Alexis Sorensen
Kant Final
James Griffith
1:30-3:00(T/TH)
11/17/12
Pure Mathematics
Immanuel Kant, a Prussian philosopher during the 1700s, examined the basis of human knowledge and its existence. Through rationalism and empiricism, Kant developed an individual model that supported the concept of pure mathematics. Kant’s logic allowed him to prove concepts that appeared unable to be proven. Pure mathematics, as an a priori cognition, can be considered to be an example of a concept that may have seemed unable to be supported through facts (Rohlf).

Mathematical principals, all consist of concepts that develop from intuitions. However, these intuitions are not able to develop as a result of prior experience. Fortunately, it is possible to intuit something a priori. An intuition of an object can occur before the actual experience of the given object occurs. One’s intuition contains the mere form of a sensory experience (Kant pp. 18-31).

Human beings, through the slightest form of sensuous intuition, are able to intuit things a priori. However, because individuals are only able to know an object as it appears to them personally, they may not be able to perceive the object as it actually exists. Mathematical concepts are constructed from a combination of intuitions, and not an analysis of concepts. Space and time are both examples of pure a priori intuitions (Rohlf). Geometry is derived from the pure intuition of space. The concept of a mathematical number developed from the addition of successive units of time. Therefore, space and time are pure a priori intuitions. Space and time exist in human beings prior to all other intuitions of objects. Therefore, both the concepts of space and time are considered to be prior knowledge of an object as it appears to an observer through the senses (Kant pp. 35-48).

Pure a priori intuition of space and time is the foundation of empirical a posteriori intuition. Synthetic a priori, in regards to mathematics, makes...

...TOK Reflection: Mathematics
To what extent is math relevant to your life and the lives of others you know and how can it become an even more viable area of knowledge.
“In mathematics I can report no deficience, except it be that men do not sufficiently understand the excellent use of the PureMathematics.”
Roger Bacon (1214-1294)
Mathematics: the abstract science of number, quantity, and space; a subject considered by many to be useless, a waste of time, and too difficult. “When am I ever going to use in real life?” or “What’s the point of learning all this?” are just some questions people ask when faced with a mathematical problem. So, what is the point of math? Paul Lockhart in his A Mathematician’s Lament stated, “How many people actually use any of this practical math they supposedly learn in school? ... So why do people think it is so important?” It isn’t important, at least from my understanding the “math” taught today isn’t important.
Lockhart says that the math taught in modern day schools is not the math that really is; it is not taught as the art that it is. “I’m sure most people use a calculator for everyday arithmetic. And why not? It’s certainly easier and more reliable. But my point is not just that the current system is so terribly bad, it’s that what it’s missing is so wonderfully good! Mathematics should be taught as art for art’s sake. These mundane...

...The importance of Mathematics
In such a modern and non-stop developing world nowadays, most people need mathematics as an important tool for their occupation, no matter what it is. Obviously, mathematics plays a vital role in daily use such as architecture, business, clerical work, etc... We even use math to balance our budget, pay bills, check our saving accounts, etc. Math is an important part of our civilization.
Most people usingmathematics recently wondered if they should learn about its history. Some claimed that the mathematics’ history itself would help nothing at all and it makes virtually no sense to study the history of this science. Because obviously in practice, they only need those formula to do the calculations. However, according to the research of Alexander (2011), he stated that “mathematics is seen not as the static skeleton of science but, instead, as a dynamic and historically evolving field in its own right – just like science itself.”
Mathematics was developed early in Babylonia since 2000 BC. The Babylonian basis of mathematics was inherited by the Greeks and independent development by the Greeks began from around 450 BC. After that, mathematics continued to flourish in some countries such as Iran, Syria, and India. In general, many famous major progresses were made by the well-known names, for examples, Galileo...

...INTRODUCTION
As the greatest Mathematician “GAUSS” has said –“MATHS IS THE QUEEN OF SCIENCE”, maths is truly the guiding force of human soul. Maths is the purest form of study of nature which comprise of a deep and rational comparison of quantities, structures, spaces, nature of change of different inter-relations and much more. Mathematicians seek out various patterns and formulate new conjectures. Mathematicians counter examines the facts by repeated and repeated logical transformations these experiments may take years and years of thorough research and dedication.
USE OF MATHS IN EVERY DAY LIFE
“I love those who love geometry”-said Plato, the famous Greek philosopher and thinker.Algebra, Arithmetic, Geometry are three major components of maths. Algebra is study of symbols, Arithmetic deals with numbers and geometry plays with figures. The structure of any railway bridge is supported with triangular shaped support system(geometry),saving is vital for ones peaceful oldage (arithmetic),the combinations of different dresses one should wear so that no continues days has same pattern(algebra),all these are nothing but some simple yet important usage of maths in ones daily life.
Where to go for shopping, choosing a shampoo or planning a holiday, all these things of our daily routine has something in common-its maths. Numerical and logical thinking plays a vital part in each of these every day activities. A good understanding of maths is essential for making sense of all...

...Semi-Detailed Lesson Plan in Mathematics (Transformations)
Level: First Year High School
Subjects: Mathematics, Geometry, Transformations
I. Objectives:
A. To recognize Euclidean transformations.
B. To recognize reflections, translations, and rotations.
C. To prove theorems related to transformations.
D. To solve problems involving transformations.
E. To apply transformations to real-world situations.
F. To create designs using transformations.
II. Materials:
papers, protractor, ruler
tangram puzzle
worksheets
III. Procedure:
A. Presentation
Activity - Folding of Paper
The teacher will give an activity that involves the folding of paper and tracing of shapes.
B. Discussion
From the activity, the teacher will point out that geometry is not only the
study of figures but is also the study of the movement of figures.
Is the original figure congruent to the other figures?
How does the second image compare to the original figure?
C. Input
Definitions:
Transformations
Reflection
Rotation
Translation
Dilation
Rigid Motion
Theorems:
Theorem 18-1
Theorem 18-2
Theorem 18-3
Theorem 18-4
C. Discussion
The above definitions and theorems will be discussed and proved. The teacher will ask the student to give examples of transformations.
D. Activity
Tangram Puzzle
The students will form six groups. Each group is going to make images of animals using tangram puzzle and they will identify the...

...Why study Mathematics?
The main reason for studying mathematics to an advanced level is that it is interesting and enjoyable. People like its challenge, its clarity, and the fact that you know when you are right. The solution of a problem has an excitement and a satisfaction. You will find all these aspects in a university degree course.
You should also be aware of the wide importance of Mathematics, and the way in which it is advancing at a spectacular rate. Mathematics is about pattern and structure; it is about logical analysis, deduction, calculation within these patterns and structures. When patterns are found, often in widely different areas of science and technology, the mathematics of these patterns can be used to explain and control natural happenings and situations. Mathematics has a pervasive influence on our everyday lives, and contributes to the wealth of the country.
The importance of mathematics
The everyday use of arithmetic and the display of information by means of graphs, are an everyday commonplace. These are the elementary aspects of mathematics. Advanced mathematics is widely used, but often in an unseen and unadvertised way.
• The mathematics of error-correcting codes is applied to CD players and to computers.
• The stunning pictures of far away planets sent by Voyager II could not have had their...

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Set (mathematics)
From Wikipedia, the free encyclopedia
This article is about what mathematicians call "intuitive" or "naive" set theory. For a more detailed account, see Naive set theory. For a rigorous modern axiomatic treatment of sets, see Set theory.
An example of a Venn diagram
The intersection of two sets is made up with the objects contained in both sets
In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}. Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.
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Definition[edit]
A set is a well defined collection of objects. The objects that make up a set (also known as the elements or members of a set) can be anything: numbers, people, letters of the alphabet, other sets, and so on. Georg Cantor, the...

...Zero in Mathematics
Zero as a number is incredibly tricky to deal with. Though zero provides us with some useful mathematical tools, such as calculus, it presents some problems that if approached incorrectly, lead to a breakdown of mathematics as we know it.
Adding, subtracting and multiplying by zero are straightforward.
If c is a real number,
c+0=c
c-0=c
c x 0=0
These facts are widely known and regarded to hold true in every situation.
However, division by zero is a far more complicated matter. With most divisions, for example,
10/5=2
We can infer that
2 x 5=10
But if we try to do this with zero,
10/0=a
0 x a=10
Can you think of a number that, when multiplied by 0, equals 10? There is no such number that we have ever encountered that will satisfy this equation.
Another example will emphasise the mysteriousness of dividing by zero.
One may assume that
(c x 0)∕0=c
The zeroes should cancel, as would be done with any other number. But since we know that
c x 0=0
it follows that
(c x 0)/0=0/0=c
This does not seem to make sense. This also means that
1=0/0=2
1=2
since 1 and 2 are both real numbers. Actually, this means that 0/0 is equal to every real number!
In effect, there is no real answer to a division by zero. It cannot be done.
In fact, if we could divide by zero, it would be possible to prove anything that we could dream of. For example, imagine a student trying to prove to his teacher that he...

...Introduction
Mathematics is an indispensable subject of study. It plays an important role in forming the basis of all other sciences which deal with the material substance of space and time.
What is Mathematics?
Mathematics may be described as the fundamental science. It may be broadly described as the science of space, time and number. The universe exists in space and time, and is constituted of units of matter. To calculate the extension or composition of matter in space and time and to compute the units that make up the total mass of the material universe is the object of Mathematics. For the space-time quantum is everywhere full of matter and we have to know matter mathematically in the first instance.
Importance of Mathematics
Knowledge of Mathematics is absolutely necessary for the study of the physical sciences.
Computation and calculation are the bases of all studies that deal with matter in any form.
Even the physician who has to study biological cells and bacilli need to have a knowledge of Mathematics, if he means to reduce the margin of error which alone can make his diagnosis dependable.
To the mechanic and the engineer it is a constant guide and help, and without exact knowledge of Mathematics, they cannot proceed one step in coming to grips with any complicated problem.
Be it the airplane or the atom bomb,...