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The Fine-Tuned Universe -- the insignificance of very small numbers

By Mark Frank

Posted December 21, 2008

Many people, especially those from the Intelligent Design community, argue that the universe has been so finely tuned for life that it must have been designed. This article by Rich Deem: Evidence for the Fine Tuning of the Universe (Deem, undated) is one of thousands of similar articles that can be easily found on the internet.

The argument is straightforward. There are many aspects of the universe, including some of the fundamental constants of physics, which if they had been even slightly different then life would not have been possible.  An extreme example is something called the space phase constant of the Big Bang which according to Roger Penrose (Penrose, 2001) has to be accurate to 1 in 101230 or the second law of thermodynamics would not be true and life would not be possible. This is easily reinterpreted as "the odds of this constant happening to take a value which would support life are 1 in 101230 ". This is so enormously improbable that the constant must have been set to this value intentionally. Hence there must be a designer.

There have been many types of response to this kind of argument. Some look for alternative explanations which reduce the apparently prodigious improbability of life. They argue that the physical constants have more leeway than supposed, or that there is an underlying physical explanation for such constants having those precise values, or that there are an infinite number of universes each with different values for the physical constants and of course we are in the one that supports life.

Others treat supernatural and natural explanations of the universe as hypotheses to be assessed using standard techniques of statistics. They then use Bayesian logic to challenge the conclusion that design is made more likely because the universe has some surprising features, for example (Perakh, 2001).

These are all powerful arguments. The point of this paper is much more limited. I want to examine  if this proposition:

(A) Constant K has to be accurate to 1 in 10a very large number to support life


(B) Without design, the probability of constant K having a value that supports life is 1 in 10a very large number

Or to offer a bit more leeway, does (A) entail

(C) Without design, the probability of constant K having a value that supports life is very small

To keep the number of zeros down to a manageable size I will use a different example of a fundamental constant - the ratio of the mass of a proton to the mass of an electron -- which is just under 2000:1. This is one of the constants that Deem identifies and he writes that life as we know it would be impossible if that ratio differed by 1 in 1037.

The design proponent must have a picture analogous to the following. Imagine a creator of universes with a set of dials -- one of which is marked proton to electron mass ratio. The creator could set the dial to a vast range of values, but with great care tunes it to a value which sustains life.  I will call the range on the dial which includes all the values which could sustain life the life supporting range. In the design proponent's mind the life supporting range is a tiny portion of the dial that corresponds roughly speaking to 2000 ± 2000 * 10-37. The alternative is that dial was twirled by some natural process and just happened to fall in the tiny life supporting range. Hence proposition B.

Given this picture, some basic questions arise. One is: "in the absence of design, is the dial equally likely to fall at any point in its range or is it more likely to land in the life supporting range". I am going to duck this one. It turns round the concept of a uniform probability distribution and there is already a lengthy on-going dispute about this. For the purposes of the rest of the essay I will concede that the dial is equally likely to fall anywhere in its range.

A second question is:  "what is the total range of values on the dial?". If the life supporting range occupies 90% of the dial's total range then the argument clearly does not hold. But how big is the non-life supporting part of the dial's range? What are the upper and lower limits? We have no other universes to compare and as far as I know there is no theory to fall back on. Here are some possible answers:

  • The dial's range includes every value from zero to infinity (Maybe even from minus infinity to plus infinity. A negative ratio is unimaginable, but in cosmology it is pretty common to talk about the unimaginable). After all if the designer is indeed God then he is omnipotent and should be able to set it whatever he wants. In this case it is not possible to give any meaning to the probability of a life supporting value. You can't have a uniform probability distribution over an infinite range. It makes no difference to the probability whether the life-supporting range is ± 10 -37 or ± 10 +37 . We might as well forget about all the figures.
  • We don't know the range of the dial. In that case we don't know if a life supporting value is improbable or not. End of story.
  • The range of the dial is the same as the value of physical constant in our universe.

I believe this last answer is the one that is being implicitly assumed. It would support both propositions (B) and (C) above. But it is an assumption which has no justification. Why should the range of possible values have any relationship to the particular value that we observe in our universe?  It is to confuse the size of a range (a dispersion) with a specific value within that range (a location). To give a mundane example: a live human body is typically at approximately 310 degrees Kelvin plus or minus one degree. This could be presented as a precision of about 0.3 per cent i.e. less than 1 in 102. Should we then conclude that the probability of someone having a normal temperature is less then 1 in 100? Of course not -- the 310 is irrelevant.

To say the same thing a bit more mathematically: suppose there is random variable which is the ratio of the mass of a proton to the mass of electron (a strange idea in itself) -- call it Μ. Its observed value in our universe is μ. Let the range of values which would support life be from μ1 to  μ(of course, μ is somewhere in this range).


The numerator is the difference between the largest value which will support life and the smallest. But what is the denominator?  It could the same size as μ -- which would give us a probability of 10 -37. It could be exactly the same as the numerator which would give us a probability of 1. It is a complete unknown.  The fact that the numerator is small compared to μ has no significance.

I appreciate that this may be counterintuitive. The idea that a precision of 1 in 101023 has no significance feels wrong. Surely such a mind blowingly small number must have implications. But if there is one thing we should have learned about modern physics it is that intuition gets us nowhere. It is world where nothing but the mathematics can be relied upon.


Deem, R. (undated). Evidence for the Fine Tuning of the Universe. Retrieved December 14, 2008, from Evidence for God.

Penrose, R. (2001, December 30). Retrieved December 14, 2008, from The Physics of the Small and Large. What is the bridge between them?

Perakh, M. (2001, July 30). The Anthropic Principles -- Reasonable and Unreasonable. Retrieved December 18, 2008, from Talk Reason.