On Spherical Cows and the Search for Truth (Part I)

Update:

This post and Part II have been edited and combined into a single essay. The full version can be found here.

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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.

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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 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 note 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 approximate it at maybe 1m in diameter, and about 1.5m long. For a cylinder, volume is given by:

V = π × r² × l

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 1.2tonnes 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.

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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.

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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 = 2 × π × r × (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 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.

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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 is actually 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.)

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We’ll look at the implications that all this has for science and the search for truth in Part II.

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Related posts:

On Spherical Cows and the Search for Truth (Part II)

Faith: reflecting on evidence

Believing and understanding

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2 thoughts on “On Spherical Cows and the Search for Truth (Part I)

  1. Pingback: On Spherical Cows and the Search for Truth (Part II) « Spiritual Meanderings

  2. Pingback: Believing and understanding « Spiritual Meanderings

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