Between Crystal Waters and Sticky Mud: The Path of Mathematical Language

‘A profound understanding of the logic of the in-between is critical to our futures’. A Mathematical Philosopher debunks the notion of ‘perfection’

LILA: Like all fields of human experience, language – the primary mode of formally structured knowledge transmission – itself is an embedment of complex mathematical properties. Will these properties expand or narrow the scope of language in the future, as knowledge ceaselessly undergoes digital transformation?

Fernando Zalamea: An appalling feature of 20th-century Western philosophy was the attempt to reduce knowledge to forms contained within the parameters of language. Analytical philosophy, in particular, tried to reduce knowledge to the languages of logic. The fact that mathematics can be rewritten in the language of Set Theory and first-order classical logic was totally misunderstood by analytical philosophers, who were trapped in the erroneous conviction that mathematical language could only manifest as the rigid axioms of rules, theorems, equations and proofs. Nothing is farther from the truth. Mathematics is the dynamic realm of imagination, visualisation, creativity, revolving around fundamental properties of space, number, and transformability. Such dynamism may be expressed in logical terms, but does not emerge from them. It is absurd to equate mathematics with fixed utterances – just as it would be nonsensical to claim that Monet’s Nymphéas, his famous water-lily paintings, are no more than a series of brushstrokes!

Thus, analytical philosophy narrowed down the scope of semiotics, in a completely orthogonal way to mathematical practice, that is looking only to syntactical claims, not to the pragmatic transformation of structures. In fact, real mathematics (topology, abstract algebra, number theory, differential geometry, functional analysis, etc.), instead of narrowing down the scope of semiotics, expands the spectrum of knowledge. Profiting from many paradigm shifts of our epoch (tech-enabled visualisation, trans-modernity, intra-localisation, etc.), mathematics will be better understood in the future, and will again help to illuminate many threads of knowledge which were obscured by reductionist perspectives. Open semiotics (akin to Charles Sanders Peirce’s semiotic theory) will expand reduced linguistics, and will merge naturally with burgeoning mathematical creativity.

LILA: What kind of paradigm shift is required to make the transition to an extensional intelligence less susceptible?

FZ: There are several paradigm shifts which may help to expand our Mind and tune it up in a better way to our evolving World. First, we need a true acceptance of diversity, in the sense of both enhancing the richness of the local, but also looking for many interconnections between the diversity of local communities. Here, the use of sheaves, as theorised by Leray (1942), Cartan (1948)1, may be very useful: they provide the most precise mathematical construction to understand the back-and-forth between the local and the global. Consequently, a shift of logics would be very important in our near future: we have to leave behind a classical, binary logic and open up to the logic of sheaves (Caicedo 1995), with its non-binary, intermediate, elastic, visual properties.

Second, we have to provide a balance between (horizontal) superficiality and (vertical) depth. If the internet has phenomenally expanded the surface of knowledge, we have also to be aware of its very thin crust. The French writer Paul Valéry had noticed in his Bilan de l’intelligence (1935) how his World was already governed by velocity and the abuse of information – imagine how this abuse has exponentially grown in our times. The leaders of the next generations will have to work both horizontally and vertically, fighting superficiality. Again, a paradigm shift to the logic of sheaves would be here interesting, since, by their very definition, sheaves work simultaneously in the foldings and unfolding of mathematical objects.

Third, we have to distrust any kind of supposed ‘pureness’ of desire, i.e., foundationalist reductionism. Humanity, knowledge, mathematics, all are essentially impure, and their very life occurs, as Bakhtin (1924) claimed, along multiple frontiers and borders. A profound understanding of logics of the ‘in-between’ will be mandatory if we are to appropriately develop our future. The paradigm shift that I am advocating towards a logic of sheaves would also be oriented in that direction. But there are also many other important desiderata for our times that we have to cherish, in order to make a softer transition for future generations: basic education for all, ecological and historical conscience, restraint in the acquisition of unnecessary luxuries, common work practices and goals beyond individualities, solidarity, social sensibility, strong ethics, aesthetic freedom, etc.

LILA: Can a mathematical perspective formulate and affirm new ontologies that would help us to survive the ongoing restlessness of our contemporary life?

FZ: The restlessness of today’s world (here, recall Valéry’s prescient 1935 admonition) is certainly something that needs to be survived. Dynamics and speed have to be understood with a range of authentic perspectives, in order to preserve the required calm and duration that the human body needs for its natural fulfillment. Mathematics here offers good guidance. In fact, from Galois (1830)2 onwards a major effort in mathematics has been to introduce variations (transformations of structures) as well as to find adequate invariants behind movement (fixed points of the transformations). This is also related to an understanding of mathematics through relational webs, beyond the primary dialectic of objects and subjects. The back-and-forth between variations and invariants, the study of relative (but not relativistic) understanding, is essential for mathematics.

This has been particularly well captured through the means of Category Theory (MacLane & Eilenberg 1945, Grothendieck 1955, Lawvere 1963), where concepts are presented as in transit, and the theory is able to find adequate universals beyond the differences of concrete mathematical regions (sets, logics, algebras, topologies, manifolds, etc.) In this way, a transitory ontology emerges (Badiou 1998), where one can talk about “relative universals”, that is, invariants which do not live in an Absolute (well torn-away in the 20th century, after Einstein’s relative physics and Grothendieck’s relative mathematics), but which can be correlated in non-arbitrary fashions in our World. Naturalness and non-arbitrariness are essential for our future. All deaths preconised by postmodernism are in fact just propagandistic (All the supposed ‘deaths’ of traditional systems, identities, beliefs, etc., scrutinised and underscored and in fact celebrated by postmodernism, are a form of propaganda deployed towards self-validation). The truth is that we live in a transmodernist era (Rodríguez Magda 1989), where we have to glue together difference and integration. In a sense, I do consider Category Theory the greatest conceptual differential and integral calculus of contemporary knowledge, and Sheaf Theory the perfect tool to study the transits and amalgamations exhibited therein. The extrapolation of Category Theory and Sheaf Theory to our cultural surroundings should help to provide an adequate ontology to survive contemporary restlessness.  

LILA: Does the principle of perfection have to be defied in order to achieve multiple transitions, via disciplined and conscious routes, to a peaceful resistance of imposed hegemonies?

FZ: Yes! We have to be very suspicious of perfection. The essence of humanity lies in its imperfection. Mathematics emerges through a complex web of obscurities, mistakes and vague ideas that, at the end, through the axiomatic method, acquire an aura of perfection – but we know that the dialectic between shadow and light, negativity and positivity, is essential for mathematical creativity. In fact, many of the most interesting features in mathematics occur often on the darker side (Galois’ impossibility theorems, Poincaré’s conjectural mistakes, Gödel’s incompleteness, Shelah’s main gap, etc.) One of the fundamental programs in our contemporary World must be then to firmly resist all irrational hegemonies, such as the various threatening claims to ‘purity’ (racial superiority, nationalism, analytical irrefutability, etc.) that are growing louder and louder in order to deafen all other discourses and modes of engagement. Multiplicity, hybridity, mixtures, imperfection, etc., are essential for a natural development of the body and the mind.

Resistance to global uniformity, resistance to rules of perfection, resistance to the centre, resistance to discourses of supposed ideological clarity, is extremely important. But this does not mean that anything and everything can and should be expressed and legitimated, i.e., the kind of extreme inclusion advocated and imposed by postmodernism. A balance between reason and sensibility is still mandatory today. Somewhere between the conceptual poles of perfection/uniformity and imperfection/diversity, somewhere between crystal waters and sticky mud, lies the very raison d’être of human knowledge. In this mediated realm, a disciplined and conscious resistance to the extremes should be a fundamental existential practice. I often use the image of the pendulum as a guiding ideal: beyond analysis and synthesis, beyond light and shadow, beyond perfection and decay; frequently, the middle paths are the ones that offer the best possibilities for the harmonious development of human intelligence.

LILA: How are mathematical probabilities going to influence the field of artificial intelligence in the future? What should be the mode of communication between human and AI, if we are to effectively enhance as well as control the apparently infinite possibilities of AI?

FZ: I sense that probabilities are going to forcefully influence not only artificial intelligence but also many other layers of knowledge. The US-based Indian mathematician Srinivasa Varadhan, who with Monroe Donsker won the Abel Prize in 1975 for the theory of large deviations, shows the way for unexpected applications in the near future. A combination of developments in big data, probabilities at large, quantum information, dynamic logics, and mathematical biology will change completely the shape of our societies in the 21st century. I recommend that interested readers seek out the Oxford-based independent research journal Collapse (edited by Robin Mackay and Reza Negarestani), where brilliant young alternative scholars are exploring all theories and forms of hybridisation in our World, opening new directions of philosophical thought.

Artificial Intelligence, as is being pursued today, will make sudden and fantastic turns when the mathematics of organic life will be coupled with the quick handling of gigantic sets of data and large deviations. I imagine that AI will be enhanced all the way, and will not be restrained at all. On the contrary, eventually the human and non-human will be much closer than we expect. And this outcome will be appropriate for our species: humanity, a miraculous drop in the still-expanding cosmos, will merge with other forms of life and understanding, and this radical transformation will completely overcome our limited present state of mind. 

Knowledge is power, and our intention is to bring the power to you. We have initiated a thought movement that aims to strengthen democracy by bringing to you direct voices of important trailblazers and pathmakers, and reclaim deep and patient reflection as an important seed for relevant and sustainable action! Help us take this movement forward. Support Inter-Actions today for as little as Rs. 100.
Donation to LILA is eligible for tax exemption u/s 80 G (5) (VI) of the Income Tax Act 1961 vide order no. NQ CIT (E) 6139 DEL-LE25902-16032015 dated 16/03/2015