Is Mathematics Real ?

“It is, as Schrödinger has remarked, a miracle that in spite of the baffling complexity of the world, certain regularities in the events could be discovered. (…) The laws of nature are concerned with such regularities.”
– Eugene Wigner

Physicist Eugene Wigner published The Unreasonable Effectiveness of Mathematics in the Natural Sciences in 1960. The mystery of this “unreasonable effectiveness” remains a profound philosophical problem and irregular kernel at the heart of contemporary physics. Our entire technological civilisation is in one way or another constructed on a foundational grid of inductive mathematical reasoning and its successful, progressive, predictive and iteratively deductive application to diverse material, spatial and temporal regularities.

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…as easy as 1, 2, 3 ?

If mathematics describes reality with uncanny effectiveness, what is it that (as Stephen Hawking opined) “breathes fire into the equations” ? An extended quote from Hawking’s A Brief History of Time (1988):

“Even if there is only 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?”

There does at least seem to exist some degree of discontinuity, dissonance, difference and distance between the lofty abstractions of mathematical models and the muddy, imperfect and uncertain reality that we inhabit. Mathematics as a product of the human brain, or rather – of the progressively and iteratively-refined cultural artefact cultivated by very many collaborating human brains over historical time, is mysterious. While as a system of thought it expresses unique and unreasonable predictive and explanatory power in describing material reality, questions as to the reality (or plausibility) of intrinsic and pre-existing mathematical entities in the world remain plagued by the very same metaphysical horizons as do all other Universals, Ideals or Forms.

In Human, all too Human Friedrich Nietzsche eviscerated the value of such metaphysical knowledge of the world as being about as useful to us for living our lives as would a knowledge of the chemical composition of water be to a boatman facing an immanent storm. In the area of mathematics, physics and diverse associated issues including (but by no means limited to) the emergence of structure, complexity and intelligence in this Universe, the metaphysical questions regarding mathematics and any notional “ultimate” explanation of reality do resurface and in non-trivial ways.

Mathematics and physics are products of brains which are themselves emergent phenomena of the same world that those theories seek to successfully explain. A rationally materialist (and intuitively unproblematic) perspective in which we assume an externally real world beyond this human experience of it suggests that mathematics and physics (or science and the production of formalised knowledge-systems more generally) are essentially sub-sets of material reality. Brains, cultures, technologies, information and communications systems are entities and iterative structures which emerge from the Universe in ways which are fundamentally secondary and after-the-fact of the existence of that Universe.

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Relax. Stare into the vortex.  Relax.  Mathematics is enigmatic.  Relax.  Enter your credit card or bank details after the tone…

Anything other than a radically (although not necessarily entirely) implausible idealism that posits the primacy of consciousness over material reality would appear to have to bias us towards asserting the pre-existence of a material world beyond our experience of it. If you have ever dipped your epistemological toes into the mysteries of participatory observation in quantum theory, you already know that there is something strange going on in regards to measurement. The mysteries of measurement and (subsequently) of the description and intuitive comprehensibility of sub-microscopic reality are fascinating but do not necessarily exclude an a priori material reality beyond our experience of it. The implication seems to be not that reality is dependent upon the observer in any fundamental way so much as that the role of observers and observation-like events are poorly understood; consequently, that quantum reality itself may be radically other than what it appears to be to macroscopic observers. The truth (and proof) of this particular matter is in any case bound to be irredeemably mischievous and astounding, should it ever be determined.

A perennial problem is that of whether or not mathematics is pre-existing in the world or if it is purely a construction of the human mind. The unreasonable effectiveness of mathematical descriptions and predictions clearly indicates a deep correlation and relationship between material reality and the intellectual abstractions of mathematics and physics. There does appear a strange series of potential questions stemming from the notion that if complex equations and mathematical systems are required to describe physical systems, what is the underlying causal neccessity of this ? A description of spacetime curvature in General Relativity can require the manipulation of complex partial differential equations and conformal diffeomorphic mappings which we might assume that spacetime itself is not required to solve to know how to continue to successfully exist within an integrated and dynamic material reality. In some way, the mathematical equations which successfully describe the entities and patterned relationships of reality appear to be decoupled from that reality: the successful description or complex model and map of a system is fundamentally _not_ identical to that system. Conversely, is a theoretical (explanatory) system that is actually identical to that which it describes still a theoretical system ?

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How would you explain a situation in which you find yourself building the ship that you are sailing on from the ocean upon which you are sailing ?

As an intuition mathematics, as a sub-set of material reality generated by materially-extended brains existing in that same reality, is in some fundamental way (and paradoxically) incompatible or incapable of fully describing that reality. This is to say: while mathematics in profoundly powerful ways describes and successfully predicts innumerable states-of-affairs in material reality, as a sub-set of logical relationships within that reality it may always be impossible to successfully and completely describe that reality. Can a sub-set of a larger set fully and accurately describe that larger set without becoming that larger set ?  Does calling upon the strange recursive loops and tangled logical hierarchies of logical incompleteness assist ?  Algorithmic Information Theory would seem to suggest that for a limited case consisting of strings of numbers and the programs required to generate those strings this may be at least superficially possible. This again begs the question of whether or not sufficiently complex numerical descriptions, matrices, vector-groups, whatever can fully and accurately describe the reality they seek to illuminate.

It is perhaps more a matter of psychological (rather than formal logical) necessity that we may even aspire to assert even the possibility of final closure on this issue.  An interesting post elsewhere sought to claim that mathematics is not a preexisting or necessary entity, but logic is.  This is, however, surely a misdirection and not a resolution – asserting the preeminence of logic over mathematics is merely to displace the chain of causal attribution to another (abstract) location.  Where does logic come from ?  Why should anything necessarily be true or false within logic ?  If we, as sentient observers, are a part of a Universe in which logical truths are necessary components of the mathematics upon which the material reality (from which our conscious minds emerge) is built in regards to physics, and we have derived those logical truths from within the fabric and patterned process of that Universe – how can we possess any final certainty or ontological and epistemological anchor from which to be certain about any of these things ?

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Hers was a rare and terrible beauty: a recursively-iterated smile of infinite complexity, but I loved her.

“No matter how far mathematics progresses and no matter how many problems are solved, there will always be, thanks to Gödel, fresh questions to ask and fresh ideas to discover. It is my hope that we may be able to prove the world of physics as inexhaustible as the world of mathematics… If it should turn out that the whole of physical reality can be described by a finite set of equations, I would feel disappointed.”

– F. J. Dyson. Infinite in all Directions. London: Penguin Books, 1990, p. 53.

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