Mathematics Begins Not with Proof, but with Vision
The stereotypical image of mathematics revolves around proof. But proof belongs to a later layer. What comes first is not a deduction, but a vision. This is precisely why mathematical thinking cannot be reduced to mechanically following rules. In its living practice, mathematics begins not with a chain of reasoning, but with the sudden discovery of a structure. This idea fits well with the way contemporary philosophy of mathematics understands the structural character of mathematical knowledge. (Stanford Encyclopedia of Philosophy)
The mathematician Vavilov writes: “You cannot know mathematics, but you can be a mathematician. Being a mathematician means, first and foremost, seeing: possessing a kind of super-vision that allows you to look through walls and over barriers—Ma chi ha gli occhi nella fronte e nella mente.”
Here, the distinction between using mathematics and thinking mathematically becomes especially clear. Mathematics can be used as a language of operations. Being a mathematician means seeing a structure before it has been fully articulated in a proof.
It is therefore more accurate to say: proof is a form of legitimization of mathematical thought, but not its source. Its source lies in a new way of seeing.
How do we actually think? We literally play out scenes in our minds over and over, performing mental actions and trying to achieve a result, trying to find that self-evident action we can be certain of. Intuitively, this involves a deep faith that movement within us is analogous to movement outside us.
All mathematical axiomatics—and therefore metaphysics and causality as well—grows out of the human scale and everyday experience. If the world were built from the basic properties of matter, it would be logical to suppose that mathematics could also grow out of less obvious axiomatic foundations. This seems to have been Russell’s starting point as he tried to kill off metaphysics in his The Principles of Mathematics. But the deeper we go into the structure of matter, the more cumbersome and sophisticated the mathematical apparatus becomes.
I see two possible reasons for this.
- An individual unit of matter has no meaning in itself, and in trying to get at it, mathematics becomes increasingly complex because it has to carry with it an ever more encompassing description of the world.
- Perhaps human beings exist at precisely this scale and in this place because it is from this position that causality becomes possible as an organization of being.
In theory, mathematics based on non-anthropic principles is possible. In practice, however, it seems unlikely that we could construct a mathematics that would be self-evident to a hypothetical being in the quantum world, founded on that being’s self-evident axioms, and from which our three-dimensional world and the natural numbers would follow. It would more likely loop back on itself.
At the same time, our mathematics grows out of basic everyday experience. From an early age, at least, we already understand such things as adding or subtracting 1, or the difference between 1 and 2. Human beings think in terms of causality and the past, carrying over some of the actions performed on material objects to objects of thought.
Mathematics is a specialized metaphysics that is simultaneously within the world of embodiment and outside it, as a rupture. Similarly, philosophy is at once the philosopher, the philosopher’s life, the time in which that life unfolds, and a rupture with that time as its stage.
The initial axiomatic foundations of mathematics originate neither in matter as such nor in pure logic, but in the stable phenomenological experience of the world.
Roman Mikhailov’s Lecture
As an example of a mathematician’s way of thinking, I would like to turn to the video lecture “Ravinagar: A State of Identification of Space, Psyche, and Grammar” by Roman Mikhailov, a mathematician and Professor of the Russian Academy of Sciences. In his eccentric, meandering manner, he talks about proliferating patterns, the psyche, his grandmother who worked as a seamstress, sects, and mathematics. But if we listen more closely, a coherent way of thinking emerges from this stream.
Mikhailov’s Patterns
My understanding of mathematics is not sufficient to fully grasp what kinds of patterns Roman is discussing in his lecture. I can only approach them indirectly. The first level of mathematical understanding here probably involves groups of symmetry transformations. Group theory allows us to consider an object through transformations that preserve its shape or essential properties. This is why it is naturally connected with symmetry, ornament, and pattern. The artist M. C. Escher, though not himself a mathematician, illustrated groups of transformations in his work.
In his research, Mikhailov explores the next “level.” In his survey Homotopy Patterns in Group Theory, he explicitly states that the difficulty of certain classical problems in group theory is actually homotopical or homological in nature: derived functors, homotopy groups of spheres, group homology, and other structures arise naturally in purely group-theoretic questions. In effect, he is saying that “pure” group theory is not enough: deeper patterns of a different kind begin to emerge within it.
This seems to be the mathematics he has in mind when he speaks of fractality, multiplicity, ruptures, and domination.
What matters for our purposes is that he seems to experience an outwardly dry, neutral, abstract mathematics as something personal, connected with concrete childhood experience. At one point in the lecture, he describes his experience as follows: the goal is always rational, but behind it there is always a manifold, fractal pattern. His mathematical-mystical performance reaches its peak when he says that he is fighting a “proliferating fractal” that will “send him to the gas chamber.” This is not madness, but rather the living, undisguised existential thinking of a professional mathematician. He offers a glimpse of how highly abstract theories come together, out of the multiplicity of lived experience of being-in-the-world, to form a total, impersonal gaze—a pattern, in his terminology.
From a philosophical perspective, behind these “patterns” and his personal vocabulary of “fascism” and “schizophrenia,” one can discern Deleuze and Guattari’s rhizome, albeit inverted in form. Mikhailov’s fractal, total pattern corresponds, in the vocabulary of postmodernism, to “grand narratives,” while the ruptures are the living proliferation of the rhizome. Mikhailov says that “you need to accept the pattern, and it needs to accept you.” This reveals how his thinking works: he experiences mathematical abstractions bodily, as fate, and lives his life through them. Mathematical structures constitute his lived perception of the world, just as his grandmother, the seamstress, perceives the world through clothing.
It is worth observing how animated he becomes when the discussion turns to “what lies beyond”—to how, “beyond infinity,” topologies are no longer equal and a “new world” opens up. It is clear that this is a significant transcendent experience for him, something of which, even if no one else understands it, he would still “not be ashamed before God.” Proof itself, meanwhile, is “technical work”: he respects its complexity and virtuosity, but still regards it as a craft rather than the central source of meaning.
I think a parallel can be drawn here with the “ontological gesture”: an impossible leap into the beyond that unfolds a new world and a new reality through a new way of seeing.
