Tok Notes
Mathematics Notes
Introduction:
* Characterized as a search for abstract patterns * The fact that if you take two of anything and another two you end up with four of that thing * Any circle no matter the size, if you take its circumference divide in by its diameter you end up with the value 3.14… * Mathematics not only gives certainty but practical value * Galileo: book of nature is written in mathematics * Mathematics also “leaves us with nowhere to hide” because in other subjects the answer is usually dependent upon interpretation, but in math there is only one answer and if you get it wrong you’re wrong. * Thinking in terms of mathematics requires a person to disregard context and consider things at an abstract level * In the expert written by Phillip Roth, Nathan’s dad gives him a math problem involving a clothing dealer trying to sell an old overcoat by reducing the price. Instead of focusing on the word problem Nathan observes the oddity of the situation and the details. This supports the idea of looking at mathematics from an abstract point of view without regarding details.
The Mathematical Paradigm: * ‘ the science of rigorous proof’ * Euclid (300 B.C.) was the “first person to organize geometry into a rigorous body of knowledge” and his theories have had a major influence on civilization. * He developed a formal system that consisted of three parts: * Axioms * Deductive reasoning * Theorems * Axioms: * “starting points or basic assumpstions” * There are requirements for a set of axioms: * Consistent: If you can deduce a variable and its opposite from a set of axioms that that is inconsistent. Inconsistency if allowed into a system causes there to be a proof for just about anything. * Independent: You should begin with the smallest number of axioms and you should not be able to deduce an axiom from another because then that
Introduction:
* Characterized as a search for abstract patterns * The fact that if you take two of anything and another two you end up with four of that thing * Any circle no matter the size, if you take its circumference divide in by its diameter you end up with the value 3.14… * Mathematics not only gives certainty but practical value * Galileo: book of nature is written in mathematics * Mathematics also “leaves us with nowhere to hide” because in other subjects the answer is usually dependent upon interpretation, but in math there is only one answer and if you get it wrong you’re wrong. * Thinking in terms of mathematics requires a person to disregard context and consider things at an abstract level * In the expert written by Phillip Roth, Nathan’s dad gives him a math problem involving a clothing dealer trying to sell an old overcoat by reducing the price. Instead of focusing on the word problem Nathan observes the oddity of the situation and the details. This supports the idea of looking at mathematics from an abstract point of view without regarding details.
The Mathematical Paradigm: * ‘ the science of rigorous proof’ * Euclid (300 B.C.) was the “first person to organize geometry into a rigorous body of knowledge” and his theories have had a major influence on civilization. * He developed a formal system that consisted of three parts: * Axioms * Deductive reasoning * Theorems * Axioms: * “starting points or basic assumpstions” * There are requirements for a set of axioms: * Consistent: If you can deduce a variable and its opposite from a set of axioms that that is inconsistent. Inconsistency if allowed into a system causes there to be a proof for just about anything. * Independent: You should begin with the smallest number of axioms and you should not be able to deduce an axiom from another because then that