The Plasticity of Functions, Information & Energy

Image Credit: Michael Taylor

Think of the word function and what comes to mind? If you are mathematically inclined, perhaps you imagine y=f(x), a graph or perhaps your mind’s thesaurus conjours up others like process, utility or meaning? Function is so deliciously amorphous that when you start to chew it over a bit more, you just can’t pin down what it is. Or can we? Imagine 2 columns of numbers. We can use the first column for a variable like x and the other for y. We can plot the points and get a graph. The points might orderly follow a line or a curve that we may recognise like a parabola right? It may even trigger us to recall the equation of the parabola. But what is the function? Is it the equation like y=x² or the graph itself? Then we think, hang on a sec, didn’t we use the table of numbers to generate the plot in the first place? Here, we start really scratching our heads – two columns of numbers – how on earth can they be a function?! The confusion grows if you slowly side-step away from the maths.

Image Credit: Michael Taylor

To a computer programmer, a function is a subroutine tucked away in the basement of a code or a separate piece called when needed. In engineering, a function is a specific process, action or task, that a system is able to perform – like a part of some wonderful machine that takes a cube of metal on a conveyor belt on one end and moulds it step by step into an masterpiece of technology at the other. This sense of a function having direction is not just special to engineering or conveyor belts either. For a biologist, a function is part of an answer to the question: why did some object or process occur – usually, in a system that evolved through a process of selection along some chain of causation toward a goal, success or environmental fitness. “Function”, apart from dinner parties and galas, also crops up in lively debates over “Structure versus Function“. But it is in mathematics, more than anywhere else, that the word function takes on many personas and accumulates many adjectives – map, continuous, bounded, closed, topological and many more …

The most mysterious and intriguing thing to me is when I return to think some more about the two column, headerless table of numbers. Pure data. No x’s or y’s. No f’s or g’s. Just digits. And yet, plot them in pairs as (x,y) coordinates and they metamorphosize into a graph, opening a whole Pandora’s Box of other cognitive triggers like algebra, equations, trigonometry, shape and form. I like to think of functions like energy. Whatever it’s form whether that be heat, gravity, motion, radiation, electromagnetic, nuclear or even dark energy – one form can be converted into any another. Energy is like an invisible Transformer in the movie sense of the word. But, energy in all its guises is measured in the same way with the Joule – one unit of energy. And here is the crux. Energy can undergo metamorphosis but there is a price to pay. Just like debt bought and sold on the stock market, eventually it will one day be called into account. Transformation of energy, like function to a biologist, or input-output process blocks to a systems engineer, follows a causal and, in the long run, irreversible arrow of time. For energy, this arrow of time is the Second Law of Thermodynamics.

Image Credit: After A. Riess, NASA

But what about data? Well, to think of data as information is not something too difficult to do. It seems logical to me to think of the table, the graph, the equation, or a map all as different ways of metamorphosizing information. Information seems to be analogous to energy. As American mathematician and engineer Claude Shannon pointed out to us in his landmark paper “A mathematical theory of communication” in 1948, information has entropy. Perhaps it is the lowest common denominator of them all? And energy is just information in another coat? What are the consequences of this? Whatever we discover in the future, one thing seems certain, there is no going back in the long run – information and its associated function looks like it is here to stay – provided that is that the universe doesn’t end in a Big Crunch. And then what happens to the notion of a function? It’s mind-boggling.

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