Post by Olivia Nephilim on Mar 25, 2006 1:33:40 GMT -5
Mathematics Primer for the Budding Scientist
This is the first part of a very basic introduction to concepts that should be understood fully before progressing further into more advanced ST physics/engineering topics. Alot of what is discussed should be common sense, but you are encouraged to explore further and reach an working knowledge of the subjects discussed.
Mathematics, which will be your best tool for discovering and understanding the unknown, should be studied inside and out. Let’s start by establishing what math isn’t:
Math is not accounting!!
Though many high school/college teachers vainly attempt to relate math to students by linking it to money, calculation is an attribute of mathematics; a means to an end. Knowing how to “plug in” numbers within an equation and compute the solution will not enable you to figure out new results, understand the applications, or imagine other possibilities.
Math is not memorization!!
It is a derivation from established truths. Where did the original truths come from? Generally they are founded in observation (i.e. analyzing speed and movement of objects) or just from abstract reasoning (i.e. understanding relationships between points and lines, which aren’t physical objects). It’s much easier to tell a child to memorize their multiplication tables then to ask them to prove the process of multiplication and derive its properties; this doesn’t mean they can’t move on to learn proofs later.
So what is math?
To put as simply as possible, mathematics is a condensed language used to allow comprehension of very abstract concepts, and for an easier method of manipulating and discovering said concepts. The ambiguous nature of spoken/written language motivated mathematicians to develop a compressed notation with strict syntax that allows us to describe information that is too abstract to convey in any other method. Modern notation allows for more precision, or rigor, than everyday speech can provide.
In order to solve a problem, you have to know exactly what you are seeking. Most riddles and simple logic problems exploit the imprecision of natural languages, as illustrated below:
You are given 6 horizontal lines, evenly spaced apart. Your task is to draw 5 additional lines to make 9. A line is defined as a straight distance between two points (either horizontal, vertical, or diagonal).
The answer is apparent once you rid yourself of assumptions made due to misunderstanding the language used. Click the following link to view the solution: www.geocities.com/midgetbobarg/nine.gif. Needless to say, such misleading terminology is avoided when trying to establish undeniable truths.
So, mathematics goes beyond the by-rote memorization of your youth, and evolves as a method of describing and analyzing our universe. Thus, this discipline is far from stagnant, and new branches are forming as the need arises. The further you delve into the subject, the more you'll understand the intrinsic beauty of a concise theorem or, perhaps, the bliss of understanding.
If you are interested in understanding the nature of mathematics further, the following disciplines should be studied in their entirety: Mathematical logic, Set theory, Category theory, Proof theory, Constructivism, and the basic foundations of mathematics.
Otherwise, start with understanding size, symmetry, and structure (Arithmetic, number theory, algebra). Then focus on branches that analyze objects in space (geometry, differential geometry, trigonometry, fractal geometry) or expressions of change (calculus, differential equations, dynamical systems, chaos theory).
en.wikipedia.org/wiki/Mathematics is an excellent starting resource.
This is the first part of a very basic introduction to concepts that should be understood fully before progressing further into more advanced ST physics/engineering topics. Alot of what is discussed should be common sense, but you are encouraged to explore further and reach an working knowledge of the subjects discussed.
Mathematics, which will be your best tool for discovering and understanding the unknown, should be studied inside and out. Let’s start by establishing what math isn’t:
Math is not accounting!!
Though many high school/college teachers vainly attempt to relate math to students by linking it to money, calculation is an attribute of mathematics; a means to an end. Knowing how to “plug in” numbers within an equation and compute the solution will not enable you to figure out new results, understand the applications, or imagine other possibilities.
Math is not memorization!!
It is a derivation from established truths. Where did the original truths come from? Generally they are founded in observation (i.e. analyzing speed and movement of objects) or just from abstract reasoning (i.e. understanding relationships between points and lines, which aren’t physical objects). It’s much easier to tell a child to memorize their multiplication tables then to ask them to prove the process of multiplication and derive its properties; this doesn’t mean they can’t move on to learn proofs later.
So what is math?
To put as simply as possible, mathematics is a condensed language used to allow comprehension of very abstract concepts, and for an easier method of manipulating and discovering said concepts. The ambiguous nature of spoken/written language motivated mathematicians to develop a compressed notation with strict syntax that allows us to describe information that is too abstract to convey in any other method. Modern notation allows for more precision, or rigor, than everyday speech can provide.
In order to solve a problem, you have to know exactly what you are seeking. Most riddles and simple logic problems exploit the imprecision of natural languages, as illustrated below:
You are given 6 horizontal lines, evenly spaced apart. Your task is to draw 5 additional lines to make 9. A line is defined as a straight distance between two points (either horizontal, vertical, or diagonal).
The answer is apparent once you rid yourself of assumptions made due to misunderstanding the language used. Click the following link to view the solution: www.geocities.com/midgetbobarg/nine.gif. Needless to say, such misleading terminology is avoided when trying to establish undeniable truths.
So, mathematics goes beyond the by-rote memorization of your youth, and evolves as a method of describing and analyzing our universe. Thus, this discipline is far from stagnant, and new branches are forming as the need arises. The further you delve into the subject, the more you'll understand the intrinsic beauty of a concise theorem or, perhaps, the bliss of understanding.
If you are interested in understanding the nature of mathematics further, the following disciplines should be studied in their entirety: Mathematical logic, Set theory, Category theory, Proof theory, Constructivism, and the basic foundations of mathematics.
Otherwise, start with understanding size, symmetry, and structure (Arithmetic, number theory, algebra). Then focus on branches that analyze objects in space (geometry, differential geometry, trigonometry, fractal geometry) or expressions of change (calculus, differential equations, dynamical systems, chaos theory).
en.wikipedia.org/wiki/Mathematics is an excellent starting resource.