The idea that the universe is fundamentally mathematical is one of the leading frameworks in modern thought. From physics to philosophy, equations are often taken to disclose reality itself. But what if this is a projection rather than a discovery? Physicist Martín López Corredoira invites us to reconsider, questioning whether mathematics describes the world as it is, or as we are able to conceive it.

There are two different methodologies for studying Nature, both inherited from different ways of thinking in ancient Greece: the rationalist–deductive method and the empirical–inductive method. The rationalist–deductive method was devised by Pythagoras (ca. 570–490 BCE) and Plato (ca. 427–347 BCE), according to whom the pure relations of numbers in arithmetic and geometry are the immutable reality behind changing appearances in the world of the senses. We cannot reach the truth through observation with the senses, they argued, but only through pure reason, which may investigate the abstract mathematical forms that govern the world. In this mode of thinking, there is a predominance of abstract theories, and mathematical modelling predominates over experimental and observational results. For Pythagoras and his followers in the 6th century BCE, the cosmos was not merely describable by mathematics, but it was fundamentally mathematical. Harmony in music, proportion in geometry, and order in the heavens were all expressions of numerical relationships.

Some historians think that the idea of a world ruled by numbers is related to the introduction of coins in the marketplace in the society in which Pythagoras lived. Anything could be reduced to abstract numbers: the value of a pot, a jar of oil, a plot of land, or a slave could be expressed by an exact number of coins, as could the wealth and worth of any citizen.

Plato also emphasised that true knowledge comes through reason alone, thus diminishing the role of the senses and the earthly realm as a whole. In his dialogue Timaeus, Plato formulates a cosmology with a creator moulding matter into approximations of these ideal shapes to create a Universe ruled by eternal mathematical laws, laws that humans can deduce only through reason. These eternal mathematical laws are the true reality, while the changeable Universe we see is a mere appearance; the observation of nature is thus unreliable.

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Mathematical elegance cannot be physics due to the lack of observational support.

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There are good cases of success using a rationalist–deductive approach. An example within modern science is Albert Einstein’s (1879–1955) General Relativity, which was posited from aesthetic and/or rational principles in a time when observational data did not require a new gravity theory. In fact, observational tests proved this theory successful.

Present-day physics and cosmology are partially Pythagorean when a theory is created before the observations. It is common among modern Pythagoreans to approve of statements such as the search for beauty in a mathematical construction describing physical reality, or the Divine plan by which the creator designed the Universe. The physicist–mathematician tries to achieve something close to a mystical approach, tries to read into the Mind of God. Also, analogously with religion, this extremely theoretical physics and cosmology can only be understood by a priestly elite able to think in four or more dimensions, or in terms of similar abstractions.

Quantum mechanics is built on linear algebra in abstract Hilbert spaces. Particles are described by wave functions, which are mathematical objects encoding probabilities. The fundamental level of nature appears less like solid matter and more like a set of mathematical relationships.