Mathematics: Invented or Discovered?

Mario Livio discusses the complementary processes of mathematical invention and discovery. While we invent some mathematical concepts—such as prime and imaginary numbers—by deciding how to define them, these concepts can lead to a plethora of mathematical discoveries.

Original Program Date: June 4, 2010

The World Science Festival gathers great minds in science and the arts to produce live and digital content that allows a broad general audience to engage with scientific discoveries. Our mission is to cultivate a general public informed by science, inspired by its wonder, convinced of its value, and prepared to engage with its implications for the future.

Subscribe to our YouTube Channel for all the latest from WSF.

[Neueregel]

*If any aliens exist in this universe, they should have the same set of axiomatic truths in their Maths. So, we just discover as much as we can. 'Invent' is a very inappropriate word. Example: Let's say, I deeply analyse a chess-opening sequence up to 30 moves. But this is not an invention, since the rules and combination of the piece movements always allowed such an opening-sequence configuration. The potential is always there. I don't invent anything, I just discover and probe deeply into the chess opening theory. The same with endgame tablebases. Computer chess engines DISCOVER more and more truthful and logically winning endgames*

[pauldhoff]

Discovered!

Writing down numbers is invented. The ratios behind it are discovered.

Paul

[bizzee]

The basic rules and concepts of mathematics are invented like we do with sports and games like baseball and chess. The intricacies of how those basic rules and concepts interact when adhered to is what is discovered.

[reallysearching]

Too cool -- I am not alone!  I remember when it occurred to me that math is not man-made, that with or without the occurrence of Homo sapiens, math would always be lying around "waiting" to be discovered, and I have been fascinated by that thought ever since!

[tombapilot]

Mathematics is a process used to make discoveries, but the process itself was also discovered - not invented.  Sure, we invented the symbols, but what the symbols represent was discovered, and the relationships between what the symbols represent were also discovered (e.g. the reality "1" was discovered, the reality "10" was discovered, and the relationship between 1 and 10 was discovered - 10 is a certain number of 1's).  Once you discover a relationship, you discover a process, and the process becomes more thorough with every discovery.  The formulations we arrive at are, as I say, arrived at rather than invented.  If they were invented, they'd just be opinion.

[avisachdeva]

I am of the opinion that math already exists. In fact, what already exists is not math. There is just patterns. We humans, in virtue of our nature to observe abstract concepts in nature, have invented a clever language that observes the patterns in the nature (which might not even be occuring intentionally but we call them patterns). Sometimes I think mathematicians force an idea to math their beliefs even if directly there is no correlation. So, math is very much invented as a language.

[danielgiegerz]

Von Neumans article, "Danger Signals" is a nice explanation of how abstraction, that useful in math, is rooted in empirical thinking. Abstract ideas, are based on abstract models, which in turn come from the empirical organization of information. That is to say, those who believe in circles, owe it to the moon.

[mosesexodus]

Mathematics, like Language, in its infinite manifestations is constantly being discovered. Mathematics, like Language, is the expression of an Intelligence that permeates All that Is, All that Was, and All that Will Be, in this and any other universe of existence. This expression facilitates self discovery of Intelligence with Itself within the 'physical' universe through the filter of us All.

[luck]

What's the difference?
Process of any invention is just a trajectory of search for the answer in some sort of configuration space. We usually agree that engines were invented. But to invent an engine, engineers were operating with imaginary objects in their mind, following rules of collisions, gas thermodynamics and mechanics. The first engineer of engine was probably juggling imaginary canons and pistons in their head until they formed some approximation of an engine, then used math to calculate optimal parameters, and got a concept of an engine. But if we say, that mind of the engineer fast wondering around different mechanical configuration untill it stumbled across an engine, then we can say that engine was discovered. We may argue, that act of invention is dedicated, whereas act of discovery is accidental. But then it depends on the methodology of a scientist, what is invented, and what is discovered. For example, a mathematician who is looking for a more succinct way of proving some theorem is inventing, but a mathematician who was aimlessly wondering about prime numbers and noticed some curious pattern, makes a discovery.

[julesskodzinski]

Math is ultimately a human concept. The world in the universe functions as it is and does not rely on equations. Math is our way of explaining and predicting these interactions

[md.fazlulkarim]

Mathematics is discovered. If any theory is mathematically proved right, then it will be proved right experimentally no doubt about that.

[winson]

If we didn't give a definition of Prime Number, it simply just does not exist.

[Daniazco]

The task is not to discover or invent is to create. For example. I did a mathematical equation to know if tomorrow will rain. Instead of that you say, I created a form to do tomorrow rain.

[emiliospowerballer]

well, the first sight of mathematics was seen as early as ancient sumer, but no one knows how they came up with this stuff. later they were further developed so the question is, who the hell made them up? they were discovered from us but they certainly werent invented from us becausd we just found them ready to be used. maybe the ancient civilizations were too smart for us

[tarunmishra]

Mathematics is language of Physics- nature. Then do we discover or invent a language or both?

[Faheemsnotes]

Wow, his idiology about iota was oxm, i confessed by his opinion

[jmerlo]

It is the same case as with Gravity or any other natural phenomenon.  Something that is there but nobody "had realized it = made it real" until someone "discovers" it.  "Inventing", however, means putting one or more of those phenomena and produce something new with them or make them do something new. Simple, isn't it?

[mathildaleon]

Türkçe alt yazısı yok mu acaba tellaffuzda zorlanıyorum bilimsel kavramlar olunca...

[ДмитрийБурбовский-мд]

#Математика определяет суть перемещения смысла последовательности устроения процесса, его все характеристики, заложенные памятью отсчёта, и активированные синтезом решения связи соответствия константы многомерности решения, в микромирном синтезе обоснования отсчёта!!!.Вы можете прекрасно создавать т.н.теории и решать их, но к математике при этом не иметь никакого отношения, ибо теории имеют связь с массовым определением участия их ВОВЛЕЧЕННОСТИ В ПРОЦЕСС, и создаются исключительно, для вышесказанного....Математика пространства являет право производить отсчет, именно ПРАВО, ИБО ОПРЕДЕЛЯЕТ КОНСТАНТУ ВАШЕГО СОУЧАСТИЯ В РЕЗУЛЬТАТЕ ПРОИЗВЕДЕННОГО СИНТЕЗА РЕШЕНИЯ ВЫХОДА!!!.#zZz#.

[xxxYYZxxx]

Theory & Mind regress to the same primordium. People love taking all the credit and accepting none of the blame; the alternative is understanding what causality, ie karma means.