Is mathematics invented or discovered?

  • Thread starter Thread starter Meden Agan
  • Start date Start date
AI Thread Summary
The debate on whether mathematics is invented or discovered centers on the interplay between the creation of symbols and rules versus the objective truths that emerge from these frameworks. Proponents of discovery argue that mathematical truths, such as Pythagoras' theorem, exist independently of human choice, evidenced by the universal convergence of cultures on certain mathematical concepts. Conversely, the invention perspective highlights how altering axioms can lead to entirely new mathematical realms, suggesting that foundational choices shape the discipline. A synthesis of these views posits that while formal languages and axioms are invented, the truths within those frameworks are discovered. Ultimately, the discussion raises philosophical questions about the nature of mathematics, suggesting it may exist in a unique category beyond mere invention or discovery.
Meden Agan
Messages
117
Reaction score
13
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.
 
Physics news on Phys.org
I am, and many mathematicians are, too, a Platonist. It is discovered. I even go as far as thinking that great compositions or paintings are discovered, or better: uncovered. However, it is sometimes hard to maintain this point of view when I see which constructions mathematicians "find" to prove theorems.

My favorite example is a circle. There is no such thing as a circle in real life. Nevertheless, we all know what is meant and can well calculate with it. Mankind may have invented the wheel, but not the circle. In the end, this is a purely philosophical question.
 
  • Like
Likes pinball1970, BillTre, Meden Agan and 1 other person
Why do we assume that one of the concepts of invention and discovery must apply to mathematics? Perhaps it's neither. Perhaps mathematics has its own category: either incovered or disvented.
 
  • Haha
Likes berkeman and Ibix
Just ONCE, I wanted to see a post titled Status Update that was not a blatant, annoying spam post by a new member. So here it is. Today was a good day here in Northern Wisconsin. Fall colors are here, no mosquitos, no deer flies, and mild temperature, so my morning run was unusually nice. Only two meetings today, and both went well. The deer that was road killed just down the road two weeks ago is now fully decomposed, so no more smell. Somebody has a spike buck skull for their...
Thread 'In the early days of electricity, they didn't have wall plugs'
Hello scientists, engineers, etc. I have not had any questions for you recently, so have not participated here. I was scanning some material and ran across these 2 ads. I had posted them at another forum, and I thought you may be interested in them as well. History is fascinating stuff! Some houses may have had plugs, but many homes just screwed the appliance into the light socket overhead. Does anyone know when electric wall plugs were in widespread use? 1906 ad DDTJRAC Even big...

Similar threads

Replies
72
Views
7K
Replies
34
Views
4K
Replies
8
Views
3K
Replies
16
Views
5K
Replies
66
Views
25K
Replies
137
Views
28K
Replies
54
Views
6K
Replies
5
Views
2K
Back
Top