“God made the integers, all else is man’s work,” stated German mathematician Leopold Kronecker in the 19th century. But is this true? Some fundamentals, such as positive integers and the 3-4-5 right triangle, are universally accepted across cultures. Almost every other aspect of mathematics is influenced by the society in which you live.
Mathematics, in this way of thinking, is like a language: it may describe real-world objects, but it doesn’t ‘exist’ outside the minds of those who use it. In ancient Greece, the Pythagorean school of thought held the belief that reality is fundamentally mathematical and philosophers and physicists are beginning to take this theory seriously after more than 2,000 years.
A central challenge for mathematical Platonists, however, is to provide an account of how mathematical explanations work. I propose a property-based account: physical systems possess mathematical properties, which either guarantee the presence of other mathematical properties and, by extension, the physical states that possess them; or rule out other mathematical properties, and their associated physical states.Sam Baron
In his paper, Sam Baron argues that mathematics is an essential component of nature that gives structure to the physical world. Take for example the ‘honeycomb conjecture’ in mathematics, which explains why beehives are built with hexagonal tiles: hexagons are the shape to employ if you want to completely cover a surface with uniformly shaped and sized tiles while limiting the perimeter length to a minimum.
Bees have evolved to utilize this design because it provides the largest cells to store honey for the smallest input of energy to produce wax, according to Charles Darwin. The honeycomb conjecture was first presented in antiquity, but mathematician Thomas Hales verified it in 1999.
It is simple to locate additional examples once we begin looking. Mathematics may be found in anything from soap operas to engine gear designs to the position and magnitude of gaps in Saturn’s rings.
It’s improbable that mathematics is something we invented if it explains so many things we observe around us. The alternative is that mathematical facts are found by insects, soap bubbles, combustion engines, and planets, not simply by humans.
But if we are discovering something, what is it? The assumption that physical objects possess mathematical properties involves a commitment to a partial form of Pythagoreanism: the belief that the universe is ‘made’ of mathematics. Plato believed that mathematics describes real-world objects. Numbers, geometric shapes, and more sophisticated mathematical objects such as groups, categories, functions, fields, and rings – mathematical objects that exist outside of space and time.
The Pythagoreans agreed with Plato that mathematics describes an object-oriented reality but disagreed with them existing beyond space-time. Rather, they believed that physical reality is made up of mathematical things, just as matter is made up of atoms. To this idea, contemporary physicist Max Tegmark argues reality is one big mathematical object, a simulation program that we perceive and discover.
Sam Baron proposes two parts to this dilemma: mathematics and matter. Matter provides mathematics its shape, while mathematics gives matter its substance. The physical universe has a structural foundation provided by mathematical objects.
Platonists should accept that physical and mathematical objects share intrinsic properties. I call this view ‘Partial Pythagoreanism’ on the grounds that physical and mathematical objects are not entirely distinct.
Baron goes beyond the beehive example to the two North American periodical cicadas subspecies that spend most of their life underground. The cicadas emerge in large swarms for roughly two weeks every 13 or 17 years (depending on the subspecies). Why are the ages 13 and 17 different? Why not 12 and 14 years old? Or maybe 16 and 18? The fact that 13 and 17 are prime numbers is one explanation, suitable to avoid predators with life cycles of 2, 3, 4, 5, 6, 7, 8, and 9 years.
Because 2, 3, and 4 divide evenly into 12, when a cicada with a 12-year life cycle emerges from the ground, the 2-year, 3-year, and 4-year predators emerge as well. Because none of the numbers 2, 3, 4, 5, 6, 7, 8, or 9 divide evenly into 13, no predators will be out of the earth when a cicada with a 13-year life cycle emerges. The same may be said with 17. These cicadas appear to have evolved to make use of basic mathematical facts.
In terms of qualitative characteristics, cicada’s example demonstrates that mathematics and matter coincide. The physical universe is partly mathematical in this, fairly weak, sense. Physical and mathematical objects have intrinsic features in common, such as having a given structure, a specific geometry or being organized in a specific way (as in a linear or partial ordering), in order to create the physical system we experience.
A physical system has a particular structure when the system instantiates it. The main challenge is to say what these structures are. Initially, this looks easy: a structure is just a collection of physical objects that stand in certain relations to one another. But this won’t do. Structures are multiply realizable: many different collections of physical objects can share the same structure. A structure therefore cannot be identified with any particular collection of physical objects. Baron argues that structures are physical phenomena forced by mathematical forces.
Structures are abstract entities; they can exist without being instantiated in space-time and they are independent of our knowledge and beliefs about them.Resnik, 1985
Mathematical Explanation: A Pythagorean Proposal, Sam Baron