The problem with mathematics by itself is that it doesn’t provide the big picture; it’s limited to a particular discipline, so it’s merely a shadow of the form of the Good.
- taken from here
Tuesday, September 30, 2008
The Problem With Mathematics
We like to think of ourselves as rational, logical creatures. We are far from it. Rational behavior, behavior directed to obtain a goal or avoid unpleasant consequences, has to be learned, studied and practiced. Logic is a recent development forced upon us by a growing awareness of the universe and is a subset of language governed by strict rules that have to be learned and obeyed for logic to function. Neither comes with the package we call language. They are both additional features that have to be purchased with hard work and study.
Mathematics is a diverse set of tools developed to help us understand the universe. The basic tool upon which all others depend is logic, for it is mathematical logic that justifies and validates the remaining tools. All of the tools that are mathematics must conform to a number of criteria, such as having consistent rulesets by which operations within the tool are defined. Any inconsistency will invalidate the tool and make it useless. Mathematicians go to great lengths to test their tools and demonstrate that they are up to the tasks set for them, the dreaded "mathematical proofs". These 'proofs' are logical meta-tool examinations of the basis and validity of the tool's ruleset.
And that statement highlights one of the biggest problems we have with mathematics: it is hard to talk about mathematics and even harder to teach to someone to whom the tools are unfamiliar.
When we teach arithmetic and basic logic in elementary and middle school, we don't stress the concept of layers of meta-language, of talking about language itself. Many of the tools of mathematics function like language. Logic is, as I have mentioned, a strict subset of natural language. When we apply logic to geometry, logic doesn't become a part of geometry. Our proofs are done in the meta-language 'logic' as applied to the tool 'geometry'. But we teach it as one subject.
Geometry is simple enough, once you learn it. So is algebra. Trigonometry is based on geometry quantified, although too little is done with imaginary numbers. Calculus is based on geometry, algebra and trigonometry with the concept of infinitesimals thrown in. They build upon each other, so that if one is learned well, the student has a solid basis for learning the next. There are lots of little fiddling details to remember and some people have enough of a problem with spacial visualization that they are unable to conquer geometry, but apart from the complexity, the tools are pretty simple. Once you learn them, they are easy to use, particularly now with the computing problem eased by the use of calculators and computers.
But mathematics isn't being sold as simple. It's being painted as being horrible difficult. For some people, it is.
Some people are incapable of learning logic. They will never admit it. In fact, should you suggest it they will shout you down or bully you into withdrawing your suggestion. Being capable only of highly emotional thought patterns, they don't have any idea what logic is and are incapable of learning it. Such people will most likely do sports while in school, then get into politics or business, where they can strive for alpha status, later. They would make poor scientists.
Still, for those capable of learning and using the tools, there is no point in confusing them by starting them off with the misconceptions we are currently generating. Mathematics is a heirarchy of little tools, each simple enough in its own way, integrated to work together. We can teach the tools individually.
We have to learn how to talk about mathematics.
- taken from here
Mathematics is a diverse set of tools developed to help us understand the universe. The basic tool upon which all others depend is logic, for it is mathematical logic that justifies and validates the remaining tools. All of the tools that are mathematics must conform to a number of criteria, such as having consistent rulesets by which operations within the tool are defined. Any inconsistency will invalidate the tool and make it useless. Mathematicians go to great lengths to test their tools and demonstrate that they are up to the tasks set for them, the dreaded "mathematical proofs". These 'proofs' are logical meta-tool examinations of the basis and validity of the tool's ruleset.
And that statement highlights one of the biggest problems we have with mathematics: it is hard to talk about mathematics and even harder to teach to someone to whom the tools are unfamiliar.
When we teach arithmetic and basic logic in elementary and middle school, we don't stress the concept of layers of meta-language, of talking about language itself. Many of the tools of mathematics function like language. Logic is, as I have mentioned, a strict subset of natural language. When we apply logic to geometry, logic doesn't become a part of geometry. Our proofs are done in the meta-language 'logic' as applied to the tool 'geometry'. But we teach it as one subject.
Geometry is simple enough, once you learn it. So is algebra. Trigonometry is based on geometry quantified, although too little is done with imaginary numbers. Calculus is based on geometry, algebra and trigonometry with the concept of infinitesimals thrown in. They build upon each other, so that if one is learned well, the student has a solid basis for learning the next. There are lots of little fiddling details to remember and some people have enough of a problem with spacial visualization that they are unable to conquer geometry, but apart from the complexity, the tools are pretty simple. Once you learn them, they are easy to use, particularly now with the computing problem eased by the use of calculators and computers.
But mathematics isn't being sold as simple. It's being painted as being horrible difficult. For some people, it is.
Some people are incapable of learning logic. They will never admit it. In fact, should you suggest it they will shout you down or bully you into withdrawing your suggestion. Being capable only of highly emotional thought patterns, they don't have any idea what logic is and are incapable of learning it. Such people will most likely do sports while in school, then get into politics or business, where they can strive for alpha status, later. They would make poor scientists.
Still, for those capable of learning and using the tools, there is no point in confusing them by starting them off with the misconceptions we are currently generating. Mathematics is a heirarchy of little tools, each simple enough in its own way, integrated to work together. We can teach the tools individually.
We have to learn how to talk about mathematics.
- taken from here
Poincaré Conjecture
This Millenium problem was recently solved and the mathematicians collected US$1 million dollars for their solution. Grigori Perelman used the Ricci flow developed by Richard Hamilton tp develop the proof.
If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is "simply connected," but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since. - taken from claymath.org
In an ordinary 2-sphere, any loop can be continuously tightened to a point on the surface. Does this condition characterize the 2-sphere? The answer is yes, and it has been known for a long time. The Poincaré conjecture asks the same question for the 3-sphere, which is more difficult to visualize. - Wikipedia
If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is "simply connected," but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since. - taken from claymath.org
In an ordinary 2-sphere, any loop can be continuously tightened to a point on the surface. Does this condition characterize the 2-sphere? The answer is yes, and it has been known for a long time. The Poincaré conjecture asks the same question for the 3-sphere, which is more difficult to visualize. - Wikipedia
The problem with religion..
... our visions and arguments concerning the "reasonable" and "the public" are infused with and are significantly and discrepantly shaped by particular histories, doctrines, perceptions, and sensibilities with which we identify - in ways that seem powerfully to elude transparency. (p.246 Beyond Gated Politics)
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