Mathematics and the Imagination: A Brief Introduction.

Neoplatonism in Late Antiquity

Chapter 8 considers the role of the imagination as it appears in Proclus’ commentary on Euclid, where mathematical or geometrical objects are taken to mediate, both ontologically and cognitively, between thinkable and physical things. With the former, mathematical things share the permanence and consistency of their properties; with the latter, they share divisibility and the possibility of being multiplied. Hence, a geometrical figure exists simultaneously on four different levels: as a noetic concept in the intellect; as a logical definition, or logos, in discursive reasoning; as an imaginary perfect figure in the imagination; and as a physical imitation or representation in sense-perception. Imagination, then, can be equated with the intelligible or geometrical matter that constitutes the medium in which a geometrical object can be constructed, represented, and studied.

2000

A metaphor is an alteration of a woorde from the proper and naturall meanynge, to that which is not proper, and yet agreeth thereunto, by some lykenes that appeareth to be in it. ... —Thomas Wilson, The Arte of Rhetorique ...

COSMIC SPIRIT, 2020

Only the linguistic expression of genuine mathematics is typical, since, as being a language, it is a human construction of symbolic/point patterns which express the earthly dimensional (Euclidean) space-time environment. These extreme spot symbols of mathematical expression are nothing else but the extreme sections of the net (intangible) mathematical universe. Namely, they are material thickenings that take place due to the function of the human brain (1). An example may analyze the above: as electromagnetism is material (electricity) and “power” (“something else”) altogether, the mathematical universe is language (material) and thought (“something else” – “energy”) together. The language should not be coincided with thought (2). On the other hand, neither the thought should be coincided with the written language. Both the oral and the written language are nothing more than a form of the net structure of thought – logic. Logic in turn, should not be coincided with mathematics (neither as being a mathematical logic), because mathematics are an immaterial – invisible “language” which is expressed as a visible material through mathematical symbols – spots. Ultimately, Mathematics themselves cannot be categorized, as we showed, in rationalism or in empiricism (“sensualism”), or in intuitionism (“intuitive mathematics”) either.

Geometries of Nature, Living Systems and Human Cognition, 2005

At the beginning, Nature set up matters its own way and, later, it constructed human intelligence in such a way that [this intelligence] could understand it" [Galileo Galilei, 1632 (Opere, p. 298)]. "The applicability of our science [mathematics] seems then as a symptom of its rooting, not as a measure of its value. Mathematics, as a tree which freely develops his top, draws its strength by the thousands roots in a ground of intuitions of real representations; it would be disastrous to cut them off, in view of a short-sided utilitarism, or to uproot them from the ground from which they rose" [H. Weyl, 1910]. Summary. Mathematics stems out from our ways of making the world intelligible by its peculiar conceptual stability and unity ; we invented it and used it to single out key regularities of space and language. This is exemplified and summarised below in references to the main foundational approaches to Mathematics, as proposed in the last 150 years. Its unity is also stressed: in this paper, Mathematics is viewed as a "three dimensional manifold" grounded on logic, formalisms and invariants of space; we will appreciate by this both its autonomous generative nature and its effectiveness. But effectiveness is also due to the fact that we reconstruct the world by Mathematics: we organise knowledge of space and language by Mathematics, and give meaning by it to their structuring. But, what is "meaning", for us living and historical beings? What does "mathematical intuition" refer to? We will try to propose an understanding of these crucial aspects of the mathematical praxis, often disregarded as "magic" or as beyond any scientific analysis. Finally, some limits of the remarkable, but reasonable effectiveness of Mathematics will be sketched, in particular in reference to its applications in Biology and in human cognition of space, is topological, while the curvature and the metric is a local property, which may depend on the "cohesive forces of matter", an amazing anticipation of Einstein's relativity 2. As a matter of fact, by the analysis of the geometric structure of space (as ether), Riemann meant to unify action at distance (heath, light ... gravitation). Clearly, Riemann greatly contributed, by his n-dimensional differential geometry, to demolish the absolute and certainty of Euclidean space, just a special case of his general approach. Yet, he tried to re-establish knowledge as related to our understanding of the (physical) world. For him, geometry is not "a priori" by its axioms, but it is its grounded on certain regularities of physical space, to be singled out and which have an objective, physical, meaning (continuity, connectivity and isotropy, for example). In Riemann's approach, we actively structure space, as manifold, by focusing on some key properties, which we evidentiate by "adding hypotheses". Thus the foundational analysis consists, for Riemann, Klein, Clifford, Helmoltz ... in making these regularities explicit and in spelling out the transformations which preserve them. However, this "relativized" neo-kantian attitude turned out to be unsatisfactory for many, since it dangerously involved an analysis of the "genesis of concepts", as apparently originating from our more or less subjective presence in the physical world. In fact, these concepts, as for instance those of differential form, group and curvature are objective because they are invariants that we actively single out of the hysical world; that is, they become concepts as a result of the interaction, at the phenomenal level, between us and the world. How to re-establish then absolute certainty and objectiveness, after the shocking revolution of non-Euclidean geometries, while avoiding this "implication of the subject" into knowledge? First, avoid any reference to space and time, the very reference that had given certainty, for so long, not only to geometry, but also to algebra and analysis. For Descartes and Gauss, say, algebraic equations or the imaginary numbers are "understood" in space: this is so in analytic geometry and in Argand-Gauss interpretation, over the Cartesian plane, of √-1 (of the complex numbers, thus). But, if our relation to space is left aside in order to avoid the uncertainties of the "many geometries" and the shaky sands of human cognition, then language remains, in particular the logical laws of thought that the English school of algebra had already been putting forward, in a minimal language of signs ([Boole,1854]). Language, considered as the locus of the manipulation of (logical) symbols, with no reference to phenomenal space, nor, in general, to forms of experience. Frege best represents this turning point, in the foundation of Mathematics. His search

One reason for the " unreasonable effectiveness of mathematics " is that it is never compared to nature itself, which is ambiguous, but to well-defined idealized versions (models). Math is consistent with nature in unfamiliar situations because it is consistent within itself. PART ONE: the Map and the Territory Nature and mathematics are divided by the same categorical gulf that has long plagued the relationship between the physical and the mental. Just as there is no mind without body, there is no mathematics without mathematicians. Mathematics and physics are alike cognitive activities undertaken by intelligent organisms. The relationship of one to the other—and of each to nature—must be considered in their context as embodied cognition. We moderns understand the universe through mathematical models, which are idealized constructs that correspond only approximately to reality. It is easy enough to mistake the model for the reality—the map for the territory—when one has learned to think of nature as literally consisting of such idealizations. This mistake parallels the naïve realism of ordinary cognition, in which we normally take our perception as a transparent window on the world. One can then even conclude that the universe is literally mathematics. It is perfectly reasonable for physicists to believe in the guidance of mathematics. Yet, if physics is a form of cognition, then it is also reasonable to believe in the guidance of cognitive theory, evolutionary psychology, and theoretical epistemology (the nascent science of possible cognitive systems). Such things are not a part of physics as we know it. The Scientific Revolution redefined natural philosophy as " first-order " science: study of the external world in strictly objectivist terms. Focus on the object excluded focus on the subject. Cognition in general is a form of map-making. So, then, are mathematics and physics. The map, however, is not the territory. At best it represents it selectively, symbolically, and adequately for specific purposes. In the case of ordinary cognition, these purposes are bequeathed by evolutionary history. Math and physics may be driven by other purposes as well, better described perhaps by sociology and anthropology. In any case, the map corresponds only grossly to the territory. There is always simplification and streamlining involved. Science tends to mask the real complexity of the world when it presumes simplicity or prefers tidy systems to messy facts. The complexity and messiness of nature are the signs of its reality, which (contra Plato) lies in its very " imperfection. " We vaunt the ability of mathematics to exhaustively represent natural systems, but the reality of nature lies precisely in its ability to resist such exhaustion. Mathematical laws generally describe ideal things and circumstances that do not occur in nature, and that could not even be stated without idealization, isolation, and experimental control. (Ellis 2002, p90-94) To single out a causal relationship, one must know how the process