I’ve long contemplates what math actually is. How can manipulating symbols on a paper according to some rules tell us how to manipulate atoms that we cannot even see, or send rockets to the moon, or create phones, and everything else we’re able to do today?
The answer occurred to me when I was considering mental models – parallels of aspects of reality in our minds that we use to predict the world. I write about this in how Phaedrus is wrong about hypothesis creation in Zen and the art of Motorcycle Maintenance.
I realized that math is simply one such model of reality. It doesn’t exist anywhere other than in our minds. It is not “real”. It is not “truth”.
It can be recreated by starting with some internally consistent concepts and rules for how to operate these concepts (run simulations) that have direct real world counterparts that can be clearly observed in the same way by different observers. Some of those concepts are basic numbers and operations.
Take for example 1 and 2, and plus. Most people have been trained to interpret and use these in the same way – and apply the corresponding real world result in the same way. So if you say “get me the sum of 1+1 chairs”, you would get two chairs by different observers.
By formally creating such concepts and rules for operation that have a direct real world counterparts, you get an internally consistent and formalized model that is a direct counterpart to a part of reality.
Then comes the magic part.
The thing with reality is, it is consistent. So a model that was created as a direct counterpart to one part of reality, with internally consistent rules that produce correct parallel results as that of reality itself in terms of these concepts, can be expanded to encompass other parts of reality too. This means that you can use the rules of your model to expand it’s scope and continue to predict other parts of reality, as long as your model continues to be internally consistent.
Thus, what is math? It is a model that has some concepts that have direct real world counterparts (such as numbers), and rules for how these concepts may be manipulated to produce real world predictions of those concepts’ counterparts. But the same rules can also be used to produce “pure math” operations, concepts and results – those with no counterpart in the real world – and then in an internally consistent way translate these results back again to the concepts with real world counterparts, and thus predict what will happen in the real world in an indirect way derived from the rules of the math model itself.