The Idea of Models

1Usually, there is an implicit inference that speculation, if it is to be taken seriously, must be based on some form of credible mathematical model. In this context, mathematical models are seen as essential  for the formulation of precise calculations and, more importantly, for making predictions that can be verified, albeit at some possibly unknown point in the future. However, there may be enough anecdotal evidence to suggest that a certain amount of care is required to ensure that a model, of any description, is not confused with reality itself, as the impossible perspective of the diagram suggests on closer inspection. For it would appear that different models, based on the same underlying data, might lead to radically different interpretations, although most models are weighted in preference towards some ‘accepted’ worldview.

So what is a model?

In practice, there are many forms of models, although we might narrow the scope to just 3 types of direct interest to this discussion:

  • Material Models:
    At its simplest, this type of model may be a simplified, scaled-down representation of an ‘object’ or ‘process’. As such, these models may be a static representation of an object or a dynamic representation of a process, where the mathematical equations are unknown or too hard to solve. As a generalisation, the limitation of such models is normally fairly obvious.

  • Mathematical Models:
    These models are realised in terms of equations, which can utilise a large set of abstracted variables typically based on a much smaller subset of physical observables. Even so, such models still tend to be a simplification of a complex system in order to facilitate some form of prediction, which may be limited in scope. However, based on the complexity of the mathematics, the limitation of the model may not be so obvious and therefore not always fully questioned.

  • Theoretical Models :
    These model might be said to underpin some overall ‘theory’ that is thought to align, and therefore describe, some observed physical phenomena. As such, theoretical models are closely related to the development of physics, e.g.  Newtonian mechanics, Maxwell’s electromagnetic, relativity and quantum physics. In this context, it might be argued that the goal of these models extends beyond the ability to just make predictions, as they also seek to explain the nature of physical reality. Within this definition, a theoretical model seeks to provide a framework of ideas and concepts that fit observations and provide what we might define as a description of nature’s laws.

In physics, there is no reason why all 3 types of models cannot be used in the acquisition and visualisation of scientific knowledge, which an ideal methodology would then subject to extensive empirical verification. Of course, as has been previously discussed, many aspects of theoretical physics now extends beyond the ability of science to directly verify and therefore the role of the model can assumed greater importance. In such cases, we have to recognise that the conclusions of a model may rest only on deductive reasoning, which in turn may reflect a preference towards some established set of ‘beliefs’ as possibly being suggested in the following quote by Willard Van Orman Quine:

"Any statement can be held true come what may, if we make drastic enough adjustments elsewhere in the system of our beliefs. The totality of our so-called knowledge or beliefs, from the most casual matters of geography and history to the profoundest laws of atomic physics or even of pure mathematics and logic, is a man-made fabric which impinges on experience only along the edges."

We might then extrapolate the inference in this quote to suggest that it may be possible to construct a model with any given feature, if we make other assumptions elsewhere in the system. Such a suggestion appears to be forwarded in the next quote by Frank Tipler:

"It is universally thought that it is impossible to construct a falsifiable theory which is consistent with the thousands of observations indicating an age of billions of years, but which holds that the Universe is only a few thousand years old. I consider such a view to be a slur on the ingenuity of theoretical physicists: we can construct a falsifiable theory with any characteristics you care to name."

2However, it may be argued that Tipler’s position only becomes increasingly possible as the ability of verification is proportionally reduced. By way of example, we might quickly review the development of the particle model over the last couple of hundred years:

  • In 1803, John Dalton’s  model assumed that an atom consisted of some uniform solid material.

  • By 1904, J.J. Thompson introduced his ‘plum-pudding’ model, where negative charged electrons were placed within a uniform, but positively charged mass like so many plums or raisons in a pudding.

  • By 1911, Ernest Rutherford model presumed that an atom was mostly empty space.

  • By 1913, Niels Bohr was to add the restriction that electrons can occupy only certain, discrete orbits.

  • Later, by 1926, Edwin Schrödinger would challenge the idea of definite orbits with the introduction of the wave function.

In part, we might describe Dalton’s and Thompsons models as essentially material models, while Rutherford’s and Bohr’s models were more mathematical in form, anchored in the limited verification of particle scattering and spectroscopy experiments. However, by 1926, Schrodinger's wave model was little more than theoretical speculation, which existed beyond any realistic form of direct verification, but which would never the less open the door for even more speculative theoretical models.

So what reality did Schrodinger’s quantum wave-function represent?

Initially, Schrödinger pursued the idea that quantum waves might correspond to physical vibrations in an electromagnetic field. However, within this model, the electron was no longer a particle, but rather a wave packet that could only be localised in space as a superposition of waves. Later, Max Born would argued that Schrodinger’s wave function only represented the probability of finding a ‘particle’ in a region of space, after the wave function had collapsed. As such, it might be suggested that Born’s model ceased to be a theoretical model of physical reality, but rather became a theoretical model of a mathematical reality that simply predicted the probability of some given outcome. While this trend would continue in the post-war era, the particle model would appear to become trapped in an apparently endless process of verifying a never-ending list of new particles. However, while some aspects of any mathematical model might be questioned, they appear to have been very successful in the field of theoretical physics.


But against what measure are mathematical models seen as successful?

It would appear that there may be two schools of thought on this issue. First, there are those like Eugene Wigner and Steven Wienberg, both Nobel-prize winners in physics, who possibly allude to something quite profound in the correlation of nature’s laws to mathematics. For example Wigner gave a famous lecture, in 1960, entitled ‘The unreasonable effectiveness of mathematics in the natural sciences’, while Weinberg later stated:

"Physicists generally find the ability of mathematicians to anticipate the mathematics needed in the theories of physics quite uncanny. It is as if Neil Armstrong in 1969 when he first set foot on the surface of the moon had found in the lunar dust the footprints of Jules Verne"

In contrast, philosopher Mark Steiner has argued that the apparent ‘mystery’, which extends beyond coincidence has to be considered in a wider context. First, there is no real assessment of the failure of mathematical models to accurately predict or describe physical reality based  on subsequent verifiable data. As a historical example, we might consider Ptolemy’s epicycle system as a mathematical model that was later proved incorrect. Later, in 1596, Kepler proposed another mathematical model of the solar system in which the spacing between the planets closely approximated the spacing of the 5 Platonic solids, when fitted inside each other in a specific order. At this time, there was no physical basis for this premise, which would eventually be abandoned with the discovery of a 6th planet, which had no corresponding Platonic solid. However, mathematician Smlomo Sternberg possibly puts Kepler’s model into better context of the modern world:

"Kepler's principal goal was to explain the relationship between the existence of five planets, and their motions, and the five regular solids. It is customary to sneer at Kepler for this. It is instructive to compare this with the current attempts to ‘explain’ the zoology of elementary particles in terms of irreducible representations of Lie groups"

However, should we really be surprised that nature’s laws appear to conform to mathematical relationship; either we assume it is ‘chaos all the way down or our perception of the natural order of things unsurprisingly conforms to some form of mathematical description. In this context, we might reflect on Dirac’s equation, developed in 1930, involving the factorial solution of quadratic polynomial using 4-dimensional matrices that yielded several unexpected solutions. While one of these solutions was the electron with spin, another implied the existence of a particle, identical to the electron, but with a positive charge, which would be called its anti-particle, the positron. As such, this mathematical model based on Clifford algebra, developed in the 19th century, for purely mathematical reasons, would appear to have helped in the subsequent discovery of one of nature’s building blocks.

But is Dirac’s equation a description of physical reality or just a mathematical model?

The concern being raised is that we might overstate the reality of the mathematical model. As previously outlined, mathematician Roger Penrose has suggested that physical reality might only be a projection of some deeper reality founded on mathematics. This idea is also supported by other scientists like the chemist Peter Atkins, who appears to go further in the suggestion that reality must ultimately be a mathematical construct. Of course, others debate this position on the grounds that it is all too easy to get carried away with the elegance and predictive power of a mathematical model, such that there is a natural desire to assume this elegance not only represents physical reality, but is the only reality. As such,  what started out as an abstract model has become elevated to the position of a mathematical reality.

How much of this debate belongs in philosophy not science?

In the current context, it can be argued that some philosophical reflection is not a bad thing, if it provides some genuine perspective of the strength and weakness of the modern scientific methodology. It is also sometimes necessarily to reflect on the role of human consciousness, which while appearing to be a physical system also appears to refute the deterministic predictions of ‘cause and effect’ that is so often implied by so many mathematical predictions. In part, we might see some of this tension in a quote taken from a book by Nobel prize-winning biologist Francis Crick:

“The Astonishing Hypothesis is that ‘You’, your joys and your sorrows, your memories and your ambitions, your sense of personal identity and free will, are in fact no more than the behaviour of a vast assembly of nerve cells…”

As such, we might perceive a circular dichotomy of ‘you’ trying to understand the nature of physical reality, while ‘you’ are only an ‘illusion’ caused by seemingly 'metaphysical' processes in the brain, but which in-turn must depend on some form of reality described in terms of the atoms and their interactions within the brain.

So do we simply end up chasing our tails in such debates?

Possibly, but it has the advantage that it makes you question the certainty of any entrenched worldview, be it theological, philosophical or scientific in orientation, and this is not necessarily ‘a bad thing’.

Doubt is not a pleasant condition, but certainty is absurd. Voltaire

So, in many ways, the main purpose of this discussion is simply to highlight some possible limits of any model as a simplified representation of reality, which should not be mistaken for ‘the real thing’, at least, not at this stage. While, Stephen Hawking has recently been quoted as saying that ‘Philosophy is dead’, although possibly being little more than a PR sound-bite for his new book, for it would seem that he is not adverse to a certain amount of philosophical musing himself, as per his book ‘A Brief History of Time’:

"Even if there is just one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe? The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing? Is the unified theory so compelling that it brings about its own existence? Or does it need a creator, and, if so, does he have any other effect on the universe? and who created him?"

In this sense, Hawking brings us back to the distinction between some ultimate physical reality and an abstract model of this reality, although in all practical terms, both will only ever be ‘perceived’ within the limits of the human ‘illusion’.