Quantum Diaries

René Descartes (1596 – 1650) was an outstanding physicist, mathematician and philosopher. In physics, he laid the ground work for Isaac Newton’s (1642 – 1727) laws of motion by pioneering work on the concept of inertia. In mathematics, he developed the foundations of analytic geometry, as illustrated by the term Cartesian[1] coordinates. However, it is in his role as a philosopher that he is best remembered. Rather ironic, as his breakthrough method was a failure.

Descartes’s goal in philosophy was to develop a sound basis for all knowledge based on ideas that were so obvious they could not be doubted. His touch stone was that anything he perceived clearly and distinctly as being true was true. The archetypical example of this was the famous I think therefore I am.  Unfortunately, little else is as obvious as that famous quote and even it can be––and has been––doubted.

Euclidean geometry provides the illusionary ideal to which Descartes and other philosophers have strived. You start with a few self-evident truths and derive a superstructure built on them.  Unfortunately even Euclidean geometry fails that test. The infamous parallel postulate has been questioned since ancient times as being a bit suspicious and even other Euclidean postulates have been questioned; extending a straight line depends on the space being continuous, unbounded and infinite.

So how are we to take Euclid’s postulates and axioms?  Perhaps we should follow the idea of Sir Karl Popper (1902 – 1994) and consider them to be bold hypotheses. This casts a different light on Euclid and his work; perhaps he was the first outstanding scientist.  If we take his basic assumptions as empirical[2] rather than sure and certain knowledge, all we lose is the illusion of certainty. Euclidean geometry then becomes an empirically testable model for the geometry of space time. The theorems, derived from the basic assumption, are prediction that can be checked against observations satisfying Popper’s demarcation criteria for science. Do the angles in a triangle add up to two right angles or not? If not, then one of the assumptions is false, probably the parallel line postulate.

Back to Descartes, he criticized Galileo Galilei (1564 – 1642) for having built without having considered the first causes of nature, he has merely sought reasons for particular effects; and thus he has built without a foundation. In the end, that lack of a foundation turned out to be less of a hindrance than Descartes’ faulty one.  To a large extent, sciences’ lack of a foundation, such as Descartes wished to provide, has not proved a significant obstacle to its advance.

Like Euclid, Sir Isaac Newton had his basic assumptions—the three laws of motion and the law of universal gravity—but he did not believe that they were self-evident; he believed that he had inferred them by the process of scientific induction. Unfortunately, scientific induction was as flawed as a foundation as the self-evident nature of the Euclidean postulates. Connecting the dots between a falling apple and the motion of the moon was an act of creative genius, a bold hypothesis, and not some algorithmic derivation from observation.

It is worth noting that, at the time, Newton’s explanation had a strong competitor in Descartes theory that planetary motion was due to vortices, large circulating bands of particles that keep the planets in place.  Descartes’s theory had the advantage that it lacked the occult action at a distance that is fundamental to Newton’s law of universal gravitation.  In spite of that, today, Descartes vortices are as unknown as is his claim that the pineal gland is the seat of the soul; so much for what he perceived clearly and distinctly as being true.

Galileo’s approach of solving problems one at time and not trying to solve all problems at once has paid big dividends. It has allowed science to advance one step at a time while Descartes’s approach has faded away as failed attempt followed failed attempt. We still do not have a grand theory of everything built on an unshakable foundation and probably never will. Rather we have models of widespread utility. Even if they are built on a shaky foundation, surely that is enough.

Peter Higgs (b. 1929) follows in the tradition of Galileo. He has not, despite his Noble prize, succeeded, where Descartes failed, in producing a foundation for all knowledge; but through creativity, he has proposed a bold hypothesis whose implications have been empirically confirmed.  Descartes would probably claim that he has merely sought reasons for a particular effect: mass. The answer to the ultimate question about life, the universe and everything still remains unanswered, much to Descartes’ chagrin but as scientists we are satisfied to solve one problem at a time then move on to the next one.

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Since model building is the essence of science, this quote has a bit of a bite to it. It is from George E. P. Box (1919 – 2013), who was not only an eminent statistician but also an eminently quotable one.  Another quote from him: One important idea is that science is a means whereby learning is achieved, not by mere theoretical speculation on the one hand, nor by the undirected accumulation of practical facts on the other, but rather by a motivated iteration between theory and practice.  Thus he saw science as an iteration between observation and theory. And what is theory but the building of erroneous, or at least approximate, models?

To amplify that last comment: The main point of my philosophical musings is that science is the building of models for how the universe works; models constrained by observation and tested by their ability to make predictions for new observations, but models nonetheless. In this context, the above quote has significant implications for science. Models, even those of science, are by their very nature simplifications and as such are not one hundred per cent accurate. Consider the case of a map. Creating a 1:1 map is not only impractical[2] but even if you had one it would be one hundred per cent useless; just try folding a 1:1 scale map of Vancouver. A model with all the complexity of the original does not help us understand the original.  Indeed the whole purpose of a model is to eliminate details that are not essential to the problem at hand.

By their very nature, numerical models are always approximate and this is probably what Box had in mind with his statement. One neglects small effects like the gravitational influence of a mosquito. Even as one begins computing, one makes numerical approximations, replacing integrals with sums or vise versa, derivatives with finite differences, etc. However, one wants to control errors and keep them to a minimum. Statistical analysis techniques, such as Box developed, help estimate and control errors.

To a large extent it is self-evident that models are approximate; so what? Again to quote George Box: Since all models are wrong the scientist cannot obtain a “correct” one by excessive elaboration. On the contrary following William of Occam he should seek an economical description of natural phenomena. Just as the ability to devise simple but evocative models is the signature of the great scientist so overelaboration and overparameterization is often the mark of mediocrity. What would he have thought of a model with twenty plus parameters, like the standard model of particle physics? His point is a valid one. All measurements have experimental errors. If your fit is perfect you are almost certainly fitting noise. Hence, adding more parameters to get a perfect fit is a fool’s errand. But even without experimental error, a large number of parameters frequently means something important has been missed. Has something been missed in the standard model of particle physics with its many parameters or is the universe really that complicated?

There is an even more basic reason all models are wrong. This goes back at least as far as Immanuel Kant (1724 – 1804). He made the distinction between observation of an object and the object in itself. One never has direct experience of things, the so-called noumenal world; what one experiences is the phenomenal world as conveyed to us by our senses. What we see is not even what has been recorded by the eye.  The mind massages the raw observation into something it can understand; a useful but not necessarily accurate model of the world. Science then continues this process in a systematic manner to construct models to describe observations but not necessarily the underlying reality.

Despite being by definition at least partially wrong, models are frequently useful. The scale model map is useful to tourists trying to find their way around Vancouver or to a general plotting strategy for his next battle. But, if the maps are too far wrong the tourist will get lost and fall into False Creek and the general will go down in history as a failure. Similarly, the models for weather predictions are useful although they are certainly not a hundred per cent accurate. However, they do indicate when it safe to plan a picnic or cut the hay; provided they are right more than by chance and the standard model of particle physics, despite having many parameters and not including gravity, is a useful description of a wide range of observations. But to return to the main point, all models, even useful ones, are wrong because they are approximations and not even approximations to reality but to our observations of that reality. Where does that leave us? Well, let us save the last word for George Box: Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful.