Mathematics: discovered, or invented?
Gordon Swobe
Mathematics: Discovered or Invented? Philosophical Perspectives
Mathematics has long been debated as either a fundamental discovery of truths inherent in nature or an invention of the human mind. Different philosophers and scientists have taken starkly different views on this issue. This discussion compares major perspectives – from ancient Platonism to modern hypotheses – focusing on key figures like Plato, Galileo, Immanuel Kant, Kurt Gödel, Eugene Wigner, and Max Tegmark, among others. We will examine each thinker’s stance on whether mathematics is discovered (a form of realism or Platonism) or invented (a form of constructivism, formalism, or empiricism), their views on the relationship between mathematics and physical reality, the epistemological implications of their views (such as the role of intuition and the limits of formal reasoning), and how these perspectives influence our approach to science and modeling.
Plato: Mathematical Realism in Antiquity (Discovered Truths)
Plato (427–347 BCE) is the archetype of the view that mathematical truths are discovered, not invented. In Plato’s philosophy, numbers and geometric forms exist in an abstract realm of Forms or Ideas, independent of human minds. Mathematical objects (like “the perfect circle” or “the number 2”) have eternal existence just as much as physical objects, only in a non-physical form. In other words, Plato’s view is a form of mathematical realism: there are objective mathematical entities and truths that exist whether or not humans perceive them . From this stance, when we do mathematics, we are uncovering or recollecting eternal truths that are “out there” in the world of Forms, rather than creating them ourselves.
Plato illustrated this idea in his dialogue Meno, where Socrates guides an uneducated slave boy to discover a geometric truth by himself, demonstrating that the boy already had implicit knowledge of mathematics. Socrates claims this is evidence that “all learning is recollection,” implying the soul innately knows mathematical truths from before birth . Epistemologically, Plato thus emphasizes intuition and recollection – the soul can recall or intuit eternal truths through reasoning. This suggests that human minds have access to a non-empirical form of knowledge.
In terms of the relationship between math and physical reality, Plato saw the physical world as an imperfect shadow of the perfect world of Forms. Mathematics, being about these perfect Forms (perfect circles, triangles, ratios, etc.), underlies reality: the physical world approximates mathematical structures. For example, any drawn circle is imperfect, but it partakes in the Form of a perfect circle. Thus, mathematics in Plato’s view is deeply inherent to the structure of reality – the laws of nature reflect the eternal mathematical Forms. This Platonic view set the stage for later thinkers to believe that studying mathematics is key to understanding the cosmos. It influenced the notion that doing science means uncovering the mathematical order of nature, an idea echoed by later figures like Galileo. The Platonic stance implies that advances in mathematics can reveal real features of the universe, thereby powerfully shaping the scientific approach of searching for mathematical laws in nature.
Galileo Galilei: Mathematics as the Language of Nature
In the early 17th century, Galileo Galilei (1564–1642) famously championed the idea that nature itself is written in a mathematical language. Galileo’s oft-quoted assertion is that the universe “is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it” . This vivid metaphor asserts that mathematics is embedded in the fabric of the natural world.
Galileo’s view aligns with the discovered/inherent perspective: mathematical truths are not mere human conventions but are out there in nature, waiting to be discerned. He insisted that to comprehend physical reality, one must decipher this mathematical language. For Galileo, then, mathematics was the key to unlock nature’s secrets – he used geometry and arithmetic to describe motion, falling bodies, planetary orbits, etc. The relationship between math and physical reality in Galileo’s view is direct and foundational: physical phenomena strictly obey mathematical relationships. In this sense, Galileo can be seen as a Pythagorean or Platonist in practice – treating mathematical relations as the blueprint of the physical world.
Epistemologically, Galileo combined empiricism with this mathematical realism. While he emphasized observation and experiment, he also held that reasoning with mathematics reveals the true laws beneath appearances. The implication is that human minds discover mathematical laws through a mix of experiment and mathematical deduction, not by inventing them. This stance greatly shaped modern science: Galileo helped inaugurate mathematical physics, showing that formulating empirical findings in mathematical terms yields universal, predictive laws. Ever since, science has assumed that the “book of nature” is mathematical, an assumption that stems from Galileo’s insight and has been spectacularly confirmed by the success of physics. It set a precedent that mathematical structure is the guiding principle for scientific modeling of reality.
Immanuel Kant: Mathematics as Synthetic A Priori (Mind-Dependent Structure)
Immanuel Kant (1724–1804) offered a nuanced perspective that sits between pure realism and pure empiricism. Kant argued that mathematical knowledge is synthetic a priori – it is necessarily true and applicable to the world (thus not derived from experience alone), yet it is not merely true by definition (not analytic) . In Kant’s philosophy, the human mind plays an active role in structuring experience. He proposed that our forms of sensibility – notably space and time – are built-in frameworks through which we perceive the world. Geometry, for example, describes the structure of space as we intuit it, and arithmetic may reflect the form of time (successive addition, etc.). Thus, mathematical truths are universal and necessary for any possible experience we have, because they stem from the mind’s structuring of phenomena .
For Kant, mathematics is not exactly “out there” in nature independently, but it’s not arbitrarily invented either. Rather, *mathematics is a human construct in the mind that remarkably applies to all natural phenomena because the mind imposes mathematical structure on raw experience. As Kant writes, “Conformity with the truths of mathematics is a precondition that we impose upon every possible object of our experience” . In other words, when we say a physical law is mathematical, it is because our cognitive framework (our way of seeing the world) is mathematical. Space and time being the “pure forms of intuition” ensure that physical objects obey geometrical and numerical relationships. Thus, the relationship between math and physical reality in Kant’s view is that reality as experienced by us conforms to mathematics – not because we discovered math in nature, but because our mind’s constitution makes it so.
The epistemological implication of Kant’s view is profound: we can be absolutely certain of mathematical propositions (e.g. the angles of a triangle sum to 180° in Euclidean geometry) and know they will hold for objects of experience, since they stem from our a priori intuitions . However, Kant also warns that this certainty only applies to the phenomenal world (the world as we experience it), not necessarily to things-in-themselves (the world independent of our mind) . This means mathematics is incredibly effective in science (since science deals with phenomena), but we cannot know if it applies beyond our human experience. Kant’s stance shaped our understanding of science by justifying why Newtonian physics (which was grounded in Euclidean geometry and arithmetic calculus) had universal validity given our mode of perception. It highlighted the role of the observer in science and suggested that if our cognitive framework were different, our mathematics and science might also differ. Later developments (like non-Euclidean geometry and Einstein’s relativity) challenged Kant’s specifics but his general insight – that the mind contributes to the structure of knowledge – remains influential in how we consider the applicability and limits of mathematics in modeling reality.
Formalism and Intuitionism: Mathematics as Human Construct (Hilbert & Brouwer)
In the late 19th and early 20th centuries, other modern contributors in the philosophy of mathematics took the position that mathematics is essentially a human-made construct – invented rather than discovered. Two influential schools here are Formalism and Intuitionism (Constructivism), exemplified by David Hilbert and L.E.J. Brouwer respectively.
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Formalism (David Hilbert): Hilbert and the formalists viewed mathematics as a creation of formal rules and symbols. In this view, mathematics is basically a “game” played with formulas on paper according to agreed-upon rules, without inherent meaning unless we later interpret it. As one summary puts it, “according to the formalist, mathematics is [the] manipulation of symbols according to agreed upon formal rules. It is therefore an autonomous activity of thought.” . In other words, mathematics does not need to represent an external reality; it is a self-contained system invented by humans. The truth of a mathematical statement is simply its derivability from axioms via rules of inference. Epistemologically, formalism focuses on consistency and coherence of the invented systems rather than any intuitive truth. For formalists, questions like “do numbers exist?” are misguided – numbers are just symbols in our game. This view implies that any application to physical reality is almost coincidental: we apply our invented math to model nature, and if the model is useful, that’s an empirical fact, not because nature is math. Hilbert aimed to solidify this by proving consistency of mathematical systems (so that our “game” would never reach a contradiction). Impact on science: Formalism encouraged a rigorous axiomatic approach in mathematics (e.g., formal set theory) and provided tools for scientists to create models without committing to their “reality.” However, it somewhat sidelines the question of why math works in nature – that puzzle is outside the formal system and left to empirical observation.
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Intuitionism / Constructivism (L.E.J. Brouwer): Brouwer’s intuitionism took a different tack on the invention idea by emphasizing the mental construction of mathematics. Intuitionism holds that mathematics is constructed by the mathematician’s mind, step by step; mathematical objects have no existence unless they can be explicitly constructed in thought. Thus, math is not a discovery of something external, but “purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality” . For example, saying “there exists an infinite set with property X” is meaningless to an intuitionist unless you can construct such a set in principle. This view denies that mathematical truths are out there waiting – we create truth by proving statements through mental constructions. Epistemologically, intuitionism gives primacy to human intuition (in a very different sense than Plato’s intuition): it stresses the subjective certitude that comes from constructing an object or a proof oneself. Brouwer even rejected classical logic principles like the law of excluded middle in math, since one cannot assert a statement is true or false unless one can construct a proof or a counterexample. Regarding math and physical reality, intuitionism would suggest that the usefulness of math in science is not because the world obeys math, but because we tailor mathematical constructions to fit our observations. Mathematics grows in tandem with experience as a human project, not as a pre-existing truth. This standpoint influenced areas like constructive mathematics and computer science (algorithms as constructive proofs). For scientific modeling, a constructivist view instills caution: one might prefer computational, finitely verifiable models and be skeptical of “existence” statements that have no constructive evidence in physics. It also means our mathematical models of reality are tools we create – very effective ones, but their success must be continually checked against experience rather than assumed as gospel truth.
Both formalism and intuitionism emphasize the invented nature of math, yet differ on method (axiomatic symbol manipulation vs. mental construction). They represent a counterpoint to the Platonic realism of thinkers like Plato and Gödel. These schools shaped 20th-century mathematics and had implications for science: for instance, they motivated rigorous development of systems (like formal axiomatic foundations and intuitionistic logic) which indirectly benefit fields like computer science and any formal modeling. However, they somewhat downplay the mystery of mathematics’ effectiveness in describing nature – something that philosophers like Wigner would later highlight as needing explanation.
Kurt Gödel: Incompleteness and Mathematical Platonism (Discovered Truths)
Kurt Gödel (1906–1978), the great logician, firmly believed in an objective mathematical reality – aligning with the Platonist (realist) camp. Gödel is famous for his Incompleteness Theorems, but philosophically he used these results to argue against the formalist notion that math is just a meaningless game of symbols. In his own words, Gödel maintained that mathematics is a descriptive science: it describes a realm of entities just as astronomy describes stars . He held “the view that the concept of mathematical truth is objective” – meaning that a mathematical statement (like Goldbach’s conjecture, or the Continuum Hypothesis) has a definite truth value independent of whether humans can prove it. For Gödel, mathematical truths are discovered, not invented, because numbers and sets exist objectively (even if not in space and time).
Gödel’s relationship between math and reality was quite literally Platonic. He wrote that “mathematics describes a non-sensual reality, which exists independently both of the acts and [of] the dispositions of the human mind.” . This non-physical reality of mathematical forms is accessible to us through a faculty of mathematical intuition. Gödel explicitly argued that we have “a way of contacting ‘reality’ other than through sense perception” – namely, through mathematical intuition that perceives mathematical objects and truths . Epistemologically, this is a bold claim: Gödel put mathematical intuition on par with sense perception as a means of knowing. He stated, “I don’t see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception” . In practice, when a mathematician “sees” that a statement must be true (even if unprovable in a given axiomatic system), Gödel takes that as genuine knowledge of an objective truth.
The implications of Gödel’s Incompleteness further reinforce his view. Incompleteness showed that no finite set of axioms can capture all mathematical truths: there will always be true statements that cannot be proved within the system. Gödel pointed out, for example, that either a statement like the Continuum Hypothesis is true or its negation is true in the objective sense – even if our axioms cannot decide which . He commented that the undecidability of such a statement by current axioms “can only mean that these axioms do not contain a complete description of reality” . In other words, the mathematical reality is richer than any formal theory. This aligns with his Platonic belief that truth exists out there and our formal systems are just incomplete attempts to catch up. The role of intuition here is vital: mathematicians may intuit new axioms or truths (as Gödel thought we would eventually intuit the truth of Cantor’s continuum hypothesis by finding new axioms ). Indeed, Gödel believed new axioms are discovered (not arbitrarily chosen) to consistently extend our knowledge .
Gödel’s stance has strongly shaped our understanding of the limits of formal systems and the nature of mind. For science and modeling, his views (and later interpretations by people like Roger Penrose) suggest that the human mind might not be reducible to a formal algorithm – since it can “see” truth beyond formal proofs . Penrose, a modern Platonist, built on Gödel’s insight to argue that consciousness has non-computable elements and that we literally “see” mathematical truths by insight, something no purely formal machine can do . In the broader scientific context, Gödel’s realism encourages scientists to trust mathematical beauty and coherence as signs of truth. It’s an almost Pythagorean attitude: if a mathematical structure is deeply true, it likely reflects something real. At the same time, his results caution us that no matter how comprehensive a mathematical model is, it may be incomplete – there could always be truths about the system that lie beyond what the model can prove. This interplay of trust in math and humility about formal methods is a lasting legacy of Gödel’s philosophy.
Eugene Wigner: The Unreasonable Effectiveness of Mathematics
Moving to the mid-20th century, Eugene Wigner (1902–1995), a Nobel-winning physicist, raised a famous philosophical puzzle regarding mathematics: Why is mathematics so unreasonably effective in the natural sciences? In his 1960 essay “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” Wigner expressed astonishment at how well mathematical concepts, developed with no regard for empirical reality, end up describing physical phenomena with uncanny precision. He noted “the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious.” . Wigner’s perspective doesn’t straightforwardly pick a side of discovered vs. invented – instead, it highlights the mystery itself as a clue.
On one hand, Wigner observed that much of mathematics consists of abstract concepts that humans invented for their own sake. He even described mathematics as “the science of skillful operations with concepts and rules invented just for this purpose.” – a phrasing that resonates with a formalist or constructivist view. Mathematicians often develop theories (like complex numbers, or group theory) with internal motivations, not because they were demanded by physical reality. Yet, on the other hand, when physicists later turned to these theories, they found them perfectly suited to formulating the laws of physics (for example, group theory underpinning quantum mechanics symmetries). This astonishing success feels like more than coincidence – Wigner calls it a “miracle”: “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift, which we neither understand nor deserve.” .
So, is math discovered or invented according to Wigner? He stops short of a definitive answer, but the implication of his essay leans toward the idea that mathematics has an uncanny life of its own beyond human whim. The effectiveness of an “invented” tool suggests that the tool was in some sense already tuned to reality. This puzzle nudges one towards a Platonic perspective – that maybe mathematical truths exist in nature and our “inventions” succeed because they uncover part of nature’s structure. Wigner’s question revitalized interest in why mathematics works so well for science. Epistemologically, it raises the possibility that our ability to do abstract mathematics is at least partly guided by reality (even unconsciously), or that there is a pre-established harmony between human reason and the physical world.
Wigner’s view greatly shaped our philosophical understanding of scientific modeling. It brought attention to the fact that the applicability of math is not a given – it’s “unreasonable” and thus demands explanation. His essay has been cited by many subsequent thinkers (both philosophers and scientists) who debate whether this effectiveness indicates mathematics is embedded in the universe (a realist argument) or whether it simply reflects how we tailor our descriptions to what works (a more empiricist argument). In practice, Wigner’s insight emboldens physicists to use even the most abstract mathematics in search of physical laws, trusting that elegant math might capture real phenomena – essentially because historically, it so often has. At the same time, it humbles us by suggesting there is a deep mystery at the intersection of math and reality, one that might point toward new philosophical or even theological insights (Wigner himself called it a “gift”).
Max Tegmark: The Mathematical Universe Hypothesis
Pushing the realism view to its extreme, Max Tegmark (1967– ) proposes that the physical world is not just described by mathematics – it is mathematics. Tegmark’s Mathematical Universe Hypothesis (MUH) posits that every aspect of reality corresponds to some mathematical structure, and in fact “our physical world is a mathematical structure” . In his words, “this means that our physical world not only is described by mathematics, but that it is mathematical (a mathematical structure), making us self‐aware parts of a giant mathematical object” . Tegmark suggests that if a mathematical structure is self-consistent, it corresponds to some possible universe – he even speculates on a multiverse of all mathematical structures. This is a radical form of mathematical Platonism (or even Pythagoreanism): mathematical existence and physical existence are equated.
For Tegmark, then, mathematics is absolutely discovered, not invented, because all mathematical structures have the same ontological status as the physical universe we inhabit. The relationship between math and physical reality in his view is identity – there is no gap. Electrons, photons, spacetime, etc., are mathematical entities in a complex structure. This hypothesis attempts to answer Wigner’s puzzle by eliminating the distinction between equations and world: math is effective in describing physics because physics fundamentally is math. Epistemologically, Tegmark’s view implies that by doing mathematics, we are directly probing the nature of reality. If our universe is a mathematical object, then learning new mathematics could potentially hint at new physics, and conversely, any physical observation must correspond to some mathematical truth. The role of intuition here is just the mathematician’s ability to explore the space of logical structures, essentially doing cosmology with pencil and paper. However, it also raises questions: if everything is math, how do we account for the perceived non-mathematical richness of experience, or why we perceive one particular mathematical structure (our universe) and not others?
Tegmark’s MUH, while speculative, shapes conversations in fundamental physics and philosophy. It encourages an almost unabashedly mathematical approach to theories (e.g., searching for an elegant “Theory of Everything” might be seen as finding the precise mathematical structure that is our universe). It also intersects with debates on the nature of reality: is mathematics the ultimate ground of being? Critics of Tegmark argue that he blurs the line between a model and the thing itself, but his supporters see a bold explanation for why math works: it works because at bottom there isn’t anything but math. In scientific modeling, if one takes Tegmark seriously, one might be inclined to consider even the wildest mathematical ideas as possibly real. It exemplifies the deep trust that certain physicists place in mathematics as the final arbiter of truth about the cosmos, an outlook that traces back through Galileo and Plato. Tegmark updates that lineage with modern cosmology, suggesting that the universe doesn’t just have some mathematical properties – it entirely conforms to a mathematical structure.
Comparative Summary of Perspectives
To clarify the contrasts between these thinkers, the table below summarizes their positions regarding whether mathematics is discovered or invented, how they relate math to physical reality, key epistemological implications, and the influence on science and modeling.
-gts
Jason Resch
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Terren Suydam
Brent Allsop
Russell Standish
Note: there was one part in the essay I originally thought was an errors where it said the axiomatic system "may be incomplete" I was going to say this ought to read "will be incomplete" but then I realized this is not true, the system can be complete if it is inconsistent. So one option for revision (for greater clarity) would be to rewrite that sentence to say: "any consistent axiomatic system will be incomplete." It is a stronger case for platonism than the "may be incomplete" but it also introduces jargon (like consistency) that wasn't defined.
do exist, just not ones capabable of representing arithmetic.
First-order Euclidean geometry is an example of a complete, consistent
axiomatic system.
https://en.wikipedia.org/wiki/Tarski%27s_axioms
Jason Resch
On Wednesday, April 16, 2025 at 10:32:25 PM UTC+10 Jason Resch wrote:Note: there was one part in the essay I originally thought was an errors where it said the axiomatic system "may be incomplete" I was going to say this ought to read "will be incomplete" but then I realized this is not true, the system can be complete if it is inconsistent. So one option for revision (for greater clarity) would be to rewrite that sentence to say: "any consistent axiomatic system will be incomplete." It is a stronger case for platonism than the "may be incomplete" but it also introduces jargon (like consistency) that wasn't defined.That's not even quite true. Complete and consistent axiomatic systems
do exist, just not ones capabable of representing arithmetic.
First-order Euclidean geometry is an example of a complete, consistent
axiomatic system.
https://en.wikipedia.org/wiki/Tarski%27s_axioms
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