Algebra: The Language of Science by 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 James Nickel, B.A., B.Th., B.Miss., M.A. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Summary 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • The misdirection of High School Mathematics Education. Biblical Christian direction (or framework) for reformation of Algebra. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Acknowledgement 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • I am grateful to Morris Kline’s Why the Professor Can’t Teach: Mathematics and the Dilemma of University Education (St. Martin’s Press, 1977) for much of the following analysis. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net 20th Century High School Mathematics: Scope and Sequence 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • • • 9th grade: Algebra I 10th grade: Geometry 11th grade: Algebra II 12th grade: Pre-calculus, Calculus, or Solid Geometry 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Analysis 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • • Note: As I analyze this Scope and Sequence, I am going to play the “devil’s advocate.” I am going to be somewhat cynical with a purpose in mind. Let’s see if we can view this material from the eyes of the typical student; i.e., what I experienced in the 1960s. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net th 9 Grade: Algebra I 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • Almost entirely a series of disconnected, atomized processes, a mélange of topics. Note, in the early 21st century, there is a move to cover Algebra I in the 8th grade. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Algebra I Topics 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • • • • Factoring. Operations with polynomials such as x2 + 9x + 20. 7x 8 Operations with fractions such as . 5x 3 Laws of exponents. Solution of equations of various degrees and with one or more unknowns. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Algebra Perspectives 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • • Algebra is a means to something else, not the end in itself. Algebra is the spelling, grammar, and rhetoric of most mathematics, but it is not literature in and of itself. This maze of letters that represent number symbols that represent numbers was fully developed in the 17th century. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Two Levels of Abstraction 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • • Level 1: Numerals are symbols of number. Level 2: Letters are symbols of numerals. For example, instead of 3x = 5, we have ax = b. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Two Levels of Abstraction 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • • But, generalizations about men and women are meaningful only to those who have had lots of experience with individual men and women. The analogue for Algebra is lots of experience with various manipulations of numbers. Unfortunately, few students have more than a nodding acquaintance with arithmetic at the time they are propelled into Algebra. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net th 10 Grade: Geometry 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • The Geometry study is Euclidean or Plane (two dimensions). Here, the student is confronted with dozens of theorems all proved in a logical sequence and, to most students, the goal seems to be to prove as many as possible. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net th 10 Grade: Geometry 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • Students are confined to learning inflexible and obtuse trains of thought that permit no derailment. Just why anyone wants these theorems and how these theorems and their proofs were ever conceived is not treated. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net th 11 Grade: Algebra II 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • • Repeats the 9th grade subject matter because students did not learn that material. Adds more algebraic processes. Introduces trigonometry, a subject in which, among other topics, the students learn many identities at the risk of losing their own. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net th 12 Grade: Various Schemes 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • Coursework has not been stable. But the material, whether solid geometry or the beginnings of calculus, is taught dogmatically and has been no more enlightening or enthralling than the preceding subjects. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Who is Responsible for this? 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • • • Who generated this curriculum? It was developed in the 19th century by university professors. Before this time, these high school subjects were only taught in college. They gradually filtered down into high school keeping the same college content. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Rationale for this Program 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • • • Utility in daily life. Value of mental discipline; i.e., mathematics teaches you how to think. Learning of higher, nobler truths; i.e., a2 – b2 = (a + b)(a – b) would uplift any soul. Preparation for college and professional life. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net State of Things since WWII 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • 1950s: the ineffectiveness of this traditional curriculum (made evident by the needs of WWII) stimulated some university professors to reform the process. This reform resulted in the “new math” of the 1960s. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net 1960s New Math 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • Emphasis on theory, generalization, rigor, and structure of number systems. Focus on set theory and symbolic logic. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net 1960s New Math 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • These “innovations” may have ensured the stability of the several mathematical edifices, but they also pulled the students so far down into the dark earth that they could no longer see the surface. e.g., 2 + 3 = 3 + 2, but does 2 + 3 = 5? It is now been recognized (in hind-sight) that this reform was not an improvement; it was a disaster. 1 • 2 4 Copyright 2008 www.biblicalchristianworldview.net Summary 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • The contents of all four years, as presented, are abstract, dull, boring, and intrinsically meaningless. It consists, at best (to use Alfred North Whitehead’s phrase), of inert ideas. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Analysis of Rationale 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 1. 2. 3. 4. Utility in daily life. Value of mental discipline. Learning of higher, nobler truths. Preparation for college and professional life. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Utility 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • • Does the average educated person use this knowledge in daily life? Do even mathematics teachers, who know the subject, ever use the quadratic formula, the Pythagorean Theorem, or the trigonometric identities outside of the classroom? The honest answer is no. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Utility 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • • Of course, the students learn that a straight line is the shortest distance between two points. That is useful. But even a donkey knows that: put some food at a distance from him and watch the path he takes. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Utility 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • • The subject matter of high school mathematics–that is, the subject matter per se–is worthless knowledge. One or two topics, such as the calculation of compound interest, may prove useful. But the exceptions do not alter the general assessment. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Thinking 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • High school courses have not taught thinking, pedagogue’s assertions to the contrary. The traditional Algebra teaches memorization of processes. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Thinking 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • Geometry purportedly emphasizes proof and therefore thinking. However, because the proofs are arranged in a logical sequence and this sequence is not natural, the rationale eludes the students and they are forced to memorize here, too. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Thinking 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • The students’ appreciation of proof in geometry is epitomized in their oftrepeated remark that geometry is where you make proofs in two columns. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Nobler Truths 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • All the preaching and rhapsodizing will not ennoble such ugly ducklings like factoring, adding fractions, and the quadratic formula. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Preparation 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • Of those who go to college, at most 10% will need the techniques and theorems now taught in the high schools. The others either take no mathematics in college–many colleges no longer require a mathematics course for the bachelor’s degree–or take a liberal arts course that does not use most of the currently taught high school mathematics. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Preparation 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • Note the growing number of “mathematical development” (i.e., remedial math) courses in most 2-year and 4-year colleges even the basics are not being learned by many high school students. About 5% to 15% of high school graduates will use mathematics professionally. 1 • 2 4 Copyright 2008 www.biblicalchristianworldview.net What to do? 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • • Don’t teach any math in high school. And the students roar, “Hip, hip, hooray!” And math teachers have to get a real job. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net The Purpose of this Cynicism 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • These arguments against the teaching of high school mathematics are arguments against the framework: – – 1 2 The kind of mathematics that has been taught. The justifications traditionally and currently given. 4 Copyright 2008 www.biblicalchristianworldview.net Reformation in Framework 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • The Bible gives direction for viewing reality. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Reformation in Framework 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • • The infinite, personal, and Triune God made the heavens the earth and they reflect a rational order based upon His Wisdom. Man, made in God’s image, can thus reason with this order. Operational science is man reasoning with this order in terms of observation and experimentation. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Reformation in Framework 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • • Biblical Christianity is a faith that is grounded in history. God commands man, male and female, to cultivate the garden; to develop the implications of creation relationally. Hence, history and culture are important pieces of the framework. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Implications for Algebra 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • Place the instruction of Algebra in the context of history, culture, and science. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Biblical Christian Education 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • What are the objectives of a Biblical Christian high school education? The courses in literature, history, science, economics, and foreign languages are intended to enable future adults to live more insightfully, wisely, and enjoyably. In short, they are an introduction to our culture and the role of Biblical Christianity therein. 1 • 2 4 Copyright 2008 www.biblicalchristianworldview.net Implications for Mathematics 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • The mathematics courses should serve the same objectives. Hence, the mathematics we teach should be worth knowing for the rest of the lives of all students. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Implications for Mathematics 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • Mathematics courses should contribute to a truly liberal arts (freedom in Christ) education wherein students get to know not only what the subject is about but also what role it plays in our culture and our society. We must teach not just what mathematics is but what it does as a Biblical Christian “tool of dominion.” 1 2 4 Copyright 2008 www.biblicalchristianworldview.net The Arabic Connection 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • • Algebra is an Arabic word. In the 7th century AD, the followers of Mohammed began ransacking the nations of North Africa, the Middle East, and Europe. In spite of some destruction of knowledge and books, Muslim leaders generally respected and absorbed the manuscripts of the ancient Greek mathematicians. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net The Hindu Connection 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • The Arabs were also responsible for introducing to Europe several valuable concepts that they borrowed from the Hindus of India: – – – – The base 10 decimal system. Positional notation. The number zero. Negative numbers. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net The Decimal System 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • The decimal system of writing numbers (e.g., 1, 2, 3, 4, etc.) proved to be much more efficient than the old, cumbersome Greek letters or Roman numerals. This method of number writing acted as an indispensable asset in later developments of European mathematics and science. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Algebra: Meaning 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • In 830 AD, the astronomer Mohammed ibn Musa al-Khowarizmi (ca. 780-ca. 850), working in Baghdad, wrote a book titled Hisab al-jabr w’al-muqabalah, which literally means “science of reunion and opposition” or more freely “science of transposition and cancellation.” 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Algebra: Meaning 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • • Al-jabr, used in a non-mathematical sense, was something a barber did in those times. A barber not only cut hair, he mended bones. Thus, an algebrista was a bonesetter (chiropractor). 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Algebra: Meaning 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • In Galatians 6:2, the Apostle Paul states, “Brethren, if a man be overtaken in a fault, ye which are spiritual restore such a one in the spirit of meekness, considering thyself lest thou also be tempted.” The Greek word for restore means “to join back together” (as in setting a broken bone) or “to mend” (as in repairing torn nets). 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Algebra: Meaning • 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • The Greek word for restore means exactly the same as the Arabic word algebrista. In a mathematical sense, solving an equation (determining values of a variable that make the equation true) is like restoring, mending, or resolving it. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Transition to Europe • This Arabic text became known in Europe through Latin translations and was shortened to the one word, “algebra,” which became synonymous with the science of equations. Arabic mathematicians did not use simple symbols like the Symbolic Algebra we used today (e.g., x, y, etc.); they used words, or at best, abbreviations of words (syncopated Algebra). 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Transition to Europe • The decimal system needed Simon Stevin (1548-1620) and François Vieta (1540-1603) to acquire the conceptual precision demanded by systematic scientific work. 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Soul of Science • The very soul of science consists in theoretical generalizations leading to the formulation of quantitative laws and systems of laws. India had practicality, craftsmanship and organizational talent, but science in the above sense did not exist in India prior to its invasion by Alexander the Great. 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Soul of Science • Tobias Dantzig (1884-1956), “Algebra... enables one to transform literal expressions and thus to paraphrase any statement into a number of equivalent forms ... it is this power of transformation that lifts algebra above the level of a convenient shorthand ...” 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Soul of Science • “the literal notation made it possible to pass from the individual to the collective, from the ‘some’ to the ‘any’ and the ‘all’ ... It is this that made possible the general theory of functions, which is the basis of all applied mathematics.” 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Soul of Science • One and the Many: Biblical view of the world enabled man to harness the incredible power of Algebra. Robin G. Collingwood (1889-1943), “The possibility of an applied mathematics is an expression, in terms of natural science, of the Christian belief that nature is the creation of an omnipotent God.” 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Culture and Science • • Christ is the Lord and Savior of both. Hence, Algebra must be seen in a Christocentric perspective. 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Gate and Key of Science • 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 English philosopher and scientist Roger Bacon (ca. 1214- ca. 1294), nicknamed the “Admirable Doctor,” said in his Opus Majus (published in 1267), Part 4: “Mathematics is the gate and key of the sciences…. Neglect of mathematics works injury to all knowledge, since he who is ignorant of it cannot know the other sciences or things of this world.” 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Gate and Key of Mathematics • 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 Algebra, for it prepares us not only to understand the sciences, but also such subjects as medicine, art, statistics, and economics. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Christ: the Gate and Key of Algebra 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • From the perspective of the Biblical Christian worldview, learning Algebra is important to our worship because Algebra is the language of science, science is the study of what God in Christ has made, and what God in Christ has made is an unveiling of His glory. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net What History Teaches 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • It is beyond dispute that, in history, mathematicians were inspired by real (i.e., physical and scientific) problems and found the meaning of mathematics through them. Equally beyond dispute is that applications to real problems are a pedagogical necessity. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Pedagogical Implications 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • However, there is some question about which applications will be interesting and meaningful to students. Only experience will enable teachers to determine the best choices. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Pedagogical Implications 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • Fortunately, the choices are so numerous and so varied, and interest in the real world is so much more widespread than interest in abstract mathematics, that attractive applications can surely be found. At the very least, it is easier to arouse curiosity about real problems than about mathematics. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Pedagogical Implications 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • The use of real and, especially, physical problems serves not only to answer the question of what value mathematics has but also gives meaning to it. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Examples 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • Negative numbers are not just inverses to positive integers under addition; they are the number of degrees below zero on a thermometer. The ellipse is not just a peculiar curve; it is the path of planets and comets. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Examples 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • Functions are not sets of ordered pairs; they are relationships between real variables such as the height and time of flight of a ball thrown into the air, the distance of a planet from the sun at various times of the year, and the population of a country over some period of years. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Implications 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • To rob mathematical concepts of their meaning is to keep the branches of a fruit tree and throw away the fruit. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Proposal 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • The proposal that the first two or three years of secondary mathematics contain all sorts of genuine applications may seem radical, but a little perspective may correct this impression. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net What History Teaches 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • In the 17th century, mathematics courses comprised astronomy, music, surveying, measurement, perspective drawing, the design of optical instruments, architecture, and the design of fortifications and machines. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net What History Teaches 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • In the intervening centuries some of these topics lost importance and were dropped from the mathematics curriculum. Also, the expansion of mathematics itself and of knowledge generally has compelled educators to drop other topics. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Isolation is Not the Answer 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • But the isolation of mathematics from all applications relevant to our times cannot be tolerated, even if the inclusion of applications necessitates covering less mathematical topography. It is better to teach a little well and with understanding than a lot poorly and with no understanding. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Ramifications 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • This approach would present what is interesting, enlightening, and culturally significant, with the inclusion of only those concepts and techniques that will serve to further the liberal arts objective. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Specifics 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • No technique for the sake of technique should be presented in the first two or three years. In other words, the material should be objective-oriented rather than subjectoriented. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Final Question 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • But what do we do for the future professional user of mathematics? 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Answer 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • Admittedly a small but appreciable percentage of the students will become mathematicians, physicists, chemists, engineers, social scientists, technicians, statisticians, actuaries, and other specialists whose work requires mathematics. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Answer 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • Of course these students, too, should know the cultural significance of mathematics. Moreover, students who are already inclined toward a specific career will certainly take an interest in mathematics if they see how the subject is involved. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net Recommendations 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • The textbooks written by Harold R. Jacobs incorporate this pedagogical thrust. – – – • Mathematics: A Human Endeavor Elementary Algebra Geometry: Seeing, Doing, Understanding 1 2 4 The only draw back is that the lifesustaining ingredient, the Christo-centric perspective, is missing from these works. Copyright 2008 www.biblicalchristianworldview.net Summary 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 • • The misdirection of High School Mathematics Education: disconnection from the world of God’s making. Biblical Christian direction (or framework) for reformation of Algebra: connection to the world of God’s making. 1 2 4 Copyright 2008 www.biblicalchristianworldview.net

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# Philosophical Principles of Mathematics