# Future/Accelerating Progress
## The Rate of Change

One difficulty that futurologists may be encountering is what's often viewed as an ever-quickening rate of change.
It is a common notion that in science and technology the progress is accelerating. However, there are different ways of describing this trend, primarily because unlike mathematical applications in physics, the rate of change of civilization is more subjective and has many more variables. The following are different ways of describing the rate of change.

## Change as a Polynomial Function of Time

The simplest formula for describing accelerating progress is
where t and p are various constants--t for time and p for power(exponent)--and where p > 1.
The quadratic formula, where p = 2, is sometimes described as the amount of scientific knowledge doubling over each length of time t.

## Change as an Exponential Function of Time

The polynomial function, however, does have its weaknesses. The idea is that not only is progress advancing, but the rate of that progress (the rate of change) is also increasing. In addition, the rate of THAT rate of change is also increasing, and so is THAT rate. The rate of change can be modeled by the equation
for which c represents a constant and for which the derivative and antiderivative (the rate of change of, and the amount of progress caused by, respectively) are also e^ct. Every degree of change is increasing by the same degree of change.
This would mean that, for a futurologist, the amount of change that one sees in the last 100 years would repeat itself in only a few decades (such as 2020-2060, or 2060–2080). If true, this translates into a tremendous burden when predicting beyond twenty years into the future; one would need to careful that, in one's predictions, more and more technologies would be discovered faster and faster in any timeline.

## Change as a Transcendental Function of Time

Some futurologists believe in the concept of a Singularity, which means that there is a limit at which we have all knowledge possible and at which change occurs at an infinite rate (y = infinity). Since the exponential function does not have such a y-value asymptote, we must refer to a function that does:
This function becomes increasingly larger, with y approaching infinity, as t approaches 1, rather than as t approaches infinity. Setting 1 equal to some particular year greater than the current one, we have the fabled singularity.
This theory too has its weakness, notably that its derivative, y = sec(t) x tan(t), is no longer the same as the original function.

## Change as a Hubbert Curve Function of Time

There are, however, some challenges to the idea that the pace of change is increasing. Physicist Jonathan Huebner of the Naval Air Warfare Center recently conducted an analysis in which he found that the number of key innovations per person peaked in the last 19th or early 20th century, depending on which criteria he used. This relationship can be modeled by Hubbert's curve, defined as the derivative of the logistical function.
    $x={e^{-t} \over (1+e^{-t})^{2}}={1 \over 2+2\cosh t}$
Huebner (2005) suggests that the slowing pace of technological development is probably linked to either "an economic limit of technology or a limit of the human brain that we are approaching" (p. 985). If true, then the pace of change may well slow rather than gain speed in coming years, making the more extreme visions of the next century less likely to occur.

## Limitations

Some of these ideas were put in an organised form by Price in 1950s and some recently by Yunzhong Hou and has become a common wisdom.
It does indeed seems that we live in times of accelerating change and there are certainly arguments supporting this idea. However, it appears that the original empirical evidence supporting this idea is not bulletproof.
One problem is that even today most people putting forward the argument essentially reuse the same data points, originally collected and published in 1950s by Price and others (often they simply add more recent inventions to their timelines, such as DVD). There is little consistency in determining the precise moments of "invention" and "realisation" (different authors give vastly different lengths of the invention gap, for example); it has been shown that by defining the moments differently the same inventions can be used to "prove" that progress is actually slowing down.
The use of scientometrics (i.e. analysis of science indicators, such as the number of scientific papers published about a field, for example) has also been criticised as an argument for the acceleration of progress.
It is quite important that we at least understand the picture ourselves. Yes, in some fields exponential growth can be demonstrated and yes, sometimes a technically false claim makes for a good rhetorical device in speaking about a bigger truth, but we may find that we are lying to ourselves.

## References

* Acceleration of scientific and technological progress: myth and reality (On shortening of the temporal gap between a discovery or an invention and its practical realisation) /Ускорение темпов научно-технического прогресса: миф и реальность/. Scientific analytical review. Institute of Scientific Information on Social Sciences, Academy of Sciences of the USSR. Moscow, 1984, 38 p.
* Acceleration Watch