Nire Bryce

Why do we have math, anyway?

Time for some extremely oversimplified handwaving. Math, very broadly, is developed for two purposes: to better understand reality, and to better understand the math we're using to better understand reality. These two motives constantly play off each other. An example:

If I throw a rock straight up, roughly how long will it take to hit the ground?

• We can approximate its position with the equation -gt²/2 + vt + h = 0. Let's call this a quadratic equation.
• We develop the quadratic formula to easily find solutions to that equation.
• We can generalize quadratics to cubic, quartic, quintic terms, giving us general polynomials. Are there corresponding formulas to get the roots of arbitrary polynomials?
• This ends up being way more complicated, so we start to discuss the abstract properties of polynomials, giving us algebraic rings.
• By studying rings as an abtract concept we develop theorems that, when applied back to polynomial rings, tells us that a general root-finding equation doesn't exist, but they do exist for special cases.
• A few of these special cases model optical lenses really well, helping us design better telescopes.
• With better telescopes, we find our existing models of celestial mechanics don't perfectly match up with reality.
• Time to develop better models...
  Okay, historically it almost certainly did not develop this way, that's not the point. The core feedback loop is real: that continuous zigzagging between advancing our knowledge of the physical world and advancing our knowledge of the abstract world. It's seeing those bridges between those two worlds that makes "knowing math" so powerful. If you can see how your real-world problem is reducible to a mathematical construct, then you can use all of the abstract machinery we've developed to analyze that problem as the mathematical construct.