Everyday Mathematics

Mathematics has long been a source of awe and mystery, praised for its precision, practicality, and universal applicability. Yet, the question of what mathematical objects genuinely are—whether they exist independently of human thought or are simply products of our cognitive processes—remains an enduring philosophical puzzle.

Theories of mathematics have historically fallen into a few significant camps, each offering a distinct view of the ontological status of mathematical entities. Formalism, for example, treats mathematics as manipulating symbols according to agreed-upon rules, independent of any external reality. Logicism reduces mathematics to a branch of logic, claiming that mathematical truths are no more than logical truths. In contrast, Platonism posits that mathematical entities exist in an abstract realm, waiting to be discovered by human minds.

Each of these theories offers valuable insights but also comes with limitations. Formalism emphasizes mathematics’s concrete and procedural nature but struggles to explain the universality of mathematical truths. Logicism grounds mathematics in logical inference but leaves open questions about the connection between logic and the natural world. Platonism accounts for mathematics’s timeless and objective nature, yet it raises the question of how humans, as physical beings, could access a non-physical realm of mathematical forms.

In response to these limitations, a more nuanced approach may be found in moderate realism, mainly expressed through the Aristotelian tradition. This view proposes that mathematical objects are neither entirely independent of the physical world nor mere inventions of the mind. Instead, they are inherent in both the world’s structures and human thought. Once abstracted from sensory experience, mathematical symbols take on a life of their own, becoming material for further abstraction. This meta-symbolization of symbols allows mathematics to operate in both concrete and abstract realms, making it uniquely suited to bridge the gap between theory and practice.

The implications of this view are wide-ranging. For mathematical research, it suggests a balance between the practical and the abstract. Researchers may focus on how mathematical models reflect real-world phenomena, but abstract mathematics continues to play a vital role by pushing the boundaries of what can be symbolically represented and manipulated. In practical settings, such as business and everyday problem-solving, this view encourages the creation of tools that allow for intuitive engagement with both concrete data and abstract relationships, making mathematics more accessible and applicable across different domains of life.

In this post, we will cover how this synthesis not only resolves some of the tensions between existing theories but also highlights the interconnectedness of mathematics with the physical world, human cognition, and culture. By understanding mathematics as a reflection of reality and a product of symbolic thinking, we gain a richer appreciation of its role in shaping our understanding of the world.

Existing Positions

In symbol-focused theories such as formalism, most famously associated with David Hilbert, mathematics is seen as manipulating symbols according to established rules. In this view, the symbols’ meaning is irrelevant; what matters is that we follow the rules precisely. For example, in algebra, one can manipulate symbols in equations—such as transforming 2x + 4 = 12 into x = 4—without needing to know what x represents in a concrete sense. Formalism underscores mathematics’s procedural, rule-based nature, allowing it to function even in purely symbolic or computational environments like computer programming. However, while formalism highlights the procedural rigor of mathematics, it also introduces a degree of relativity since we can adopt different sets of rules depending on the system.

By contrast, logic-focused theories like logicism, introduced by figures such as Frege and Russell, posit that mathematics is reducible to logic. According to this view, mathematical truths are essentially logical truths derived from fundamental axioms. A classic example is how we can reduce the principles of set theory to logical propositions. Consider the logical foundations of arithmetic: “There exists a number such that adding it to one equals two” is not just a mathematical claim but a logical proposition that we can express through formal logic. Logicism emphasizes the precision and necessity of mathematical reasoning, showing how we can build complex mathematical truths from simple logical foundations. However, it needs help explaining how this purely logical structure connects to practical uses of mathematics in the empirical world.

Transcendent-realist theories, such as Platonism, inspired by Plato’s work, take a different approach. Platonists argue that mathematical entities exist independently of the physical world and human minds. A classic example here is the concept of a perfect circle. No perfect circle exists in everyday life—every drawing or physical representation is only an approximation. Yet, mathematicians can reason about the properties of a perfect circle as though it exists in a realm of ideal forms. In this view, mathematicians discover truths about perfect mathematical objects, like circles or prime numbers, which exist timelessly in a non-physical realm. While this perspective explains the seemingly eternal and unchanging nature of mathematical truths, it raises the question of how human beings, as material entities, access this abstract realm of mathematical forms.

A Proposed Synthesis of Inherence in Mathematics

To develop a more comprehensive view of mathematics, we seek to capture its practical effectiveness, precision, and cross-cultural validity. By addressing the limitations of the existing theories, this synthesis aims to ground the rules of mathematics, explain how its logical necessity translates into practicality, and understand how the human mind apprehends mathematical objects.

Moderate realism serves as a synthesis, particularly in Aristotelian realism. Unlike Platonism, which posits a separate, non-physical realm for mathematical entities, Aristotelian realism holds that these entities are real but inherent in both the physical world and human thought. Mathematical truths are discovered within the structure of reality rather than existing in an abstract, transcendent realm.

The effectiveness of mathematical rules corresponds to their ability to represent real-world actions and operations. A mapping exists between structures in the world, including the mathematical symbols themselves, and mental representations. This inherent connection grounds the rules of mathematics in physical reality. For instance, in Where Mathematics Comes From, George Lakoff and Rafael E. Nuñez argue that mathematical ideas correspond to conceptual metaphors, such as sets being analogous to containers. Similarly, in Mathematics, Form, and Function, Saunders Mac Lane links mathematical objects like bijections to activities such as rearranging. While these authors may not identify as moderate realists, their work supports a view of mathematics rooted in particular practices, showing how mathematical concepts emerge from human interactions with the world.

Mathematics also stands apart in its precision; unlike empirical inferences, which are conditional and based on evidence that we might revise with new data, mathematical theorems, once proven, apply universally and without exception. For example, a theorem in Euclidean geometry remains valid for all instances, regardless of future discoveries. Unlike in scientific experiments, we can verify mathematical truths through examples, but no new fact could alter a proven mathematical result.

This logical necessity translates into practical effectiveness because mathematical objects represent physical structures in their idealized forms. While the empirical sciences deal with highly complex objects that require abstraction, mathematics idealizes these objects in a way that retains their most essential properties. This abstraction allows mathematical models to be so effective in diverse practical applications.

The cross-cultural validity of mathematics suggests that mathematical objects are the discovery of actual structures rather than inventions. However, moderate realism proposes that this discovery is more akin to finding a new species than uncovering hidden, transcendent truths. The discovery process is grounded in the physical and mental world rather than an abstract realm. Different cultures converging on similar mathematical structures shows that while math is not wholly mind-independent, it is independent of any particular individual mind. This inherence in nature explains why mathematical systems can arise across different societies, even when culturally contingent.

The symbolic nature of mathematics also plays a crucial role in managing human cognitive limitations. Symbols are culturally contingent but possess a universal quality in their ability to represent other symbols, facilitating abstract thinking. For example, we can move from simple markings (III) to numbers (3), to variables (n), to sets (N), and further to collections of sets ({N, Q, R,…}). Each symbol allows us to work with increasingly abstract mathematical objects without overwhelming our cognitive faculties. This nested structure of symbols reflects the human need to work within our cognitive limits while still grappling with higher levels of abstraction.

In this view, mathematics is both a product of the mind and a reflection of the inherent structure of the world. It bridges the gap between abstract thought and physical reality, allowing for its broad applicability across cultures and profound effectiveness in the natural sciences.

Case Study: The Role of Infinity

The concept of infinity has long been a topic of debate in the philosophy of mathematics, with different schools of thought offering distinct interpretations of what it means and how it is used in practice. In symbol-focused theories, such as formalism, infinity is treated as a symbol within the rules of a mathematical system. It doesn’t have to represent anything “real” in the external world; it is simply a placeholder that allows mathematicians to manipulate formulas and carry out computations. In these theories, infinity is not tied to any external or physical reality, but rather to the internal consistency of the mathematical rules themselves.

In logic-based theories, such as logicism, infinity is viewed through the lens of logical structure. For example, a set can be described as infinite if there is no upper bound to the number of elements in the set, and logical systems can be constructed to handle these infinite structures. In this view, infinity exists as part of the logical framework of mathematics, and proofs about infinite sets or sequences follow from basic axioms of logic. However, the connection between this logical infinity and the physical or real world is often not directly addressed.

On the other hand, Platonism posits that infinity, like other mathematical objects, exists in a separate, abstract realm. Platonists argue that mathematical infinities are real objects—infinite sets, for instance, are actual entities that exist independently of the physical world. Mathematicians, according to Platonism, discover these infinite entities through reason and logical inquiry, much like they discover other timeless, abstract truths. In this framework, infinity is as real as any other mathematical object, even if we cannot physically observe or interact with it.

Moderate realism, particularly in the Aristotelian tradition, takes a different approach to infinity. In this view, mathematical objects are real but are grounded in the physical world or cognitive processes. As a result, infinity is treated not as an actual entity but as a potential infinity. This means that infinity is understood as a process—something that can continue indefinitely but is never fully completed. For instance, we can keep counting numbers forever, but we never actually “arrive” at an infinite number. In moderate realism, infinity is not something that exists in the world but rather a conceptual tool that describes ongoing, unbounded processes.

A clear example of how infinity is interpreted in mathematics can be seen in the definition of a limit in calculus. In the standard epsilon-delta definition, a function f(x) is said to approach a limit L as x approaches a, written as: lim⁡ x → a f(x) = L. When x approaches infinity as in lim⁡ x → ∞, the basic idea of an epsilon-delta proof remains the same—you show that for every ϵ > 0, there exists some large N such that whenever x > N, the value of the function f(x) is within ϵ of the limit. This proof structure does not require the use of an actual infinite set. Instead, it relies on the idea that, no matter how small we make ϵ, we can always find an N big enough that satisfies the condition. It is an ongoing process, capturing the idea of potential infinity.

In this view, the concept of infinity in the limit definition does not refer to something real in the physical world, nor does it exist in an abstract realm as it might for Platonists. Instead, infinity represents an unbounded process—a way of reasoning about how close the function gets to L, rather than a completed infinite sequence. The potential infinity describes the possibility of making the function as close as we like to the limit, but it never assumes that we reach an actual infinity.

These proofs rely on syntax and inference process in classical logic that remain the same, but the semantics—the interpretation of what the limit is describing—differs. For Aristotelian realists, the epsilon-delta process is grounded in the real-world behavior of functions and their continuous change, abstracted from physical phenomena but always tied to a potential infinity rather than an actual one.

Implications and Further Considerations

The inherence view of mathematics, which holds that mathematical objects are embedded in both the physical world and human cognition, has significant implications for how mathematics research is conducted and how mathematical concepts are applied in practical settings. This perspective encourages a grounded, context-sensitive approach to mathematical inquiry. Still, it also preserves the importance of abstract or “pure” mathematics in expanding our understanding of symbolic structures beyond immediate sensory experience.

In mathematical research, the inherence view supports a balance between practical applications and more theoretical exploration. While mathematical objects often begin as abstractions from real-world phenomena, once they are symbolized, the symbols themselves become material for further development. For example, the structure of a basic expression like 2 + 2 = 4, grounded in everyday experience, can evolve into a more general equation like x + 2 = 4, where x represents an unknown quantity. This capacity to extend mathematical concepts from symbols allows for more profound, more sophisticated theories that are not directly tied to physical reality but push mathematical knowledge’s boundaries. Thus, abstract mathematics remains essential, contributing to theoretical progress and practical applications.

At the same time, the inherence view emphasizes the importance of real-world grounding in mathematical research. Scholars might increasingly explore how mathematical structures represent physical systems, focusing on applied mathematics, engineering, and computational sciences. Even in these fields, the power of abstract thought remains central to creating models we can apply across multiple contexts. Interdisciplinary collaboration will likely flourish as mathematicians work with professionals in other fields to study how mathematical principles operate within both the material and symbolic layers of the systems they examine.

This perspective also draws attention to the cognitive processes behind mathematical thinking, which are critical to pure and applied mathematics. Understanding how the mind naturally grasps mathematical structures could lead to more effective teaching methods, helping students bridge the gap between everyday reasoning and more abstract concepts. Moreover, research into mathematical cognition could reveal new insights into how we process concrete and symbolic information, informing the development of new mathematical frameworks.

In practical applications, this view encourages the design of mathematical tools that reflect real-world scenarios and mathematics’s symbolic nature. For example, decision-making systems incorporating visual and symbolic representations in business would allow users to navigate complex data more efficiently. Mathematical models can help organizations simplify intricate systems without losing sight of critical details in the finance, logistics, and technology industries. The ability to represent patterns in a way that aligns with human cognition ensures that mathematical thinking remains accessible and effective in everyday contexts. For example, businesses could benefit from decision-making tools that use visual or symbolic representations, allowing users to interact with complex data in natural and intuitive ways. 

The ethical implications of this view are also significant, particularly in the realms of business and technology. Since mathematical models are used to inform decisions in real-world environments, it becomes essential to recognize their limitations and potential biases. By acknowledging the role of human thought in shaping these models, businesses and technologists can create more inclusive and fair systems that better serve diverse communities and a human-centered approach to developing algorithms, data models, and AI systems.

Lastly, the inherence view highlights the universality of mathematics while acknowledging its adaptability across cultures. Mathematical systems are not merely inventions of the mind but discoveries within the fabric of reality, with different cultures arriving at similar principles through distinct paths. This flexibility allows for more inclusive approaches to teaching and applying mathematics across various contexts, making it a tool for fostering global collaboration.

The inherence view supports a comprehensive understanding of mathematics in both research and practical settings. It values both theoretical exploration and real-world application, allowing mathematics to evolve in ways that respect its symbolic and practical foundations. Whether advancing pure mathematics or solving everyday problems, this perspective underscores the power of mathematics to reflect both the structure of the world and the capacities of the human mind.

The brilliance of mathematics, particularly in limits, is that it abstracts infinite processes in a way that makes them understandable and manageable. You don’t have to “reach” infinity to know exactly where the function is heading. This kind of reasoning is powerful because it combines practical computation (we only need finite steps) with abstract certainty (we know exactly what happens as the process goes on forever), giving you the best of both worlds—certainty without the need for completion. It speaks to the deeper connection between human activity, trusting the process and fidelity to certain ideals.

Further Reading

1. Thinking About Mathematics: The Philosophy of Mathematics by Stewart Shapiro. This book offers a comprehensive introduction to the philosophy of mathematics, covering major debates and theories such as formalism, logicism, Platonism, and more. Shapiro explores how philosophers have conceptualized mathematical objects what it means to “know” mathematical truths, providing a great overview for anyone interested in the foundational issues in mathematics.
2. Mathematics: Form and Function by Saunders Mac Lane. Saunders Mac Lane, one of the key figures in 20th-century mathematics, discusses the nature and structure of mathematical concepts in this seminal work. The book connects abstract mathematical ideas with their practical applications, showing how different branches of mathematics relate to real-world activities. This book is valuable for understanding how mathematics can be seen as a reflection of the operations and structures we encounter in the world.
3. Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being by George Lakoff and Rafael E. Núñez. Through the lens of conceptual metaphor, the authors show how basic operations like addition are rooted in physical interactions, such as moving objects together in a container. While Lakoff and Núñez’s view differs from an inherence perspective by framing mathematics as constructed through human cognition rather than inherent in reality, their emphasis on how abstract concepts emerge from embodied experiences contributes valuable insights. This cognitive approach helps illuminate how humans naturally engage with mathematical structures and can complement the inherence view by showing how physical experience gives rise to symbolic abstraction.
4. An Aristotelian Realist Philosophy of Mathematics by James Franklin. Franklin provides a modern take on Aristotelian realism, arguing that mathematical objects exist in the physical world, inherent in real structures and processes rather than in an abstract realm. This book defends a realist approach to mathematics that differs from Platonism, grounding mathematical truths in the observable world. It’s an essential read for those interested in a more naturalistic approach to the philosophy of mathematics.
   

Credits

This post was edited with the help of Grammarly and AI-assisted tools to ensure clarity and grammatical accuracy. The featured image is by Tsunami Green on Unsplash.