- 1. University of Mindanao Arellano St. Tagum City = History of Math (2:30-3:30) Submitted By: Junila A. Tejada Submitted To: Prof. Paulino Tado
- 2. 1. Define and exemplify each philosophical thoughts of Mathematics. The three philosophical school of thoughts in Mathematics were introduced and emphasized during the report of Jamia Mangotara entitled Three-isms. Three-isms were created because of the beliefs of the Mathematicians that philosophy plays a great role in mathematics and it provides a systematic and secure foundation for mathematical knowledge and mathematical truth. Different Mathematicians emerged with different philosophy such as the Logicism, Intuitionism, and Formalism. The major proponents of logicisms are G.Leibniz, G.Frege (1893), B.Russell (1919), A.N.Whitehead and R.Carnap (1931). In logicism, the beliefs are commonly focused on proving Mathematical concepts by using axioms and rules of logic alone. Through it, the broad mathematical concepts can be reduced into logical concepts. Logicism is also applicable by proving a certain assumption through the use of axioms, postulates, and logic in order to support the statement which sometimes leads to theorems. In short, logicism extends the importance of actualizing logical reasoning in order to support concepts. Likewise, the scope of intuitionism is more focused on the approach of reconstructing mathematical knowledge in order to secure that the significance of the ideas are more emphasized and to avoid contradictions from other Mathematicians. The raw concepts of Mathematics are very important because it serves as the main basis of all mathematical knowledge which allows the mathematicians to collect ideas and construct it into more improved and refined ideas. Furthermore, formalism believes that all mathematical truths should be supported by formal theorems and proofs. A formalist believes that in proving statements with formal theorems is to demonstrate a safety results from consistency. But in the real context of Mathematics, not all truths in Math can be represented by theorems in formal systems just like Euclid’s postulates. Euclid’s postulates are his assumptions which are accepted to be true without a formal proof. By that instance, it is not a guarantee that ideas in Math should be proven formally.
- 3. 2. Among the three philosophical thoughts of Mathematics, which do you believe most and why? The thought of Intuitionism is for me the best among the three philosophical thoughts in Mathematics. In Intuitionism, it allows the newly emerged mathematicians in collecting, rechecking, and revisiting ideas that is already provided in the past years. The raw concepts afforded by the past mathematicians require reconstruction in order to specify and highlight important concepts that was not elaborated well. Through this belief, mathematical concepts become more coherent and legitimate which leads the readers to a more detailed and specific information feeding. 3. Why do you think that Newton and Leibniz have been given much credit as the inventor of calculus where in fact so much of Calculus was developed by others ahead of them? In Calculus, many mathematicians imparted concepts and ideas but Newton and Leibniz who were recognized because for me they provide specific and coherent ideas in Calculus. Leibniz also introduced the rational notational system, algorithms and gave the name “Calculus”. By reconstructing ideas like creating new derivatives and denoting functions in Calculus they unified consistent ideas which make them recognized. For me, their fame was only due to their improved ideas and unique perspectives in Calculus which made them as the greatest inventor in Calculus. 4. Give an outline of the history of Greek mathematics from the time of Thales to the collapse of the University of Alexandria. During the 600 century B.C., Greek mathematics was founded by Thales with his concepts and approaches in Mathematics. Thales is known for his adventures and explorations wherein every culture he observed some of the mathematical concepts and he applied it into Greek mathematics. He also founded the Ionian school wherein he taught mathematics and astronomy to a number of pupils. One of his students was Anaximander.
- 4. Around 5th century B.C. , Pythagoras along with his subordinates created a society dedicated to mathematics and logic of numbers which improved the ideas of Thales. During the period of 500-300 B.C., Greek mathematics was only on slow development wherein minor advances and only improved ideas are created. But around 300 B.C., the University of Alexandria was established and founded and the Greek mathematics began to bloom. The peak of success during the 300 B.C. was the creation of Euclid’s elements. Another boost of knowledge came from Archimedes wherein his famous inventions were the lever, the compound pulley, the catapult and the use of parabolic mirrors to focus lights. He also provides works in the field of Mathematics and Sciences. Centuries follows was the time that other mathematicians like Erastosthenes, Appolunius, and Hipparchus. Appolunius was known for his 3-d solid geometry and for his elaboration on his complete theory of conics and conics sections. But during the 46 B.C, Caesar accidentally caused the university extremely damaged and burned causing some books to burn and vanished. After the fire, the University continued their works and improved their works. And around 410 A.D. the Arabs invade some part of Greece and destroyed the University. Due to this destruction, the fruitful mathematical ideas was ceased since some of their basis are destroyed by the invaders. 5. Discuss the history of Transformation Geometry and define, explain and illustrate each by construction. Transformation Geometry was based from the ideas and concepts that were already introduced by the previous mathematicians. In 17th century, Rene Descartes invented the use of Cartesian coordinate system and provides concepts in analytical geometry. Descartes contribution shows a great link between geometry and algebra. Also Pierre de Fermat was Descartes buddy in developing concepts in analytical geometry and he made more direct approach which is similar to the system currently used. In short, Fermat revised some of the concepts used in analytical geometry. Also, Felix Klein is a German geometer who showed importance of groups in geometry. He devised the
- 5. Enlanger Program which is designed to provide a cohesive system in rearranging unrelated geometries. Geometric designs are also involved in the arts and cultures of different races. The composition of designs serves as our guide in determining the artifacts belong and how long it is made. Arabesque and appliqué is a good example of geometric designs during the ancient times. Arabesque originated from the belief that creating a living objects at designs is a mortal sin so they use geometric designs as a substitute from their creation and art. Transformation in appliqué was believed due to lack of materials which enable the woman to devise a technique called appliqué. Applique is a process of sewing wherein geometric transformations where typically symmetric. Transformation geometry has its background and history which allows Klein to form four types of transformation namely as reflection, translation, rotation and dilation. Reflection simply the reflection of a figure using a line or a point and the image is also the same. Translation involves the shape in moving by sliding it up , down, sideways, or diagonal which makes the shape bigger or smaller. Rotation involves a clockwise or a counterclockwise angle and change the position of the shape but the figure stay the same. Lastly, dilation involves change the size of a figure and not its shape. 6. Enumerate each renowned mathematicians of the 20th century and discuss their contributions in the field of mathematics. The period of the 20th century mathematics is more complex and complicated even if the concepts are more detailed and specific. The emergence of the mathematicians during this period allows them to provide and discover more technology driven concepts in mathematics. The mathematicians emerge in this period are David Hilbert, Srinivasa Aiyangar Ramanujan, Kurt Goedel, Wolfgang Haken and Kenneth Appel, Andrew Wiles, and Johann Gustav Hermes. David Hilbert is the mathematician who invented and developed a broad range of fundamental ideas in invariant theory, the axiomatization of geometry, and with the notion of foundations of functional analysis. He is also considered as one of the founders of proof theory, mathematical logic and the distinction between mathematics
- 6. and meta-mathematics. Hilbert’s contribution in mathematics provides new concepts which are considered as advanced and detailed. In his invariant theory, he work into the systems with more than two variables but he failed to prove Gordan’s work since the computational difficulties were too great. Srinivasa Aiyangar Ramanujan is the mathematician who made contributions in the areas of mathematical analysis, number theory, infinite series and continued fractions. He also independently compiled nearly 3900 results during his short lifetime, and proved himself to be one of the most brilliant minds of the century. He is considered as phenomenal because of his short life he contributed many significant compilation of ideas which gathered important data and basis in Mathematics. Kurt Goedel created the two incompleteness theorems which proved the unthinkable problems that there could be solutions to mathematical statements. He devised strategies in finding for clues which helps him in solving problems. He also showed that continuum hypothesis cannot be disproved from the accepted axioms of set theory, if those axioms are consistent. Wolfgang Haken and Kenneth Appel contributed a lot in solving the four color theorem wherein they solve the mystery and the pattern behind the coherence of the colors and pattern. Four color theorem is said to be one of the hardest theorems ever proved because of its complexity but Haken and Appel proved that four color theorem provides challenging computation method and logical reasoning. Andrew Wiles is a mathematician who specialized for number theory. Number theory was first introduced by Pierre de Fermat wherein he exposed the nature and the derivation of the numbers. Due to his specialization, he was eager to discover and study the number theory presented by Fermat and enable him to prove Fermat’s Last Theorem which is considered as his greatest contribution. Johann Gustav Hermes was a mathematician who was basically inspired by the works and contribution of Euclid. Due to this inspiration, he took almost 10 years in completing the construction of his regular polygon with 65,537 sides just using compass and
- 7. straight edges. He allotted his 10 years in sketching his multisided polygon which amazed me because he is very imaginative and creative. 7. Define and compare theorem, axiom and postulate by giving relevant illustrations and examples for better understanding. The defined and important terms in geometry are theorems, axioms and postulates. Through the different theorems, axioms and postulates a certain assumption or statement can be prove which highlight that the main usage those defined terms are use to support statements in Mathematics. Theorems are defined as the important statements that are proved to be true by using illustrations or other statements that are meant to support the proposed statements. Also theorem is defined as a logical consequence of the axioms wherein rules are afforded in proving statements. In Geometry, the "propositions" are all theorems: they are derived using the axioms and the valid rules. In geometry, Theorem 1.1 states that “if two lines intersect, then they intersect in exactly one point”. In proving this theorem, we are bound to illustrate the two lines intersect and emphasized that their intersection is only one point. In that process, we are supporting and proving the statements in order to prove it right. Based on logic, an axiom or postulate is a statement that is considered to be self- evident. Both axioms and postulates are assumed to be true without any proof or demonstration. Basically, something that is obvious or declared to be true and accepted but have no proof for that, is called an axiom or a postulate. The main differences between these two concepts are axioms are defined as self-evident assumptions, which are common to all branches of science, while postulates are related to the particular science. Postulates and axioms are stated in Euclidean geometry such as: given any two distinct points, there is a line that contains them and any line segment can be extended to an infinite line. The statements are obviously true that don’t require proofs.
- 8. 8. State, define, apply and illustrate Fermat’s last theorem In Fermat’s last theorem, he stated that, "it is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain." As a result of Fermat's marginal note, the proposition that the Diophantine equation 𝑥 𝑛 + 𝑦 𝑛 = 𝑧 𝑛 where x, y,z , and n are integers, has no nonzero solutions for n>2 has come to be known as Fermat's Last Theorem. Note that the restriction n>2 is obviously necessary since there are a number of elementary formulas for generating an infinite number of Pythagorean triples (x,y,z) satisfying the equation for n=2. A first attempt to solve the equation can be made by attempting to factor the equation, giving 𝑥2 + 𝑦2 = 𝑧2 Since the product is an exact power, (𝑧 𝑛/2 + 𝑦 𝑛/2 ) ((𝑧 𝑛/2 − 𝑦 𝑛/2 ) = 𝑥2 Solving for x and y gives 𝑧 𝑛/2 = 2 𝑛−2 𝑝 𝑛 + 𝑞 𝑛 𝑦 𝑛/2 = 2 𝑛−2 𝑝 𝑛 − 𝑞 𝑛 which give
- 9. 𝑧 = (2 𝑛−2 𝑝 𝑛 + 𝑞 𝑛)2/𝑛 𝑦 = (2 𝑛−2 𝑝 𝑛 + 𝑞 𝑛)2/𝑛 9. Trace the history of algorithm briefly. How has the use of it changed? Why is it still important for students to be familiar with? The word algorithm comes after the name of the father of Algebra Abdullah Muhammad ibn Musa Al-khwarizmi. Algorithm refers to the rules of performing arithmetic using Hindu-Arabic numerals. As time goes by, Mathematicians used the concepts in Algebra in proving number relations and statements in other field of mathematics. Leibniz is one of the mathematicians who employed algorithm in determining and creating with his concepts in Calculus. He certainly used logic in order to prove statements and concepts in Calculus. By the time of Al-khwarizmi algorithm was a separate topic from all concepts in Algebra and time passed, algorithm was used as a medium of proving identities and statements in Math. Students are in need to familiarize algorithm in order to proceed with other higher concepts in Algebra because algorithm is considered as one of the basics in understanding concepts in Algebra. Algebra is also considered as the basics of all Maths, therefore algorithms should be mastered by the students so that they can proceed with other topics. 10. Discuss the role of Euclidean constructions and the three classical problems in the history of Greek Mathematics. Shapes and figures created by a compass are the known and common Euclidean constructions. By using this type of instruments, many illustrations of angles, lengths, and different types of shapes was invented. The invention of compass was later improved in order to address the three classical problems in the history of Math which includes the trisection of an angle, squaring a circle, and doubling a cube. In trisecting an angle, their goal is to divide the angle into three equal parts, constructing a square with the same area in a given circle for squaring a circle and constructing a cube involves the process of making the twice volume of the cube. The attempts of solving these problems challenged the Greeks in improvising new and more complicated instruments. Due to their perspective that the compass can solve the problems they
- 10. kept on improving the compass so that they will able to solve the problem. And almost 2000 years of use, the limitations of compass was discovered by abstract and symbolic areas in Mathematics. The development of abstract algebra proves that these constructions were impossible.