Claremont Colleges Scholarship @ Claremont CMC Senior Theses CMC Student Scholarship 2011 The aM thematical Landscape Antonio Collazo Claremont McKenna College Recommended Citation Collazo, Antonio, "The aM thematical Landscape" (2011). CMC Senior Theses. Paper 116. http://scholarship.claremont.edu/cmc_theses/116 This Open Access Senior Thesis is brought to you by Scholarship@Claremont. It has been accepted for inclusion in this collection by an authorized administrator. For more information, please contact [email protected]. Claremont McKenna College The Mathematical Landscape A guided tour Submitted To Professor Asuman G. Aksoy and Dean Gregory Hess By Antonio Collazo For Senior Thesis Spring 2011 April 25, 2011 Abstract The intent of this paper is to present the reader will enough information to spark a curiosity in to the subject. By no means, is the following a complete formulation of any of the topics covered. I want to give the reader a tour of the mathematical landscape, hoping maybe to see them again. There is plenty of further details to explore in each section, I have just touched the tip the iceberg. The work is basically in four sections: Numbers, Geometry, Functions, Sets and Logic, which are the basic building block of Math. The first sections are a exposition into the mathematical objects and their algebras. The last section dives into the foundation of math, sets and logic, and develops the \language" of Math. My hope is that after this, the read will have the necessarily (maybe not sufficient) information needed to talk the language of Math. Antonio Collazo What is Mathematics? Take a random sample from our population, and ask them \What is Mathematics?" Their answer, most likely, is that Mathematics is about numbers; some will add that Mathematics is the study of numbers and their arithmetic. This view of math has been held, incorrectly, by society for almost 2500 years; dating back to the Greeks. However, to fully appreciate the beauty and the power of Math, one must dismiss this view as a definition of math. A clever joke goes: \What is Mathematics? Well, mathematics is what a Mathematician does?" This is clearly does no better in a our search for a definition for math, it just moves the burden from defining Math to defining what a mathematician does; equally as formidable. So then, what is a useful definition of math? I propose that Mathematics is the study of Structures and their Relationships. But, this definition again moves the burden from defining Math, to defining two things: Structures and Relationships. The rest of the section will discuss definitions for these two terms; from which the section's title can be answered. Also, I hope I can convince the reader that this definition does encompass the true essence of math, in a way that the \old definition" could not. For instance, if Math is \numbers", then where does geometry settle in? The \old" definition plainly states that geometry is not Mathematics, for it's is not the study of numbers; yet, this is unsatisfying situation. If geometry isn't a Math, then what? However, by way of the proposed definition: Geometry is the study of the structure of Shapes: triangles, squares, polygons, etc, and their relationships: congruent, similarity, length, area, etc. More specifically, Geometry, as defined by the Erlanger Program, is the study of shapes, and the invariant properties under a group of transformations. Either way, there is a clear notion of the structure and of the relationships being studied, thus geometry is a math. The true power of Math is that all one needs to do Math is some thinking, maybe a pencil and paper. Although most mathematicians, will argue no math worth doing is done without a pencil. To which I do not argue, insofar as convergent thinking practice is a necessary component to creative thinking. However, the ability to endure a race through the landscapes of your mind, is a powerful and necessary tool for math. This is the other component of creativity: divergent thinking. As an exercise of such thinking, lets construct an abstract notion of numbers. 2 The Mathematical Landscape Antonio Collazo Natural Numbers, N What is the common pattern in this collection of objects: three sheep, three cows, three dogs? Although there are several answers to the question, as in a collection of four-legged animals (mammals) or animals found on farm, all have in common the number three. The recognition of this pattern is the creation of an abstract concept of a number. Three is property that each element in the collection satisfies. However, the collection: three ties and three shirts, also has the common pattern of three; yet, none of the elements in the collection are the same in anyway. This abstract notion of a number devoid of anything else but the pattern mentioned above. We find some other patterns that hold for this abstract notion of a number. For instance, if we take three cows and then get two more cows, we will have five cows. The same holds true for dogs or sheeps instead of cows, and so suppose that this pattern is a pattern of Numbers. When we take a number then add another number the result is also a number. This is called closure, written x + y 2 N 8x; y 2 N. The property of associativity can also be inferred by similar reasoning. Given three numbers, it does not matter if I add the first two and then add the third, or if I add the last two and then the first. Much more precisely stated, (a + b) + c = a + (b + c). Effectively this property states that there is no ambiguity in writing a + b + c, since it does not matter which addition is done first. One also can test that a + b = a + b, which the Commutativity law. Consider a bag of marbles and some containers, the above mentioned properties can easily be verified. Suppose, one marble represents the number one. The addition 3 + 4 can be visualized by putting three marbles in a container, then putting four marbles in the same container. The resulting number of marbles in the container is the answer to the calculation. The commutativity law is easily verified, putting three then four is the same as putting four then three marbles into the container. Associativity is also easily checked, (2 + 4) + 7 = 2 + (4 + 7) represents putting two then four marbles in one container, then putting seven marbles into that container. Which is the same as putting four then seven marbles into a container, then putting two more marbles in. There is an equally as appealing representation of multiplication using the bags of marbles and containers. If there are three cups, each with four marbles, then how many marbles are there? Twelve, but how did I calculate that? I could pour all the marbles 3 Antonio Collazo in one cup, and then count them one-by-one; however we could define a new operation: multiplication to be answer to this question. 3 × 4 = 12. Using the same ideas as with addition, we want to consider if this new operation has the properties of closure, associativity, and commutativity. Each can be verified by similar means, for instance commutativity: there are the same amount of marbles if there are two containers each with five marbles, and if there are five containers each with two marbles. The next logical question is how do addition and multiplication relate? Symbolically, what is 2 × (4 + 6)? Pictorially, how many marbles are there if I double the amount of marbles in a container with four marbles and a container with six marbles? Notice, this question is not incalculable, even without a formal law of how multiplication and addition interact. 4+6 = 10 and then 2∗10 = 20. Yet, there is value in recognizing the pattern of the Distributive Law a×(b+c) = a×b+a×c. Note that 2×4+6 = 8+6 = 14 6= 2×(4+6) = 20, the parenthesis are important in defining the order of operations. Collecting terms, we have inferred that Numbers obey the above mentioned properties, at least in every case we've tried. There is no guarantee, however intuitive it seems, that under the evidence presented the numbers must obey these properties. Here is the first rather disappointing/unsatisfying step, although we will find this dissatisfaction can never truly be overcome. We assume that Numbers must obey these properties. Any system must have some terms and relationships that must be assumed rather deduced; if not, the resulting is circular logic. i.e.: All men are male. All males are men. (Although one could argue the need a certain maturity factored needed to socially be a \man"). A more sufficient treatment of why this is so will be handled later. The main point is that any system must start with some rules that must be assumed to be true; and consequently, the system developed from these assumed rules is dependent on the validity of these rules to be valid. I.e.: if one of the assumed axioms is invalid, then so is the developed system and if all of the assumed axioms are valid then so is the developed system. In this light, the dissatisfaction in just plainly assuming these properties subsides, insofar as the validity of theses assumed rules is plainly obvious. In the case of the above mentioned properties, the assumed axioms are sufficiently self-evident that one can rest assured that any system developed from the properties will have some valid value. To be more specific the \numbers" just described are the Natural Numbers, denoted N.