The Nature of Mathematics: Creation or Discovery?
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Understanding the Essence of Mathematics
Consider this thought-provoking quote from Leopold Kronecker: “God made the integers; all else is the work of man.” Next time you find yourself among mathematicians, and the conversation stalls (although how could that possibly happen?), try posing this question: “Is mathematics a construct of our minds, or is it something we uncover?”
Physicists handle tangible matter. Chemists analyze substances. Biologists study living organisms and their environments. Their method typically involves:
- Formulating a hypothesis about the material.
- Conducting experiments to observe its reactions.
- Revising the hypothesis based on the findings.
However, in mathematics, the universe is an internal landscape. There are no flawless circles in the external world. When I draw what I call a circle, whether with a compass or digitally, that representation serves merely as a model. The essence of the circle exists in my mind. Yet, we discuss concepts like the unit circle and the golden triangle as if they were singular, shared entities that exist in both our minds.
Mathematics often generates the very concepts it examines, frequently disregarding the external reality. Remarkably, the two realms still echo each other. Physicist Eugene Wigner famously noted the "unreasonable effectiveness of mathematics" in explaining the physical universe. A mathematical construct, birthed from aesthetic considerations, may later prove to be beneficial in the tangible world. Conversely, a model deemed unappealing might be resurrected centuries later to elucidate a new scientific discovery.
Let's delve deeper. Grab a pen or any writing tool, and sketch a square. Connect one pair of opposite corners. Your drawing should resemble this:
Now, reflect on your creation: Does the diagonal appear longer than each side? How much longer is it? If this square represents a city block, what is the distance between the corners? Is it consistent with the squares others have drawn? Why or why not?
One fascinating aspect of our squares is the ability to define our own measurement system. If we designate this as a unit square, then its sides measure 1, and its area is 1. Why? Simply because we declare it to be so. Once we establish this, we must adhere to it throughout our discussion.
You might cleverly deduce that the diagonal is represented by “the square root of two,” either through memory or by applying the Pythagorean theorem to one of the triangles formed. Here's how that looks:
If the Pythagorean theorem was unfamiliar to you, how would you tackle this query? Perhaps you'd experiment with various lines and curves until you decided to draw another square and pose an additional question: How much larger is this new square?
Have we complicated matters, or are we inching closer to clarity? With just a couple more lines, it becomes evident: the smaller square can be divided into two triangles. Four of those triangles will form the larger square, making it twice the size of the smaller one.
We can rearrange two of the smaller squares in any manner we choose and fit them perfectly into the larger square, thus establishing that the area of the larger square is 2.
Returning to the diagonal, we can best describe it as: the side length of a square with an area of 2. This phrase can become quite handy. So, we adopt a symbol—a shorthand for notation—and a term—a shortcut for conversation: √2. It’s pronounced as "the square root of two," a detail you likely already knew.
We approached the diagonal from two different perspectives. One treated it more as an algebraic challenge, while the other leaned toward geometric reasoning. Was this primarily an algebraic puzzle we illustrated through geometry, or was it a geometric puzzle modeled using algebra?
Chess and its Mathematical Nature
Even if you don't play chess, you're likely familiar with it. The game has specific rules governing how pieces move: White goes first, the bishop moves diagonally, and the knight jumps in an L shape. The rook, resembling a castle, slides vertically and horizontally.
In contrast to card games, chess is about moving pieces across a board. While cribbage uses a board and pegs merely to keep score, chess is more akin to football in its spatial dynamics.
If you're curious, you can download an inexpensive chess program and start playing right away. But is your computer genuinely playing chess, or simply simulating it? Set your software to play against itself. Now, turn off the screen.
The chess game persists, yet there are no physical pieces or colored squares—only electrical pulses in a circuit board, represented by ones and zeros.
Have you ever engaged in a video game, only to realize, "I'm just sitting here moving my thumbs and pressing buttons, and I'm really getting excited about it"?
Does physics utilize mathematics to represent the universe, or are mathematical structures the core reality, reflected in varying degrees of accuracy by clumsy physical objects? Why not consider both perspectives? Max Tegmark, in his paper "The Mathematical Universe" and his book "Our Mathematical Universe," proposes a fascinating idea: perhaps the universe is not merely mathematical; perhaps the universe itself is mathematics.
Chapter 2: Perspectives on Mathematics
The first video titled "Is Math Discovered or Invented?" by Jeff Dekofsky delves into this philosophical debate, exploring whether mathematics is an invention of the human mind or a discovery of truths that exist independently in the universe.
The second video, "Roger Penrose - Is Mathematics Invented or Discovered?", offers insights from a renowned physicist and mathematician, further examining the nature of mathematics in our understanding of the universe.