Notes |
“Was mathematics invented or discovered?” This is an age old question that has yet to be definitively answered. Some people believe that the human mind is the creator of mathematics, like a painter or poet, is a maker of patterns. [N, p. 2027] Others believe that mathematics exists independently of the human mind and therefore was discovered. This thesis will support the position that mathematics was discovered, not invented. When making a discovery, we are obtaining knowledge about something that already previously existed, just hadn’t been found yet. This paper will develop the ideas of previous philosophers and mathematicians to support my own mathematically philosophical based theory. It will take basic elements and prove they exist by showing their constructability. Then proceed to show that every branch of math is based on these constructible elements by the use of deductive reasoning. With constructability, comes existence, and with existence, comes discoveries. The following is a list of the basic elements this paper will use as the foundation of mathematics; a point, a line, a line segment, a surface, a plane surface, addition, commutative property of addition, associative property of addition, multiplication, commutative property of multiplication, associative property of multiplication, distributive property for multiplication over addition, subtraction, division, square roots, the natural numbers, the integers, the rational numbers, the irrational numbers, the real numbers, pi, Euclid’s 5 axioms, Euclid’s 5 propositions, area of a rectangle, area of a triangle, area of a parallelogram, area of a trapezoid, area of a circle, volume of a 3-dimensional object, perpendicular lines, parallel lines, proportionality, similar triangles, bisection of an angle, bisection of a finite line, angle-sum theorem, right triangles, equilateral triangles, the Pythagorean theorem, inscribing a regular polygon in a given circle, circumscribing a regular polygon about a given circle, the coordinate plane, coordinate space, fundamental theorem of arithmetic, Euclidean algorithm, a prime number, Euclid’s lemma, divisibility, the division theorem, sets, and infinity. Mathematics is rooted in these basic elements, upon which the rest grew. After these basic elements are proven to exist independently of us, the remainder of this thesis will demonstrate deductively how the rest of mathematics also exists. Mathematics was discovered and is still being discovered today. |