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A similar conclusion struck me when I was taking a linear algebra course in my 1st year. I had a very smart friend, and sometimes I studied with him. He usually
by darkmighty 13y ago
A similar conclusion struck me when I was taking a linear algebra course in my 1st year. I had a very smart friend, and sometimes I studied with him. He usually studied a lot more than myself, and naturally fared a lot better (and more consistently).
However there were some problems which were very unusual which I would look at, think for just a few seconds, and have a delightful path to the solution -- and often times, many. My friend, otoh, would usually look into the question, think for quite some time, and move on to the more straightforward ones, although with adequate time he would definitively solve it, and with adequate study, he would get consistent, as he did. But at once I discovered his whole process was fundamentally different. He found the solutions thinking algebraically, not geometrically. This algebraic skills allowed him to plow through most problems with speed, but when he needed to think creatively, to have a geometric intuition on something, he was slowed down.
He is a wonderful programmer however, and I'm sure he is "algotihmically creative".
For me this indicates that a lot of science lends itself a lot to have a spatial intuition to be effectively creative. Einstein famously came up with relativity through thought experiments, not algebraic reasoning of the laws of nature. Planck however, with equal notoriety introduced the quanta as an algebraic "trick", and Maxwell formulated an electromagnetic theory of light by manipulating differential equations. Computer science and a fraction of math, lend themselves to algebraic reasoning and I believe what drives those is an algebraic type of creativity, in contrast to say physics, architecture and mechanical engineering -- although I believe we need both types of thinkers everywhere.
It's a funny Yin and Yang aspect of our minds!
- rsmith05 13y agoWould you be able to elaborate on "algebraical vs geometrical" thinking? I've noticed myself that there are two types of "math" that people tend to be good at: 1) Algorithmic - Following a set of steps to achieve a particular result. Algorithms, discrete math, "compsci" math, and procedural and OO programming. Coders tend to be good at this kind. 2) "Abstract" - "Pure Math", what I would label as the harder kind of math, like calculating the intersection of planes, calculus, linear algebra, etc. Pure math majors and theoretical computer science folks are good at this kind. I have met folks that are good at 1 or 2, or both. Forgive me for my poor use of terms, I'm one of those folks that is not particularly great at math (mostly due to lack of practice) but I was always curious if my observation was backed up with any "real" terms or if there has been any research into this kind of thing. What you describe seems to be relatively close to what I've observed myself.
- bcbrown 13y agoI know that I personally am much stronger with algebraic math than geometrical math. For me, that means that integrals, linear algebra, etc are something that I'm strong at, but I'm poor at anything that requires spatial reasoning. That's different from your two types. For example, take your "calculating the intersection of planes." I would approach that by first writing the algebraic equations that define the two planes, then trying to find the manipulation that will let me solve for the intersection. Someone who is a geometric thinker might start by plotting the two planes, and reasoning spatially. So, that distinction is wholly separate from your algorithmic/abstract distinction, and to me, is a distinction within the abstract realm.
- darkmighty 13y agoYea, this is a good illustration. For example, for an intersection of two planes going through origin, the plane equation coefficients are (a multiple of) the of the normal vector coefficients. Now, the intersection vector must be perpendicular to both planes normals, so if you take the cross product of the two vectors, you get the intersection vector. This in this case I would just look at it and instantly do a cross product of the coefficients, whereas you would eventually do precisely the same, but reasoning algebraically, perhaps just as fast. In this case, thinking algebraically doesn't seem to yield the intimate relationship with your object of study required for creativity, just because of the geometric nature of it. But this reasoning wouldn't be much use in -- for instance -- fiddling with discrete mathematics, where you may get more intimate precisely by reasoning algebraically!
- jclos 13y agoThere was an essay from an extremely famous mathematician that I read not a long time ago on analytical vs geometrical oriented thinking in mathematics and I can't recall where it is and it is driving me crazy. If anyone knows what it is, please link it.
- bcgraham 13y agoYou mean Planck, not Maxwell, right?
- darkmighty 13y agoRight, Planck! (It stands corrected)