@ Leonardo Araujo

**Math is the formalization of a human idea** Mathematics is a broad field of study that involves the investigation of patterns, quantities, structures, and changes in the abstract form as well as their real-world applications. It is foundational to a variety of disciplines including science, engineering, medicine, and the social sciences, providing a framework for reasoning, problem-solving, and understanding the universe. <iframe title="The Map of Mathematics" src="https://www.youtube.com/embed/OmJ-4B-mS-Y?feature=oembed" height="113" width="200" allowfullscreen="" allow="fullscreen" style="aspect-ratio: 1.76991 / 1; width: 100%; height: 100%;"></iframe> Mathematics is composed of many subfields, including but not limited to: 1. **Arithmetic:** The study of numbers and the basic operations on them: addition, subtraction, multiplication, and division. 2. **Algebra:** The study of symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. 3. **Geometry:** The study of shapes, sizes, and properties of space. 4. **Calculus:** The study of change in mathematical functions and models, dealing with limits, derivatives, integrals, and infinite series. 5. **Statistics:** The study of data collection, analysis, interpretation, presentation, and organization. 6. **Number Theory:** The study of properties and relationships of numbers, especially the integers. 7. **Topology:** The study of properties that remain constant through continuous deformations, such as stretching and bending, but not tearing or gluing. 8. **Applied Mathematics:** Uses mathematical methods and reasoning to solve real-world problems in business, science, engineering, and other fields. Mathematics is both ancient and modern; it has a rich history stretching back thousands of years, yet it continues to develop and evolve today, with new theories, discoveries, and applications constantly emerging. It is both a rigorous discipline in its own right and an essential tool used throughout the sciences and beyond. ## Pure and Applied Mathematics Pure mathematics and applied mathematics represent two broad categories within the field of mathematics, each with its focus, methodologies, and applications. The distinction between them lies in their primary objectives and the way mathematical theories are utilized. [[Attachments/01233372a59d2dc17b937c38319d672f_MD5.jpeg|Open: Pasted image 20240405125055.png]] ![[Attachments/01233372a59d2dc17b937c38319d672f_MD5.jpeg]] #### Pure Mathematics Pure mathematics is concerned with the study of mathematical concepts independent of any application outside mathematics. It is motivated by a desire to understand abstract principles and the properties of mathematical structures. The pursuit in pure mathematics is knowledge for its own sake, not necessarily aiming to find immediate practical applications. Pure mathematicians often focus on proving theorems and exploring theoretical frameworks, driven by curiosity and the aesthetic appeal of mathematics itself. Key areas within pure mathematics include: - **Algebra:** The study of symbols and the rules for manipulating these symbols. - **Geometry:** The investigation of the properties of space and figures. - **Analysis:** The rigorous formulation of calculus, focusing on limits, continuity, and infinite series. - **Number Theory:** The study of the properties of numbers, particularly integers. - **Topology:** The study of properties preserved through deformations, twistings, and stretchings of objects. #### Applied Mathematics Applied mathematics, on the other hand, is focused on the development and practical use of mathematical methods to solve problems in other areas, such as science, engineering, technology, economics, business, and industry. Applied mathematics is deeply connected with empirical research and the application of mathematical models to real-world situations. It involves the formulation, study, and use of mathematical models and seeks to make predictions, optimize solutions, and develop new approaches based on mathematical theory. Key areas within applied mathematics include: - **Differential Equations:** Used to model rates of change in applied contexts. - **Statistics and Probability:** The study of data, uncertainty, and the quantification of the likelihood of events. - **Computational Mathematics:** The use of algorithmic techniques for solving mathematical problems more efficiently, especially those that are too large for human numerical capacity. - **Mathematical Physics:** The application of mathematics to solve problems in physics and the development of mathematical methods for such applications. ## Is Mathematics discovered or Invented The question of whether mathematics is discovered or invented is a philosophical one that has sparked debate among mathematicians, philosophers, and scientists for centuries. Both viewpoints offer compelling arguments, and the distinction often hinges on one's perspective on the nature of mathematical objects and the universality of mathematical truths. <iframe title="Is math discovered or invented? - Jeff Dekofsky" src="https://www.youtube.com/embed/X_xR5Kes4Rs?feature=oembed" height="113" width="200" allowfullscreen="" allow="fullscreen" style="aspect-ratio: 1.76991 / 1; width: 100%; height: 100%;"></iframe> ### Mathematics as Discovered Those who argue that mathematics is discovered believe that mathematical truths exist independently of human thought and that mathematicians uncover these truths through investigation and reasoning. This viewpoint suggests that mathematical concepts like numbers, geometrical shapes, and even more abstract ideas have a reality that transcends human invention. The consistency of mathematical laws across cultures and times, and their applicability in accurately describing the natural world, supports the notion that mathematics is a universal truth waiting to be discovered. According to this perspective, mathematical structures exist in some abstract realm, and humans merely uncover aspects of this pre-existing world. #### Mathematics as Invented On the other hand, the viewpoint that mathematics is invented centers on the idea that mathematical concepts are human creations, designed to describe and understand the world. According to this perspective, mathematical theories and structures are the products of human thought, created to serve specific purposes in science, engineering, and other fields. This view emphasizes the role of creativity and invention in the development of mathematical ideas, suggesting that different cultures or species might develop entirely different mathematical systems depending on their needs and experiences. Proponents of this view point to the variety of mathematical systems (such as different geometries or number systems) that have been invented to solve particular types of problems, arguing that this diversity is evidence of mathematics being a human invention. #### A Middle Ground Some argue for a middle ground, suggesting that while the basic elements of mathematics are discovered, the development of complex mathematical theories and the choice of which aspects to study or develop further involve human invention and creativity. This perspective acknowledges the intrinsic properties of mathematical objects while also recognizing the role of human ingenuity in shaping the field of mathematics. #### Conclusion The debate between discovery and invention in mathematics may never be conclusively resolved, as it touches on deep philosophical questions about the nature of reality and the human mind's relationship to it. Whether one views mathematics as discovered or invented often reflects deeper beliefs about the world and our place within it. ## Axioms a particular mathematical system or theory. They serve as the foundational building blocks from which theorems and other mathematical truths are derived. Axioms are assumed to be self-evident, and their selection is based on their ability to produce a coherent and logically consistent framework for a body of mathematical knowledge. In the context of different branches of mathematics, axioms can vary significantly: - **In Euclidean geometry,** one of the most famous sets of axioms are Euclid's postulates, which include statements like "A straight line segment can be drawn joining any two points," and "All right angles are congruent." - **In algebra,** the field axioms define the properties of operations like addition and multiplication over sets of numbers. - **In set theory,** Zermelo-Fraenkel axioms (with the Axiom of Choice) are a set of axioms used to establish a foundation for much of modern mathematics. The role of axioms has evolved throughout the history of mathematics. Initially, they were considered self-evident truths, but as mathematics has developed, the emphasis has shifted to viewing axioms more as arbitrary starting points chosen for their usefulness in building a mathematical theory. This shift allows for the creation of different, sometimes non-intuitive, mathematical frameworks such as non-Euclidean geometries, which arise from altering Euclid's original postulates. **Everything in maths is constructed based on Axioms, not observation of the scientific method!** It's based only on the human logic reasoning. **And the universe doesn't five a fuck to fit inside human logic reasoning.** <iframe title="The paradox at the heart of mathematics: Gödel's Incompleteness Theorem - Marcus du Sautoy" src="https://www.youtube.com/embed/I4pQbo5MQOs?feature=oembed" height="113" width="200" allowfullscreen="" allow="fullscreen" style="aspect-ratio: 1.76991 / 1; width: 100%; height: 100%;"></iframe> ## Mathematics is not a Science Mathematics and science are deeply interconnected, but they are distinguished by their fundamental approaches, methodologies, and objectives. The distinction between mathematics as a formal science and other natural or empirical sciences like physics, biology, and chemistry lies in the nature of their inquiry and validation methods. **Maths doesn't need to prove itself through the scientific method, it only needs axioms, logic, and previous definitions.** - **Mathematics:** Uses deduction as a primary tool. Starting from axioms and definitions, mathematicians use logical reasoning to derive theorems and propositions. The validity of mathematical statements is determined through proofs, which are arguments that demonstrate their truth within the context of axiomatic systems. - **Science:** Employs the scientific method, which involves hypothesis formation, experimentation, observation, and the modification of hypotheses based on empirical evidence. Scientific theories and laws are validated by their ability to predict and explain phenomena in the natural world, and they are always subject to revision in light of new evidence. ### Objectives - **Mathematics:** Aims to create a coherent set of rules and structures that can explain and predict outcomes within abstract systems. Its primary goal is not to describe the physical world but to explore the properties and possibilities of mathematical structures. - **Science:** Aims to understand and describe the universe. The goal is to produce a body of knowledge that explains natural phenomena and can predict outcomes based on empirical evidence. In summary, while mathematics is often used as a tool in science to model and solve problems, its focus on abstract reasoning and logical proof distinguishes it from the empirical methodologies of the natural sciences. This fundamental difference in approach and objective is why mathematics is considered a formal science or a branch of knowledge distinct from natural or physical sciences, which are based on empirical evidence and experimental validation.