-and-answer format (i.e. it approaches it from the perspective of a programmer looking at a piece of Scheme code and going "What's with all these parenthesis? Where are the variable definitions? What in the name of all things digital is up with that lambda keyword?!") The book answers each question with a fairly blunt, concise answer without going into much detail of the nuances of why the lambda represents a function (or even what that means). In terms of the Dreyfus model of skill acquisition, it seems to be fantastic for someone at the novice or advanced beginner level who needs lots of rules and isn't particularly context aware, because it gives a lot of rules without providing much in the way of context (it slowly and subtly builds that up through the book).
That aside, page 31 is mostly blank (it's reserved for doodling, which is another clever learning trick that the authors hit upon) but at the top it asks the question, "Do you believe all this?" which at first struck me as an odd question to be asking in a book that is supposed to be teaching me. It seemed to me that a better question would have been "Do you understand all this?" However, after a bit of consideration, it seems to me like "Do you believe all this?" is a more appropriate question to be asking. It has a subtly but importantly different focus from the question of understanding, and it also strikes me as encouraging in that it implicitly removes the burden of understanding from the reader (at least for the time being). It gives me a sense of reassurance; "You don't have to understand it all yet, you just have to believe it."
It seems to be an oddly important question beyond that, though, and I think the reason for that is related to the core of what it means to believe something and, strangely enough, possibly to mathematics as well (after my algorithms course, I have the sneaking suspicion that programming is nothing but a branch of math, which in turn holds echoes of incredibly significant linguistic implications, but I think I'll save that exploration for another post). Belief, I think, is only possible for things that are consistent with one's existing worldview. Most people are either taught or instinctually have as an axiom of their worldview the premise that two contradictory pieces of information cannot both be true (I don't know where it comes from, and I don't remember the first time that I was aware of it, but it is definitely a key component of the core of my belief system). Thus when someone is presented with a new piece of information they must check it to ensure that it does not contradict any of the pieces of information that they believe to be true. If it does, they will either reject the new information as necessarily false due to the aforementioned axiom, or they will inspect their current beliefs in more depth and try to determine whether they are in error and this new piece of information is actually the true one.
I suspect, though I have no way of proving this, that there is an inclination in the human mind to accept new pieces of information that have neither supporting nor contradictory counterparts in the mind already; i.e. the mind wants to learn and is probably perfectly happy to do so as long as there is nothing apparently contradictory between the things it is learning and the things it has already learned. That, or perhaps certain people's minds are "wired" that way, while others' are more inclined to reject new information unless there is already sufficient supporting information present.
An interesting thing about math is that a huge portion (if not all) of it can be taught as a completely independent collection of information. If I understand correctly, it does not rely at all on the physical world or our observations of any real phenomena; if necessary, it could be (and largely has been) constructed and verified entirely from a collection of axioms (which are pieces of information that have neither supporting nor contradicting pieces of information except in the form of other axioms, i.e. axioms are the core of a belief system) that are completely self-contained.**
To get back to Scheme, the question "Do you believe all this?" is asking whether you perceive that all of the pieces of information presented so far are non-contradictory both to themselves and to all of the information that you already believe. If so, then you're probably in good shape, and while you take a break to doodle your mind is simultaneously working to learn them better and create additional connections between them. If not, your mind is either trying to hold two contradictory ideas simultaneously and twiddle them around until it can see a way that they are non-contradictory, or it has rejected one of them. Either way, if you are trying to learn Scheme, it would probably be most productive if you go back to the parts that you are having trouble believing (i.e. the parts that seem to contain contradictions with either each other or your other beliefs) and consciously try to resolve them, since the contradictions posed by such information is almost certainly the sort of thing that your L-mode*** would be suited for.
As a novice or advanced beginner that belief is crucial, as is a willingness and ability to suspend the need to understand everything right away. That belief enables your subconscious mind to begin creating a network of connections between the different pieces of information and other related pieces of information it already knows. As you believe more and more related pieces of information, and your mind puts together the different ways that they are related, you will find that understanding naturally develops. The best way to have that happen is to practice, to work with those bits of information, to try them against each other, to play with them and watch what happens when you combine two, three, ten in different configurations.
And with that I make the assertion that understanding is a strong belief in how different pieces of information are related to each other.
That is, understanding is itself a collection of pieces of information (each of the relationships between beliefs is a piece of information) which you believe.**** The reason why it works so well to develop an understanding through playing is because the strongest beliefs are those that come about from directly experiencing the information. You could be told about these relationships, but you would believe and remember them much more strongly if you experienced them yourself.
*Actually, looking at it that way, I should try to pick it up and read a single page each day. I'd be farther along now if I had done it that way... ~wry grin~
**Obligatory reference to Plato's Allegory of the Cave. It's been awhile since I read it, but I think that he was trying to take it one step further: that the physical world we observe is due to something else, like the two-dimensional shadows cast on a cave wall by three-dimensional objects with a light behind them. Whether he explicitly said so or not, the physical world that we perceive might well be such a shadow of what we call mathematics.
***L-mode is what I'm using to refer to the more logical/linguistic operations of the brain (also known as "left brained", though my understanding is that term contains implications that aren't entirely accurate).
****~blink~ I think I just reduced Understanding to Graph Theory. Or at least I made a claim of some kind of analog between the two.
The book is arranged in an almost-entirely question-and-answer format (i.e. it approaches it from the perspective of a programmer looking at a piece of Scheme code and going "What's with all these parenthesis? Where are the variable definitions? What in the name of all things digital is up with that lambda keyword?!") The book answers each question with a fairly blunt, concise answer without going into much detail of the nuances of why the lambda represents a function (or even what that means). In terms of the Dreyfus model of skill acquisition, it seems to be fantastic for someone at the novice or advanced beginner level who needs lots of rules and isn't particularly context aware, because it gives a lot of rules without providing much in the way of context (it slowly and subtly builds that up through the book).
That aside, page 31 is mostly blank (it's reserved for doodling, which is another clever learning trick that the authors hit upon) but at the top it asks the question, "Do you believe all this?" which at first struck me as an odd question to be asking in a book that is supposed to be teaching me. It seemed to me that a better question would have been "Do you understand all this?" However, after a bit of consideration, it seems to me like "Do you believe all this?" is a more appropriate question to be asking. It has a subtly but importantly different focus from the question of understanding, and it also strikes me as encouraging in that it implicitly removes the burden of understanding from the reader (at least for the time being). It gives me a sense of reassurance; "You don't have to understand it all yet, you just have to believe it."
It seems to be an oddly important question beyond that, though, and I think the reason for that is related to the core of what it means to believe something and, strangely enough, possibly to mathematics as well (after my algorithms course, I have the sneaking suspicion that programming is nothing but a branch of math, which in turn holds echoes of incredibly significant linguistic implications, but I think I'll save that exploration for another post). Belief, I think, is only possible for things that are consistent with one's existing worldview. Most people are either taught or instinctually have as an axiom of their worldview the premise that two contradictory pieces of information cannot both be true (I don't know where it comes from, and I don't remember the first time that I was aware of it, but it is definitely a key component of the core of my belief system). Thus when someone is presented with a new piece of information they must check it to ensure that it does not contradict any of the pieces of information that they believe to be true. If it does, they will either reject the new information as necessarily false due to the aforementioned axiom, or they will inspect their current beliefs in more depth and try to determine whether they are in error and this new piece of information is actually the true one.
I suspect, though I have no way of proving this, that there is an inclination in the human mind to accept new pieces of information that have neither supporting nor contradictory counterparts in the mind already; i.e. the mind wants to learn and is probably perfectly happy to do so as long as there is nothing apparently contradictory between the things it is learning and the things it has already learned. That, or perhaps certain people's minds are "wired" that way, while others' are more inclined to reject new information unless there is already sufficient supporting information present.
An interesting thing about math is that a huge portion (if not all) of it can be taught as a completely independent collection of information. If I understand correctly, it does not rely at all on the physical world or our observations of any real phenomena; if necessary, it could be (and largely has been) constructed and verified entirely from a collection of axioms (which are pieces of information that have neither supporting nor contradicting pieces of information except in the form of other axioms, i.e. axioms are the core of a belief system) that are completely self-contained.**
To get back to Scheme, the question "Do you believe all this?" is asking whether you perceive that all of the pieces of information presented so far are non-contradictory both to themselves and to all of the information that you already believe. If so, then you're probably in good shape, and while you take a break to doodle your mind is simultaneously working to learn them better and create additional connections between them. If not, your mind is either trying to hold two contradictory ideas simultaneously and twiddle them around until it can see a way that they are non-contradictory, or it has rejected one of them. Either way, if you are trying to learn Scheme, it would probably be most productive if you go back to the parts that you are having trouble believing (i.e. the parts that seem to contain contradictions with either each other or your other beliefs) and consciously try to resolve them, since the contradictions posed by such information is almost certainly the sort of thing that your L-mode*** would be suited for.
As a novice or advanced beginner that belief is crucial, as is a willingness and ability to suspend the need to understand everything right away. That belief enables your subconscious mind to begin creating a network of connections between the different pieces of information and other related pieces of information it already knows. As you believe more and more related pieces of information, and your mind puts together the different ways that they are related, you will find that understanding naturally develops. The best way to have that happen is to practice, to work with those bits of information, to try them against each other, to play with them and watch what happens when you combine two, three, ten in different configurations.
And with that I make the assertion that understanding is a strong belief in how different pieces of information are related to each other.
That is, understanding is itself a collection of pieces of information (each of the relationships between beliefs is a piece of information) which you believe.**** The reason why it works so well to develop an understanding through playing is because the strongest beliefs are those that come about from directly experiencing the information. You could be told about these relationships, but you would believe and remember them much more strongly if you experienced them yourself.
*Actually, looking at it that way, I should try to pick it up and read a single page each day. I'd be farther along now if I had done it that way... ~wry grin~
**Obligatory reference to Plato's Allegory of the Cave. It's been awhile since I read it, but I think that he was trying to take it one step further: that the physical world we observe is due to something else, like the two-dimensional shadows cast on a cave wall by three-dimensional objects with a light behind them. Whether he explicitly said so or not, the physical world that we perceive might well be such a shadow of what we call mathematics.
***L-mode is what I'm using to refer to the more logical/linguistic operations of the brain (also known as "left brained", though my understanding is that term contains implications that aren't entirely accurate).
****~blink~ I think I just reduced Understanding to Graph Theory. Or at least I made a claim of some kind of analog between the two.
