Mathematics is a set of precise languages comprising such topics as arithmetic (numbers) geometry (shapes) Algebra (symbols) and calculus (concepts).
Mathematics is an edifice, built from the ground up, assembled, definition by definition, from scratch. Those of you who studied geometry in school will remember its never-ending series of theorem proofs. Geometry, we were told, is one of the oldest branches of mathematics, taught by Pythagoras in the sixth century BCE, and used by the Pharaohs’ surveyors to restore field boundaries each time the Nile flooded. Geometry starts with the very simplest statement, a definition of a point, a line or a circle, then looks for the extensions and connections that are logically implicit within these definitions. The whole process is repeated each time a new definition (that of a parallelogram, for instance) is introduced.
Mathematicians have been adding new definitions to geometry for centuries. At the same time, they have been busy constructing other branches of mathematics: algebra, calculus, trigonometry, topology, set theory, and so on. Each function, each definition, and each statement in every branch has to be very carefully assessed for logical consistency when introduced, then again every time it is used to link to something newly added, and once more whenever it is put to theoretical or practical use. This is done, because each newly added feature introduces more relationships, and it is these relationships that determine if the whole assembly makes sense. Mathematics, then, is held together just as precisely as the universe itself seems to be held together.
Through these means, mathematics is created to be internally sound and rational, self sufficient to the extent of possessing its own reality, dependent upon the real world only in as much as it is built from a language defined in the real world, and connected to the real world in meaning only if we choose to make such a connection. By itself, mathematics is abstract, pure and complete; it does not need to be given any link to the universe (other than that necessarily implicit in its nomenclature). In fact, it is not uncommon for mathematicians to explore the properties of creations such as multidimensional space or imaginary numbers—fancies which no one has experienced first hand.
However, we can, and very often do, link our mathematical understandings to the real world. We do this, for instance, when demonstrating to children that three fingers plus two fingers equals five fingers. Remarkably, it is becoming more and more certain that the mentally constructed world of abstract mathematics contains the ability to describe, explain and predict the very concrete behaviour of the real universe we inhabit.
Pythagoras showed this over two and a half thousand years ago, when he described mathematically a property of two dimensional space (the relationship between squares formed on the sides of a right-angled triangle). Newton demonstrated the same connection between mathematics and reality over four hundred years ago, when he showed mathematically that the force holding planets in orbit is related to the involved masses and distances between them.
Einstein confirmed this connection when he discovered and proved, again mathematically, that the properties of the four (space-plus-time) dimensions prohibit matter from moving faster than the speed of light. Mathematicians continually push the boundaries and today routinely use complex number theories to define the properties of multidimensional space, a reality which some think may actually exist (perhaps within black holes, or defining fundamental "superstring" properties, or building a universe external to our own).
Because mathematics has been rigorously and logically constructed to be an abstract entity, mathematicians think that its various domains will be considered to be as true in a million years time as they would have been a million years ago, long before they could have been understood by any sentient being living upon this planet. (Moreover, because scientists can use mathematics to predict and explain events occurring billions of light-years distant.
Mathematicians also consider that these mathematical statements hold true in other galaxies, and are therefore discoverable by life forms living upon planets in those regions.) To pure mathematicians, it is often a subsequent (and, possibly, less important) finding that the mathematical properties they uncover have meaning in the real world. They prefer to solve problems within the bounded beauty of a fully discoverable, self-consistent, abstract world. Be this as it may, the many connections between abstract mathematics and the practical realities of the real world have allowed us to solve countless complex problems, and have led to a multitude of discoveries in arenas as diverse as economics, sociology, epidemiology, space flight, nuclear physics, genetics, cosmology and medicine, to name just a few.
That logically generated mathematics describes and defines the universe so accurately reinforces a fact that has already been stated: the universe must be causal and rational, for, if it were not, the intrinsic fit between mathematics and the universe’s functioning would not exist.
- And perhaps by the same formula—something like: the universe’s underlying causality enforces its rational behaviour, while mathematics underlying rationality enforces its causal interrelationships.
- Of course, mathematics must describe the real world because each of its many terms has been precisely defined using language, a language that has itself been constructed from our knowledge of the real world (as Thinking And The Universe and the previous paragraph pointed out).
- See Gödel’s Theorem, General Systems Theory, and The Conservation Laws.
- A light-year is the distance that light travels through space in one year, about 9.5 x 1012 kilometres, or 5.9 x 1012 miles.