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RE: Is mathematics discovered, developed, or both?
November 22, 2016 at 1:53 pm
That's why I always taught my students to check their answers make sense, especially when they've been using a calculator. If they end up with a length of -5cm, they've probably hit a wrong button.
But of course, it's possible to model a real situation with an equation which then produces extra solutions that don't apply to reality. In other words, it's not a one-to-one mapping.
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RE: Is mathematics discovered, developed, or both?
November 22, 2016 at 4:51 pm
(This post was last modified: November 22, 2016 at 4:52 pm by Angrboda.)
Humans identify things by common perceptions. An apple is round and red, with a skin and a core. Looked at another way, it's just a bunch of stuff that melds from one thing into another. We identify parts of the whole because our eyes separate things out into light and dark, boundary and substance. Our ability to recognize objects as having the same generic features or specific type lies at the foundation of mathematics. Without the ability to sort things by generality, we wouldn't have numbers. Without the mental ability to represent the world as a collection of objects, we wouldn't have math. The circle is a perfect example of how our mental representation of the world is composed of pure objects. When we see line segments lying around the periphery of a circular shape, our mind mentally connects the lines to form a circle. It is this idealized circle in our minds eye which generates the ratio Pi. So numbers, math, and geometry are built up out of lower level precepts. We didn't so much discover math as we did evolve a "mathematical mind." Our ability to separate, combine, sort, and compare are all traits which served the animals which became us in negotiating their environment. The ability to abstract things apart from our sense experience is another key feature, as is the development of robust natural languages. Mathematics was neither invented nor discovered; it evolved.
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