Preface
Mathematics as a Construct is a series of writings that aim to provide certain foundations for the study of math, it is a passion project derived from the need to intuitively understand what lies behind math, we intend to demonstrate how we could construct naturally all the conclusions we are faithfully taught, so that they make sense.
Mathematics is a broad scientific field, the reader cannot expect to know everything, hence, if at any point it is neeeded to do deeper research or re-read certain topics, they are more than encouraged, constantly going back and forth between concepts, objects, and more, is common when learning mathematics.
Intuition is one of the most important notions when trying to comprehend any subject, despite the title indicating that we are going to construct mathematics, constructivism as a philosophical point of view for mathematics, isn’t the main topic for discussion, this writing work is not based upon the idea of building witnesses that represent our theorems, but more so, building the intuition needed to easily comprehend the foundations and results that make up modern mathematics, without it feeling like a mindless procedure of proving theorems or solving exercises, understanding the foundations first, allow us to tackle different problems and outlooks more effectively.
You can already start, go ahead into .
How to use this book
There is no recommended way to read this book, because each reader will digest information at their own pace, each chapter is divided in sections that follow a linear development, with a section for important results at the end, one can read specific sections as they find fit, or go along the designated order, there is no shame in going back and forth, in the end, mathematics is one big entangled web of knowledge.
Theorems are always provided with proofs, although, these proofs aren’t necessary for building the intuition and mindset math can provide, they greatly enhance how we can perceive results and constructions, proofs are specially useful for learning how to read, follow and process mathematics, just reading proofs can improve how we interpret and think around math, but they can also be nice for reassuring our understanding.
Various exercises are left per section, but none of them are required for further reading, the interested reader might attempt to solve them or use them as practice sets for multiple purposes, this is not to undermine their value as learning devices, but problem-solving isn’t the main purpose of the book, such an ability can be acquired through different methods, after comprehending how our objects and relations work, which is the actual main purpose of the book.
Notation can be a complicated entry point for math, most of it comes from the necessity to write, thus, notation usually doesn’t follow any general rules, however, notation has a lot of thought and care put into it, anytime the reader doesn’t comprehend or forgets some symbol, the Notation Index has all the symbols used in this book.
On Sources
Mathematics are taught faithfully at introductory levels, and rarely covered in such a way that we can fully understand where each result comes from, whilst it is unreasonable to define all the important foundations that are required for an individual course or book, there is still some consideration on where and how the information is sourced.
Most academic settings list the bibliography used in creating a course, books usually have a section of recommended readings, and scientific writings cite their sources carefully, one can use books and courses only as a quick reference for specific concepts, so the sources cited for this work serve as references to very specific concepts, with a few exceptions for the ones the author has thoroughly read, the reader is encouraged to check these materials anytime a question can’t be answered by the text, or they can do further research by themselves on other places.
The bibliography used for this book is listed down below:
- McWeeny, R. (2011b). Number and symbols: From counting to abstract algebras.
- McWeeny, R. (2011c). Space: From Euclid to Einstein.
- Sullivan, M. (2006). Algebra y trigonometria (7.a ed.). Pearson Educación.
- Lipschutz, S. (1991). Teoría de conjuntos y temas afines. McGraw-Hill.
- Rosen, K. (2011). Discrete Mathematics and Its Applications. McGraw-Hill Education.
- Kolman, B., & Hill, D. R. (2006). Algebra lineal (8.a ed.). Pearson Educación.
- Shafarevich, I. R., & Remizov, A. (2012). Linear Algebra and Geometry. Springer.
- Stewart, J. (2018). Cálculo: Trascendentes tempranas (8.a ed.). Cengage Learning Mexico.
- Kojima, H., Togami, S., & Ltd, B. C. (2009). The Manga Guide to Calculus. No Starch Press.
- Kaplan, W. (2003). Advanced Calculus. Pearson.
- Shoenfield, R. (1967). Mathematical logic. Addison-Wesley.
- Mendelson, E. (2015). Introduction to mathematical logic (6.a ed.). CRC Press.
- Munkres, J. (2013). Topology (2.a ed.). Pearson Education.
- Awodey, S. (2010). Category theory (2.a ed.). Oxford University Press.
- Ziemer, W. P. (2017). Modern Real analysis. Springer.
Motivations for this primer
A Pragmatic paradigm for teaching is the norm at the basic and intermediate level of academics, straight forward and easy for teaching, one only needs to worry about how to do things, not why they work or what they are, we are taught how to do basic arithmetic and then algebra, but we are never given an explanation of how things work, why they are that way and what precisely do they mean, how to interpret them, or why do we even care about them, at least not directly.
This primer1 aims to teach the foundations of mathematics, so we can first think and understand, which is the hardest part of any scientific field, forgetting all mechanized procedures instilled in our brain, and beginning to shape our mind, so it aligns with the intuition that allows us to model the world, using nothing but pure math.
Footnotes
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A primer is an introductory writing work for a subject, however, even if that is the purpose of this text, we will go beyond just introductory matters, that sadly, will also lack sufficient depth in most topics covered. ↩