'Is mathematics discovered or invented?' – My thoughts on the topic

Mathematics, rightly viewed, possesses not only truth, but supreme beauty — cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.

— Bertrand Russell, British logician

Introduction

Many a mathematician would at some point in their mathematical career hear this puzzling question: “Is mathematics invented or discovered?”.

Anecdotal evidence suggests that the majority believes the latter, that mathematics is “discovered”. After all, how would we do research and get new results in mathematics if not by discovering new properties of mathematical objects? While there is some merit and truth to this idea, I have thoughts of my own on this question, which I would like to discuss in this more casual blog post (compared to my other drafts and posts).

First, to be clear, mathematics can be and is both invented and discovered, as they are not mutually exclusive. Besides, this kind of question is usually intentionally left vague to stimulate further discussion more than to get a definitive conclusion. However, for psychological ease, I will assert that my thesis is that mathematics is invented, or at least it is a lot more invented than it is discovered, to better illustrate some of my points in this post.

Introduction
Table of Contents
Terminology: “discovery” vs. “invention”
Inductivism: Building Knowledge from the Ground Up (Literally!)
- The Scientific Method and Inductivism
- But Here’s Where It Gets a Bit Messy (and More Interesting!)
Mathematics: A Different Beast Altogether?
The Great Divide: Why Mathematics is Largely Invented, and Physics Largely Discovered
Conclusion: Two Paths to Understanding

Terminology: “discovery” vs. “invention”

Before we begin, let us discuss some semantics.

As you might have already sensed, the question itself is phrased in a way that’s, well, rather open-ended and, some might say, extremely vague. What does it truly mean for something like a mathematical concept or theorem to be “discovered”? Are we stumbling upon pre-existing truths as an explorer might find a new continent? And what about “invented”? Does that mean mathematics is purely a human creation, akin to a new language or a symphony, conjured entirely from our own intellect?

The core of the issue is that there aren’t any universally agreed-upon, pre-established definitions of “discovery” and “invention” that neatly apply to the unique, abstract nature of mathematics. When people take a stance in this debate, their interpretation of these key terms is often deeply personal and significantly influenced by their own philosophical outlook, their specific area of work (especially if they are mathematicians), and their broader beliefs about what knowledge is and how it relates to reality. It’s less about applying a strict, objective dictionary definition and more about articulating a particular worldview.

So, as we explore the different arguments and perspectives in the sections that follow, it’s useful to keep this terminological flexibility in mind. The “right” answer to whether mathematics is discovered or invented might be less about a definitive, one-size-fits-all verdict, and more about appreciating the different, deeply considered ways we can conceptualize humanity’s unique and profound relationship with the extraordinary world of mathematics.

Now, with that framing in place, let’s start by looking at how science often approaches understanding, which can provide a useful contrast to the world of mathematics.

Inductivism: Building Knowledge from the Ground Up (Literally!)

The Scientific Method and Inductivism

So, how do scientists traditionally go about trying to understand the world? One classic idea is inductivism. At its heart, inductivism proposes that scientific knowledge is meticulously built upon a solid foundation of observed facts. Think of it as a bottom-up approach. The traditional view, famously championed by thinkers like Francis Bacon, envisioned a systematic process:

Observe and Record (The Ideal of “Pure” Data): Scientists were to begin by neutrally and objectively gathering a vast amount of data about a specific phenomenon. This meant observing the world with an open mind, free from preconceived notions, often aided by instruments to extend the senses, and meticulously recording every detail. The more comprehensive the observations, the better.
Infer Laws (Spotting the Patterns): With a rich collection of facts in hand, the next step was to engage in inductive reasoning. This involves moving from these specific, individual observations to broader, general statements or laws. Scientists would look for regularities, for patterns that repeated consistently, and then make the “inductive leap” to infer that these patterns represented underlying natural laws. For example, after observing countless apples (and other objects) falling downwards from trees, one might inductively infer a general law of gravity acting on all objects.
Confirm and Predict (Testing the “Truth”): Once a general law was inferred, it would be used to make specific predictions about future occurrences. If these predictions consistently held true when tested against further observations and experiments, the law would gain more support and be considered increasingly confirmed, edging closer to what was hoped to be the “true” theory governing that aspect of the natural world.

The grand ambition of this inductivist picture was to discover the objective truths that govern the universe, all starting from what we can directly see, touch, and measure.

But Here’s Where It Gets a Bit Messy (and More Interesting!)

Now, while this inductivist model sounds like a wonderfully straightforward and logical way to “do” science, it’s not without its philosophical challenges and practical hiccups. For centuries, it was a dominant ideal, but thoughtful critique eventually revealed some tough questions:

The Problem of Induction (Hume’s Thorn): This is a big one, famously articulated by philosopher David Hume. Just because we’ve observed something happening a particular way, even a million times (like the sun rising every day), does that logically guarantee it will happen exactly the same way the next time? Hume pointed out that our belief in such regularities is more a matter of custom or psychological expectation than strict logical necessity. There’s no absolute logical contradiction in imagining the future being different from the past. So, our inductive conclusions, however well-supported by evidence, remain at best highly probable, never absolutely certain. Certainty is elusive.
Theory-Laden Observation (Are Our Senses Truly Neutral?): Can we ever truly observe the world in a completely neutral, unbiased way? Critics argue that our observations are often “theory-laden.” Our existing knowledge, our cultural background, our expectations, the scientific paradigms we work within, and even the theories we’re trying to test can subtly (or not so subtly) influence what we pay attention to, how we interpret what we see through our senses (in the Cartesian sense of all incoming perceptions), and what we even consider relevant data. If you’re looking for evidence of a specific particle, you might interpret a blip on a screen differently than someone with no such expectation. So, the ideal of “pure,” unadulterated observation might be more of a myth than a reality.
Underdetermination (Too Many Stories for One Set of Facts): Often, the same set of observed data points can be explained by, or support, multiple, even conflicting, theories. Imagine plotting data points on a graph; you might be able to draw several different curves that pass perfectly through all of them. Which curve represents the “true” underlying law? If the observations alone are our only guide, it can be difficult, if not impossible, to definitively choose the “correct” theory from among several empirically equivalent alternatives. The data, by itself, underdetermines the theory.
The Leap to the Unseen (Beyond Direct Sight): Modern scientific theories are replete with concepts and entities that we can’t directly observe: think of quarks, electrons, genes (before advanced microscopy), black holes, or dark matter. These are often powerful theoretical constructs whose existence is inferred indirectly through their effects. How do we get from simple generalizations based on directly observable phenomena (like falling apples) to positing these radically different kinds of unobservable entities and their complex properties through induction alone? The leap often seems too great for simple generalization to bridge.

Because of these profound challenges, the strictly inductivist view has largely given way to a more nuanced understanding of scientific methodology, often described as hypothetico-deductivism. This doesn’t throw observation out the window – far from it! Instead, it recasts its role.

In the hypothetico-deductive model, scientists might develop a hypothesis through various means – perhaps inspired by existing observations, a flash of intuition, creative guesswork, or even an analogy. This hypothesis is not necessarily “derived” directly from data in a mechanical way. Then, from this hypothesis, they deduce specific, testable predictions. The crucial step is then to design experiments or make observations to see if these predictions hold true.

If the predictions are confirmed, the hypothesis gains credibility and support. If the predictions are falsified, the hypothesis must be revised or discarded. So, observation and empirical testing are absolutely critical, but they function within a more dynamic cycle of creative conjecture, logical deduction, and rigorous testing. It’s a sort of dance between theory and evidence, where leaps of “imagination” are just as important as careful observation.

Mathematics: A Different Beast Altogether?

This brings us to mathematics. It operates in a fundamentally different way: while natural scientists are out there getting their hands dirty with empirical data, mathematicians are, in a sense, exploring a world of pure thought.

Mathematical truth isn’t confirmed by looking through a microscope or running an experiment. It’s established through deductive proof. Starting from a set of foundational assumptions (axioms), mathematicians use logical steps to arrive at conclusions (theorems). If the axioms are accepted and the logic is sound, the theorem is considered true within that system – absolutely and infallibly. We, of course, might use intuition and empirical observation, but not to confirm the statements per se, rather we use them to help our understanding of the abstract concepts we are grappling with. No matter how good a person is at mathematics or abstract reasoning, without specific instances in the real world where a concept shows up, it is very difficult to reach a deep understanding of some mathematical abstraction, e.g. functions, integration, vector spaces, groups…etc.

Mathematics is different from science in that it does not rely on our “senses” (in the Cartesian sense of the word). The truth of a statement is absolute, provided we rigorously proved it, as it is dependent only on our axioms, and whether such a proof exists.

The Bedrock of Math: Axioms – Discovered or Invented?

Sure, one could argue, that this is still discovery, since the axioms that we come up with are based on our material conditions (wink wink Marxism). We discover properties of the systems that we invent.

This is the crux of a long-standing debate: Is mathematics discovered or invented?

The Platonist View (Discovery): This perspective, echoing Plato’s realm of ideas, suggests that mathematical truths exist independently of human minds, in some abstract realm. We don’t create “2+2=4”; we discover it. The surprising effectiveness of mathematics in describing the physical universe lends some weight to this idea. Why should these abstract, human-independent rules map so well onto reality?
The Formalist/Constructivist View (Invention): Others argue that mathematics is a human invention, a complex game played with symbols and rules that we create. We invent the axioms, and then we explore the consequences of those inventions. The fact that we can create different, equally valid mathematical systems (like non-Euclidean geometries, which were initially mind-bending but proved essential for Einstein’s theory of relativity) supports this. Intuitionistic logic, which we’ll touch on next, explores systems where the principle of double negation, i.e. ¬¬A ⇒ A, doesn’t universally hold! We generally stick to classical logic because, simply put, it would screw up everything we have already built so far. In this sense, axiomatic systems are nothing more than convention.

Perhaps the most satisfying answer is that it’s both. Humans invent mathematical concepts by abstracting from their experiences of the world (shapes, quantities, patterns). Then, they discover the intricate relationships and necessary consequences that flow from these invented concepts.

For Culture: Intuitionism: Mathematics as a Mental Construction

A fascinating and quite distinct perspective in the philosophy of mathematics is Intuitionism, pioneered by the Dutch mathematician L.E.J. Brouwer. Drawing some inspiration from Kantian ideas about the mind’s role in structuring experience, Intuitionism posits that mathematics is essentially an activity of construction, an internal creation of the human mind. Natural numbers, real numbers, proofs, theorems – these aren’t discovered in some external Platonic realm, nor are they mere formal symbols; they are all mental constructions.

The intuitionistic mathematician, often conceived as an idealized “creative subject,” operates free from physical limitations or errors in reasoning but is still fundamentally a finite being whose constructions unfold in time. This finitude has profound consequences, most notably leading intuitionism to reject the concept of a “completed” or “actual” infinity. We can mentally enact the process of counting natural numbers going on forever (a potential infinity), step by step, but we can never grasp or survey the totality of all natural numbers as a single, finished, existing set.

For an intuitionist, a mathematical statement is considered true if, and only if, the mathematician has experienced or can carry out a mental construction that verifies it. The existence of a mathematical object is synonymous with its constructibility. This often means that to prove something exists (an “existence proof”), you must provide a method or algorithm for actually constructing it or finding an instance.

This constructive requirement starkly contrasts with methods common in classical mathematics. For instance, classical mathematicians often use proof by contradiction to establish existence: if assuming an object doesn’t exist leads to a logical contradiction, then it must exist. Intuitionists generally don’t accept this as a valid way to prove existence without a direct construction.

As Brouwer himself argued, this constructive viewpoint led to a radical re-evaluation of fundamental logical principles, particularly the Law of Excluded Middle (LEM) – the principle that any statement P is either true or false (P ∨ ¬P). Intuitionists question its universal applicability in mathematics. For them, to assert “P is true or P is false,” one must either have a proof of P or a proof that P leads to a contradiction (a proof of ¬P). For certain mathematical statements, especially those involving infinite collections where a complete survey is impossible, we might not (yet) possess either type of proof. In such cases, the intuitionist would refrain from asserting that LEM holds. Logic, from this perspective, is not an a priori set of rules that mathematics must obey, but rather a collection of principles that are observed to be valid within the activity of correct mathematical construction.

The Material World’s Imprint on Abstract Thought

Finally, it is difficult to deny that mathematics is to some extent rooted in the physical world. But the reason why this is the case is simply because our consciousness is limited, determined by our material conditions (a little Marxian term here and there can’t hurt right?): we cannot conceive of anything other than the things we observe in reality.

When we speak of “material conditions” in this sense, we’re referring to the concrete, tangible realities of human existence: the socio-economic environment we are born into, the tools and technologies available (the means of production), the nature of our work, the social relations we form in order to survive and reproduce our lives, and the practical challenges and necessities posed by our physical environment. These aren’t just background details; they are the very soil in which human thought, culture, and consciousness grow. Our ideas, beliefs, and even our most abstract concepts don’t spring fully formed from a vacuum; they are responses to, reflections of, and attempts to grapple with these lived realities.

This perspective suggests that we cannot easily conceive of anything that is utterly divorced from the things we observe, experience, and interact with in reality, either directly or through complex chains of abstraction ultimately grounded in that reality. As a human being, I cannot truly picture or come up with something that has no conceivable basis, however remote, in the forms and processes that could exist or be constructed within the real world. One cannot, for instance, reasonably picture ℝ⁴ (four-dimensional Euclidean space) in its full, intuitive entirety; our sensory and cognitive apparatus, shaped by a three-dimensional existence, forces us to make do with analogies, projections, and a somewhat feeble, reductive three-dimensional understanding of such a space.

While mathematics can soar to incredible heights of abstraction, its seeds are often sown in our experience of the material world. Our brains, our cognitive tools, evolved within this world, facing its particular challenges and opportunities. The patterns we instinctively recognize, the structures we find initially intuitive (like basic counting or simple geometric shapes), the very way we conceptualize number and space – all bear the indelible imprint of our environment and our societal development. It’s not that mathematics is the physical world, or that every mathematical concept has a direct, one-to-one physical counterpart. Rather, our journey into mathematics, the initial signposts and the foundational metaphors, often begins there. Even the most abstract concepts are ultimately grappled with by minds that have been shaped, constrained, and enabled by material reality.

So, while the natural scientist gazes outward, using inductivism (and other tools) to model the observable world, the mathematician often gazes inward, constructing and exploring ideal worlds of logic and structure. Yet, both are profoundly human endeavors, shaped by our experiences and our unceasing quest to understand. And sometimes, as in the uncanny and often astonishing effectiveness of mathematics in describing the workings of the physical sciences, these two distinct paths of inquiry meet in truly profound and revealing ways.

Example: The Curious Case of “1 + 1 = 2”

Consider the statement “1 + 1 = 2.” From the perspective of everyday experience, this feels like a physical, factual observation. We’ve seen it happen countless times: you take one object, add another identical object, and voilà, you have two objects. This is “common sense,” born from repeated, inductive, scientific-like observation of the real world. We trust it because our senses, in the Cartesian sense, have confirmed it over and over.

However, the mathematical statement “1 + 1 = 2” is a different creature entirely. It isn’t primarily about those physical objects. It’s an abstract truth derived rigorously from a pre-defined set of rules and starting points – axioms. Within a system like Peano arithmetic, “1 + 1 = 2” is a logical consequence, a theorem proven through accepted methods of construction. It’s true because the system is defined that way, not because of our initial empirical observation of the idea.

This begs the question: if mathematical axioms are somewhat arbitrary starting points we define, why do they so often lead to results that perfectly describe our natural world?

The Art of Choosing Axioms: Crafting the Playing Field

The answer lies in the goals and methods of mathematicians. They don’t just pluck axioms out of thin air. There’s a craft to it. Often, axioms are chosen carefully to:

Formalize Known Truths: Crucially, axioms often serve to provide a rigorous foundation for mathematical ideas and results that were already known or suspected to be true, perhaps through intuition or application. They level the playing field, ensuring that established knowledge stands on solid logical ground.
Ensure Consistency: The system built upon them shouldn’t lead to contradictions.
Generate Interesting Results: They should allow for a rich and complex theoretical landscape.

Simply put, one could roughly divide mathematicians into 2 categories (some of them being in both or even none):

Mathematicians working on the cutting edge of mathematical research in fields such as PDEs, analytic number theory, algebraic geometry, and so on and so forth. This is what we picture the typical mathematician to be, the stereotype of the academic working in pure mathematics.
Mathematicians working on the foundations of mathematics to give basis to the advancements that are made by the cutting-edge mathematicians in fields such as set theory, type theory, model theory, formal logic, and so on and so forth. This gets interesting because their job is not necessarily to prove “new results” (even though to some extent they do), but rather to build up the logical basis, the frameworks on which the remaining body of mathematics will stand. Studying axiomatic systems, how the axioms fit together, and whether this will lead to a consistent foundation for mathematics are among the questions they study.

Those who lean towards mathematics being “discovered” might often be focused on the seemingly objective truths unearthed at the cutting edge. Those who lean towards “invention” might be more attuned to the constructed nature of the foundational systems. An applied mathematician using game theory might see it differently than a category theorist exploring abstract structures.

Mathematics: An Invented Language, Rooted in Abstraction

Many mathematical concepts are far from intuitive at first glance. Think of tensors, commutative algebra, or infinite-dimensional spaces. Yet, these abstract ideas find powerful applications in describing real-world phenomena. Why? Because they are powerful abstractions of ideas and patterns that can be used to model reality. This power to abstract and model echoes, in a way, Plato’s ancient concept of a ‘realm of ideas’ or Forms. For Plato, the physical world we perceive was but a shadow of a higher, more real realm of perfect, eternal archetypes – the Form of ‘Triangle,’ the Form of ‘Goodness,’ and so on. While modern mathematical abstractions aren’t necessarily these mystical Forms, they share that quality of being idealized, perfect patterns. Real-world phenomena then ‘participate’ in these mathematical structures, or at least can be described as approximations of them. It’s this resonance with underlying, pure patterns that perhaps gives these human-invented abstractions their surprising descriptive power.

But the crucial point is this: the real-world phenomenon is NOT what we describe it as. Reality is what it is, a collection of facts. Our mathematical models are sophisticated languages we’ve invented to represent, understand, and predict that reality. Nature didn’t hand us the Axiom of Choice, or define what a “set” or a “function” must be. We came up with these concepts, refining them over time (it took decades, for instance, to settle on the modern axioms of a group) to build a consistent and useful descriptive framework.

Physics: Discovering Facts, Inventing Models

This is where physics (and natural science more broadly) takes a different path. Physics is the observation of fact. As discussed previously, scientists observe the universe and then, yes, they invent theories to explain those observations. They use the language of mathematics to build models that can predict future events.

But here’s a critical distinction: no physical theory is “true” in an absolute, final sense. This isn’t up to us to declare. A theory is “good” or “useful” insofar as it accurately describes the phenomena it aims to describe and makes testable predictions.

General Relativity is an incredibly successful theory of gravity, but it’s not “true” in an ultimate sense.
Quantum Electrodynamics describes the interactions of light and matter with breathtaking precision, but it’s a model.
The various interpretations of quantum mechanics offer different ways to understand its bizarre implications, but none claim to be the final “truth.”

When a physical theory encounters phenomena it can’t explain, we don’t typically scrap everything (unless absolutely necessary). Instead, we add more theories, refine existing ones, or develop new frameworks to cover the gaps. For example:

Newtonian gravity worked beautifully for everyday objects at regular speeds.
It struggled with electromagnetic phenomena, leading to the development of Maxwell’s theory of electrodynamics.
Electrodynamics had issues at the subatomic level (like predicting an electron should have infinite self-energy, which it clearly doesn’t). This spurred the development of quantum mechanics.
Now, we face a major challenge: quantum mechanics and general relativity, our best theories for the very small and the very large, are fundamentally incompatible in their current forms. This is the frontier of modern physics.

The Great Divide: Why Mathematics is Largely Invented, and Physics Largely Discovered

That iterative process of modeling, testing, and refining physical theories based on observed reality is absolutely core to how science operates. It starkly highlights a fundamental characteristic: physical theories, no matter how successful or well-confirmed, remain provisional. They are forever subject to the arbitration of new empirical evidence, constantly aiming for an ever-closer, yet perhaps never perfect, map of the complex reality we observe.

Mathematics, however, charts a different course towards its truths. While physical theories are tethered to the ever-unfolding complexities of the observed world, a mathematical theorem, once rigorously proven from its accepted axioms, possesses a distinct kind of permanence and certainty within its defined system.

This mathematical truth doesn’t rely on our senses or direct empirical validation in the way physical theories do; it’s a consequence of the axioms and the agreed-upon rules of logic. Imagine traveling to another universe with entirely different laws of physics, or even slightly altered physical constants – our current understanding of physics would likely face a catastrophic breakdown. Yet, as long as you keep the axioms of a mathematical system intact (think of Euclidean geometry or Peano arithmetic), its theorems remain steadfastly valid. This shows how even the foundational axioms of logic itself are part of this carefully human-constructed mathematical framework.

Consider concepts like a perfectly fair coin or a flawless normal distribution. Can you point to one in the physical universe? They don’t exist in such pristine, ideal forms. These are mathematical abstractions, powerful tools we’ve created to reason about the world with precision, allowing us to build models and draw conclusions without getting bogged down in the messy, imperfect data of “common sense” or raw observation at every single step. Mathematics provides the tools for this precise, abstract reasoning.

It is this fundamental difference in how “truth” is established and the relationship each discipline has with empirical reality that leads many to see mathematics as a profound human invention – a language and a set of tools designed for internal consistency and descriptive utility – while physics is more akin to a process of discovery, an ongoing effort to uncover the pre-existing facts and operating laws of the physical universe around us.

Conclusion: Two Paths to Understanding

So, while the natural scientist gazes outward, using observation and inductive reasoning (among other tools) to model the observable world, the mathematician often gazes inward, constructing and exploring ideal worlds of logic and abstract structure.

Both are essential human endeavors, shaped by our experiences and our unceasing quest to understand. And sometimes, remarkably, as seen in the “unreasonable effectiveness of mathematics” (to quote Eugene Wigner) in describing the physical sciences, these two distinct paths of inquiry meet, revealing the deep and often mysterious connections between the world within our minds and the universe around us.