One night, I was sitting in the office trying to grok linear algebra. A wonderfully lucid textbook served as my guide: Introduction to Linear Algebra, written by Gilbert Strang. But I just wasn’t getting it. I was looking at various definitions — eigen decomposition, Jordan canonical forms, matrix inversions, etc. — and I thought, “Why?” Why does everything look so weird? Why is the inverse defined this way? Come to think of it, why are any of the matrix operations defined the way they are?While staring at a hopeless wall of symbols, a flash of lightning went off in my mind. I had an insight: math is a design. Prior to that moment, I had approached mathematics as if it were universal truth: transcendent in its perfection, almost unknowable by mere mortals. But on that night, I realized that mathematics is a human-constructed tool. Math is designed, just like software programs are designed, and using many of the same design principles. These principles may not be apparent, but they are comprehensible. In that moment, mathematics went from being unknowable to reasonable.
Mathematics is a system of objects, operations, and shorthand representations. It is designed to model real-world phenomena. Like all designs, there are certain degrees of freedom. The system could have been constructed in one way, or another. A matrix could have been designed as a round ball, in polar coordinates. It doesn’t matter, as long as the operations are consistent; it’s just a shorthand. At some point, someone made those design decisions. They picked the objects and the operations, and laid down rules of organization. Based on these fundamental decisions — if they are designed well — a number of other useful, provable properties then follow, and the whole thing can be used to model the things that we experience in the real world: the way that a tossed ball travels through space, the way sound waves dash across the ether, the rise and fall of stock prices. Physical reality contains layer upon layer of complexity. Well-designed mathematical systems offer clean and concise tools to represent physical reality at every layer.
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These are but a few examples of mathematical design at work. Our culture instills the strange notion that “math is hard.” Math is seen as too abstract, too impenetrable, too difficult to digest and impossible to know. But from an alternative perspective, mathematics contains striking parallels with software engineering. Both disciplines are heavy on jargon and notation. But once we parse through the jargon, we can begin to see the flesh and bones of mathematics. Understanding the design principles within mathematics provides us with an inlet into this strange land of hierarchical objects and changing representations. By becoming more familiar with the landscape of mathematics, we can help with the cross pollination of ideas between mathematics and software engineering. Maybe we can even begin to make modifications and come up with new designs of mathematics. Hey, that real number system is getting pretty old and cumbersome. Ready for something new?
- Striking parallels between mathematics and software engineering
Mathematics is a system of objects, operations, and shorthand representations. It is designed to model real-world phenomena. Like all designs, there are certain degrees of freedom. The system could have been constructed in one way, or another. A matrix could have been designed as a round ball, in polar coordinates. It doesn’t matter, as long as the operations are consistent; it’s just a shorthand. At some point, someone made those design decisions. They picked the objects and the operations, and laid down rules of organization. Based on these fundamental decisions — if they are designed well — a number of other useful, provable properties then follow, and the whole thing can be used to model the things that we experience in the real world: the way that a tossed ball travels through space, the way sound waves dash across the ether, the rise and fall of stock prices. Physical reality contains layer upon layer of complexity. Well-designed mathematical systems offer clean and concise tools to represent physical reality at every layer.
[---]
These are but a few examples of mathematical design at work. Our culture instills the strange notion that “math is hard.” Math is seen as too abstract, too impenetrable, too difficult to digest and impossible to know. But from an alternative perspective, mathematics contains striking parallels with software engineering. Both disciplines are heavy on jargon and notation. But once we parse through the jargon, we can begin to see the flesh and bones of mathematics. Understanding the design principles within mathematics provides us with an inlet into this strange land of hierarchical objects and changing representations. By becoming more familiar with the landscape of mathematics, we can help with the cross pollination of ideas between mathematics and software engineering. Maybe we can even begin to make modifications and come up with new designs of mathematics. Hey, that real number system is getting pretty old and cumbersome. Ready for something new?
- Striking parallels between mathematics and software engineering
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