The Unreasonable Effectiveness of Mathematics (1980)

Wild speculation: Perhaps one could also attempt to explain the effectiveness of mathematics the other way around: Why are the rules of physics simple enough so that we can describe them fairly accurately? I think this stems from the fact that complex systems are often the substrate for simpler systems. There are plenty of examples for this: Simple lifeforms emerge from biochemistry, planets are on elliptical orbits around stars as the result of countless particle interactions and so on. Perhaps this can be explained by the minimum total potential energy principle, that simple rules happen to be the ones which are stable or at least metastable and more complex systems would need more energy to maintain the relationships between its constituent elements. Simple systems can be abstracted from the substrate to the degree they are a reliable phenomenon. The human brain is a rather reliable information processing system, but it's also rather limited in capacity (it certainly has many times less capacity compared to the systems we are able to reason about). However, since the rules of most physical systems happen to be rather simple and since the language of mathematics allows us to compress rules into efficient chunks which fit into the working memories of the brightest humans, we are able to describe physical systems with counter-intuitive effectiveness.

When I mentioned I wouldn&#x27;t be continuing with Economics after I graduated, one professor asked my why. I said that I had wanted to find a theory with the same mathematical elegance as classical economics (i.e. general equilibrium theory) but found that there didn&#x27;t seem to be such a theory left to discover. Almost everything beyond classical economics was mathematically shallower (not to say it was wrong or poorly thought out, just that deeper mathematics wasn&#x27;t applicable).<p>So while mathematics is very effective in Physics, and I believe it will remain so, I am highly skeptical whenever I see people trying to apply deep mathematics outside physics and chemistry. Some things just don&#x27;t seem amenable to mathematical laws.

This piece always irritates me. I don&#x27;t think it&#x27;s that interesting of a question.<p>Firstly, Mathematics is a language for studying structure.<p>The reason it works is because we won&#x27;t call something &quot;mathematics&quot; unless it is a reliable&#x2F;reproducible&#x2F;communicable tool for representing structure. As it is a language, it employs linguistic structure in an act of mimicry of experience. This is called analogy.<p>Secondly, the brain -- the basic function of a brain -- is to differentiate between experiences -- to assign meaning or to separate a signal from noise. This is how a brain develops a notion of &#x27;appropriateness.&#x27; We apply this concept to linguistic analogies, and from that you can recover a notion of &#x27;truth.&#x27; (An &#x27;appropriateness engine&#x27; could be a suitable term for a moral algorithm. &#x27;Utility function&#x27; is also often used in this context.)<p>It is not unreasonable that math works. Math works because if some language doesn&#x27;t work, we just won&#x27;t call it math. The mystery here is that anything works at all, and to answer that you&#x27;d have to explain why anything even exists.