The History of Mathematics: Necessary knowledge for Christian school mathematics students

1. The History of Mathematics: Necessary knowledge for Christian school mathematics students 2004 Teachers Convention Brian Kuiper

2. What is Mathematics? • “the science of quantity and space” (Davis and Hersh as quoted by Huber, 2) • “a means to define and render intelligible the various relationships between creatures” (Hanko, 21) • “human experience and activity with two aspects of God’s created reality: numbers and space” (Van Brummelen)

3. Why study (teach) mathematics? • Teaches students to know God • See His attributes in creation – faithfulness, order, structure, law, patterns • Students must realize the fact that mathematics is a changing field of study – adapting to fit needs of the day (technology) • Serve God as His friend-servants in the calling He has for us as students and pilgrims/ strangers/ employees

4. How to teach from a Christian perspective? • NOT simply “I found certain Bible passages where numbers and measurement are found.” • Some writing on subject by Christian mathematicians – see list at end of this presentation • References:-Tom Bergman’s masters project “Mathematics Education in a Reformed, Christian School” -R. Koole Convention Speech 5? years ago • Takes a conscious effort (as it does in other fields)

5. Rationale for teaching mathematics history • “The history of mathematics should also be a part of the curriculum. Students should see how mathematicians unfold new concepts and techniques, often in informal, intuitive, and creative ways…Historical approaches can make clear how mathematical problem posing and solving has contributed to cultural development…can show how different worldviews have led to different approaches to mathematics.” (Van Brummelen)

6. Rationale for teaching mathematics history • “Can help increase motivation and helps develop a positive attitude towards learning” • “Past obstacles in the development of mathematics can help explain what today’s students find difficult.” • “Historical problems can help develop students’ mathematical thinking.” • “History reveals the humanistic facets of mathematical knowledge.” • “History gives teachers a guide for teaching.” (Po-Hung Liu)

7. Rationale for teaching mathematics history • “I definitely believe that the historical sequence is an excellent guide to pedagogy…every teacher of secondary and college mathematics should know the history of mathematics. There are many reasons, but perhaps the most important is that it is a guide to pedagogy.” (Morris Kline)

8. Rationale for teaching mathematics history • “Another serious drawback to our present approach is that it deprives students of the sense that mathematics is a process. On a small scale, modern textbooks and typical methods of instruction fail to illustrate the way mathematicians actually think about and work on problems. On a larger scale, we deprive students of the long-term process by which a mathematical theory emerges from struggling with one or more central problems, often over many centuries. These processes, however, are the very things that we want them to understand.” (Laubenbacher, et al)

9. Rationale for teaching mathematics history • Only reference in NCTM materials – “help students learn the value of mathematics” (1989) and “all students should develop an appreciation of mathematics as being one of the greatest cultural and intellectual achievements of humankind.” (2000) • No clear guidelines in MI Curriculum Frameworks

10. Rationale for teaching mathematics history • Since mathematics is a part of God’s created world, we must be sure to teach it as such. Part of this teaching will include information regarding what has been done in the field of mathematics. I have come to think that this is as important as our need to give applications for the theories and concepts that we are teaching in the mathematics classroom. Maybe it won’t take the same amount of time, but it should be a foundation for all of our mathematics classes. (Kuiper’s opinion)

11. Rationale for teaching mathematics history • “There is probably no other science that presents such different appearances to one who cultivates it and one who does not, as mathematics. To [the noncultivator] it is ancient, venerable, and complete; a body of dry, irrefutable, unambiguous reasoning. To the mathematician, on the other hand, his science is yet in the purple bloom of vigorous youth, everywhere stretching out after the ‘attainable but unattained,’ and full of the excitement of nascent thoughts; its logic is beset with ambiguities, and its analytic processes, like Bunyan’s road, have a quagmire on one side and a deep ditch on the other, and branch off into innumerable by-paths that end in a wilderness.” (Bell, v)

12. How can I do this? • Bulletin boards • Videos – REMC (Educational Media) • Introduce to students when introducing a new concept. Example: History of Pythagoras leads in to a lesson introducing the Pythagorean Theorem. • Reports, biography reviews, and presentations of students • Reports/ presentations on mathematical symbolism

13. Examples of what to include in teaching the history of mathematics • What follows is an attempt to summarize some of what I think we can include in our mathematics lessons. I welcome your input concerning gaps or oversights. • Try to include rationale for ideas as well as their philosophy of mathematics. I think that this will lend the most insight to our students as to what their motivation was for teaching and learning certain concepts. • Important to include discussion regarding whether mathematics is “created” or “discovered” • Need to have: • “Father of…” • Founders of certain key concepts • Inventors of methods or technology

14. Periods of mathematics history • Creation - Babylonians - Egyptians • The Greeks 600BC-300AD • Oriental and Semitic peoples from 300AD to the 14th Century • Europe (Renaissance and Reformation) during the 15th and 16th Centuries • The 17th and 18th Centuries • The 19th Century • The 20th Century (Bell, 16)

15. Creation - Babylonians - Egyptians • Babylonian and Egyptian reliance on agriculture leads to need for calendar -360 days per year arranged according to the phases of the moon -Egyptian seasons determined by the cycles of the Nile -Had to add a month every so often to correct months and seasons (or 5 feast days) -Some records preserved on papyrus (Egypt) or clay tablets (Babylonian)

16. Creation - Babylonians - Egyptians Numeral systems of this era

17. Creation - Babylonians - Egyptians • Ahmes: (1680 BC) Scribe noted for the writing of the Ahmes papyrus (sometimes called Rhind). Contains much of the information about Egyptian mathematics • Contains division of 2 by the odd numbers 3 to 101 in unit fractions and the numbers 1 to 9 by 10. Also has 87 problems on the four operations, solution of equations, progressions, volumes of granaries, the two-thirds rule etc.

18. Creation - Babylonians - Egyptians • Problem 63. 700 loaves are to be divided among recipients where the amounts they are to receive are in the continued proportion

19. Creation - Babylonians - Egyptians • Not much known or written about the Babylonian mathematicians. • Important for their number system based on the number 60 – make reference to the fact that this has bearing on our system of time and degree measure. • Sun dials attributed to Egyptian and Babylonian cultures

20. The Greeks 600BC-300AD • The abacus • Responsible for many discoveries in the area of Geometry. • Early Greek mathematicians influenced by the work of the Egyptians (Thales and Pythagoras) • Applied mathematics to natural phenomenon and developed a deductive system.

21. The Greeks 600BC-300AD • Probably the most recognizable names of ancient mathematicians. • Thales, Pythagoras, Plato, Euclid, Archimedes, Heron, Diophantus, Hypatia, etc.

22. The Greeks 600BC-300AD • Concentrated on 2 areas of numbers: -logistic which dealt with techniques of calculation in trade, commerce, and science. -arithmetica, or theory, which concerned itself with properties of numbers • Developed idea of proof by deduction • Conjectured that nature can be understood through mathematics

23. The Greeks 600BC-300ADTHALES • First known Greek philosopher, scientist and mathematician although his occupation was that of an engineer. • However, none of his writing survives so it is difficult to determine his views or to be certain about his mathematical discoveries…unclear whether he wrote any works at all...On the other hand there are claims that he wrote a book on navigation but these are based on little evidence…quite probable that Thales did indeed define the constellation Ursa Minor. • Theorems of Geometry attributed to Thales: -A circle is bisected by any diameter. -The base angles of an isosceles triangle are equal. -The angles between two intersecting straight lines are equal. -Two triangles are congruent if they have two angles and one side equal. -An angle in a semicircle is a right angle. • Predicted a solar eclipse in 585BC • Measured the heights of pyramids by measuring shadows at a time when his shadow matched his height. Precursor to similar triangles?

24. The Greeks 600BC-300ADPYTHAGORAS • (1) that at its deepest level, reality is mathematical in nature,(2) that philosophy can be used for spiritual purification,(3) that the soul can rise to union with the divine,(4) that certain symbols have a mystical significance, and(5) that all brothers of the order should observe strict loyalty and secrecy. • Famous for right triangle theorem • Led to thinking on irrational numbers

25. The Greeks 600BC-300ADEUCLID • Wrote 13 books on Mathematics called The Elements. • Content included plane geometry, number theory, and irrational numbers. • Not credited with many discoveries, but with “collecting” the ideas of others and writing them out. Proved some of their postulates. • Criticized for use of parallel lines postulate which he couldn’t prove yet used in many proofs. Studies in this area led to non-Euclidean geometries.

26. The Greeks 600BC-300ADDIOPHANTUS • “Father of algebra” although much of his work is derived from Babylonian discoveries • Worked especially with linear and quadratic equations. Considered only rational solutions – all others were useless. • Recognized three types of quadratic equations ax2 + bx = c, ax2 = bx + c and ax2 + c = bx. The reason why there were three cases to Diophantus, while today we have only one case, is that he did not have any notion for zero and he avoided negative coefficients by considering the given numbers a, b, c to all be positive in each of the three cases above. • “... his boyhood lasted 1/6th of his life; he married after 1/7th more; his beard grew after 1/12th more, and his son was born 5 years later; the son lived to half his father's age, and the father died 4 years after the son.”

27. Oriental and Semitic Peoples-300AD to the 14th Century • Chinese fond of magic squares – led to matrices possibly before birth of Christ • Worked with fractions and especially reduced to lowest terms • Sine tables produced by the Hindus in 4th C. or so • Work done on quadratic equation where there were 2 roots including 1 negative root.

28. Oriental and Semitic Peoples-300AD to the 14th CenturyAl-Khwarizmi • He first reduces an equation (linear or quadratic) to one of six standard forms: • 1. Squares equal to roots.2. Squares equal to numbers.3. Roots equal to numbers.4. Squares and roots equal to numbers; e.g. x2 + 10 x = 39.5. Squares and numbers equal to roots; e.g. x2 + 21 = 10 x.6. Roots and numbers equal to squares; e.g. 3 x + 4 = x2. • Began work on completing the square. • Mohammad Kahn “In the foremost rank of mathematicians of all time stands Al-Khwarizmi. He composed the oldest works on arithmetic and algebra. They were the principal source of mathematical knowledge for centuries to come in the East and the West. The work on arithmetic first introduced the Hindu numbers to Europe, as the very name algorism signifies; and the work on algebra ... gave the name to this important branch of mathematics in the European world...”

29. Europe (Renaissance and Reformation) during the 15th and 16th Centuries • Began to develop 3 C. earlier with many translations of famous works (Elements and Algebra of different authors) • 1200’s Fibonacci sequences • Began applying algebra to geometry • Solutions to cubic and quartic equations leads to complex numbers (numeri ficti - Candano) • Much work done with maps and cartography (Mercator projection)

30. The 17th and 18th Centuries • Cavilieri – logarithms and telescope lenses • Viete – pled for the use of a decimal system and began using vowels for unknowns and consonants for givens • Napier - Plaine Discovery of the Whole Revelation of St. John (1593). Also, continued to develop logarithms • Galileo – astronomy and physics led to applications for mathematical reasoning – forerunner to calculus • Pascal – projective geometry – first digital calculator • Lagrange – calculus of probabilities, dynamics • D’Alembert – work with limits • Development of the metric system of measurement

31. The 17th and 18th CenturiesDESCARTES • Writings included discoveries in Geometry and optics • First to study meteorology from a scientific basis • I should look upon Des-Cartes as a man most truly inspired in the knowledge of Nature, than any that have professed themselves so these sixteen hundred years... (theologian Henry More)

32. The 17th and 18th CenturiesEULER • ... after 1730 he carried out state projects dealing with cartography, science education, magnetism, fire engines, machines, and ship building. ... The core of his research program was now set in place: number theory; infinitary analysis including its emerging branches, differential equations and the calculus of variations; and rational mechanics. He viewed these three fields as intimately interconnected. Studies of number theory were vital to the foundations of calculus, and special functions and differential equations were essential to rational mechanics, which supplied concrete problems. • We owe to Euler the notation f(x) for a function (1734), e for the base of natural logs (1727), i for the square root of -1 (1777), π for pi, ∑ for summation (1755), and many others. • Studied and developed works of Bernoulli, Leibniz, Goldbach, Fermat and many others of his time

33. The 19th Century • Considered by some to be the “Golden Age” of mathematics (Boyer, 496) • Additions to field total more during this 100 years than all preceding periods combined • Non-Euclidean geometries, n-dimensional spaces, infinite processes, etc. • No longer were famous mathematicians “clustered” – spread all over the globe • Much support for applicational mathematics in surveying and navigation and reduced emphasis on pure mathematics

34. The 19th Century • Gauss – differential geometry, statistics, theories of probability, bell-shaped curve • Lobatchevsky – non-Euclidean geometry, roots of equations • Cantor – much work with sets of numbers and infinite members of these sets • Cauchy – determinants, differential equations • Abel, Jacobi, Galois, Riemann, Boole, Cayley • Babbage – “Calculating Engines”

35. The 20th Century • More communication between mathematicians than ever before • Emphasis on relationships between areas of mathematics not realized before • Example: topology – relationships betweens geometric shapes became algebraic especially in regards to group theory and sets of points • Focus by some on disproving prior postulates “Formerly, when one invented a new function it was in view of some practical goal; today one invents them expressly to point out flaws in the reasoning of our fathers and one will never derive anything from them but that” (Poincare quoted by Boyer, 617)

36. The 20th CenturyPOINCARE • Differential equations • Multiple integrals • Group theory • Poincare disk – hyperbolic geometry • optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory, theory of relativity and cosmology

37. The 20th Century • Einstein, Hilbert, Bohr, Godel and others had a huge influence on the fields of science, physics, and mathematics with their discoveries. We can use the history and discoveries of these men and others to teach the history of many discoveries in many different fields and emphasize the relationships between these disciplines. They also continued to emphasize pure mathematics at certain times although the emphasis of the 20th C. had to do with applications

38. Differences in philosophy • Perhaps the most important part of the history of mathematics that we can teach our students has to do with the philosophy of different mathematicians or their rationale for doing what they did. Probably a good idea to try to find this information on each historical figure that we incorporate into our lessons. • However, it is a good lesson all by itself to at least bring this to the attention of our students if we do nothing else with the history of mathematics. • “Man-made” VS “Created” • “Invented” VS “Discovered”

39. Differences in philosophy • “the study of mathematics and its contributions to science exposes a deep question. Mathematics is man-made. The concepts, the broad ideas, the logical standards and methods of reasoning, and the ideals which have been steadfastly pursued for over two thousand years were fashioned by human beings. Yet with this product of his fallible mind man has surveyed spaces too vast for his imagination to encompass; he has predicted and shown how to control radio waves which none of our senses can perceive; and he has discovered particles too small to be seen by the most powerful microscope. Cold symbols and formulas completely at the disposition of man have enabled him to secure a portentous grip on the universe. Some explication of this marvelous power is called for.” (Morris Kline, ix)

40. Differences in philosophyMan-made (humanistic) • Most likely a product of the teachings of Aristotle – mathematics begins by the activity of the mind • Morris Kline “”Mathematicians had given up God and so it behooved them to accept man…Nature’s laws are man’s creation. We, not God, are the lawgivers of the universe. A law of nature is man’s description and not God’s prescription.” (97) • “Mathematics is a man-made, artificial subject” and “We just don’t know why mathematics works as it does. We’re faced with a mystery” (Kline from US News and World Report quoted by Zimmerman, 36)

41. Differences in philosophyMan-made (humanistic) • Quotes from Zimmerman in “Mathematics: Is God Silent” an article he published in The Biblical Educator -“the subject in which we never knew what we were talking about, nor whether what we are saying is true” (Bertrand Russell) -“a body of knowledge containing no truths” (Morris Kline) -“a meaningless, formal game” (David Hilbert) -“Integration is not possible in mathematics. In mathematics, God’s revelation is silent. There is nothing to integrate…the mathematician is not seeking for truth…as far as mathematics goes, there ain’t nothin’ there.” (Dr. David Neu, Professor of mathematics at Westmont College) -G.H. Hardy boasted that ‘he had never done anything useful and regarded God as his personal enemy’

42. Differences in philosophyMan-made (humanistic) • Some mathematicians try to combine humanism with God in some way. They usually believe that mathematics is the product of the human mind. Then, they allow God to have some part in the process of mathematics. • Examples (to one extent or another) are Kant, Descartes, and Pascal. They were pure thinkers who felt that God wouldn’t allow mathematicians to invent things untrue so there may have been some way he guided the mind in these matters.

43. Concerning these the Bible says: • II Timothy 3:2a and 7, “For men shall be lovers of their own selves…Ever learning, and never able to come to the knowledge of the truth” • Romans 1:20-22, “For the invisible things of him from the creation of the world are clearly seen, being understood by the things that are made…they glorified him not as God…their foolish heart was darkened…Professing themselves to be wise, they became fools.”

44. AND YET… “The relationship between mathematics and science to Kline is ‘mysterious’ and ‘miraculous’; to Wigner, ‘unreasonable’; to von Neumann, ‘quite peculiar’; to Bourbaki, ‘unexpected’; to Bell it is ‘curious’; to Whitehead, ‘a paradox’; and Einstein asked, ‘How can it be?’ If the ‘peculiar’ usefulness of mathematics in natural science cannot be rationally accounted for from the perspective of mathematics-as-art or human invention, those of this persuasion should ask themselves if their position is realistic.” (Zimmerman, 36)

45. Differences in philosophyDiscovered through Creation • Platonistic – mathematics is discovered – “what evolves is not mathematics but our knowledge of mathematics.” (Huber, 30) • A number of famous mathematicians have made connections between mathematics  and God , often likening God to a mathematician. • The Greek study of mathematics was closely related to that of religion. Plato is quoted as saying "God ever geometrizes" and Pythagoras  as saying "Numbers rule the Universe". • Johannes Kepler  stated that "The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics.“ • Isaac Newton  became extremely religious in his old age, and devoted the rest of his life to the study of religion. • Leopold Kronecker  is quoted as saying "God made the integers , all the rest is the work of man." • James Jeans  said "From the intrinsic evidence of his creation, the Great Architect of the Universe begins to appear as a pure mathematician".

46. Differences in philosophyDiscovered through Creation • According to Henri Poincare , "If God speaks to man, he undoubtedly uses the language of mathematics." • Georg Cantor  equated what he called the Absolute Infinite  with God. He held that the Absolute Infinite had various mathematical properties, including (if I recall correctly) that every property of the Absolute Infinite is also held by some smaller object. St. Anselm's ontological argument  sought to use logic  to prove the existence of God. A more elaborate version was given by Gottfried Leibniz ; this is the version that Gödel studied and attempted to clarify with his ontological argument. • Kurt Gödel  created a formalization of St. Anselm's ontological argument for God's existence known as Gödel's ontological proof . While Gödel was deeply religious, he never published his argument because he feared that it would be mistaken as establishing God's existence beyond doubt. Instead, he only saw it as a logical investigation and a clean formulation of Leibniz' argument with all assumptions spelled out.

47. Differences in philosophyDiscovered through Creation • Newton believed that God created the world mathematically. However, he believed that God needed to intervene on occasion to keep the universe functioning properly. • Galileo said “the study of nature was on a par with the study of Scriptures as a means of learning a revelation of God. ‘Nor does God less admirably reveal Himself to us in nature's actions than in the Scriptures’ sacred dictions’” (Huber, 34)

48. Differences in philosophyDiscovered through Creation • “the mere spectator of mathematical history is soon overwhelmed by the appalling mass of mathematical inventions that still maintain their vitality and importance for scientific work, as discoveries of the past in any other field of scientific endeavor do not, after centuries and tens of centuries.” (Bell, 11) • Jacques Hardamond, “Although the truth is not yet known to us, it pre-exists, and inescapably imposes on us the path we must follow.” (Huber, 31)

49. What should we teach? • Expose students to these differing views of the philosophy of mathematics. • Teach a sovereign God, completely in control of mathematics as well as all other aspects of His creation. • “And God saw all that he had made, and it was good.” (Genesis 1) • Psalm 19 – The creation proclaims the glory of God.

50. What should we teach? • Demonstrate the usefulness of mathematics in other subject areas especially science. • Point out where discoveries were made that had later applications in different areas. G.H. Hardy, that personal enemy of God, discovered things necessary for work being done today in genetics and Rh-blood groups and the treatment of diseases