Is mathmatics invented or discovered?
Does Michelangelo create statues from marble or find them already existing inside the marble?
No matter what Michelangelo says, we know he didn’t find David inside marble. He made it out of marble. The only reason there’s any doubt in our minds of this fact is that the physical act of sculpting marble happens to be removing pieces of marble from a single block. If he had built the statue by casting bronze or sculpting clay, we would clearly see the act of construction. It’s merely a clever illusion that David was an act of creation.
And now on to mathmatics. 1 + 1 = 2. Was that invented or discovered. You may say, well of course it was discovered. If I have one thing and other thing, then I have two things. That has always been the case. By definition!
Ok, but who made the definiton? For that matter, who made the words or symbols for “1”, “one”, “+”, “plus”, “=”, and “two”? Clearly those smybols and words were invented. It’s a more clear case with the number 0. While many people have had none of a particular class of items since there were people, the abstraction 0 didn’t exist for a long time until someone thought it up and used it. How about imaginary numbers. They first disqualified as numbers and then included, but only with the nomer of imagionary, which is literally to say, “these are creations of the human imagionation, not real math-things,” which I would argue is ironic through our current lens that sees imagionary numbers as real as it gets for mathmatics. All math is human invention.
No, that can’t be right, you may say, because what about the beautiful way that it all fits together? Or what about the way in which we are able to leverage mathmatics to understand our phsysical world and build things in it, like architecture or engineering?
Firstly, it all fits together because that’s how much of mathmatics works. You design a system of symbols and the rules to produce more symbols and then you use the rules upon the axiomatic symbols to produce more symbols. We literally have a rule – proof by contradiction – that says if you are able to find a rule that break the cohesiveness of the system, that rule is not allowed in this mathmatical system.
Secondly mathmatics is able to help us understand and tame our phsyical world, again, because we build it that way. We have constructed 1 + 1 = 2 because in the physical world that’s how quantities work. That doesn’t have to be how quantities work in a mathmatical system. However mathmatics is often used as an abstraction for reality because we designed it to model reality.
Same goes for e=mc2 and for the lightbulb. These were constructions, not discoveries.
This is why I strongly dislike the phrase “find yourself” or “find myself,” because there is no “you” out there to find. Searching is the wrong metaphor. It sends young people out wondering the earth looking for… What? What could you hope to see with your eyes that would mean that you’ve “found yourself?” Clearly no such moment of epiphany is possible.
While we may not be born as blank slates, we are born with little understanding of our world or ourselves. As Piaget says, we then construct knowledge of our world inside our own heads. We build small concepts through trial and error, and then we test those concepts out in the real world like mini-scientists. Then we build bigger concepts out of the smaller concepts. We are constructing how the world works, and in turn, we are constructing our very minds.
As Hosfader says, analogy is the core of cognition. In the most literal sense, we, adults and children, can only understand those things that we can relate to the concepts that are already inside our brain. We build new structures in our brain by combining existing structures.
The way we understand anything in our world is through constructing knowledge about it. There is no trasmission of knowledge. You cannot pour your knowledge into my brain because the way the knowledge is hooked up to other ideas inside your brain may not fit within the configuration of ideas inside my brain. Thus to get an idea from your brain to my brain we must first find relevant ideas we share in common and relate the concept you are trying to transfer to each of those ideas in turn. The trasmission of knowledge is a series of analogies built upon knowledge already inside the recipient’s brain.
This is why it’s of utmost importance to know one’s audience when trying to communicate a point. It informs which analogies we can and cannot use. (Relevant points here: Chicken soup for the teenage soul and i18n.)
We learn about everything and everyone through construction. We, ourselves, are no exception. We learn about ourselves by constructing knowledge about ourselves in our heads. The proccess of learning about onesself is literally an act of “constructing” or “building oneself” inside one’s own brain. It’s the act of relating yourself to ideas that you already have inside your brain. A clear correlary of this idea is that you can literally only be as good as the ideas inside your brain.