Why is math powerful?

Question

I've been having this thought for days now, and I haven't been able to come up with a satisfactory answer.

It seems to me that one can arguably caricature mathematics as an impoverished natural language, with two provisos:

1. Only declarative sentences are allowed, like "5+7=12" (consider mathematical conditionals to be declarative by nature).
2. The discourse is restricted to highly idealized abstracta.

Why is it, then, that mathematics is such a powerful tool/language when it looks so poor on paper given 1 and 2?

Answer 1

Good afternoon!

The utility of math has many facets, but I'd list four off the top of my head.

• Math is abstract. And why is that important, from a psychological perspective it allows one to find patterns as well as well as psychologically manipulate information. Having two distinct, but equivalent differential equations might be hard to even memorize, but remembering the identity property in algebra is pithy. Think about how complex the data can be for real-world laminar flow. And then think about how approximating that data into a computationally tractable set of symbols can allow for a simplification that is useful for an engineer or a machine learning routine. Which is easier to work with:

The hypotenuse of a right triangle is equal to the positive square root of the sum of the legs each squared.

OR

c² = a² + b²

Natural language has many advantages over artificial languages, but simple, formal languages make it easier to use and recall.

• Math can be used for modeling and prediction. Models are useful things. Currently, drug manufacturers looking for solutions to treat diseases are moving away from testing in vitro and are moving to simulations using mathematical models. In fact, mathematical modeling revolutionized science starting arguable with Galileo. No longer were physical objects just discussed, they started being measured and became pertinent to one of the major goals of science, explanation. In fact, here's Stanford's article on the explanation and mathematics. Along with mathematical explanation from models also comes prediction. With Newton's Laws, along came the power to predict motion and eventually cruise missiles. The parabola has been a central occupation of militaries for hundreds of years. The canon is not just military power, it is political power.

• Mathematics can be automated in very sophisticated ways including machine learning (ML). The heart of a computer is the ALU which is a set of atomic mathematical and logical operations. Any well-designed computer can be purchased and extensive mathematical libraries can be installed enabling even a cellphone to have a better command of algorithmic mathematical processes. Think about how much math is done in the world by the billions of microprocessors. Not only are computers doing mathematical proofs, but they are learning and making engineering decisions. Here's a fascinating example of how ML is being used to squeeze efficiency out of the engineering in the renewable energy sector.

• The formal systems of mathematics are largely culturally neutral. A mathematician in Beijing can collaborate with one in Berlin, and submit their article to a journal in the US and then be shared with the world. The abstraction makes it easier to understand and do no matter what a person's first language is. Andrew Wiles's proof is one that is studied by the international mathematical community, and it has little regard for politics or nation. With the objection of certain schools of philosophical thought, Mathematics is widely to believed to be an objective discipline and psychologists have shown that certain mathematical abilities are psychologically innate and universal.

So, one can view the formal systems of mathematics as being an impoverished natural language, or one can see them as distilled reason, capable of being used to model, predict, automate, and communicate, a potential available in some form to every living human being to further technology and advance the human condition.

Answer 2

In fact, math is powerful precisely because it is an "impoverished version" of natural language. By restricting what we say, we can guarantee certain properties of our overall system. For example, natural languages allow us to state liar paradoxes and set theoretic paradoxes. Your foundation of choice, say ZFC, imposes many restrictions on what can be said so that these paradoxes do not arise.

Further mathematicians generally insist that all symbols refer, that there is no vagueness- no amphiboly or sorites predicates,etc. All these consitute heavy restrictions on the language of mathematics, but are "necessary" to guarantee that everyday mathematics functions "the way it should".

(In fact, we can formalize many of these different notions, but they are technical and greatly complicate the overall structure, hence why most mathematicians do not use them)

Answer 3

Your statement is basically a variation of Eugene Wigner's seeking to explain 'The Unreasonable Effectiveness of Mathematics in the Natural Sciences'.

I argue here that conservation laws in physics arise from symmetries (Noether's theorem), and that number lines are an abstraction of translational symmetries: The Unreasonable Ineffectiveness of Mathematics in most sciences By this similarity in form, maths is very powerful for physics. But biology, takes a lot more words, a lot more conceptual units. Causality in biology needs to be understood in a different way, because it is not generally directly grounded in conservation laws. We form heuristic explanatory layers, which supervene over physics with causes in the terms of their layer, but are reducible to physics in principle: Is the idea of a causal chain physical (or even scientific)? The layer of intentions and identity, is a far more effective tractable way of predicting a human's behaviour, than knowing the position and momentum of their particles.

Language and the sharing of abstractions, also relates to a symmetry: inviting others into our subjectivity, and mentally travelling into theirs. This is intersubjectivity, and following the Private Language argument, it's a critical groundwork to beginning to systemise abstractions: According to the major theories of concepts, where do meanings come from? Language forms 'salience landscapes', overlays which we project on top of our experiences, that sort them into useful groupings, and foreground where and how we can act, in relation to achieving our goals. We can understand that as arising from narratives, that abstract 'the moral of the story', and causality as a form of narrative that aims to support it's story by direct observations, making the 'lessons' very transferable to other circumstances where what is modelled is known to be the same - Cartwright's 'How The Laws Of Physics Lie' goes in to how physics can only be as valid as our abstractions are sound, and we must not mistake physics for mathematics.

I argue Socrates was paradigmatic in defining philosophy. Plato was paradigmatic in defining academia, and to create his Academy he fused Socratic Dialogue, with the math-mysticism of Pythagoras, successfully combining disruptive thinkers into a workable 'cult'. Plato believed in a cosmos of planets defined by platonic solids, with associated 'music of the spheres'. Mathematical Platonism is still a popular stance, eg Tegmark, with people still getting misty-eyed imagine a 'more real' world of Forms, as embodied by mathematics. That's backwards though, maths is the systemising of abstractions, emergent from examining the world, and intersubjectivity. Don't fall for the math-cult propaganda.

Answer 4

Math is powerful. Saw a video on math a coupla moons ago where this lady mathematician says everything is math. I was completely bowled over if only as a mere fan of math having no real skill/knowledge in the field.

Currently, I'm, when I can, workin' on a post-math world where math is an ancient language cum paradigm. We go beyond numbers and I'm sure there's a mathematician out here somewhere who could fill us in on the latest developments in the field. Are the most advanced topics in math losing their numerical "quality"? Less numbery, is math becoming?