Mathematics realism
Mathematics realism is a philosophical stance that raises important questions about the nature of mathematical truths and the processes involved in mathematical discovery and invention. Central to the debate is whether mathematicians uncover pre-existing mathematical laws or create them through human activity. This discourse has roots in the ideas of ancient philosophers like Plato, who believed that mathematical concepts exist independently of human thought, a view often associated with Platonism. Proponents of mathematical realism assert that mathematics represents a timeless science of logic, identifying universal constants such as π and the principles governing mathematical structures, which they argue exist regardless of human awareness.
Conversely, critics of mathematical realism, including social constructivists, contend that mathematical knowledge is a sociocultural construct. They argue that mathematics is akin to institutions like law and religion, formed by human society and consciousness rather than discovered in an abstract realm. This perspective emphasizes that mathematical truths evolve with societal influences and conventions. Other viewpoints, such as those presented by philosopher Ludwig Wittgenstein, suggest that mathematics is a form of communication shaped by human interaction and cognitive processes. These diverse perspectives contribute to an ongoing, rich dialogue about the essence and role of mathematics in understanding our universe.
Mathematics realism
SUMMARY: One of the central questions of the philosophy of mathematics is that of mathematical realism.
Mathematicians engage in a great many activities, including investigating and extending old and new concepts within the field, as well as developing new techniques to solve problems in mathematics and other disciplines. The question is, when they carry out this activity, do they discover existing laws or do they invent and create? If invention is involved, is it individual or is it social? This question is a polemical topic that has been subject to strong controversy and refers to ideas that have emanated everywhere from ancient Greek personages, such as Plato, up to modern advocates of artificial intelligence (AI).
Platonists
Those who subscribe to the discovery position are usually classified as Platonists. Plato expressed that mathematical ideas are discovered, existing independently of human observation or changes of a physical nature. However, the general trend known as “mathematical realism,” which includes formalism and logicism, also catalogued within the discovery perspective. Mathematics is seen as the science of logic with its laws based on enduring truths, whether they have been discovered or not. Those who subscribe to this position cite, for example, the existence of universal constants, such as π, ϕ, Euler’s e, or Feigenbaum’s α and δ in bifurcation theory. It is put forth that the circumference of a circle has always measured π times diameter, whether or not that fact had been discovered by a particular society or culture.
It is also claimed that the discovery of mathematical laws, objects, and relations occurs simultaneously, or over time, in distant places. The most famous examples include the simultaneous, but independent, discovery of calculus by Isaac Newton and Gottfried Leibniz in the seventeenth century and the independent discovery of the universal constant π by the Babylonians, Greeks, Chinese, and others at different historical moments. Many of the structures from very abstract areas of mathematics are often found to model phenomena in the physical world, such as the case of Cantor’s set, originally an abstract construct, which serves as a model for error distribution of the noise in transmission lines (for example, electric power lines or telephone wires). This case is also taken as evidence that mathematics is, apart from a consistent logical system when accepting the axioms, a language that describes the physical universe, whether or not that description was intended by the mathematician who discovered the pattern, technique, theorem, or other relevant mathematical object.
Criticisms of Platonists
This idea adds another element to the discussion. For the realists, it is important to distinguish between mathematics itself, as a timeless science of logic, together with the laws that govern its existence, and the practice of mathematics, which includes many aspects that are language-like and that, they agree, are created, such as particular symbolism, notation, formalization, and nomenclature. Often the Platonists are dismissed by arguments that ridicule or simplify Plato’s allegory of the cave to an alleged discovery of an almost physical mathematical realm. This simplification seems because of a literal, instead of a metaphorical, interpretation of the way that many working mathematicians refer to their subject, a way of expression that reflects the actual feeling of “concreteness” that is provoked by daily contact, manipulation, and struggle with their abstract objects. Roger Penrose, for example, who identifies with the Platonist perspective, speaks of the Mandelbrot set as a structure whose constant surprises, within its self-similarity, are waiting to be explored.
Diversities of Non-Platonists
On the other hand, those that challenge Platonism and mathematical realism in general are not a homogeneous group.
One of these positions asserts that the existence of mathematics can be understood only as part of human culture. It is argued that the reality of mathematics is a sociocultural and historical phenomenon and that mathematics exists only because there are human beings who create it. Advocates of this position argue that mathematics is in the same category as law, religion, and money. It is only human consciousness and society with its conventions that makes them real.
Philosopher Ludwig Wittgenstein regarded mathematics as a type of “. . . communication; people play ‘language-games’ and ‘sign-games’ to invent, rather than discover, mathematics.” The Social Constructivists, supporters of this position, argue that mathematical development is guided by fashions and trends in human societies. They claim that mathematical truth is invented and depends on the sociocultural context.
The term “quasi-empirism” is used for the type of modern mathematical research that relies on computers and other quasi-experimental methods that seem to contradict the deductive nature of mathematics and question the existence of absolute and eternal mathematical truth. The Social Constructivists assert that this activity demonstrates the fallibility of mathematical activity and removes it from the realm of any absolutes, thus supporting their claim that mathematics is “man-made.”
The embodied theories consider mathematics as an exclusively human endeavor, invented according to the physical and cognitive human reality. Exponents of this position privilege the biological evolution of the human brain and consider mathematical objects as a reflection of human cognition. Hence, according to this perspective, mathematics is constructed by the human brain, and its apparent truths were created because they actually work efficiently in the universe in which we find ourselves.
Bibliography
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