Philosophy and Science 2: Singularities

Following on from my previous post about science and philosophy, in this post I introduce the mathematical concept of a singularity as it relates to science and philosophy.  You can read my earlier post here: Philosophy and Science.

What is a singularity?

In mathematics, a singularity is a point within a function for which its value is undefined.  You can read a further discussion of mathematical singularities on Wikipedia here http://en.wikipedia.org/wiki/Mathematical_singularity.

In the abstract sense, there is not much more to say about a singularity, but when considered in relation to time things are more interesting…

Singularities involving time

The following graph is intended to be illustrative only:

Graph of a time-based singularity (illustrative)

Graph of a time-based singularity (illustrative)

 

The function f(t) merely represents the state of whatever is being examined, whether an individual experiment or the entire universe!  At the time T, the value of the function is undefined (represented in this case by positive infinity).

With a time-based function, there is a unique perspective in that we exist at a point on the time-line as well.  All our observations are made at a relatively small number of discrete moments along that time-line, although we also make use of records (to tell us of previous observations) and extrapolation to hypothesise about what might have occurred previously (and what might occur in the future).

Now, if we suppose that a singularity exists along our timeline at a previous but unknown time T, then we have to include a consideration of singularities in those hypotheses.

The significance of a singularity is that the value of the function is undefined.  That means we know absolutely nothing (not just nearly nothing) about the state of whatever it is that we are observing at that time.  Worse, extrapolation is based on the theory of a continuous function and so it is impossible to use extrapolation to tell us anything about what occurred before the singularity.

In other words, extrapolation (and historical observations) can only tell us about the state of something since the most recent singularity.

Common singularities involving time

This discussion would be entirely inconsequential if it was not for the fact that discussions involving singularities are common, whether or not people are aware of it.  Common singularities include:

  • The big bang
  • The big crunch (one possible outcome in the big bang theories)
  • Creation

However, it is important to consider singularities in other scenarios too, because much of science is based on the presumption of the lack of a singularity.  If a singularity were to occur (and it only has to be a singularity with respect to the phenomena being observed) then extrapolation would be meaningless.  So, any theory that involves significant extrapolation is wholly dependent on a presumption of a lack of a singularity.  Because this presumption is not based on a testable hypothesis, the theory is dependent to a large degree on an unproven belief.  Whether that belief is reasonable is a matter for the reader :)

One passing thought for those who want to take this to its logical conclusion… There is no way of proving that a moment ago you existed, because the theory of Creation is not unreasonable in itself (whether you accept it or believe it is an entirely separate matter).  If you assume Creation then it is perfectly plausible that you were created already in the process of reading this article, and because Creation is a singularity you cannot utilise extrapolation to tell you otherwise!  I recommend that you do not let this worry you too much, though :)  It is only a theory, like any other.

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