From Intuition to Math
Isaac Newton is often attributed as the person who “discovered” gravity, but this is a blatant misunderstanding. The existence of gravity is an obvious fact to every animal with a sufficiently large brain to comprehend a constant downward force.
Newton didn’t “discover” gravity; he discovered an equation to describe it quantitatively.
Broadly speaking, our knowledge of the world roughly follows a path from the unknown, to the intuitive, to the rigorous. Our brains are incredible at taking observations of the world and transforming them into a mental model that can be surprisingly accurate, even if we can’t explain it in much explicit detail.
Many animals can be said to have an intuitive understanding of many principles of basic physics, even if us humans have only developed mathematics for describing it precisely in the past few centuries.
A fundamental problem with intuition however is that our brains aren’t perfect. We often develop intuitive models that are innacurate or biased. It can be very easy to find a story of how something works that has some hidden logical inconsistencies that can lead us astray in our reasoning. The fuzziness of language can make that problem significantly worse. If we lack a rigorous foundation, arguments about how to solve problems related to these models can easily devolve into word games with no real basis in reality.
In some cases, we have extremely deep models of the world, allowing us to build devices of incredible complexity, such as the device you’re reading this on. In other cases, extremely important aspects of the world, like politics and history for example, are very poorly understood with any rigorous basis. The models we have are of questionable accuracy (economics and psychology), and the people we put in charge of making decisions (politicians and voters) rarely understand even these models well.
In the case of politics, we’re left making decisions based on ideologies that perhaps individually model some minor aspect of the world well, but are overall far too simplistic to accurately describe anything beyond that. These ideologies often aim for completeness over consistency, and a lack of consistency can make it easy to use them to justify anything that is politically convenient, regardless of if it really is a good idea.
An intuitive model of some aspect of the world can be surprisingly valuable, and is definitely much better than having no model. However, whenever possible, we should be actively striving to formalize these models into something more rigorous and quantitative.
When solving hard problems, it is critical that everyone means exactly the same thing when using the same word. It is critical to be able to tell precisely what the consequences of something will be. It is critical to know what the limits of the model are and when a new model is necessary to explain the world further. If we lack these properties, we can easily wander off into misleading narratives that increasingly lack any relationship to reality.
In some of my coming articles, I will be exploring these ideas further. What is the value of intuition? What are its limitations? How do we develop more rigorous models of things?
I’ll also be sharing some ideas I have on building more rigorous foundations for some problems, such as programming language design.
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