On Spherical Cows and the Search for Truth

Science is great.

It lets me play with cool toys, it pays my bills, it helps me understand the world.

But I have to phrase that last part carefully. It doesn’t completely explain the world, it helps me understand the world.

That’s because science deals with models.

And models have assumptions.

And assumptions lead to limitations.

This may not be a bad thing. Depending what we are trying to understand, and the level of detail at which we are trying to understand it, the assumptions may greatly simplify our work without interfering with our objectives. But a clear statement of the assumptions is vital for anyone trying to assess the usefulness of a model in addressing a particular question.

Let me illustrate all this with cows.



Modelling a cow

Suppose we need a quick estimate of the mass of a cow. (Imagine we’re in a rural setting far from a WiFi signal and can’t use Google). With just a pencil and piece of paper, how would we get a quick first-order approximation?

Well, we can’t lift it up, so we have to come up with an indirect route to get the mass. We know that:

Mass = Volume x Density

…and we know that most animals are approximately the density of water (hence the fact that they float at the surface of water but are mostly submerged). And we know from high school that water is 1000kg per cubic metre. (For the USA readers, sorry, but I’m going to use SI units for this. One of the beauties of a rational measurement system is that it makes this mental arithmetic a lot easier).

So all we have to do is estimate the volume of a cow and we’re home free.

Now, since a cow is an awkward shape for which we can’t calculate a volume, we’ll approximate it to something simpler. Such as a sphere:

Or, if that seems a little too abstract, try a cylinder:

Better? OK.

You’ll notice that the legs, tail, ears and head and neck are all drawn in lines: that’s because we’re going to ignore them in our calculation. If we make the cylinder a little bigger than the cow’s body, we’ll be able to safely assume that the “small skinny bits” could fit in the left-over spaces, and the overall volume will be about right. Remember this is just a first-order approximation.

So now we walk over to the cow, and try and gauge the dimensions of our cylinder.

We’ll assume an average-sized cow, and let’s say we estimate it at maybe 1m in diameter, and about 1.5m long. For a cylinder, volume is given by:

V = πr²l

Crunching the numbers, (and assuming π=3.2 to make the maths easier), this comes out at 1.2m³. Recalling the density of water, our final estimate is 1.2 tonnes for the mass of a cow. Which is actually pretty decent: steers are about 750kg, the heaviest bulls are about 1750kg, so 1200kg is the right ballpark.


The modelling process

Note what we did in the exercise: we built a model to organise our thinking, and we did it in following way:

1. Define your objectives. (Give a rough estimate the mass of the cow).

2. Make assumptions in light of the objectives. (The skinny/pointy bits can be ignored. The body can be represented by a cylinder. The density can be approximated by water).

3. Build a model incorporating those assumptions. (A simple cylinder of density 1000kg/m³)

4. Extrapolate from the model results back to the real world. (Our cylindrical model weighs 1.2 tonnes » we estimate that an actual cow weighs approximately 1.2 tonnes).

Thus our model of reality helps us to understand reality by simplifying it and then extrapolating the results back to reality.

Einstein famously said that an explanation should be as simple as possible, but no simpler.

How do we decide how simple to make it? By understanding our objectives and making assumptions in light of those objectives. The assumptions are all valid based on the starting objective that we only need a rough estimate. If we need an accurate mass (ie., our objectives change), those assumptions don’t hold anymore.

Now let’s watch it all go wrong.


Tripping on the next step

Suppose we now ask ourselves: “What is the surface area of a cow?” (Don’t ask why we’re pondering mathematical questions in a cow paddock, just run with it).

Well, we think to ourselves, we have a model of a cow. We know how to calculate the surface area of a cylinder:

A = 2πr(r + l)

…so we’ll take our cylinder and crunch the numbers again. This is easy!

Unfortunately, it’s also incorrect.

In using the cylinder, we are mistaking our model cow for an actual cow.

In modelling terms, we have modified our objectives without revisiting our assumptions.

The assumption that “all the pointy bits don’t make much difference” is true for volume, but it is not true for surface area: they make a very significant contribution to that value. Thus our cylindrical cow is a very poor model for estimating surface area.

Modifying the objectives of a model will generally require a new model. At the very least, all the assumptions must be revisited and re-evaluated if our model results will retain any relevance in the real world.

This is also true, of course, of any unconscious assumptions we may have made.


The verification problem

Science is basically a giant collection of models. The process that I’ve just described is analogous to the entire scientific method.

What we do in science is look at data, try and imagine an underlying process which could explain it, and then build a conceptual model. (The models are often mathematical – but not always – because mathematics allows us to express concepts simply and clearly in a well-defined system). We then try and imagine what other observations would be consistent with that model, and we look for support for it. If it reliably predicts actual observations (or in scientific jargon, if it has good explanatory power), we might regard the model as having been confirmed. This is the stage at which we may move from regarding it as an hypothesis to calling it a theory.

What we cannot do in science is verify a model. Verification (from the Latin “verus”, meaning “truth”), implies that the model actually represents the truth.

A classic paper by Naomi Oreskes, Kristin Shrader-Frechette and Kenneth Belitz in the journal Science phrased this point particularly succinctly:

“Verification and validation of numerical models of natural systems is impossible. This is because natural systems are never closed and because model results are always non-unique. Models can be confirmed by the demonstration of agreement between observation and prediction, but confirmation is inherently partial. Complete confirmation is logically precluded by the fallacy of affirming the consequent and by incomplete access to natural phenomena. Models can only be evaluated in relative terms, and their predictive value is always open to question. The primary value of models is heuristic.” (“Verification, Validation, and Confirmation of Numerical Models in the Earth Sciences”, Science 263 (5147), 1994)

Models are useful. The whole point of a model is to help us understand what is otherwise incomprehensible. But at all times we must remember that any model (including a scientific theory) is not truth.

Oreskes et al. continue:

“A model, like a novel, may resonate with nature, but it is not a “real” thing. Like a novel, a model may be convincing – it may “ring true” if it is consistent with our experience of the natural world. But just as we may wonder how much the characters in a novel are drawn from real life and how much is artifice, we might ask the same of a model: How much is based on observation and measurement of accessible phenomena, how much is based on informed judgment, and how much is convenience?” (Ibid.)


Methodological naturalism and the limitations of scientific models

Now that we understand how models work, I want to look at the limitations of science as a method of pursuing truth. With our newly-acquired perspective on how science actually works, I hope that it will be clearer what those limitations are.

Science, as a collection of models (termed theories or hypotheses according to their level of confirmation), is built on a set of assumptions. These are broadly grouped under the philosophy of methodological naturalism, and could be summarised as:

  • The world we observe actually exists and is consistent.
  • We can use our reason and senses to explore it.
  • The material world is all that there is.

So we must ask ourselves: how useful is naturalism as an assumption?

The general opinion amongst philosophers of science is that it is a useful simplification. That is not to say that it is true, only that it is useful. Steven Schafersman, a geologist and prominent advocate against Creationism, writes that:

“… science is not metaphysical and does not depend on the ultimate truth of any metaphysics for its success … but methodological naturalism must be adopted as a strategy or working hypothesis for science to succeed. We may therefore be agnostic about the ultimate truth of naturalism, but must nevertheless adopt it and investigate nature as if nature is all that there is.”

Philosopher of science Robert Pennock, also a prominent voice against Creationism (and Intelligent Design), is more explicit. In his 1997 paper for a conference on “Naturalism, Theism and the Scientific Enterprise”, he states that science “makes use of naturalism only in a heuristic, methodological manner.” He also argues against even the theoretical possibility of using scientific methodology to explore supernatural issues:

“Methodological naturalism itself … follows from reasonable evidential requirements in science, most importantly, that hypotheses be intersubjectively testable by reference to law-governed processes.”

Why does this preclude the supernatural? In the same essay, Pennock writes:

“Experimentation requires observation and control of the variables. We confirm causal laws by performing controlled experiments in which the purported independent variable is made to vary while all other factors are held constant and we observe the effect on the dependent variable. But by definition we have no control over supernatural entities or forces.”

Pierre-Simon Laplace.


The usefulness of models (and theories)

In their correct place, of course, models are very useful. The great French mathematician Pierre-Simon Laplace used Newton’s model of gravity to calculate the motion of the heavens (as well as for predicting ballistics) in his masterpiece Mécanique céleste. Napoleon asked to see the manuscript, being greatly interested in ballistics. According to the story, after perusing the equations Napoleon turned to Laplace and asked, “Where is God in your book?” To which Laplace famously replied, “Je n’avais pas besoin de cette hypothèse-là.” (“I had no need of that hypothesis.”).

Laplace was perfectly correct. He was using calculus to predict the motions of celestial bodies and objects moving through air, and it is not useful to incorporate theological complications into that  prediction. Remember: as simple as possible, but no simpler. Of course, Laplace also didn’t include gravitational attraction from other stars in calculating the orbits of the planets. In the real world, we believe that other stars do exert gravitational attraction, but it is a useful simplification in our model that we ignore them at the scale of our solar system.

Laplace’s model does not correspond perfectly to reality, but it does allow us to make sense of data and make predictions, provided that we stay within the limits of its assumptions. The influential philosopher Karl Popper comments on the usefulness of the Darwinian evolutionary synthesis, despite the great limitations of that theory:

“Darwinism  is not a testable scientific theory, but a metaphysical research program … And yet, the theory is invaluable. I do not see how, without it, our knowledge could have grown as it has done since Darwin …  Although it is metaphysical, it sheds much light upon very concrete and very practical researches … it suggests the existence of a mechanism of adaptation, and it allows us even to study in detail the mechanism at work.”

But let us never confuse useful with true.


Science and truth

So what can science really tell us, if not truth? Well, within the limitations of its assumptions, it can give us great insight into the processes and nature of the material universe. But it cannot, by definition, tell us anything about the immaterial – including the supernatural, philosophical reasoning and morality.

The great Stephen Jay Gould, in his essay Nonmoral Nature, commented thus on the limitations of science:

“Our failure to discern a universal good does not record any lack of insight or ingenuity, but merely demonstrates that nature contains no moral messages framed in human terms. Morality is a subject for philosophers, theologians … indeed for all thinking people. The answers will not be read passively from nature; they do not, and cannot, arise from the data of science. The factual state of the world does not teach us how we, with our powers for good and evil, should alter or preserve it in the most ethical manner.”

Indeed, science cannot even comment on the validity of its own assumptions: they must simply be accepted at face value for any science to be done at all. As per Gödel’s Incompleteness Theorem, they are postulates which cannot be proven by the system itself.

In our search for insight into the supernatural, we’re out of the territory of science. And recall the fundamental principle of modelling: we cannot change our objectives without re-evaluating our assumptions. So we can’t even adapt any current science to deal with these questions: science is simply not equipped for the task.

I do not propose allowing supernatural explanations into science. But I do suggest that it is very misleading to imply that science in any way supports a materialist worldview. This is mere question-begging: scientific theory, by its very assumptions, operates within a materialist worldview.

But we do not live in “science”. We live in reality.

Are we searching for truth, or are we searching for a theory nested in unprovable assumptions?

If the supernatural exists, it is beyond the tools of science. But if we have a supernatural aspect to our existence, it is not beyond our experience. To limit ourselves wholly to a materialist view may deprive us of fully experiencing a part of ourselves.

Philosopher Alvin Plantinga argues strongly for this line of thinking. He wrote:

“If you exclude the supernatural from science, then if the world or some phenomena within it are supernaturally caused – as most of the world’s people believe – you won’t be able to reach that truth scientifically.”

Are you missing out on something important by clinging to rigid materialism, perhaps because of a mistaken belief that such a worldview has scientific justification? Is there anything more to life?

Not to science. To life.

C. S. Lewis, certainly, had no doubt about the importance of our supernatural aspect. In Mere Christianity he described the human condition thus:

“You don’t have a soul. You are a soul. You have a body.”

What are you missing out on?



Additional notes: The pursuit of data

Assumption are fundamental to understanding the usefulness of the outputs of a model. But the assumptions underlying the scientific method will also influence the data that we subsequently look for.

This limitation has been noted by Karl Popper and historian of science Thomas Kuhn, who observes that the “route from theory to measurement can almost never be traveled backward”. Philosopher of science John Lennox points out that:

“In common with the rest of humanity, scientists have preconceived ideas and worldviews that they bring to bear on every situation. Even observations tend to be “theory-laden”; we cannot take a temperature without having an underlying theory of heat. … Furthermore, at the quantum level the very process of observation gives rise to disturbances that cannot be ignored. By its very nature, therefore, science possesses a certain degree of provisionality.” (“Challenges from Science” in Beyond Opinion, edited by Ravi Zacharias)

Theories also tend to build on each other, usually without revisiting the underlying assumptions. Popper examines this problem of nested assumptions in his critique of naturalism:

“I reject the naturalistic view: It is uncritical. Its upholders fail to notice that whenever they believe to have discovered a fact, they have only proposed a convention. Hence the convention is liable to turn into a dogma. This criticism of the naturalistic view applies not only to its criterion of meaning, but also to its idea of science, and consequently to its idea of empirical method.” (The Logic of Scientific Discovery)

Note again the emphasis (in the second sentence) on the problem of confusing model confirmation with verification. This self-reinforcement of theory dominates most of science. Kuhn writes:

“Once it has been adopted by a profession … no theory is recognized to be testable by any quantitative tests that it has not already passed.” (The Structure of Scientific Revolutions)

We will never find what we do not seek and are unwilling to see.




This essay is an updated and collated version of two older posts. The original articles (with comments) can be found here and here.