- #1
Meden Agan
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The question ‘is mathematics invented or discovered?’ is a much-debated issue in the philosophy of mathematics. It is a classic. I would like to know what you think about it.
Below is my view.
IMO, the question ‘is mathematics invented or discovered?’ pits two equally plausible intuitions against each other. On the one hand, mathematicians choose symbols, definitions and axioms; on the other, once the rules are set, they encounter facts that cannot be changed at will. To bring some order to this, it is useful to distinguish between three levels: syntax (the signs and rules of deduction), semantics (the abstract structures to which those signs refer) and application (the relationship with the physical world). “Inventing” mainly concerns the first level; “discovering” concerns the second; effectiveness in science concerns the third.
On the discovery side, mathematics shows objectivity and necessity. Once Euclidean geometry has been established, Pythagoras' theorem does not depend on the taste of those who prove it. Fermat's Last Theorem didn't become true because Wiles proved it: the proof only revealed a truth already implied by the axioms of arithmetic and set theory used in the proof. The historical convergence of distant cultures on concepts such as natural numbers, divisibility, or symmetry suggests that there are robust structures that our cognitive activity encounters rather than creates.
On the invention side, however, foundations matter. Non-Euclidean geometry arises from modifying an axiom; topology arises from introducing a new notion of neighborhood; functional analysis invents spaces and norms; category theory reorganizes mathematical objects in terms of arrows and composition. These choices open up entire continents of otherwise inaccessible results. Furthermore, some statements are independent of the most commonly used axioms: the continuum hypothesis cannot be proven or disproven starting from Zermelo–Fraenkel–Choice (ZFC); the same is true for many statements about large cardinals. Here, the discipline shows pluralism: there are several coherent “mathematical worlds”, each with its own truths. This is strong evidence in favor of the constitutive role of invention in establishing the framework within which we then reason.
A rigorous way to bring the two approaches together can be as follows. Mathematics studies possible structures. We invent formal languages and choose axioms that isolate a class of structures; within that class, truths are no longer invented, they are discovered. Logic makes this transition precise: “provable” means obtainable syntactically from axioms and rules; “true in a model” means satisfied by a structure that interprets those symbols. The first-order completeness theorem guarantees that what is true in all models of an axiomatic system is also provable, and vice versa. At the same time, the incompleteness theorems say that, for sufficiently rich systems, there are arithmetic truths that cannot be proven within the system: a sign that discovery is not exhausted in a single formal framework and that sometimes axiomatic extensions guided by internal mathematical criteria (fertility, consistency, explanatory power) are needed.
Concrete examples clarify the intertwining. Non-Euclidean geometries are ‘invented’ by modifying the parallel postulate; but once the axioms are chosen, the theorems describing triangles on surfaces with positive or negative curvature are ‘discovered’ and do not depend on arbitrariness. The notion of computability has been introduced with different definitions (Turing machines, recursive functions, lambda calculus), but they all converge on the same class of computable functions: this convergence is an indication that the concept understood is not a pure artefact, but rather a stable aspect of the notion of effective computation. The arithmetic of natural numbers is often seen as particularly realistic: in second-order logic, Peano axioms characterize natural numbers “up to isomorphism”, i.e. they describe an essentially unique structure; in first-order logic, however, non-standard models appear, reminding us that the result also depends on the logical framework adopted.
The question remains as to why mathematics works so well in science. A non-mystical answer is that the physical world has relational structures that can be homomorphically represented by mathematical structures. We do not “discover” mathematics by looking at nature, nor do we “invent” it at random: we select, from among many possible theories, those that capture observable regularities with the maximum ratio of predictive power to simplicity. When the choice is successful, mathematics is applicable because there is a structural correspondence between the model and the phenomenon. Here, too, the invention lies in proposing the model; the discovery lies in drawing the necessary conclusions that are then empirically verified.
This perspective also explains the authority of rigor without falling into metaphysical dogmatism. Rigor is the discipline of the second step: once the framework has been invented, it is proven. Mathematical creativity, on the other hand, lives in the first step: defining concepts, conjecturing axioms, changing points of view. The objectivity of the conclusions doesn't deny the historicity of the premises; the historicity of the premises doesn't diminish the necessity of the conclusions. In central and stable cases (elementary arithmetic, real analysis), the various approaches converge, and the sense of “discovery” is strong; at the exploratory margins (foundations, advanced set theory), the weight of inventive choices and pluralism grows.
Let me know.
Below is my view.
IMO, the question ‘is mathematics invented or discovered?’ pits two equally plausible intuitions against each other. On the one hand, mathematicians choose symbols, definitions and axioms; on the other, once the rules are set, they encounter facts that cannot be changed at will. To bring some order to this, it is useful to distinguish between three levels: syntax (the signs and rules of deduction), semantics (the abstract structures to which those signs refer) and application (the relationship with the physical world). “Inventing” mainly concerns the first level; “discovering” concerns the second; effectiveness in science concerns the third.
On the discovery side, mathematics shows objectivity and necessity. Once Euclidean geometry has been established, Pythagoras' theorem does not depend on the taste of those who prove it. Fermat's Last Theorem didn't become true because Wiles proved it: the proof only revealed a truth already implied by the axioms of arithmetic and set theory used in the proof. The historical convergence of distant cultures on concepts such as natural numbers, divisibility, or symmetry suggests that there are robust structures that our cognitive activity encounters rather than creates.
On the invention side, however, foundations matter. Non-Euclidean geometry arises from modifying an axiom; topology arises from introducing a new notion of neighborhood; functional analysis invents spaces and norms; category theory reorganizes mathematical objects in terms of arrows and composition. These choices open up entire continents of otherwise inaccessible results. Furthermore, some statements are independent of the most commonly used axioms: the continuum hypothesis cannot be proven or disproven starting from Zermelo–Fraenkel–Choice (ZFC); the same is true for many statements about large cardinals. Here, the discipline shows pluralism: there are several coherent “mathematical worlds”, each with its own truths. This is strong evidence in favor of the constitutive role of invention in establishing the framework within which we then reason.
A rigorous way to bring the two approaches together can be as follows. Mathematics studies possible structures. We invent formal languages and choose axioms that isolate a class of structures; within that class, truths are no longer invented, they are discovered. Logic makes this transition precise: “provable” means obtainable syntactically from axioms and rules; “true in a model” means satisfied by a structure that interprets those symbols. The first-order completeness theorem guarantees that what is true in all models of an axiomatic system is also provable, and vice versa. At the same time, the incompleteness theorems say that, for sufficiently rich systems, there are arithmetic truths that cannot be proven within the system: a sign that discovery is not exhausted in a single formal framework and that sometimes axiomatic extensions guided by internal mathematical criteria (fertility, consistency, explanatory power) are needed.
Concrete examples clarify the intertwining. Non-Euclidean geometries are ‘invented’ by modifying the parallel postulate; but once the axioms are chosen, the theorems describing triangles on surfaces with positive or negative curvature are ‘discovered’ and do not depend on arbitrariness. The notion of computability has been introduced with different definitions (Turing machines, recursive functions, lambda calculus), but they all converge on the same class of computable functions: this convergence is an indication that the concept understood is not a pure artefact, but rather a stable aspect of the notion of effective computation. The arithmetic of natural numbers is often seen as particularly realistic: in second-order logic, Peano axioms characterize natural numbers “up to isomorphism”, i.e. they describe an essentially unique structure; in first-order logic, however, non-standard models appear, reminding us that the result also depends on the logical framework adopted.
The question remains as to why mathematics works so well in science. A non-mystical answer is that the physical world has relational structures that can be homomorphically represented by mathematical structures. We do not “discover” mathematics by looking at nature, nor do we “invent” it at random: we select, from among many possible theories, those that capture observable regularities with the maximum ratio of predictive power to simplicity. When the choice is successful, mathematics is applicable because there is a structural correspondence between the model and the phenomenon. Here, too, the invention lies in proposing the model; the discovery lies in drawing the necessary conclusions that are then empirically verified.
This perspective also explains the authority of rigor without falling into metaphysical dogmatism. Rigor is the discipline of the second step: once the framework has been invented, it is proven. Mathematical creativity, on the other hand, lives in the first step: defining concepts, conjecturing axioms, changing points of view. The objectivity of the conclusions doesn't deny the historicity of the premises; the historicity of the premises doesn't diminish the necessity of the conclusions. In central and stable cases (elementary arithmetic, real analysis), the various approaches converge, and the sense of “discovery” is strong; at the exploratory margins (foundations, advanced set theory), the weight of inventive choices and pluralism grows.
Let me know.
