The essence of mathematics cannot be easily discerned. This intellectual pursuit lurks behind a murky haze of complexity. Those that are fortunate enough to have natural ability in this field are able to manipulate algebraic equations as easily as spoken word. However, for the vast majority of the population, mathematical expertise is elusive, receding away at each desperate grasp and attempt at comprehension. What exactly is this strange language of numerical shapes, with its logical rule-sets and quirky laws of commutativity? It seems as though the more intensely this concept is scrutinised, the faster its superfluous layers of complexity are peeled away. But what of these hidden foundations? Are mathematical formulations the key to understanding the nature of reality? Can all this complexity around which we eke out a meagre existence really condense into a single set of equations? If not, what are the implications for and likelihood of a purely mathematical and unified ‘theory of everything? These are the questions I would like to explore in this article.

The history of mathematics dates back to the dawn of civilisation. The earliest known examples of mathematical reasoning are believed to be from some 70,000 years BC. Geometric patterns and shapes on cave-walls shed light onto how our ancestors may have thought about abstract concepts. These primitive examples also include rudimentary attempts at measuring the passage of time through measured, systematic notches and depictions of celestial cycles. Humankind’s abilities progressed fairly steadily from this point, with the next major revolution in mathematics occurring some 3000-4000 years BC.

Neolithic religious sites (such as Stonehenge, UK and Ġgantija, Malta) are thought to have made use of the growing body of mathematical knowledge and an increased awareness and appreciation of standardised observation. In a sense, these structures spawned appreciation of mathematical representation by encouraging measurement standardisation. For example, a static structure allows for patterns in constellations and deviations from the usual to stand out in prominence. Orion’s belt rises over stone X in January, progressing towards stone Y; what position will the heavens be in tomorrow?

Such observational practices allowed mathematics, through the medium of astronomy, to foster and grow. Humanity began to recognise the cyclical rhythm of nature and use this standardised base to extrapolate and predict future events. It was not until 2000BC that mathematics grew into some semblance of the formalised language we use today. Spurred on by the great ancient civilisations of Greece and Egypt, mathematical knowledge advanced at a rapid pace. Formalised branches of maths emerged around this time period, with construction projects inspiring minds to realise the underlying patterns and regularities in nature. Pythagoras’ Theorem is but one prominent result from the inquiries of this time as is Euclid’s work on geometry and number theory. Mathematics grew steadily, although hampered by the ‘Dark Ages’ (Ptolemic model of the universe) and a subsequent waning interest in scientific method.

Arabic scholars picked up this slack, contributing greatly to geometry, astronomy and number theory (the numerical system of base ten we use today is an adoption of Arabic descent). Newton’s Principia was perhaps the first wide-spread instance of formalised applied mathematics (in the form of generalised equations; geometry had previously been employed for centuries in explaining planetary orbits) in the context of explaining and predicting physical events.

However, this brings us no closer to the true properties of mathematics. An examination of the historical developments in this field simply demonstrates that human ability began with rudimentary representations and has since progressed to a standardised, formalised institution. What essentially are these defining features? Building upon ideas proposed by Frayn (2006), our gift for maths arises from prehistoric attempts at grouping and classifying external objects. Humans (and lower forms of life) began with the primitive notion of ‘big’ versus ‘small’, that is, the comparison of groupings (threats, friends or food sources). Mathematics comprises our ability to make analogies, recognise patterns and predict future events; a specialised language with which to conduct the act of mental juggling. Perhaps due to the increasing encephalic volume and neuronal connectivity (spurred on by genetic mutation and social evolution) humankind progressed beyond the simple comparison of size and required a way of mentally manipulating objects in the physical world. Counting a small herd of sheep is easy; there is a finger, toe or stick notch with which to capture the property of small and large. But what happens when the herd becomes unmanageably large, or you wish to compare groups of herds (or even different animals)? Here, the power of maths really comes into a world of its own.

Referring back to the idea of social evolution acting as a catalyst for encephalic development, perhaps emerging social patterns also acted to improve mathematical ability. More specifically, the disparities in power as individuals become more elevated compared to their compatriots would have precipitated a need to keep track of assets and incur taxation. Here we observe the leap from singular comparison of external group sizes (leaning heavily on primal instincts of flight/fight and satiation) to a more abstract, representative use of mathematics. Social elevation brings about wealth and resources. Power over others necessitates some way of keeping track of these possessions (as the size of the wealth outgrows the managerial abilities of one person). Therefore, we see not only a cognitive, but also a social, aspect of mathematical evolution and development.

It is this move away from the direct and superficial towards abstract universality that heralded a new destiny for mathematics. Philosophers and scientists alike wondered (and still wonder today) whether the patterns and descriptions of reality offered by maths are really getting to the crux of the matter. Can mathematics be the one tool with which a unified theory of everything can be erected? Mathematical investigations are primarily concerned with underlying regularities in nature; patterns. However it is the patterns themselves that are the fundamental essence of the universe; mathematics simply describes them and allows for their manipulation. The use of numerals is arbitrary; interchange them with letters or even squiggles in the dirt and the only thing that changes is the rule-set to combine and manipulate them. Just as words convey meaning and grammatical laws are employed with conjunctions to connect (addition?) premises, numerals stand as labels and the symbols between them convey the operation to be performed. When put this way, mathematics is synonymous with language, it is just highly standardised and ‘to the point’.

However this feature is a double-edged sword. The sterile nature of numerals (lacking such properties as metaphor, analogy and other semantic parlour tricks) leaves their interpretation open. A purely mathematical theory is only as good as the interpreter. Human thought processes descend upon formulae picking apart and extracting like a vulture battles haphazardly over a carcass. Thus the question moves from one of validating mathematics as an objective tool to a more fundamental evaluation of human perception and interpretation. Are the patterns we observe in nature really some sort of objective reality, or are they simply figments of our over-active imagination; coincidences or ‘brain puns’ that just happen to align our thoughts with external phenomena?

If previous scientific progress is anything to go by, humanity is definitely onto something. As time progresses, our theories become closer and closer to unearthing the ‘true’ formulation of what underpins reality. Quantum physics may have dashed our hopes of ever knowing with complete certainty what a particle will do when poke and prodded, but at least we have a fairly good idea. Mathematics also seems to be the tool with which this lofty goal will be accomplished. Its ability to allow manipulation of the intangible is immense. The only concern is whether the increasing abstractivity of physical theories is outpacing our ability to interpret and comprehend them. One only has to look at the plethora of alternative quantum interpretations to see evidence for this effect.

Recent developments in mathematics include the mapping of E8. From what can be discerned by a ‘lay-man’, E8 is a multi-dimensional geometric figure, the exact specifications of which eluded mathematicians since the 19^{th} century. It was only through a concerted effort involving hundreds of computers operating in parallel that its secrets were revealed. Even more exciting is the recent exclamation of a potential ‘theory of everything’. The brainchild behind this effort is not what could be called stereotypical; this ‘surfing scientist’ claims to have utilised the new-found knowledge of E8 to unite the four fundamental forces of nature under one banner. Whether his ship turns out to hold any water is something that remains to be seen. The full paper can be obtained here.

This theory is not the easiest to understand; elegant but inherently complex. Intuitively, two very fitting characteristics of a potential theory of everything. The following explanation from Slashdot.org is perhaps the most easily grasped for the non-mathematically inclined.* *

*“The 248-dimensions that he is talking about are not like the time-space dimensions, which particles move through. They describe the state of the particle itself – things like spin, charge, etc. The standard model has 6(?) properties. Some of the combinations of these properties are allowed, some are not. E8 is a very generalized mathematical model that has 248-properties, where only some of the combinations are allowed. What Garrett Lisi showed is that the rules that describe the allowed combinations of the 6 properties of the standard model show up in E8, and furthermore, the symmetries of gravity can be described with it as well.” *Slashdot.org, (2007).

Therefore, E8 is a description of particle properties, not the ‘shape’ of some omnipresent, underlying pervasive force. The geometric characteristics of the shape outline the numbers of particles, their properties and the constraints over these properties (possible states, such as spin, charge etc). In effect, the geometric representation is an illustration of underlying patterns and relationships amongst elementary particles. The biggest strength of this theory is that it offers testable elements, and predictions of as yet undiscovered physical constituents of the universe.

It is surely an exciting time to live, as these developments unfurl. On first glance, mathematics can be an incredibly complex undertaking, in terms of both comprehension and performance. Once the external layers of complexity are peeled away, we are left with the raw fundamental feature; a description of underlying universals. Akin to every human endeavour, the conclusions are open to interpretation, however with practice, and an open mind free from prejudicial tendencies, humanity may eventually crack the mysteries of the physical universe. After all, we are a component of this universe therefore it makes intuitive (if not empirical) sense that our minds should be relatively objective and capable of unearthing a comprehensive ‘theory of everything’.

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24 December, 2007 at 12:22 pm

AnnaWell, I find it extraordinarily interesting.Good luck

to all of you. And I’m sure you’ll do fine. Really.

Just fine.