Maybe I was born without an intuition...
My mom used to say that, as a kid, I found the things that most people found easy to be highly perplexing. However, if you gave me something complicated that no one could understand, I would be able to figure it out with minimal effort. (Prime example: learning how to use our new computer at nine-years-old. No one in the house had any clue what to do with it, and I ended up learning how to use it as well as teaching myself how to program in Basic.)
Today I had a conversation with a friend about something (in my mind) highly related. I struggle getting the right mix of "concept" and "mathematical rigor" in a course. Specifically, I don't feel like I have gained much if the instructor only talk about things in terms of concepts. I like to see the mathematical development. Likewise, sitting in a course in engineering or physics can be frustrating unless I happen to have a professor who is good at saying things like, "What the math is telling us is..." I seem to have a strong need for a well-developed mathematical grounding for a concept to stick. On the other hand, one of my undergrad instructors told me that I seemed to need to work with physical systems (which implied, to me, that mathematical abstraction was definitely not a strength). I don't doubt the veracity of this statement: working out problems is often the best way for me to gain insight into a concept, which means abstraction must be at least somewhat limited.
My friend told me that I use math as a crutch and that I need to develop my intuition. I'm not sure that I agree with that statement. I could easily counter that most people who rely on intuition don't rely enough on a mathematical foundation. However, it got me curious about how much people rely on logic versus intuition. While looking on the web, I came across an excellent essay by Poincaré about different approaches in math. Specifically, he grouped mathematicians into either 'logicians' or 'geometers'. The logicians rely strongly on mathematical rigor, while the geometers use intuition. His argument was that people have a natural tendency to fall into one group or the other.
It is impossible to study the works of the great mathematicians, or even those of the lesser, without noticing and distinguishing two opposite tendencies, or rather two entirely different kinds of minds. The one sort are above all preoccupied with logic; to read their works, one is tempted to believe they have advanced only step by step, after the manner of a Vauban who pushes on his trenches against the place besieged, leaving nothing to chance. The other sort are guided by intuition and at the first stroke make quick but sometimes precarious conquests, like bold cavalrymen of the advance guard.
The method is not imposed by the matter treated. Though one often says of the first that they are analysts and calls the others geometers, that does not prevent the one sort from remaining analysts even when they work at geometry, while the others are still geometers even when they occupy themselves with pure analysis. It is the very nature of their mind which makes them logicians or intuitionalists, and they can not lay it aside when they approach a new subject.
Nor is it education which has developed in them one of the two tendencies and stifled the other. The mathematician is born, not made, and it seems he is born a geometer or an analyst.
I keep pondering this question, wondering what group I would be in. I'm sure my friend would say I'm a logician. However, I actually do use intuition a lot. I seldom rely on it and very often don't trust it. I don't feel comfortable making any type of assertion unless I have a way to prove it mathematically, mostly because I've been wrong following that path on many occasions. (Of course, on other occasions, I've been right, as well.)
Thinking back to the computer example, I know I must've had some insights that allowed me to figure out how to do what some of the adults around me couldn't. However, using this insight, I figured out how to program, which is, IMO, one heck of an exercise in logic.
Fortunately, Poincaré makes the point that one cannot fall purely into one category or the other. He also makes the point that both types of approach are necessary for the advancement of science and math. It makes me hopeful, looking at his list of examples of both types, that one can be effective at something regardless of which class they have been born into.
Today I had a conversation with a friend about something (in my mind) highly related. I struggle getting the right mix of "concept" and "mathematical rigor" in a course. Specifically, I don't feel like I have gained much if the instructor only talk about things in terms of concepts. I like to see the mathematical development. Likewise, sitting in a course in engineering or physics can be frustrating unless I happen to have a professor who is good at saying things like, "What the math is telling us is..." I seem to have a strong need for a well-developed mathematical grounding for a concept to stick. On the other hand, one of my undergrad instructors told me that I seemed to need to work with physical systems (which implied, to me, that mathematical abstraction was definitely not a strength). I don't doubt the veracity of this statement: working out problems is often the best way for me to gain insight into a concept, which means abstraction must be at least somewhat limited.
My friend told me that I use math as a crutch and that I need to develop my intuition. I'm not sure that I agree with that statement. I could easily counter that most people who rely on intuition don't rely enough on a mathematical foundation. However, it got me curious about how much people rely on logic versus intuition. While looking on the web, I came across an excellent essay by Poincaré about different approaches in math. Specifically, he grouped mathematicians into either 'logicians' or 'geometers'. The logicians rely strongly on mathematical rigor, while the geometers use intuition. His argument was that people have a natural tendency to fall into one group or the other.
It is impossible to study the works of the great mathematicians, or even those of the lesser, without noticing and distinguishing two opposite tendencies, or rather two entirely different kinds of minds. The one sort are above all preoccupied with logic; to read their works, one is tempted to believe they have advanced only step by step, after the manner of a Vauban who pushes on his trenches against the place besieged, leaving nothing to chance. The other sort are guided by intuition and at the first stroke make quick but sometimes precarious conquests, like bold cavalrymen of the advance guard.
The method is not imposed by the matter treated. Though one often says of the first that they are analysts and calls the others geometers, that does not prevent the one sort from remaining analysts even when they work at geometry, while the others are still geometers even when they occupy themselves with pure analysis. It is the very nature of their mind which makes them logicians or intuitionalists, and they can not lay it aside when they approach a new subject.
Nor is it education which has developed in them one of the two tendencies and stifled the other. The mathematician is born, not made, and it seems he is born a geometer or an analyst.
I keep pondering this question, wondering what group I would be in. I'm sure my friend would say I'm a logician. However, I actually do use intuition a lot. I seldom rely on it and very often don't trust it. I don't feel comfortable making any type of assertion unless I have a way to prove it mathematically, mostly because I've been wrong following that path on many occasions. (Of course, on other occasions, I've been right, as well.)
Thinking back to the computer example, I know I must've had some insights that allowed me to figure out how to do what some of the adults around me couldn't. However, using this insight, I figured out how to program, which is, IMO, one heck of an exercise in logic.
Fortunately, Poincaré makes the point that one cannot fall purely into one category or the other. He also makes the point that both types of approach are necessary for the advancement of science and math. It makes me hopeful, looking at his list of examples of both types, that one can be effective at something regardless of which class they have been born into.