# Is mathematics invented or discovered **URL:** \ **Category:** Great Debates\ **Created:** [November 15, 2018, 3:33am UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542 "2018-11-15T03:33:33Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![Mijin](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mijin/32/9369_2.png) [@Mijin](https://boards.straightdope.com/u/Mijin)\ **Post date:** [November 15, 2018, 1:29pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/21 "2018-11-15T13:29:04Z")
> [@GreenWyvern](#): > > But _why_ is mathematics incredibly useful in the real world? This is the whole essence of the question in the OP. Right, and I did cover it, albeit briefly. Mathematics is a set of operations for manipulating information. It’s sound because it’s based on a foundation of formal logic and our universe is logical. And it’s useful because formalizing information processing, adding to our intuition about what facts can be derived is basically augmenting our reasoning ability. > [@](#): > > Either - > Mathematics exists as an external attribute of the universe, independent of the human mind - in which case it is discovered, not invented. > > Or - > The human mind is so fundamentally a part of rest of the universe that our pure creative thought can predict how the universe behaves. False dichotomy. Consider English. The English language has been used in many predictions and observations about the universe. So is the English language “out there”, floating in the ether or are our minds somehow in tune with the universe when we speak English? > [@](#): > > This question has been debated in detail and at length for decades by top scientists and mathematicians, without any consensus emerging. And that’s the problem. There are certain philosophical questions which have been talked about so long, and by so many eminent names, that it becomes impossible to point out that the Emperor has no clothes and this problem may well have a simple solution. Note that for all the quoting of Einstein, you have not added any arguments to the two I outlined in the OP (in fact you haven’t addressed the second one). ---
**Author:** ![GreenWyvern](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/greenwyvern/32/2834_2.png) [@GreenWyvern](https://boards.straightdope.com/u/GreenWyvern)\ **Post date:** [November 15, 2018, 1:48pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/22 "2018-11-15T13:48:36Z")
> [@Mijin](#): > > Consider English. The English language has been used in many predictions and observations about the universe. So is the English language “out there”, floating in the ether or are our minds somehow in tune with the universe when we speak English? No, the English language does not predict natural laws and phenomena. It only describes them. This is completely different from mathematics. It you think otherwise, then please give an example. And, no, ‘someone wrote a story about flying machines before they existed’ is NOT an example. ‘The laws of aerodynamics are predicted by English grammar’ would be an example. ---
**Author:** ![Nava](https://avatars.discourse-cdn.com/v4/letter/n/da6949/32.png) [@Nava](https://boards.straightdope.com/u/Nava)\ **Post date:** [November 15, 2018, 1:57pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/23 "2018-11-15T13:57:47Z")
> [@Mijin](#): > > The inside angles of a square are 90 degrees. Does this number “exist” out there as some universal constant? No. In fact, those same angles are 100 degrees and/or pi/2 radians. pi/2 radians OTOH is a constant, or at least it is one within Euclidean geometries (I barely know that non-Euclidean ones exist, so don’t ask me what happens to circles in them). ---
**Author:** ![rat\_avatar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/rat_avatar/32/255_2.png) [@rat\_avatar](https://boards.straightdope.com/u/rat_avatar)\ **Post date:** [November 15, 2018, 2:08pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/24 "2018-11-15T14:08:02Z")
The problem is that the term “math” is so broad that there will never be a definitive answer on either side, the topic is just too broad. Complex multiplication by i rotates you by 90 degrees, and complex numbers are fundamental to higher physics and spacial orientation in a way that would not match some claim of “philosophical commitment to materialism” Solutions to problems like the wavefunction inherently involve complex numbers as an example. But PEMDAS is not going to arise out of nature and is thus similar to English, it is purely convention. To repeat, this question is only hard to answer because the way it is framed is overly broad. ---
**Author:** ![magellan01](https://avatars.discourse-cdn.com/v4/letter/m/45deac/32.png) [@magellan01](https://boards.straightdope.com/u/magellan01)\ **Post date:** [November 15, 2018, 2:08pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/25 "2018-11-15T14:08:47Z")
> [@Thudlow\_Boink](#): > > I think mathematical truths do exist, but they do not exist “in nature.” > > I think some people are reluctant to saying that mathematics is discovered because of their philosophical commitment to materialism: the belief that only the material world can be said to exist or be discoverable. I’m not sure I understand your point. Isn’t all mathematics describing a physical world, even if it’s a hypothetical one. Even the most basic numbers, 1, 6, 153 are referring to _things._ When a volcano spews a boulder into the sky, it’s trajectory comports to the laws of physics. That was true a at the time of the first volcano on earth and it’s true for cannonballs. The laws of physics operated perfectly well before man understood them. But, again, maybe I’m not understanding your point. ---
**Author:** ![Left\_Hand\_of\_Dorkness](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/left_hand_of_dorkness/32/7156_2.png) [@Left\_Hand\_of\_Dorkness](https://boards.straightdope.com/u/Left_Hand_of_Dorkness)\ **Post date:** [November 15, 2018, 2:11pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/26 "2018-11-15T14:11:33Z")
> [@Mijin](#): > > But it would be trivial to define a branch of mathematics that doesn’t “work” – that gives incorrect predictions about the external universe. The reason our mathematics does not do this is that new mathematics must follow certain rules of basic logic e.g. self-consistency. > So indeed it is a fact about the external universe that it follows fundamental logic (e.g. it’s self-consistent) but that’s a much smaller claim than saying every branch of mathematics actually exists “out there”. Can a person come up with a branch of mathematics that _does_ give correct predictions about the external universe but that’s different from ours? Consider English. English is invented, and it describes the world pretty well. Quechua, despite sharing very few words with English, also describes the world pretty well, and is also invented. It’s not like the Inca, in trying to talk about water, said, “Yaku, yaku…hmm, that’s not a particularly apt descriptor for this liquid, let’s try saying ‘water’ instead, that’s a lot closer to the reality.” If you can come up with an entirely different system of math that “works”, then it seems to me a lot more like a language, and hence invented. But if there’s only one set of consistent axioms that ends up describing the world accurately, and other consistent sets don’t describe the cosmos accurately, then it seems to me a lot more like a science, and hence discovered. ---
**Author:** ![rat\_avatar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/rat_avatar/32/255_2.png) [@rat\_avatar](https://boards.straightdope.com/u/rat_avatar)\ **Post date:** [November 15, 2018, 2:13pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/27 "2018-11-15T14:13:55Z")
> [@Left\_Hand\_of\_Dorkness](#): > > Can a person come up with a branch of mathematics that _does_ give correct predictions about the external universe but that’s different from ours? > > Consider English. English is invented, and it describes the world pretty well. Quechua, despite sharing very few words with English, also describes the world pretty well, and is also invented. It’s not like the Inca, in trying to talk about water, said, “Yaku, yaku…hmm, that’s not a particularly apt descriptor for this liquid, let’s try saying ‘water’ instead, that’s a lot closer to the reality.” > > If you can come up with an entirely different system of math that “works”, then it seems to me a lot more like a language, and hence invented. > > But if there’s only one set of consistent axioms that ends up describing the world accurately, and other consistent sets don’t describe the cosmos accurately, then it seems to me a lot more like a science, and hence discovered. Physics is descriptive, not prescriptive and it also tends to follow perturbation theory. But some core concepts would arise in any theory. As an example Newtons universal law of gravitation is really just the inverse square law. While Newton’s theory may not arise in some other population the inverse square law would. ---
**Author:** ![msmith537](https://avatars.discourse-cdn.com/v4/letter/m/d9b06d/32.png) [@msmith537](https://boards.straightdope.com/u/msmith537)\ **Post date:** [November 15, 2018, 2:26pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/28 "2018-11-15T14:26:18Z")
> [@Left Hand of Dorkness](#): > > View Post Can a person come up with a branch of mathematics that _does_ give correct predictions about the external universe but that’s different from ours? Certainly. We don’t even use one form of mathematics. When most people think of math, they think of our base 10 numerical system (0,1,2,3,4,5,6,7,8,9). But in computer science they use binary, which is a base 2 (0,1) and hexadecimal, which is a base 16 (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F). Only in base 10 does 1+1=2. In binary, 1+1 = 10. In hexadecimal, 1+1 still =2 but 9+1=A, not 10. The point being, these are all different mathematical systems, but they all grounded in their ability to describe something observable in nature. In this particular example, describing the number of objects in a set when objects are added or removed from it. And I’m not even getting into mathematical constructs that can’t be expressed in physical terms. Like imaginary numbers (I^2 = -1) for example. > [@Isamu](#): > > Here: Pi > > This argument is prefectly circular. 😛 Classic:D ---
**Author:** ![John\_Mace](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_mace/32/185_2.png) [@John\_Mace](https://boards.straightdope.com/u/John_Mace)\ **Post date:** [November 15, 2018, 2:30pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/29 "2018-11-15T14:30:48Z")
> [@Isamu](#): > > Here: Pi > > This argument is prefectly circular. 😛 It appears to be irrational. ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [November 15, 2018, 2:31pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/30 "2018-11-15T14:31:39Z")
Stand back far enough in one’s perspective, and I’m not sure there is a good criterion to distinguish inventions from discoveries. Let’s contrast the design of a Frisbee, which most would call an “invention,” with the Monster Group, a “discovery.” Yet both of them can be described with marks of pencil on paper describing relationships. The rules from which the Monster Group builds may be more rigid and/or beautiful than the Frisbee rules, but at some level of abstraction they are similar. An alien civilization which discovered the Monster Group might also have discovered the Frisbee design. Yes, it would have little chance of duplicating certain types of human “inventions” — so what? ---
**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\ **Post date:** [November 15, 2018, 2:33pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/31 "2018-11-15T14:33:10Z")
> [@OldGuy](#): > > I guess I’d say the axioms are invented and the theorems are then discovered. This is my viewpoint as well. ---
**Author:** ![rat\_avatar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/rat_avatar/32/255_2.png) [@rat\_avatar](https://boards.straightdope.com/u/rat_avatar)\ **Post date:** [November 15, 2018, 2:35pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/32 "2018-11-15T14:35:50Z")
Imaginary numbers are only imaginary due to our conventions. Neither General Relativity nor QED would exist without complex numbers. While I get that it is commonly taught that imaginary numbers are not re-presentable in the real world that only applies if one sticks to the convention on number lines where integers are places and not distances. Euler’s Identity and Euler’s formula are all over the physical world and are one of the easiest ways without any other knowledge to derive the periodic nature of sin and cosin. Heck you can’t even do a proof of the the fundamental theorem of algebra, which is probably not universal but convention without complex numbers. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [November 15, 2018, 2:47pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/33 "2018-11-15T14:47:00Z")
> [@magellan01](#): > > Isn’t all mathematics describing a physical world, even if it’s a hypothetical one. It appears to me that this is false, unless you define “a physical world” in such a way as to make it trivially true. Are the Riemann Zeta function, or the Monster Group, or the Dirichlet function, or the Cantor set, or an n-dimensional sphere, physical things? ---
**Author:** ![Mijin](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mijin/32/9369_2.png) [@Mijin](https://boards.straightdope.com/u/Mijin)\ **Post date:** [November 15, 2018, 4:07pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/34 "2018-11-15T16:07:22Z")
> [@septimus](#): > > Stand back far enough in one’s perspective, and I’m not sure there is a good criterion to distinguish inventions from discoveries. > > Let’s contrast the design of a Frisbee, which most would call an “invention,” with the Monster Group, a “discovery.” > > Yet both of them can be described with marks of pencil on paper describing relationships. The rules from which the Monster Group builds may be more rigid and/or beautiful than the Frisbee rules, but at some level of abstraction they are similar. > > An alien civilization which discovered the Monster Group might also have discovered the Frisbee design. Yes, it would have little chance of duplicating certain types of human “inventions” — so what? Yes I was having the same thought. Consider if I find an effective combination in a particular chess position. Is it a discovery or an invention? Well it’s hard to say…the process for thinking up such a combination has elements of “searching” and elements of “constructing”. In this situation though most people would be happy with the idea that that combination didn’t “exist” in the universe prior to a human describing it (or even prior to humans inventing the game). Perhaps the invent / discovery framework is not the right framing. ---
**Author:** ![Kimstu](https://avatars.discourse-cdn.com/v4/letter/k/ecd19e/32.png) [@Kimstu](https://boards.straightdope.com/u/Kimstu)\ **Post date:** [November 15, 2018, 4:17pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/35 "2018-11-15T16:17:38Z")
> [@rat\_avatar](#): > > A few places. > > 1. The trace of a rotation of any polygon in 2D space :rolleyes: Now you’ve just replaced the abstraction of a perfect circle with the abstractions of a perfect polygon and a perfectly traced rotation. **No ideally precise geometrical object actually exists in perfect form in nature.** Which is why I think it’s epistemologically dubious to claim that our ideally precise geometrical abstractions are merely “discovered” from natural objects. ---
**Author:** ![DSYoungEsq](https://avatars.discourse-cdn.com/v4/letter/d/c6cbf5/32.png) [@DSYoungEsq](https://boards.straightdope.com/u/DSYoungEsq)\ **Post date:** [November 15, 2018, 4:18pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/36 "2018-11-15T16:18:49Z")
“There are only two ways in which we can account for a necessary agreement of experience with the concepts of its objects: either experience makes these concepts possible, or these concepts make experience possible.” Immanuel Kant (Datalinks) Seems to me there is no way for either _invented_ or _discovered_ to be exclusively applied. ---
**Author:** ![John\_Mace](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_mace/32/185_2.png) [@John\_Mace](https://boards.straightdope.com/u/John_Mace)\ **Post date:** [November 15, 2018, 4:19pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/37 "2018-11-15T16:19:55Z")
“Pi”, but itself is not math. It’s a measurement, or a ratio. If we go by the entry in wikipedia: > [@](#): > > Mathematics has no generally accepted definition.[33][34] Aristotle defined mathematics as “the science of quantity”, and this definition prevailed until the 18th century.[35] Starting in the 19th century, when the study of mathematics increased in rigor and began to address abstract topics such as group theory and projective geometry, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions.[36] Some of these definitions emphasize the deductive character of much of mathematics, some emphasize its abstractness, some emphasize certain topics within mathematics. Today, no consensus on the definition of mathematics prevails, even among professionals.[33] There is not even consensus on whether mathematics is an art or a science.[34] A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable.[33] Some just say, “Mathematics is what mathematicians do.”[33] then it’s going to be hard to categorize something unless we agree on its definition. But if we look at academic mathematicians and what they do, it would appear to be more invention than discovery. It’s more a form of philosophy than of science. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [November 15, 2018, 4:23pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/38 "2018-11-15T16:23:04Z")
> [@Mijin](#): > > Perhaps the invent / discovery framework is not the right framing. I agree that the distinction between discovery and invention is a fuzzy one. For example, Alexander Graham Bell **invented** the telephone, but this means that he **discovered** a way to transmit and receive sound over a distance through a wire. Does it make sense to use “discover” for things that would have been the same no matter who discovered them? In this sense I think much of math is discovered, though of course things like notation and terminology are invented. ---
**Author:** ![rat\_avatar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/rat_avatar/32/255_2.png) [@rat\_avatar](https://boards.straightdope.com/u/rat_avatar)\ **Post date:** [November 15, 2018, 4:27pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/39 "2018-11-15T16:27:08Z")
> [@Kimstu](#): > > :rolleyes: Now you’ve just replaced the abstraction of a perfect circle with the abstractions of a perfect polygon and a perfectly traced rotation. > > **No ideally precise geometrical object actually exists in perfect form in nature.** Which is why I think it’s epistemologically dubious to claim that our ideally precise geometrical abstractions are merely “discovered” from natural objects. This is just pure Sophistry. The fact that one cannot draw a perfect circle, and that one never under your definition never exists. Effects like circular polarization of EM radiation do exist, and the unit circle is far from a random quirk of axioms. A circle arises from rotation being possible, not from some mythical idealized drawing implement. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [November 15, 2018, 4:28pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/40 "2018-11-15T16:28:33Z")
> [@John\_Mace](#): > > “Pi”, but itself is not math. It’s a measurement, or a ratio. It’s not a measurement or ratio of anything that actually exists in the physical world. > [@](#): > > But if we look at academic mathematicians and what they do, it would appear to be more invention than discovery. Why do you say that? > [@](#): > > It’s more a form of philosophy than of science. But is philosophy invented or discovered? [Previous page](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542.md?page=1) [Next page](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542.md?page=3)