Hello friends,

This is an interesting question that I’ve been asked many times: “How does math relate to the real world?” Recently, while explaining algebra to my kids, this question came up again. It’s fascinating because we constantly use math in the real world, whether in arithmetic or algebra, and there are many engineering marvels that would be impossible without understanding basic theories of calculus and algebraic equations.

Mathematics in Engineering and System Design

Let’s begin with the fundamental role of mathematics in engineering and system design. Calculus, for instance, is integral to understanding the behavior of systems and designing complex structures. Famous examples include the Golden Gate Bridge and the architecture of skyscrapers. These structures are engineered based on precise mathematical models that ensure their stability and durability.

In the realm of cybersecurity, we encounter mathematical models like the Bell-La-Padula (BLP) model, the Biba model, the Clark-Wilson model, and the Lipner model. These models are designed using foundational principles of discrete mathematics, including sets, relations, and functions. Understanding these models on a deeper level can be challenging due to their mathematical complexity, often presented in academic research papers.

Bell-La-Padula Model

The Bell-La-Padula model, for example, is detailed in the March 1976 research paper “Secure Computer Systems: Unified Exposition and Multics Interpretation” by Bell and LaPadula. This 137-page paper delves into the mathematical models used to achieve a secure system design. For those preparing for the CISSP exam, it’s not necessary to understand these mathematical foundations in depth. However, for those interested, a deeper understanding reveals the intricate logic and mathematical constructs behind these security models.

Mathematics and Natural Laws

To answer the broader question about the relevance of mathematics, we need to recognize that our world, including the universe, is governed by natural laws. These laws are foundational principles that dictate how the natural world operates. Famous works, such as Isaac Newton’s Philosophiæ Naturalis Principia Mathematica, highlight the principles of motion and universal gravitation, illustrating how mathematical equations can describe the physical interactions between objects.

For example, Newton’s law of gravitation states that every mass exerts an attractive force on every other mass. This force is proportional to the product of their masses and inversely proportional to the square of the distance between them. The equation F=G*m1*m2/r2 succinctly encapsulates this relationship, where ( G ) is the gravitational constant.

Mathematical Logic and Principles

Natural laws are governed by underlying principles, such as the principle of material interaction. This principle explains how objects with mass attract each other due to gravitational force. Similarly, the principles governing electric fields describe how electric charges interact. James Clerk Maxwell’s A Treatise on Electricity and Magnetism is a seminal work that elaborates on these principles and their mathematical formulations.

Mathematics as a Tool for Understanding

Mathematics is the tool we use to understand these principles and the logic behind them. Through equations and models, we can visualize and quantify these natural laws. For instance, the movement of planets is explained through Kepler’s laws of planetary motion, which were later confirmed and expanded upon by Newton’s laws. This logical structure is detailed in the works of Johannes Kepler and further analyzed in Newton’s Principia.

Practical Examples of Mathematics in Everyday Life

1. Number System: We use numbers to quantify objects. For instance, we define the number of mangoes as 1, 2, 3, etc. This discrete mathematics helps us understand quantities and their properties.
2. Units of Measurement: Associating numbers with units, like meters or kilograms, helps us measure length, mass, and derived quantities such as velocity and force. This understanding leads to technological advancements, from cars to airplanes and satellites.
3. Gravity and Planetary Motion: The gravitational force between celestial bodies follows a mathematical formula, allowing us to predict planetary movements accurately. This concept is elaborated in works like Stephen Hawking’s A Brief History of Time, where he explains complex astrophysical phenomena using mathematical principles.

Conclusion

Mathematics is not just a subject studied in isolation; it is deeply interwoven with our understanding of the natural world and technological advancements. From the laws of gravity to cybersecurity models, mathematics provides the framework for understanding and designing the world around us.

In upcoming discussions, we will delve into specific models like Bell-La-Padula, exploring how discrete mathematics and logical relationships underpin these constructs. Stay tuned for more insights, and best of luck with your CISSP exams!

References

1. Newton, Isaac. Philosophiæ Naturalis Principia Mathematica. London: S. Pepys, 1687.
2. Maxwell, James Clerk. A Treatise on Electricity and Magnetism. Clarendon Press, 1873.
3. Hawking, Stephen. A Brief History of Time. Bantam Books, 1988.
4. Bell, D. E., & LaPadula, L. J. Secure Computer Systems: Unified Exposition and Multics Interpretation. MITRE Corporation, 1976.

I hope this explanation helps you see the profound relevance of mathematics in our daily lives and the natural world. Stay curious and keep exploring the wonders of math!