#### Topic: Mathematics isn't unbelievable.

Mathematics is not intuitive for the very reason mathematics works so well: Abstraction. Many believe we evolved or were "given" an innate ability to understand mathematics. Some argue that mathematics is unreasonably effective because we can understand it. I disagree. It may be unreasonably effective, but I reject the notion that it is unreasonably effective because it was "made for us" or that it's intuitive. Here's a very simple example:

\(

{ x^2 } = 4 \\

=> { \sqrt{x^2} } = { \pm \sqrt{4} } \\

=> x = \pm { \sqrt{4} } \\

x = 2 \\

or \\

x = -2

\)

But, remember that:

\(

\sqrt{4} = 2 \\

and \\

- \sqrt{4} = -2

\)

There's no plus or minus, positive is implied outside of the context of quadratics, except when we explicitly add a negative in front. Try it on your calculator, if you like. There is no intuitive way to know that quadratics have two solutions. You have to know that ahead of time. That is, at minimum you have to know that this is the case:

\(

\sqrt{ x^2 } = | x |

\)

Which means we don't know weather x is going to be positive or negative, because it will always *present* as positive in this context. How can that be intuited? How can one derive that? The only reason quadratics work at all is because when "we" developed math, we tweaked it so it works with the rest of mathematics. In this case, if we failed to interpret x as an absolute value, then it would break the definition of a function. We would get -x and x, two outputs given the same input. There is no proof for the outcome of an absolute value, it is simply defined in mathematics as so. [1]

There are plenty little rules in mathematics similar to this, in that one can't possibly "just know" without either being taught them, or going through a long series of mental gymnastics to make an educated guess as to what the standard definition is. To my knowledge, the difference between principle square root and square root wasn't mathematically derived, so much as defined so that the rest of mathematics could work. Allow me to explain.

\(

(-3)^2 = 9 \\

but \\

-3^2 = -9

\)

Why? Because:

\(

(-3)^2 = 9 \\

=> -3 * -3 = 9 \\

and \\

-3^2 = -9 \\

=> -3 * 3 = -9

\)

So the parenthesis make all the difference here. The square only applies to 3 in the first example, and it treats the negative as a separate term, whereas the square applies to both the negative and the 3 using the parenthesis, because -3 is basically -1* 3. Note the difficulty and mental gymnastics one would have to jump through to simply intuit all of this. One cannot know intuitively how to interpret parenthesis, they are simply mathematically defined. In order to pull something like that off, you would need to conceptually re-invent mathematics in a sense. We skipped over something, however. In the above example, how do we even know -3 * -3 is equal to positive 9? Again, it's simply a definition, and that definition wasn't the product of a proof, but instead the product of necessity:

So, in my opinion, mathematics isn't unbelievable. It isn't an intuitive part of human nature. It is the product of thousands of years of analytic, grueling, step by step thinking. Mathematics was and is built on previous steps, and when mathematics was originally being constructed, sometimes we simply had to define rules to construct a consistent language. These days, new rules are rarely needed, as the consistent framework of mathematics has already been built. The lower levels of logic and definitions allow the higher levels to flow and create accurate, consistent, logical and provable mathematics. To those who are adept at math, to experts, things may seem to work *too* well. Everything fits together perfectly, it all makes sense, and it produces useful results. Consider, however, how irrational it all seems to someone who doesn't know the underlying rules or the motivation behind these rules. It seems totally arbitrary, and, in a sense, that argument could be made. The motivations for underlying definitions being necessity for consistency.