As a PhD student who’s also doing algebraic topology coincidentally, I refer to the techniques that I use in proofs as things I invented, while the fact that I’m proving as discovered.

There’s an interesting connection to Turing-Completeness here. When a computer is Turing-Complete, it can simulate any other Turing-Complete system. So I think that while we may start in a particular axiom system, many other axiom systems will emerge from it or be dual to it. If our universe is itself a Turing-Complete system, then we are involved in this interlinked network. That’s why math is so effective in physics.

Ok discovered vs invented is probably too simplistic, so share your full ideas down below. And if you want a great office chair, use the code C750 for an extra $50 bucks off the C7 chair. ►bit.ly/3YXBW4v

Awesome explanation. Thank you (from Brazil ) for the video. I have a question. Wronskian = 0 doesn't imply necessarily that the funcions are linearly dependent, right? But Wronskian = 0 imply two functions are linearly dependent only if these functions are solutions to the same differential equation?? Thanks in advance, professor.

math is discovered by invention and invented by discovery

i think both

the systems of math are invented the resulting interactions are discovered conway invented his game of life but its resulting interactions were discovered the imaginary unit i was invented but its effects on math were discovered

Math can model nonsense just as closely as observations. It's invented.

I see them both correct, it depends on the perspective : Invented : we invented a way to manipulate the abstraction of nature rules, logic,laws, patterns... Discovered : we dicovered how quantities interact with eachothers ( numbers system is invente) Sorry my english is bad.

The mathematical realm exists as its own entity, even when humans aren't present, where we are able to access it through invention.

I'd say Invented. My reasoning being, Math is a language which we use to understand the universe. So just like an apple is discovered but the word "apple" Is invented, similarly all the laws of universe are discovered, but the numbers and formulae we use to represent then are invented.

We invented the game of chess, we didn't invent the universe. We set axioms to explore possibilities that fit reality

Chat gpy could even solve a simple gcd(), it kept confidently getting the wrong answer and even when I gave it the right answer it wrote the right answer but the wrong work to get there.

Is math invented or discovered? Yes.

Mathematician Descartes duality body-soul. Math is software thus invented language by humans but Nature speaks in language of math so all inanimate (body) is also written in the language of math. Thus our reality is based on math. That is philosophy of formalism. But no proof exist to separate soul and body. Quantum physics have proved duality fact that all matter (soul-body) is in wave and particle state. That is closest proof that God wrote reality in language of math where at micro level matter is just vibrating wavelet.

If axiom is invented and otherwise is discovered, why we can go to "some axiom" and "discover" another axiom, while the latter is just specification? So i do not think axiom is invented, it kinda discover through observation, and when observation become more deep, it relevelead another "weak axiom". Like euclid axiom. And then fith one.

Both. You invent the axioms and discover the consequences of your actions

Math is neither invented nor discovered. I will not elaborate

Hey thats me!

When going down math rabbit holes, you figure out it's kind of both. Math is just a formalized logic system where you start out with certain assumptions and make logical deductions from those assumptions. Regular mathematics (counting, adding, multiplication, algebra, trig, calc, basically anything to do with arithmetic and coordinate space) is about finding and measuring quantities. The reason that math has so much to do with the real world is because the initial assumptions about math are from the real world, and the stuff we strive to discover is related to real life problems. For example, the area of a square is the side length squared, however we first half to define what area is, what a square is, what a side of a square is, what length is, what the squaring operation is, what a number is, etc. and only from rigorously defining the problem can we discover an answer, which only has relation to real life because we defined everything from our real life experience. A good example of this are graphs, due to how we don't typically learn about them in basic education. A graph is a collection of nodes and their connections to each other, and from defining just those terms, we can define other terms (edges, faces, planar etc.) and then we can make logical deductions and assumptions about those things. A graph isn't some everlasting idea that somehow permeates the universe, it was created by humans. However, people don't fully understand such abstract ideas fully, even though they defined them. Deriving the logical conclusions about ideas that we defined is what math is really about.

axioms are invented, theorems are discovered

Discovered, nobody makes 1 + 1 = 2, we just invented the notion and symbols to express it but math is built off axioms which are discovered, and built upon it are theories that are discovered.

Chess was invented. Checkmates are discovered.

We can invent new chemicals by combining different molecules, but we do not create the elementary particles out of which they are composed and we cannot just will them to have certain properties. Whether a chemical cures or causes cancer depends on the laws of nature. If we want to invent a cure for cancer, we must discover what combination of molecules will do so. We cannot just choose for a mix of salt and sugar to cure cancer. Likewise, we can invent new mathematical objects and operations, such as matrices and their multiplication, by combining different definitions and axioms. However, this does not mean that we create the fundamental logical concepts out of which they are composed or can just will them to have certain properties. Once we define matrix multiplication as we do, the laws of logic determine that their multiplication is not commutative.

The iPhone isn't an invention; it' a collaboration. TSMC has the chip technology, for example. I think Tim Cook calls it "curated technology." Like, Apple decides which phone models have OLED screens, or not. Even fixed price data plans were part of the launch of iPhones. note: to me, the word "invention" is usually tied to people's ego.

The concept of Math is logic and logic was discovered and is a natural thing. The system of math itself was invented

In the the first chapter of the book 'Is God a Mathematician' by Mario Livio, he paraphrases remarks concerning Roger Penrose's three worlds hypothesis. They are: 1) the world of our conscious perceptions, 2) the physical world and 3) the Platonic world. The first is how we view everything with all of our biases, the second is 'physical reality' and the third is the world of mathematical forms (prime numbers, laws of physics etc). 'Physical reality' seems to follow the norms of the Platonic world. Second our conscious perception emerged from the physical world. The circled is closed by the fact that our conscious perception allowed us to discover the Platonic world. All arguments become an infinite loop, as does the invented versus discovered debate.

graph --> hyper-graph polyhedra --> abstract polytope integer --> complex numbers --> inf dimensional complex-like number integer --> surreal number

Down with platonism!

Mathematics, at the foundation, is purely invented. The interactions that arise from that foundation can produce emergent phenomena that are discovered rather than invented. Farther along there are other bits of invention too. There are parts of mathematics where the standard conventions are not logically consistent. For example calling the 0th term of a series 1, even though that isn't what the Oth term should be if you calculate it the same way as all the other terms. Usually you could make a working expression for the 0th, but it would be more work to calculate, so carving out a special case where the rules don't apply is easier. If you're faced with a choice between consistency and laziness in calculating, and laziness in calculation wins out in pretty much every case where it wouldn't seriously break other operations, then that's pretty compelling evidence of invention rather than discovery. Ease of use is a design feature, and one that was deliberated on. Mathematics is a generalized set of tools for modeling observed things. Models, sort of by definition, are invented. You can build a model and subsequently find out that it had features that you didn't realize were there, but it is constructed. We can 't see how all of the bits are connected to each other while constructing, so there's a lot of scope for surprises or discovery, but the structure as a whole is constructed.

As hoc I would claim all the following (though I can't proove all of that): We try to discover all (or art least as many as possible) real existing structures (structures that are instanciated in reality) by guessing some basic truth in the form of true propositions (which we call axioms) and try to discover as much derived truth (also in form of true propositions, which we call hypothesis, theorem, ...) as possible. We use definitions to describe structures that we want to study and we use an invented notation to reference and/or represent objects. We call a specific system of basic, derived truthes, definitions and notation an '(sub)area of Mathematics' and name it, to be able to differ between multiple ones. We call all areas of Mathematics combined Mathematics. If all axioms are real (in the sense that they are instantiated in reality), then we discovered all axioms, and discover all derived truthes. If at least one axiom in a set of axioms is not real (in the above sense), then we might derive truthes that are not instanciated in reality; I would call those truthes (axioms and derived) invented, though we might also discover real truthes (and we may not be able to differ between invented and discovered ones). However, I believe that invented axioms always lead to contradictions; that also seems to be an historical assumption, because as far as I know, whenever axioms lead to contradictions, mathematicians seemed to have either completely dropped such bad axioms (at least those they could identify) or they have replaced it with a more restricted axiom (which often came as a 'you can't do that specific thing').

Concepts are invented ("odd integer", "field") but properties are discovered.

Both. Basic math discovered. Imaginary numbers were definitely invented.

Math invents correspondence to nature from principles of numbers.

Euclid, Pythagoras etc. only proved why certain (systems, math's, etc.) worked. At school we were taught about these Theories & we didn't think much about it. Assuming that these things were invented by these people. Having a think later (now) these things must have been used by people for a long, long time, before these scientists decided to find out why they worked. E.g. Jenner & Smallpox. Jenner noticed that people would inoculate their children with Cowpox & had done so for a long, long time. He, himself was saved by inoculation at a young age.

I'd like to see mathematicians get into a cage match to work out their differences! 🤣

Everything exists in a vacuum until humans put numbers and words to it. Math is invented. Yet that snake example is a good one. We invented the concept of a snake, but nature invented the snake based on how we categorize it.

It's framed. Neither invented or discovered. Basically, it just gets more and more clear the more we learn about it. Every equation gets more focused the more advanced the mathematics become, or the more we need to do with it, so it's like framing a picture, and cropping parts of it. Like, all principles in math are just built from other smaller principles, starting with the basic operations, all the way to a shape that could describe Reiman's Hypothesis. You don't discover the shape, or invent it, but describe it.

To me, the most interesting part of mathematics is that pure mathematics so often becomes applied mathematics.

I would say math describes relationships, the relationships we find is what we discover. But the symbols and the way we approach addition, subtraction, multiplication, division, etc. I would say are invented, since you can use the Roman numerals for example, or a different method to represent addition. I like to think of it like Celsius or m/s. They are not the only way to describe reality, so they must be invented, we have for example Kelvin instead of Celsius. But their relationship to reality is real, and one could say is discovered. Another way to think about it is like natural language. The word Rock describes something real, something discovered. But the word itself is invented and a multitude of other words can be used from different languages to describe the same thing that the word Rock describes.

Is that a 3blue1brown pi creature on your desk?

Math is the language of science. Science is discovered and math is invented to describe it. The number 1 is not a prime. Why not? Choice or discovery?

infinity primes exist because of unit incremental number system. if you change the number system to only made up of primes or sqrt irrationals then ypu have erased prime definition to a new number system.

The "invention" position is self-contradictory. This needs to be teased out.

Imho We invented mathematical notation, but all the relationships that we study are discovered.

High school senior here. My take on this topic will probably sound very childish compared to some of you guys who have degrees. I strongly believe math was discovered. Before quadratic equations were written down, flying balls in the air were already obeying projectile motion. When the first cavemen walked on Earth, their heights were already following a normal distribution. Heck, our planet was already obeying circular motion when in orbit around the Sun. The hours of sunlight we get in a day throughout a year was already in the shape of a sine/cosine wave. Even the light rays themselves was following laws from linear algebra. I like to think of math as this ancient entity that transcends language, conflict, and time. It's abstract art in my eyes. You ever wonder why math has to be written down with symbols and notation? It's because math technically is nowhere in our universe. Physics phenomena, interest rates, height distributions, and card games are all manifestations of math. We never invented mathematical laws, we discovered them, buried below our very noses.

Isn't the concept of numbers an invention though? Yes, it represents some way that we explain what we see in reality e.g. there are 10 apples on the ground, but at the heart of it we invented the numbers that we use, and as such any derivative work that comes from that is also an invention. Either way, I'm not that hung up on the idea that I care that much to go any further with this line of thinking 😂

Is math invented or discovered? Yes!

The first person to discover that making something round reduces friction on a flat surface must also be the person who invented the wheel

Math, as we know it, is a human construct. It does try to abstract ideas beyond the human experience and in the process we discover new results inside this framework, but it is human made, not merely human made, but it is a eurocentric construct (not saying that europeans have all the credit, on the contrary, many things in math comes from eastern civilizations), and saying that it is "the language of nature" and that we're just "discovering it" puts eurocentric civilizations on a high pedestral. Yes, it is extremely potent as a tool to describe reality, but is not perfect, and neither is universal, believing that aliens would arrive at the same mathemathical structure as we did is assuming eurocentric culture is somehow superior for "figuring out nature's language" Edit: no, this is not the opinion of someone outside the highly mathemathical sciences, this is the opinion I've arrived as a theorethical physicist who works with quantum gravity, so I do study A LOT of high-level abstract math, and the more I study the more clear it is that this is a human-made language as any other