A Simple Guide to the Difference Between Algebra and Analysis

A Simple Guide to the Difference Between Algebra and Analysis Algebra and Analysis are Similar, but Different It might not always be necessary to separate the fields of mathematics, but doing so can sometimes be useful. Through junior high school, the curriculum is standardized as part of compulsory education. Then, in high school, students are split into different tracks. For science-track students, subjects like Mathematics 1, 2, 3, A, B, and C are required, while liberal-arts-track students might study a subset of these or choose based on their university entrance exam needs. Is that how it was? Looking back now, I'd like to know the reason for dividing the subjects in such a way. Upon entering university, it becomes clearer. For science majors, Calculus and Linear Algebra are almost always required courses. In my time, statistics was an elective. It's interesting that the term "Calculus" (微分積分学) is used instead of "Analysis" (解析学). Similarly, it's "Linear Algebra" (線形代数学) and not "Algebra" (代数学) itself, which is also curious. Both Calculus and Linear Algebra are, in fact, subfields of Analysis and Algebra, respectively. Approaching from an elementary level, one might divide mathematics into geometry, algebra, and analysis, and perhaps also (with some reservations) probability, number theory (does this include arithmetic?), and foundations of mathematics. While there are many ways to classify them, I will try to explain the difference between algebra and analysis. The Difference Between Algebra and Analysis I will summarize the difference between algebra and analysis in roughly three points. ① Algebra does not deal with infinity; Analysis does. ② Algebra deals with addition, subtraction, multiplication, and division (and roots); Analysis handles a wider variety of operations. ③ It is Analysis, not Algebra, that deals with "functions" (in algebra, other terms like "mapping" or "correspondence" are used). At the introductory, fundamental, or elementary stages, I believe it's fair to classify them based on these perspectives. Looking at the terms "algebra" and "analysis" through these three lenses may provide a clear and intuitive clue as to why algebra is called "algebra" (代数学) and analysis is called "analysis" (解析学). What is "Algebra"? Starting in junior high school mathematics, the image of algebra might be that it's about using symbols like x and y to handle "variables." I believe calculus was not taught in junior high. The main subjects were likely geometry and algebra, and perhaps algebraic geometry which combines them. Students solve equations, and some functions are introduced. These variables x, y, and z are unknown numbers until an equation is solved, so perhaps the Japanese term "代数" (daisū), which means "substitute number," is used because they act as stand-ins for numbers. This is likely a translated term from the late Edo or Meiji period. In English, it's "algebra," a word of Arabic origin. In ancient Greece, Diophantus worked on algebra, but its existence was largely ignored during Europe's Dark Ages. It's common knowledge that the legacy of ancient Greece was preserved by the Islamic world, with only a fraction remaining in Europe. As you delve deeper into algebra, the term "algebra" takes on another meaning. In set theory, when you give elements of a set specific properties or attributes, they become the numbers we know. But if you give them different properties, they become different things. In some cases, you can consider these other things as a more generalized form of number called an "algebra." The numbers we know—natural, integer, rational, real, irrational, imaginary, and complex—are just a part of this more general concept of "algebra." In this sense, "algebra" could almost be rephrased as "the elements of a set." Abstract mathematics develops in this direction. Addition, Subtraction, Multiplication, Division, and Polynomials We learn various functions like sine, cosine, tangent, as well as exponential and logarithmic functions. These are called transcendental functions. At the introductory, basic, or elementary level, these do not appear in algebra. Such functions appear in analysis. Algebra fundamentally deals with addition, subtraction, multiplication, and division. Therefore, the expressions it handles are polynomials. For a single variable, an expression that can be formed using real numbers and 'x' with addition, subtraction, multiplication, and division is either a polynomial or a rational expression. A rational expression looks like a division of polynomials, but let's set that aside for now. To put it plainly, at the risk of being misunderstood, the expressions algebra deals with are polynomials. They are not expressions mixed with sines, cosines, or exponential functions. Nor are they functions themselves. And the number of terms in a polynomial is finite; infinite terms are not handled. Incidentally, if you use infinite polynomials, you can actually express sine, cosine, tangent, as well as exponential and logarithmic functions. But, following rule ①—that algebra deals with the finite and not the infinite—it should clarify your thinking to consider that trigonometry and exponential functions are left to analysis and are generally not handled in algebra. What is Analysis? In junior high school, students learn algebra, geometry, and algebraic geometry. Some students might be moved by the power of mathematics just from this, as it allows them to solve many problems. At the same time, other students might feel a sense of constraint or limitation. For example, you cannot find the exact area or circumference of a circle. Archimedes found approximations, but not the exact values. The question of "what is an exact value?" requires some unpacking, but I will skip that here. In fact, using analysis, you can find the exact values. You just need to perform Archimedes' method an infinite number of times. His method was to approximate a circle using inscribed or circumscribed regular polygons. The idea is that as the number of corners (or sides) approaches infinity, the polygon becomes almost identical to the circle. However, Archimedes did not adopt this method of infinity. He was wary of it. The same goes for Newton, the discoverer of calculus. In his book Principia Mathematica, he explained his own classical mechanics and astronomy using the law of universal gravitation, but he deliberately avoided using calculus and wrote it in the style of Euclidean geometry. He, too, was likely cautious about infinity. However, subsequent physicists and mathematicians began to use infinity freely. As a result, the natural sciences became incredibly fruitful. The question of whether it was acceptable to use infinity so casually was resolved thanks to the foundational work of mathematicians like Cauchy and Weierstrass (Husserl's teacher and the so-called "king of analysis"), who made it possible to handle infinity formally and axiomatically. This was just before the era of modern mathematics, so their approach might not have been as thorough as Hilbert's, but I think one can imagine the practical handling of it through tools like the epsilon-delta argument that we use today. The Battle Between Cantor, Algebra, and Analysis However, infinity is not straightforward. A mathematician named Cantor discovered that using infinity leads to various unnatural and counter-intuitive phenomena. For example, Cantor found that while the number of rational numbers and irrational numbers are both infinite, the infinity of irrational numbers is "larger" than the infinity of rational numbers. Because of such strange results, the algebraist of the time, Kronecker (who was also a teacher of Husserl), criticized Cantor. Kronecker was a great algebraist who, while retaining some realist aspects due to the era he lived in, axiomatized algebra. The Kronecker delta is famous (though it is taught in analysis, not algebra), and he is known for the quote, "God made the integers, all else is the work of man." Perhaps due to the influence of Hilbert and Bourbaki, we tend to think of axiomatism, formalism, and the use of undefined terms and concepts as a set, but these are independent ideas. Axiomatism, formalism, and structuralism can coexist with or without the use of undefined terms, and they are all separate concepts. If Kronecker axiomatized algebra, he likely did so while still acknowledging the real existence of natural numbers or integers. The latter half of Kronecker's quote expresses constructivism. Kronecker is saying that if you have natural numbers or integers, humans can construct all other numbers. (In reality, humans can construct natural numbers and integers as well, but Kronecker was a transitional figure, so this can't be helped.) This intuitionist and constructivist stance is something algebraists may be prone to, and it has deep roots. Here, it might be seen as a confrontation between algebra and analysis regarding infinity. It is the philosophy of dealing only with things that humans can intuit and construct. The opposing stance is that if something is logically sound, it is acceptable to deal with it, even if it cannot be constructed by humans. This means that if it's logically consistent, it's fine to handle infinity. A similar conflict arose a little later. Today, they are not in conflict, but both streams can coexist within a single field or area, or a field might choose one over the other. The Cantor-Kronecker conflict was less of a conflict and more like one-sided bullying, but it is known in the history of mathematics as a precursor to the conflict between the Hilbert school's project (led by Hilbert, the father of modern mathematics and originator of axiomatism, formalism, and the use of undefined terms) and the opposing intuitionist and constructivist Brouwer. Does the Name Fit the Subject? Any discipline can be used in various ways, but as the proverb "the name reflects the reality" suggests, algebra has an aspect of a foundational journey concerning numbers—what is a "number" or an "algebra," and what relationships do they have? On the other hand, analysis, perhaps because it grew hand-in-hand with physics, seems to have a strong practical and applied aspect, serving as a tool for analysis within mathematics, physics, and other sciences. A function feels like a computer or a measuring device—easy to understand, with one output for every input. I wrote that analysis can derive the correct values for a circle's circumference and area, which algebra cannot. However, there is an issue with this "correct value." The circumference is 2πr and the area is πr 2 , but this leads to the question, "What is π?" In algebra, this is a number that cannot be derived through algebraic operations. This means it is not a solution to an algebraic equation. On the other hand, in analysis, one can feel like they understand π, but if asked to "write down the actual specific number," you cannot; you would have to continue writing forever with the help of a supercomputer. I don't know if this should be called possible or impossible, but it means that "if you are asked to calculate it to a certain point, you can, but it will only ever be an approximation." The epsilon-delta argument was created to settle things by deciding that "this kind of thing is acceptable." The term "transcendental number" is also interesting. Science aims for omnipotence, but transcendence, like that of gods or buddhas which is beyond human reach, is not what science desires. There might be an underlying feeling in science that "one must not transcend." If so, the person who used the word "transcendental" may have intended it ironically. A transcendental number is a real number like π or e that is not a solution to any algebraic equation. It is also a number related to infinity, as even in analysis, if you were asked to write it down, you would have to calculate and write forever. Trigonometric, exponential, and logarithmic functions are called transcendental functions, and these are also defined by infinite polynomials. From a standpoint skeptical of infinity, the casual use of it might seem improper. Conclusion I have attempted to summarize the rough difference between algebra and analysis. In more advanced domains, it becomes difficult to explain things in such a simple, clear-cut manner. In fact, one discovers surprising and moving relationships between different fields. However, I thought that first "knowing the differences" between these "different academic fields" might be useful in getting people interested in mathematics. The way mathematics is divided in elementary, junior high, and high school, using numbers and letters, can be confusing. University-level general mathematics often only touches on math as an applied and practical tool. While that may be fine in its own right, I felt it might not stimulate intellectual curiosity and the spirit of inquiry. Therefore, I tried to devise a way to understand the two fields from various angles, yet with the clarity of the three points ①, ②, and ③. I hope this can serve as a catalyst for someone to become interested in mathematics.