Self-Reinforcement and Exponential Functions

Special relativity is really kind of mean. Not only it prohibits anything from going faster than the speed of light (therefore destroying our Star Trek-inspired dreams of interstellar travel) but also threatens with extreme adverse effects should anyone dare to even come close to the impenetrable barrier of c. Assuming we can deal with the steady increase of mass as the speed goes up, there is always this issue of time dilation. While you are taking your short (i.e. few years-long) trip to nearby star, time passed on Earth could very well be measured in centuries. Having a millennium to catch up might prove cumbersome, and rather frustrating. Just think of all the iPhone models you would have missed!

As a solace, though, you could get quite a pile of cash waiting for you to pick up. Let’s say you’ve put 10,000 dollars (or euro, or your favorite currency) into investment with a yearly interest rate of 10 percent. Every year, this deposit will therefore increase by one tenth, and this will happen continuously over the next 1000 years. Could you quickly tell how big the final amount will be, compared to the initial one? How many times will it increase?…

You shouldn’t be very hard on yourself if you answered instinctively with e.g. 100 times or something similar. I mean, such figures are totally, utterly wrong by many orders of magnitude because the actual value is bigger than 10⁴⁰. But it’s absolutely common to have problems with grasping exponential functions intuitively. In many ways this is quite pitiful, for they accurately describe many phenomenons that occur in nature, civilization, technology and culture. Yet they often escape understanding, leading to unfulfilled predictions, incorrect extrapolations, and plain old cognitive biases.

What is so bizarre about these functions that they tend to confuse a significant fraction, if not the majority of people?…

A derivative problem

First, let’s clarify that the exponential function is a very specific one. It can be defined as the only function which is its own derivative. We can write it down as:

where e (equal to approximately 2.71) is typically known as Euler’s number and is easily the most important constant in all of mathematics. Yes, much more significant that , , or whatever the current fashion in geometry dictates.

In practice, we often deal with exponentials of different base. For example, computer scientists are especially fond of base 2 for variety of reasons. While the above equality isn’t satisfied for those functions in exact manner, the general principle still stands. Differentiating an exponential function always leaves us with another exponential function.

And this, I believe, might be one of the roots of the problem. A significant portion of human brain – the sensory cortex – is basically a very complicated differentiator that operates on incoming neural signals. We experience it as Weber-Fechner law, which basically states that our senses detect changes in stimuli only in proportion to the already perceived one. In other words: the brain is, for the most part, conveniently ignoring the absolute values of sensory data as it’s mostly interested in their proportional changes. It would seem that applying differentiation across the board makes the data more manageable for cognitive processing.

How this relates to our poor intuitions towards exponential functions? Well, we could speculate that some equivalent of before-mentioned law works on deeper level: the one related to our “perception” of numbers, measures, quantities and trends – or just mathematics for short. If that’s indeed true, then we can easily see how exponentials fail to comply. Because they are immune to differentiation, we are unable to subconsciously transform them into more “natural” form which would be easier for our mind to act upon. Instead, they appear as conceptual black box that we might never really grasp on a fundamental level.

Reinforced by itself

Regardless of whether the above supposition is sound, it is certainly true that exponential trends mess badly with one of our basic intuitions: that future will be more or less similar to the past. Despite countless fluctuations of human condition over the centuries, this heuristic held ground because significant and overarching changes almost always took longer than a single lifetime. It is only through history (written or spoken) that we can learn of the curious and alien ways of living in the distant past.

But cultural and technological development is a self-reinforcing process. This means that any amount of progress not only has an immediate effect in the present but also results in further improvement of speed and effectiveness of the process itself. In physics, we encounter many phenomena that also exhibit this property (e.g. air resistance), and they are described using certain class of differential equations. Their solutions always take a form of – you guessed it – exponential functions.

Maybe this is why we fail so hard at making any kind of mid- to long-term predictions. Linear extrapolation of exponential trend is bound to diverge from reality at very fast pace. On the other hand, this can be a reason why some things remain almost invisible until they “explode” into ubiquity in very short time frame. If they feed upon themselves – being self-reinforcing processes – this is no coincidence, and we could see that happen in case of social networks and mobile applications. I suppose this is also what founders of technological startups should make sure their idea is like.

Adjusting perception

It would appear that exponential functions are quite a pain to comprehend on intuitive, subconscious level. But, as I believe to have shown above, developing more effective intuitions towards them offers plenty of benefits. I’m just not entirely sure what’s the best approach to achieve this end. For one, though, I think that gaining awareness of how exponential trends are interleaving our daily lives seems like a good start.

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