# 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:** 12\ **Page:** 5
**Author:** ![Jackknifed\_Juggernaut](https://avatars.discourse-cdn.com/v4/letter/j/9de053/32.png) [@Jackknifed\_Juggernaut](https://boards.straightdope.com/u/Jackknifed_Juggernaut)\ **Post date:** [November 19, 2018, 9:40pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/81 "2018-11-19T21:40:34Z")
Perhaps this thread has gone way past this, but I liked this Nova special on the mystery of how mathematics describes the universe so well… [![]( " - YouTube") ](https://www.youtube.com/watch?v=JiH7lMGW60A) ---
**Author:** ![Half\_Man\_Half\_Wit](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/half_man_half_wit/32/21766_2.png) [@Half\_Man\_Half\_Wit](https://boards.straightdope.com/u/Half_Man_Half_Wit)\ **Post date:** [November 20, 2018, 4:51pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/82 "2018-11-20T16:51:14Z")
> [@Nava](#): > > Well, QM being more intuitive in terms of Clifford algebras requires one to know what a Clifford algebra is 🙂 I don’t, so in my case I’ll need either to stick to imaginary numbers or for someone to explain Cliffords. > > (mind you, it’s possible that I know them but just not by that name) Well, it’s not terribly important—basically, they’re a sort of generalization of complex numbers in a similar way to how the complex numbers are a generalization of real numbers. I was just intending to point out that just because some particular formalism makes a description of a certain set of phenomena ‘easier’ in some sense, doesn’t really lend itself to a good argument for the independent reality of those concepts in particular. The more general form of this argument is the Quine-Putnam indispensability argument: mathematical entities are indispensable to our description of reality, and thus, we ought to be committed to their existence just as well as we are committed to the existence of other theoretical items, like electrons, quarks, or spacetime. But I don’t believe this really holds water, either: see the reference to Hartry Field’s nominalization of Newtonian mechanics above. Mathematical entities aren’t really indispensable—although they can make things much more convenient. In general, to put it into a slogan, I think mathematics is the science of structure, and structure basically is the sets of relations the elements of the world (or some domain of interest) may enter into. For another, more intuitive definition of ‘structure’, I submit ‘that which carries over from a system to a model of that system’. To see this, consider the set of your paternal ancestors. They can be ordered according to lineage: who is whose ancestor, basically. Furthermore, consider the set of books on your shelf: they can be ordered according to thickness. The fact that both sets admit of such an ordering relation (i. e. both have the same structure) means that one can be used as a model of the other. All we need for that is a mapping between them: say, _The Old Man and the Sea_ is mapped to Zachariah, while _Moby Dick_ is mapped to Jeremiah. Now, instead of laboriously looking up who is who’s ancestor in old records and whatnot, a look at the books on your shelf convinces you that Jeremiah is Zachariah’s ancestor—you can answer questions about the original domain by looking at the model, because both implement the same structure (the ordering relation). The same goes for other models—an orrery models the solar system because the little metal beads stand in the same spatiotemporal relations to one another as do the planets. Model-building is implementing the relevant structure of an object system within a model; and, if the implementation is faithful, you can use the model to answer questions about the original, and even make novel predictions. The structure that made this possible is mathematical: both the books and the ancestors form a well-ordered set. Once you thus get the hang of how modeling works, you may want to study the sorts of structures that enable this modeling in their own right. To do so, you craft a standardized modeling system—say, a set of symbols on paper, plus rules governing their manipulation (or, alternatively, a certain mental framework, or a physical instantiation, like a computer). This way, you can construct structures that have never before been realized in nature. But you don’t pull them out of some Platonic ether; rather, you simply do the analogue of building models, with the building blocks dictating what sort of things you can build. It’s ultimately not all that different from Lego: you can use Lego to build things that exist in the real world—a model of the Statue of Liberty, for example—but just as well, you can use it to build things that have never before existed. That’s what you do with mathematical symbols (or computer programs). Indeed, computers play a special role here: they are, in a sense, the ultimate models—the phenomenon of computational universality means that you can essentially instantiate any structure whatsoever within a computer. They’re sort of the ultimate modeling clay. That goes just as well for our brains, to the extent that they compute: that’s how we come up with new mathematics that doesn’t describe anything in nature, that has never been realized in any concrete form, without having to postulate Platonic shadow realms. That new math is simply instantiated in the structure of the stuff of our brains—neuron firing patterns, perhaps, or whatever else underlies cognition in the end—in the same way that a simulation of some non-existent universe may be instantiated in a supercomputer’s ones and zeros. ---
**Author:** ![bump](https://avatars.discourse-cdn.com/v4/letter/b/7c8e57/32.png) [@bump](https://boards.straightdope.com/u/bump)\ **Post date:** [November 20, 2018, 5:36pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/83 "2018-11-20T17:36:50Z")
> [@Nava](#): > > Proposal: > > mathematical relationships exist independently of any observer. They can be discovered. > > How to express them is something which requires someone to express them. These are invented. That’s very close to what I was thinking, except that I was thinking of it more like we have to invent a language of sorts (mathematics) to describe the relationships that exist in nature. I mean, the ratio of a circle’s diameter to its circumference just IS, but we had to come up with a way to describe that: 22/7, 3.14159, π, pi, 4(arctan(12)+arctan(13)) , 4(1−13+15−17+⋯) , and so on. ---
**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 20, 2018, 6:53pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/84 "2018-11-20T18:53:06Z")
> [@bump](#): > > That’s very close to what I was thinking, except that I was thinking of it more like we have to invent a language of sorts (mathematics) to describe the relationships that exist in nature. > > I mean, the ratio of a circle’s diameter to its circumference just IS, but we had to come up with a way to describe that: 22/7, 3.14159, π, pi, 4(arctan(12)+arctan(13)) , 4(1−13+15−17+⋯) , and so on. Why do you specify “exist **in nature** ”? ---
**Author:** ![bump](https://avatars.discourse-cdn.com/v4/letter/b/7c8e57/32.png) [@bump](https://boards.straightdope.com/u/bump)\ **Post date:** [November 20, 2018, 7:07pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/85 "2018-11-20T19:07:40Z")
> [@Thudlow\_Boink](#): > > Why do you specify “exist **in nature** ”? Why not? I just meant that exist in the universe, regardless of whether or how we perceive them. That’s why I used pi as an example- it’s a specific ratio that basically is what it is- there are a whole lot of ways we’ve devised to describe or note it, but the actual ratio is independent of our notations or formulas. ---
**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 21, 2018, 5:30pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/86 "2018-11-21T17:30:52Z")
After a phone call with my brother, who happens to be ABD in complex math…and after falling into the time old pit of trying to decide if we use the math or the physics concept of a vector we realized he had never seen a good visualization of curl and how that apply to Maxwell’s equations. 3blue1brown has a great video that helps with this [Divergence and curl: The language of Maxwell’s equations, fluid flow, and more](https://youtu.be/rB83DpBJQsE) While he doesn’t go into enough detail to explain it fully, at around the 10:00 mark he does cover how the electrical and magnetic components work and that may explain why I think that complex numbers are well suited for explaining EM. The 90 degree turn and the “imaginary” component are important here, as they are simply effects of the same fundamental property but from two different frames of reference. If you aren’t familiar how relativity relates the two here is a video from veritasium that may help. [![](https://img.youtube.com/vi/1TKSfAkWWN0/maxresdefault.jpg "How Special Relativity Makes Magnets Work") ](https://www.youtube.com/watch?v=1TKSfAkWWN0) The thing I failed to describe above on why the _imaginary_ concept is good is because based on your frame of reference you can only interact with one part of the EM relationship. You have to change your frame of reference to experience the other which works perfectly with the “imaginary” line/plane/hyper-sphere. While you can still dismiss that as being emergent from how it is defined, it is an amazingly useful and applicable _invented_ form. You can create math that will come up with the same answer but typically those lose the “always perpendicular” property of the imaginary model that allow for isomorphic translations and actually accurately describe the change in frame from the concept of space-time. ---
**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 21, 2018, 6:52pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/87 "2018-11-21T18:52:14Z")
> [@Thudlow\_Boink](#): > > Why do you specify “exist **in nature** ”? > [@bump](#): > > Why not? I just meant that exist in the universe, regardless of whether or how we perceive them. > > That’s why I used pi as an example- it’s a specific ratio that basically is what it is- there are a whole lot of ways we’ve devised to describe or note it, but the actual ratio is independent of our notations or formulas. But pi is the ratio of the circumference to the diameter of a _perfect circle_, and there are no perfect circles in nature. Although they are, of course, idealizations or abstractions of things that do. You could say that perfect circles exist within the human mind; if you’re a Platonist, you could say they exist in the Platonic world of Forms; you could say they exist within certain mathematical systems, like Euclidean geometry; or in the sense that a mathematician might use the word “exists,” like in a statement like “For every epsilon \> 0 **there exists** delta \> 0 …” ---
**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 21, 2018, 7:30pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/88 "2018-11-21T19:30:52Z")
> [@Thudlow\_Boink](#): > > But pi is the ratio of the circumference to the diameter of a _perfect circle_, and there are no perfect circles in nature. Although they are, of course, idealizations or abstractions of things that do. > > You could say that perfect circles exist within the human mind; if you’re a Platonist, you could say they exist in the Platonic world of Forms; you could say they exist within certain mathematical systems, like Euclidean geometry; or in the sense that a mathematician might use the word “exists,” like in a statement like “For every epsilon \> 0 **there exists** delta \> 0 …” There are lots of perfect circles in nature, there are no perfect circles on _graph paper_. You are conflating two human creations and the those limitations while ignoring that lots of models absolutely require _rotational symmetry_ for us to even describe. Rotational symmetry simply does not exist without “perfect circles” ---
**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 21, 2018, 7:40pm UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/89 "2018-11-21T19:40:44Z")
As a simplified example, Newton’s law of universal gravitation would be impossible. Under this model the gravitational potential does not depend on anything but the distance between two bodies. This is due to perfect circles or really spheres existing and is an example of _Rotational Symmetry_ in nature. Nature really doesn’t care that we can’t draw a perfect circle on a piece of paper, in fact nature doesn’t care that we developed the concept of drawing at all. Nature however, does force us to use the idea of rotational invariance to describe how nature works. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [November 26, 2018, 1:06am UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/90 "2018-11-26T01:06:30Z")
Having spent the last 55 years of my life doing mathematics, I can only affirm that it has been a voyage of discovery. It is most like exploring an abstract landscape and discovering what is in it. Most (though not all) mathematicians will agree with this. ---
**Author:** ![cerberus](https://avatars.discourse-cdn.com/v4/letter/c/71e660/32.png) [@cerberus](https://boards.straightdope.com/u/cerberus)\ **Post date:** [November 26, 2018, 4:02am UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/91 "2018-11-26T04:02:53Z")
Yes ---
**Author:** ![cerberus](https://avatars.discourse-cdn.com/v4/letter/c/71e660/32.png) [@cerberus](https://boards.straightdope.com/u/cerberus)\ **Post date:** [November 26, 2018, 4:04am UTC](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542/92 "2018-11-26T04:04:24Z")
There are merits to both Platonist and formalist perspectives. [Previous page](https://boards.straightdope.com/t/is-mathematics-invented-or-discovered/824542.md?page=4)