A JOURNEY TO THE HEART OF MATHEMATICS
By: JAYATRA SAXENA
Edited By: SCISTEMIC TEAM
31.03.2021
Table of Contents
“If numbers aren’t beautiful, I don’t know what is.”
-Paul Erdos (Schechter & Pal, 1998, p.14)
Introduction
Mathematics is the backbone of almost all societal development today. Advancements in the field have helped us improve almost every major field of research such as quantum mechanics, genetics, law and investment. When we ask people about math, we realize that the importance of math is well-known, however, there’s relatively less public information available on the subject itself. What exactly is math? What do mathematicians do? What does new research in math look like? And how is it important? Let’s try to answer these fundamental questions about the subject.
Math itself is difficult to define because of the vastness of the subject, however, we can try to define it in a way that best describes what the study of math (namely, pure math) is like. In math, we try to get results about different schools of thought. For example, the Pythagorean Theorem is a result about shapes, the infinitude of prime numbers is a result about natural numbers, and the irrationality of the square root of 2 is a result about the real numbers. In its simplest form, mathematics is the science of understanding and producing results about any given abstract well-structured system of thought. This definition is rather vague and confusing, moreover one might question the necessity of doing math this way. To understand this definition and answer various questions we may have about it, let’s look at this definition in detail.
The Need for Abstraction
Most of the mathematics we do at an academic level is abstract, full of things like definitions, axioms, proofs, et cetera. Hence, before we dive deep into understanding how math works, we need to look at why we do math abstractly in the first place. Pure math is more like art, abstractness helps us express ideas that are deeper, and more complex than what reality can provide. Take a look at one of the most popular mathematical results for example eiπ + 1 = 0. This equation is so appealing to mathematicians when they see it for the first time because it is a very elegant combination of 5 fundamental numbers to the study of mathematics. This is, of course, an important result but that isn’t the primary motivation for most mathematicians to study it.
The motivation for studying math abstractly can be its artistic beauty, however, that doesn’t mean that the results’ value is decreased by it. Studying math abstractly also helps us apply our results to any number of problems that are mathematically similar questions. For example, take the number 2, the number 2 can represent a lot of things in our reality, number of chairs around a table, number of beers safe to drink before an exam, number of protagonists in the Rush Hour movies, and so on. When we separate our thoughts from reality while maintaining their structure, we gain a lot of independence on how that structure could be applied to different scenarios. To understand the strength of this abstraction, imagine or hold a ball. Now mark two points far away from each other on the ball, and draw the smallest possible line between the two. With understanding this smallest possible line concept in an abstract way, you can find the smallest possible line on any ball-like object. We use this idea while planning a flight around the globe. As the earth is spherical, when planes fly routes such as Dubai to San Francisco they fly through the north pole. Hence, abstraction in math is what increases its importance significantly.
Figure 1
Flight plan from Dubai to San Francisco passing almost through the north pole due to earth's spherical shape
(Wolfram Alpha, n.d.)
Building abstract systems
After understanding why math is studied abstractly, we can start exploring how to build the abstract system in math mentioned before. Many abstract systems mathematicians create are inspired by reality, however, while creating an abstract math system, a mathematician needs to get rid of all of its reliance on reality, while preserving the essence of the properties observed. This careful construction of the system of thought constitutes four major elements: definitions, axioms, logic, and theorems.
Definitions
When doing something in pure abstractness, we need to redefine most terms we already know based on pure words and entities already existing in our system of thought. This is necessary to make the system self-sufficient and independent. For example, in mathematics, we wouldn’t define a circle to be a flat object that looks ball-shaped, because this necessitates the reader to refer to reality to understand the object. In mathematics, we would define a circle to be an aggregate of points with the same distance from a specific point in two-dimensional space. This is an example of a definition that uses the pre-existing parts of the system to introduce a concept. Another key component of any definition in mathematics is precision. Since these definitions are the building blocks of the whole system, it is necessary to have very precise definitions to avoid any confusion or contradiction to a mathematician using them in the future. Hence, any definition used by a mathematician should have the ability to be interpreted in only one way. ‘A nice number is a number divisible by a large number’ is a bad definition because the word large can mean anything from 10 to 1,000,000 according to the person using the definition. On the other hand, ‘A nice number is a number divisible by a number bigger than 10,000’ is precise, and can be interpreted in one way only.
Figure 2
Image describing the definition of a Circle as a locus of points equidistant from the center.
(Basic-mathematics.com, n.d.)
Axioms
After defining the meaning of elements of a given system its only natural to think about deriving facts about the system. But we can’t derive anything out of nothing. Hence, to construct a system capable of development, we need to establish certain facts about the systems. These facts or things we assume to be true about a given system are called axioms. Any mathematician who wishes to explore the given mathematical system has to hold these axioms to be self-evident. A mathematician’s job is generally to use these axioms to develop the system, they don’t have to care about the axioms themselves. To understand how axioms work let’s look at an example of the simplest system of axioms the Peano Axioms. The Peano axioms construct the set of Natural numbers (0, 1, 2, 3, 4, 5, and so on).
The Peano axioms:
There exists an element called zero, this element belongs to the set of natural numbers.
For each element in the set of natural numbers, there’s another element called its ‘next number.’
For any element in the natural numbers, the next number is not zero.
If any two elements in the natural numbers have the same next number then they are the same element.
For any subset (a part of the natural numbers) of the set of natural numbers, if the subset contains zero and for any element in the subset, the subset contains its next number, then the subset is the natural numbers itself.
These axioms construct the natural numbers in a concise and mathematically powerful way. The first three axioms are facts about how the elements (or numbers) of the natural numbers relate to each other. The fourth axiom explains how this relation is like a line and not any other structure, hence we can understand things like the magnitude of numbers or the distance between them. The last axiom combines all of the previous axioms to construct the whole set of natural numbers. We have 0 and we have all the next numbers means we have 1, 2, 3, 4, and everything else. These five statements are written in a way that they never have to expect any knowledge from the reader to interpret them, they only use a single element (zero) for constructing the set of all the natural numbers. Moreover, these statements are so powerful that by using them we can build whatever ideas we need about natural numbers. For example, addition by an element (number) n can be just going to the next number n times. So, adding 2 to 4 is just going to the next number 2 times, that is 4 to 5 and 5 to 6. Axioms are a fundamental part of any given abstract mathematical system. A careful and consistent (contradiction-free) set of axioms are essential to building any abstract mathematical system. An inconsistent set of axioms in an abstract system can have ideas that are both true and false, which fails the system itself. Axioms give a mathematician power to build concepts about an abstract system and exploit them, and hence develop and understand the system to an unimaginable level of complexity.
Theorems, Proofs, and Results
Now that we have a good grasp of the base of our abstract system of thought, i.e. definitions and axioms, we can start developing our understanding of the system. We do this by guessing a concept (or statement) we think is true for our system and then making an unbeatable case about the truth or falsity of the concept. The guessed concept we think is true is called a conjecture, and the unbeatable case for the correctness of a conjecture combined with the statement is called a theorem. A concept shown to be true or false is a result about the mathematical system, and with each result, we increase our understanding of the system. One uses axioms, definitions, previously obtained results in the system, combined with the art of argument (logic) to obtain a new result in a system. Mathematicians use a lot of different techniques to prove a statement. Let’s look at an example to understand the process of proving better.
Definition: A prime number or a prime is a number greater than one only divisible by one and itself. For example, 2, 3, 5, 17, et cetera.
Theorem. Euclid: There is an infinitude of prime numbers.
Proof: Assume there are only finitely many prime numbers and they all exist in a collection, say collection A. We multiply all these prime numbers and add one to the product (so it’s indivisible by all numbers in collection A), let’s call it k. Now we have obtained a number that is not divisible by any prime number in collection A. This means that either k is a prime itself or there’s a prime outside collection A that is a prime. Either way, we have reached a contradiction since we assumed that all primes were in collection A. Hence, our assumption was wrong. Hence there are infinitely many primes.
This is one of the oldest theorems known to mankind, dating back to ancient Greece. It uses a method of proof called contradiction, which involves assuming our statement is false to generate a contradiction in the system, thus implying our assumption of the falsity of the statement was incorrect. This implies that our statement is true. To prove our theorem, we use definitions of divisibility and prime numbers, axioms about natural numbers, and a previous result about natural numbers stating all numbers are products of prime numbers. All these elements are used in a proof; however, they are not always explicitly written so that the reader can follow the argument easily. Contradiction is among the many methods mathematicians use to prove a statement. But regardless of the method used, a mathematical proof is logically irrefutable and undeniable (in that system). Because of this, after a theorem is published without any errors it can be used to support or refute any statement made about the system. Definitions, axioms, and theorems give our mathematical system a robust structure, because of which we get to build more complex ideas and apply them without worrying about their correctness.
Nature and Importance of Mathematical Research
“Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. “Immortality” may be a silly word, but probably a mathematician has the best chance of whatever it may mean.”
-(Hardy, 1992, p.81)
Theorems, axioms, and definitions make up the abstract structured systems of thought. A mathematician’s job is to explore, understand, and develop these systems. Most mathematical research is fundamentally different from scientific research. The aforementioned methods used in building up mathematical systems make the facts obtained within that particular system permanent. Mathematical results are not based on observations that statistically hold to be true, they are deduced from some very obvious axioms about a given system using logic. The statement ‘smokers have a higher chance of getting lung diseases’ is non-mathematical, while ‘Every smoker has smoked a cigarette’ is a mathematical one. The first statement is a result obtained from data, and it’s been ratified with scientific research to a very reasonable degree but it is not ‘obvious’. The second statement is a mathematical one because it's ‘obviously’ true, it is impossible to change its truth value without changing what ‘smoker’, ‘to smoke’, and ‘cigarette’ means. A theory about physics, biology or any other science can be proved false or at least be changed with contrary evidence or an improvised theory, but a theorem in a mathematical system is permanent. Our understanding of how planets in our solar system interact with each other has changed over time, but in a Euclidean geometric system, the Pythagoras theorem can never change. Any result a mathematician comes up with remains true forever, this permanence is the primary power and beauty of mathematics.
Mathematical permanence is certainly amazing, but one may ask the importance of doing math or the reason for going deep into the world of abstractness to find and demonstrate what’s true. First, it is important to note that math is extremely important to any field of study. Studying anything involves objects, quantities, ideas and patterns, and mathematics is the tool we use to describe and develop them. This means that obtaining results about anything is impossible without mathematics. While mathematics is quite abstract it’s in a way inspired by reality, this means that oftentimes mathematical results have unpredictable applications while studying other fields. There are countless examples of this happening, for example modern quantum mechanics and cryptography use results from fields of mathematics which were conventionally thought to be non applicable. This reflects on the fact that almost all fields of math research are equally important due to the unpredictability of the applicability of mathematical research.
It is also important to note here that mathematicians don’t just study math for its importance and usefulness, they study it for its beauty. As explained earlier mathematical abstractness is more like art, and just like every true artist mathematicians study math for the beauty they observe in their subject and the compassion they have for it. There’s a level of satisfaction in understanding mathematical systems and theorems let alone advancing them. There’s a comparison that can be made between mathematics and music; music is certainly important, but it is not only used for advertising products; music is important on its own because it is beautiful. Mathematics is the same, as studying it adds value to the world, yet it is beautiful on its own.
Fields of Mathematical Research
Understanding the power of permanence and importance, we can now take a look at different aspects and fields of mathematical research, One of the most common types of research is done by coming up with conjectures and proving them. For example, Fermat’s Last theorem, a very important statement in number theory, remained unproven for more than 300 years until it was proved by Andrew Wiles in the 90s (Singh, 1997). The twin prime conjecture, a statement that hypothesizes the infinitude of primes separated by one number (3, 5; 11, 13; 17, 19; and so on), remains unproven to date and many mathematicians are still working on it.
Mathematical research is not just limited to conjecturing and proving theorems in one system. Often, mathematicians like to combine different systems to obtain a result that develops all the fields together or combines them to start a new more complex system. Rene Descartes, combined the concepts of basic algebra and geometry to find a way to represent algebraic ideas geometrically and vice-versa. The concepts and methods introduced by Descartes remain central to many fields of mathematics and science such as Analytic Geometry, Calculus, Mechanics, and more.
Instead of working on more complex systems, some mathematicians like to explore more fundamental bits of mathematics such as understanding logic (the art of proving) or Set Theory. Results in these fields make sure that the structure in the mathematical systems is robustly maintained. Research done in this relatively new field has helped formalize mathematics to a greater extent and it has removed many misconceptions that mathematicians’ intuition can create.
Many mathematicians like to combine the work done in abstract mathematical systems to solve problems in other fields. This can include producing mathematical results directly related to a problem or exploring the uses of preexisting research for a specific problem. A simple example of this is using the cartesian coordinate system to describe the position of an object concerning time, and obtaining facts like the object’s velocity or acceleration from it. The use of partial differential equations in thermodynamics, the use of linear algebra and representation theory in quantum mechanics, and the use of number theory in cryptography are some examples of popular research topics in applied math.
Conclusion
Mathematics is central to almost everything we do. It is composed of abstract well-structured schools of thought that include definitions, axioms, theorems, and more. Mathematicians work towards developing and understanding these structures by obtaining various results about them. Mathematics is certainly an important subject, but its significance on its own is often overlooked. Mathematical beauty can be seen in the order found in complex ideas of a mathematical system, and in the fact that the complex arguments developed are irrefutable. Mathematics helps us explore ideas and concepts beyond our visual imagination, and establish facts about them. One can always ask the purpose of studying these abstract systems of thought when we can see no possible use for them. The truth is these complex ideas and systems help us understand systems present in the universe and the systems present within ourselves. We can never predict when society will reach a point that requires the complexity found in mathematical research, however studying math ensures when we get there we know how to process the ideas found in that level of complexity.
“The difference between the poet and the mathematician is that the poet tries to get his head into the heavens while the mathematician tries to get the heavens into his head.”
― G.K. Chesterton (as cited in Osterlind, 2019)
Acknowledgements
It is not possible to work on a task such as writing an article without the help of other people, there are some people that I would like to thank for helping me write this article. I would first like to thank the members of the Scistemic team for guiding me and supporting me throughout the writing process. I would also like to extend my appreciation to Prof. Burak Kaya and Prof. Tolga Karayayla of the Middle East Technical University Math Department for helping me write this article and providing me with their insights. Finally, I would like to thank my Instagram (@jayatra21) followers for answering my questions about the perception of mathematics.
References
Schechter, B., & Pál, E. (1998). My brain is open: The mathematical journeys of Paul Erdős. New York, NY: Simon & Schuster.
dubai+to+los+angeles - Wolfram: Alpha. WolframAlpha computational knowledge AI. (n.d.). https://www.wolframalpha.com/input/?i=dubai%2Bto%2Blos%2Bangeles
What is a circle? Basic. (n.d.). https://www.basic-mathematics.com/what-is-a-circle.html
Singh, S. (1997). Fermat's last theorem. London: Fourth Estate.
Hardy, G. H., & Snow, C. P. (1992). A mathematician's apology. In A mathematician's apology (Canto ed., Ser. 2000, pp. 80-82). Cambridge: Cambridge University Press.
Osterlind, S. (2019). The error of truth: How history and mathematics came together to form our character and shape our worldview. In 1277445093 942793276 S. J. Osterlind (Author), The error of truth: How history and mathematics came together to form our character and shape our worldview. Oxford, United Kingdom: Oxford University Press.
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