Mathematics, as it is usually taught, is boring and dull for most students. But, if we can look at mathematics as a philosophical pursuit into the order of the universe itself, it can be seen as one of the most elegant and beautiful areas of human research.
Math is the language of nature and it is in us, our thoughts, our world, and our systems of communication. Everything can be seen mathematically and math is everything. It is the same across imagination. It is purely of mind applied to a reality that may or may not be mind itself.
Mathematics is the study of numbers and space without necessary reference to empirical observation, making it fundamentally different than science and more akin to logic.
Although science could not progress without mathematics, mathematics can proceed without science. Humans use mathematics to approximate and quantify the observable world and describe pure, a priori, mathematical worlds in various types of space.
We generally hold that most of what constitutes mathematics (or all of it) is consistent across all possible worlds, or, in other words, that mathematics is necessarily the way it is. We can not, I think, imagine, in any coherent way, that one thing and one thing are anything other than two things. Particular forms of space, such as Eclidean and non-Euclidean are contingent.
Mathematics is a discipline that largely requires understanding of each foundational level of understanding. Toward that end, I have created a number of tutorial and explanatory pages that are each designed to cover a basic concept.
My tutorials are extremely dense and summary but, if followed closely, attempt to cover all of the most relevant and central aspects of the mathematical topics being described. My tutorials that are linked in numerical order below are designed to cut out the extranneous materials found in many modern textbooks.
It is my belief that mathematics is not necessarily difficult but the way in which students are often taught mathematics, through memorization rather than conceptual development, makes math both largely intractable and boring to the average student.
The following is a new structure I am creating towards a better understanding of mathematical concepts. There are infinite ways to teach mathematics but any that do not focus on concepts are doomed to fail. Mathematics is a beautiful philosophical system.
I am no great mathematician of any sort, but I am trying to teach myself higher mathematical concepts based on conceptual parts or a conceptual mathematical bedrock of understanding. So often, we are taught to memorize rules and results rather than deeply and truly understand why this or that equation, graph, or series is the resultant solution to a given problem.
Mathematics education in the United States in the last five or six years of primary school (through high school) generally consists of Algebra 1, Geometry, Algebra II, Pre-Calculus, and Calculus. Trigonometry is somewhere between Geometry and Calculus and has largely been adopted into Pre-Calculus.
I can not be certain that everything that follows is 100% accurate as all texts typically contain at least some errors. There are only so many hours in the day and I do my best to proof all of this. But, always check that what is being shown makes sense and is right. I've seen some textbooks that are horrendous in their first editions and very accurate by the third or fourth. This does nothing to help the student who wonders, perhaps indefinitely, why two plus two should equal five.
I have only briefly here touched upon mathematical concepts but I hope that a more philosophical treatment of these topics will aid in both interest and understanding. Something (or some lack of some thing) as simple as a point, for instance, upon further inspection, is deeply fascinating to me. Most teachers simply say "here is a point" and "this is point P" and that's it..
The following pages are somewhat rough drafts of what is to come and are under editing and construction. I am constantly working to update these and other pages on this site and things like mathematics, logic, and computer science, for instance, take a lot of time to explain well and in detail.
[1.00-01] Points - the philosophy of points in space and/or time.
[1.00-02] Symbols - the terms used in mathematical languages are all symbols.
[1.01-01] Numbers - naming the addition of points creates numbers.
[1.02-01] Lines - the space between two points is a line.
[1.03-01] Angles - the space between intersecting lines are angles.
[1.04-01] Variables - the space(s) where numbers are stored are variables.
[1.05-01] Shapes - lines and points in space, in certain arrangements, make shapes.
[1.05-02] The Coordinate Plane - a 2d or 3d grided shape used in mathematics is a coordinate plane.
[1.06-01] Triangles - three different points and the lines between them make a triangle.
[1.07-01] Circles - the collection of all points equidistant from one point is a circle.
[1.08-01] Operations - the ways in which we manipulate numbers with signs are operations.
[1.09-01] Properties - the rules for grouping and arranging numbers are their properties.
[1.10-01] Equations - when two sides of an expression are the same it is an equation.
[1.11-01] Inequalities - when two sides of an expression are not the same it is an inequality.
[1.11-02] Basic Algebra - variables needed to be solved for in equations or inequalities begins basic algebra. (coming soon)
[1.12-01] Functions - any process by which to manipulate or calculate a number is a function.
[1.13-01] Trigonometry - the functions of angles related to lengths are studied in trigonometry. (coming soon)
[1.13-03] Gaphing Trignometric Functions - (coming soon)
I hope to continue this tutorial with more advanced topics and concepts as well as "fill in" between these items any concepts left out. Mathematics could never be completely explained because its topics are always unfolding, being discovered, or in progress.
