Invention or Discovery?

A scientist searches for knowledge of the natural world through a specific methodology: the scientific method of hypotheses and experimental validation. Scientists have a clear goal: to describe natural phenomena with rational explanations based on repeatable experimentation and consensus. Speculation is allowed only insofar as it leads to verifiable predictions.
— Marcelo Gleiser (“The Island of Knowledge”, 2014)

Excerpted from Marcelo Gleiser's book: "The Island of Knowledge", wherein he discusses whether mathematics is an invention or a discovery and why it matters:

“The display of wonder and regularity of Nature – day and night, seasons and tides, a Moon with phases, planets that return, the life and death cycle of plants and animals, gestation periods – requires a methodic counting and organizing as a means to gain some level of control over what is otherwise distant and unapproachable, the trends of a world evolving in ways clearly beyond human power. How else would pattern-seeking humans order their sense of reality if not through a language capable of describing these patterns, of analyzing them, of exploring their repetition as a learning tool? The mathematization of Nature, and the ordering of observed trends in terms of laws, is one of the distinctive achievements of our species.

“The power of mathematics comes from its being detached from physical reality, from the abstract treatment of its quantities and concepts. It starts in the outside world, the world as it is perceived by our senses, when we identify approximately circular and triangular forms in Nature, or learn how to count and measure distances and time. But then mathematics takes a simplifying step and lifts these asymmetric shapes from Nature and idealizes them as symmetric, so that we can more easily construct mental relations with them. These relations and their progeny may or [may] not be applicable back to the study of Nature. If they are, they may be used in a scientific model of some kind. If not, they may remain forever locked in the abstract realm of ideas they inhabit. This transplanting of forms and numbers from Nature, which allows for the abstract manipulation of number and form, is also why mathematics is always an approximation to reality and never reality as it is.

   In the mind’s eye: A beautiful image is a poor reflection of truth.

In the mind’s eye: A beautiful image is a poor reflection of truth.

“Nature’s creative power often hides behind asymmetries and not symmetries. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. The richness is found not in isolating order above everything else, but in contrasting order and disorder, symmetry and asymmetry, as complementary players in the ways we describe Nature. Symmetry … are excellent approximations to what we are attempting to describe. The danger, and the origin of the Platonic fallacy, is to believe that the symmetries are an imprint of Nature instead of an explanatory device we conceived to describe what we see and measure.

“There is a very productive alliance between the human brain and its mathematical attempts to make sense of reality. Mathematical results are not snapshots of some transcendent truth but a very human invention. The nexus of our quest for knowledge is not to be found outside of us but within us. Theorems in abstract mathematics, even if apparently completely disconnected from immediate reality, are the products of logical rules and concepts constructed with our minds. Our minds function in specific ways that reflect the embodiment of cognitive tools, which facilitate the development of abstract conceptual tools. We create the mind games of pure mathematics in the convolutions of our neocortex. And our neocortex is the result of eons of evolution driven by the pressures of natural selection and genetic variability, where the link between creature and environment is essential.

“The discussion of mathematics being an invention or a discovery, … points more to the importance of the human brain as a rare and wondrous oddity in the Universe than to the elusive truths written in some imponderable abstract realm. The cause of celebration is not “out there” or “up above” or in the “mind of God” but in this small mass we humans carry within our cranial cavity.

References:

  1. Gleiser, Marcelo (2014). The Island of Knowledge: The Limits of Science and the Search for Meaning. Basic Books.
  2. Lakoff, George and Nuñez, Rafael (2001). Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. Basic Books.
  3. Baum, Eric B. (2006). What is Thought? A Bradford Book.