A Mathematician’s Manifesto
Mathematics is a language of science. All languages are necessarily symbolic conventions that refer to an objective reality. What differentiates mathematics from all other conventional languages is the belief that the essence of its symbolic narrative is timeless and incontrovertible. While the artistic essence of Homer’s Odyssey or Shakespeare’s Twelfth Night lies in their shapeless resonance with the zeitgeist of each and every generation, a mathematical proof is often understood to represent an incontrovertible truth that every subsequent generation is obliged to uphold.
If I were to argue that Telemachus, the prince of Ithaca in Odyssey, is an artistic prototype of Jesus in Christian faith, my interpretation, though controversial it may be in certain religious circles, would find a sympathetic audience. If I were to argue that Shakespeare’s Malvolio is a tragic hero blindly searching for the meaning of life, a caricature of the Modern Man in a senseless world, the argument may stretch a conventional understanding of Shakespeare’s comic play, but it is nevertheless a valid perspective from which the play can be enjoyed.
Take, for an example, Euclid’s proof that there are infinitely many prime numbers. If I were to suggest that the proof is a logical circularity, I would likely face an irate professor demanding a proof of my egregious claim. I’d be an object of mockery and condescension for not understanding an idea that even an elementary-school student knows to be true. I am likely to face a fate not so dissimilar to Malvolio’s, if I were to persist in my dogged insistence on the absurdity of the proof.
Now, suppose that there was a logician who shared the same opinion and that his name was Ludwig Wittgenstein.
The narrative essence of each account is necessarily truthful: a narrative, by construction, is a representation in shared symbols of one’s authentic experience of the unknown. In mathematics, narratives (and all truthful objects in general) are time signatures that reference felt experiences of an objective reality as witnessed by a private observer in space. Every time measure is a function that translates real events to their subjective significance in the metaphorical eyes of the observer. The same reality can give rise to a multitude of time measures, if the spatial structure of the reality permits a diversity of perspectives from which distinctly unique observations can be made.
When we speak of an objective reality, we imagine a maximal collection of time measures permitted by the topology of space. It is a non-trivial problem when the space of reality contains elements of dubious provenance, such as axioms and hypotheses, whose coordinates cannot be resolved definitively by any of the existing measures. The presence of such singularities creates asymmetry in inferential logic; statements involving an indeterminate proposition q, such as p → q and q → r, cannot be conjoined to yield their synthesis, p → r. In plain English, it means that a logical statement cannot exist simultaneously as a hypothesis (p → q) and an axiom (q → r).
Together with reflexivity and symmetry, transitivity constitutes one of the three pillars essential to the conception of objective identity in mathematical logic. Reflexivity and symmetry demarcate, respectively, boundaries of objects and their binary associations. As such, to the extent that we are concerned with a universe that contains objects and their movements, reflexivity and symmetry of logical relations are essential and indispensable. Transitivity, on the other hand, asserts something more, something extraneous to the structure of the observable universe itself. It hypothesizes a universal perspective that implicitly commutes with every known perspective, thereby validating every subjective experience as an objective reality.
Although approximation of an objective reality is the goal of any theoretical endeavor, adopting objectivity as a premise of a theory feels like putting the cart before the horses. An objective reality is, and will always remain to be, a limiting concept without a verifiable existence. For an act of verification cannot be separated from the uniquely subjective context that authenticates a perception as an admissible form of evidence.
In a world where we let go of relational transitivity, a successful concatenation of two inferential statements is not always a guaranteed outcome, even if we grant that both accounts are, by nature, truthful. Time signatures may extend indefinitely without ever resulting in an event of coincidence.
An event has no intrinsic quality that explains its origin; it merely is. The meaning of an event is that it is given as a condition for perceptual comprehension. On the condition that an event is created by a coincidence of time measures, mathematics seeks to build the most parsimonious structure of language that is capable of preserving the shared experience of subjectivity.
The grammatical syntax of such a language emulates the observable symmetry that exists in Nature. For example, the notion of a homogenous metric on a space is expressed by the syntax of an orthogonal group that consists of distance-preserving transformations. The goal of mathematics is a complete alignment between spatial symmetry and the syntax of relativistic association.
