Math: Invented or Discovered?

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I couldn't find a thread like this anywhere (I searched through a number of pages), so I decided to make it. If I am wrong, then please merge; I've been known to be blind while exhausted D:

So, is math invented or discovered?

Be advised that math means different things to different people; for some it's a process, others it's a fact, others it's a concept, etc. and so on and so forth. So I suppose the easiest way for anybody to start here is to define exactly what they think math is. After that, do you think it is invented, discovered, or otherwise? Why? What leads you to this conclusion, and what do you feel is part of your background that led you to think in this way?

This is a debate thread, so I would advise more than just stating your beliefs; if you find something you disagree with written by somebody else, then say so! Talk, discuss, debate, but flame and you'll be toasted like a marshmallow over my firey passion.

So what is math to me? It is, to me, a process by which we interpret the world around us. In essence, we see something, and then we add that something together with our learning and make an interpretation. My actual belief is far more in-depth than this (I just finished writing a discussion folder for a Math class that involved this very question), but that is it in a nutshell. This arises mainly from the way I've come to experience other people; it is the way I see the world act around me that leads me to believe that math is a process of understanding. More than this, though, I feel that math is a process of creating. The best mathematicians don't just solve problems, they ask questions and look for ways of innovating the world around them. All hail Gauss!

So is math invented or discovered? To be honest...I don't know. And to be even more frank, I do not see the importance of knowing such, and by extension really do not care. Math is like religion in that if it has any origin, the origin itself is irrelevant to the present state. Don't get me wrong; history is MASSIVELY important when discussing both math and religion, but origins are not. When we discuss math and religion, we discuss it in such a way that we acknowledge it as a fact: it exists. Regardless of whether we believe it, the fact is that somebody else does. Unlike religion, however, math is present in all cultures of the world in some form; it's not like the triune god vs. pantheon vs. non-personal reality in that it is contested by other forms of itself; through formulas, identities, and other tools we can say that the ancient Mayan math can be translated into ancient Babylonian, which in turn can be translated into Egyptian and then into Greek and so on and so forth until it comes to our modern day concept of math, which all the ancient forms can also be translated into directly. We can, therefore, argue that math exists in the absence of compelling evidence to the contrary. Regardless of whether it is discovered or invented, it does exist.
 
Math is like religion? What? The historical aspect of mathematics isn't relevant to mathematics because it works on principles of logic and premises that exist today even if they were conjured thousands of years ago. For religion, history is very important. I mean the Bible is itself (well at least for the majority of books) a very large history book. Apples and oranges imo.

As for it being invented or discovered...I don't think those are very good terms to be used. It's invented in the sense that language was first invented. Mathematics is its own language to describe our natural world. But it's just a representation, and nothing more. In that sense you can also say it was discovered when a person first intuitively saw that certain natural principles tend to follow again and again. This is going to turn into a very semantical argument, so I agree with you when you pretty much said "who cares" about the origin of mathematics. Because in math, what matters are the principles. 2+2=4 was true in 2000 B.C. and 2+2=4 is true today. Who gives a damn about who actually figured that 2+2=4. That's not at all what mathematics is about. But don't confuse history of mathematics with study of mathematics. These are two different things. (And again, like I said before, this is very different from its comparison to religion as a form of belief and the justification underlying such).
 
Interesting one. If it was discovered, then who made it? God? Fuck off. If it was invented - how could humans have fathomed so much?

Is maths even right? If we changed certain equations would we discover more about life?

I haven't the foggiest but it seems a bit much for me to believe it was anything other than created to serve us. I have 2 foot. You eat foot. I have 1 foot ... etc. We manipulate numbers to serve our individual needs. Definitely something we've made up with the help of pretty big minds.
 
We've invented a system for interpreting mathematical concepts, which are based on the world around us. If there are two dogs, there are two dogs, no matter what word you use for "two" or how you convey that concept. Extraterrestrial civilizations thousands of star systems away might not use the same numerical systems that we use--they're probably not going to use base ten, for example--but that doesn't mean that the underlying concepts based upon the world around us aren't universal.

So, in the end, asking whether we "invented" or "discovered" math is a meaningless word game. The fact of the matter is that we discovered the way the world works and we invented a system to measure it (Which is probaby still very, very flawed; we still have much more to learn about how our universe works).
 
I couldn't find a thread like this anywhere (I searched through a number of pages), so I decided to make it. If I am wrong, then please merge; I've been known to be blind while exhausted D:

So, is math invented or discovered?

Be advised that math means different things to different people; for some it's a process, others it's a fact, others it's a concept, etc. and so on and so forth. So I suppose the easiest way for anybody to start here is to define exactly what they think math is. After that, do you think it is invented, discovered, or otherwise? Why? What leads you to this conclusion, and what do you feel is part of your background that led you to think in this way?

I distinguish abstract concepts from concrete things in that the concrete things don't exist until there is evidence for them. You may have an idea for a kind of hovercraft that runs on electricity, but until you can actually create one like it, it doesn't actually exist; it's just a concept. It's harder to talk about this in math because math doesn't exist in the concrete sense; it's an abstract concept. In that sense, it's always existed; we just haven't been aware of it. It's actually quite obvious how that works because there are often mathematicians who make discoveries about certain theorems or math ideas, but either they never publish it, or other mathematicians never read their works, so we never attribute the discovery to them, even though the discovery existed well before the public world knew about them. Who knows of all the things Archimedes discovered; perhaps he knew about 17-sided polygon constructions, but we wouldn't know because we don't have all of his written works. There was no time in history in which you couldn't construct a 17-sided polygon; the only reason we didn't was simply because we didn't know how to. It's the same with Gauss. The guy made tons of discoveries in math, but because he was so cryptic about it, his discoveries weren't made public until later mathematicians discovered the same ideas.

Another interesting perspective works like this: I have a conjecture about a mathematical theorem, but because it is only a conjecture, I have no solid proof for it, and it could be true or false. If it were true, then it's existed all along the entire time; I just had no way of showing that it works without fail no matter what. And during all the times I thought about this conjecture, it might actually have been true the entire time. The only reason why we never use it in practice is because it's not reliable enough for us to use since the proof does not exist. And I think it was like this with the four color theorem for a long time. Interestingly, the proof was made with the help of a computer.

So what is math to me? It is, to me, a process by which we interpret the world around us. In essence, we see something, and then we add that something together with our learning and make an interpretation. My actual belief is far more in-depth than this (I just finished writing a discussion folder for a Math class that involved this very question), but that is it in a nutshell. This arises mainly from the way I've come to experience other people; it is the way I see the world act around me that leads me to believe that math is a process of understanding. More than this, though, I feel that math is a process of creating. The best mathematicians don't just solve problems, they ask questions and look for ways of innovating the world around them. All hail Gauss!

What I actually find most amusing about math is that we often consider it to be an abstract concept, so if it doesn't make sense or we don't quite understand how something works, why would we worry about it? It doesn't have anything to do with the real world, does it? And the irony is that it does; for such a highly abstract concept as math, it gets used an awful lot in many academic fields, including statistics, economics, businesses, sciences, computing sciences, security, etc. The possibilities are limitless. I suspect that people in those fields find math useful because as an abstract concept, it is good at explaining models that end up being used in practice.

Math is one of the few abstract concepts that are designed to be consistent and be compatible with logic. Although I should phrase that as any abstract concept which we categorize as math should be consistent and compatible with logic (since we haven't actually created any ideas). I suspect that's why it's universal for many things, and that's also where its success comes from.

So is math invented or discovered? To be honest...I don't know. And to be even more frank, I do not see the importance of knowing such, and by extension really do not care. Math is like religion in that if it has any origin, the origin itself is irrelevant to the present state. Don't get me wrong; history is MASSIVELY important when discussing both math and religion, but origins are not. When we discuss math and religion, we discuss it in such a way that we acknowledge it as a fact: it exists. Regardless of whether we believe it, the fact is that somebody else does. Unlike religion, however, math is present in all cultures of the world in some form; it's not like the triune god vs. pantheon vs. non-personal reality in that it is contested by other forms of itself; through formulas, identities, and other tools we can say that the ancient Mayan math can be translated into ancient Babylonian, which in turn can be translated into Egyptian and then into Greek and so on and so forth until it comes to our modern day concept of math, which all the ancient forms can also be translated into directly. We can, therefore, argue that math exists in the absence of compelling evidence to the contrary. Regardless of whether it is discovered or invented, it does exist.

That's one of the most appealing aspects of mathematics--that you can use it over and over in several different forms, and it will still be the same thing. Whether you're counting bytes in binary or decimal, you're basically doing the same thing. Or if you change an algebraic equation to something else, it's really just the same thing. Or solve a complicated math problem a different way--the solution is the same. That's why it's so beautiful, and that's also why it's infinitely more plausible than religion. It's got consistency, and it's actually useful.
 
Both, but leaning toward the discovered side.

The design was clearly laid out beforehand, but we've invented ways to manipulate mathematics and use it to our advantage, just like everything else that was discovered in nature. Everything in the world is discoverable. That's what makes it awesome.

Water, sugar, and lemons already existed, but lemonade was invented from them. Basically, God gave us the tools, we make unique creations from his creations. Math is no different. While we didn't invent it, we can use it to make something new, something specific.
 
To be fair, I did not read any of the responses on top.
In my perspective, I believe that anything in general has to be discovered in ordered to be invented.
For Math, it is the same thing. My explanation for this is because if you invent something first, and then say you discovered it, it would not make any sense. But for something to be discovered and then you claim it to be invented from your discoveries, that would make a lot more sense.
 
Math itself has never been proven "wrong", nor can it.

Well, it can be taken out of context, or justified on faulty logic, but then that's our problem; not math.

Something most people don't realise when they question math or "changing equations" is what is required for any mathematical claim to be proven. For this to happen someone has to discover something called a mathematical proof. At the most basic level this is an equation that proves that it is physically impossible for any other result to take place - in this way math can prove a number (for instance pi) will continue to an infinite number of decimal places.

Well actually, I'm not sure if it always says other things can't happen; that depends on the theorem. There are some theorems that make specific statements on when something is possible (eg, if an equation has a particular characteristic, such as being linearly independent), and if it satisfies those conditions, then something is true; but it doesn't have to say anything about the converse. For example, there are theorems on the uniqueness of fixed points and whether or not they converge, and while these theorems will guarantee that certain equations will have fixed points that are unique and converge, it does not guarantee that other equations which do not match the conditions in the theorem are not unique or converge. That is to say that some theorems only suggest conditions that are sufficient; not necessarily necessary.

For an interesting study in how these proof work, try reading about Fermat's last theorem. This was a statement made (for instance a + b can never equal a - b). It took 350 years for mathematicians to actually be able to provide a proof for it (It was only solved in the 1990's). Be warned - it's not easy reading and the proof itself will probably be completely unreadable to you.

Yes, it was proven by a guy called Andrew Wiles, but I think he found a hole in his proof at some point. He had to go back and fix it. The interesting thing is that Fermat's last theorem would probably have worked in those last 350 years; again, we just don't consider using it because we don't know if it always works.

Angelus will probably be better as far as specifics goes, my area of interest is physics, she is the math expert and is more knowledgable about it than me by several orders of magnitude. :P

Well, physics and math are related, so I'm sure you would know some of it anyways. I guess I'm just more keen on the abstract part of it.
 
So, is math invented or discovered?

Yes.

The tricky part of this question lies in the fact that most mathematical concepts are human constructs...starting back with the Arabic numerals and basic set theory.

Someone (no clue who) had to decide that N={1,2,3,4...}, Z={...,-2,-1,0,1,2,...}, etc. If the basic sets of numbers that dominate all the other fields of mathematics were not "created", then we would have no basis for constructing proofs. While it may seem basic to us, we must remember that at some point in history, even the concept of a prime number did not exist...as far as humankind was concerned.

Think about a system like the differential calculus. Newton (or Leibniz, whatever) derived the calculus. He invented the calculus based on his observations (note that he was something of a physicist as well), but he still created the rules. Before Newton there was no calculus of the like we know today.

Lastly, there are the axioms to consider. Axioms are statements much like theorems, but they serve as the basic logical building blocks since they do not require proof, unlike theorems. For example, it is axiomatic that x=x in any mathematical structure. There isn't any way to "prove" that's true, but without the idea of equality, you would have trouble proving much of anything in all of mathematics. Again, someone had to formulate these axioms (like Zermelo formulated the Axiom of Choice), but the statements of the axioms is again based on careful observation of mathematical phenomena.

So, to answer the question, I think it's a bit of both. Humans formulated the formal structure of the mathematics we use today...but their formulations were based on observations of how the world actually behaved.
 
We've invented a system for interpreting mathematical concepts, which are based on the world around us. If there are two dogs, there are two dogs, no matter what word you use for "two" or how you convey that concept. Extraterrestrial civilizations thousands of star systems away might not use the same numerical systems that we use--they're probably not going to use base ten, for example--but that doesn't mean that the underlying concepts based upon the world around us aren't universal.

So, in the end, asking whether we "invented" or "discovered" math is a meaningless word game. The fact of the matter is that we discovered the way the world works and we invented a system to measure it (Which is probaby still very, very flawed; we still have much more to learn about how our universe works).


I'm going to have to sum my thoughts up to this as well.

M1ghty Mous3
MOD EDIT: Can you please put a little more effort into your post please? This is a spam post in a post count section. Just elaborate a little more on your thoughts. Thank you.
 
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Let me first start out by saying I am not a mathematician but a statistician. So while I have some (alot?) of overlap I do not do pure mathematics.

I can confidently say that the notation we use for math was invented. But I do think that in general a lot of the rules & relationships have been discovered.

For instance anyone who knows statistics knows that you can compare groups through a number of different approachs- comparison of means (via t-tests & anova, which looks at within group variation to between group variation) or via regression (which predicts a variable from a set of other variables).

These two approaches were developed sort of side by side and were used to answer different research questions until R. A. Fisher showed them to be the exact same.

I have other examples where methodsconverge but they are even more abstract unless you have taken pretty advanced stats.
 
^^
You see, the problem with saying mathematics is "invented" is really dependent on how you view mathematics itself.

Now I've taken mathematics past advanced calculus @ university level, so I can tell you that upper-level mathematics looks nothing like arithmetic, algebra, or calculus that's done in highschool/intro-college level. You stop working with numbers and rote calculation, and begin to work with ideas. The notation or numbers used is not mathematics, it's just a representation of an idea, that in turn is a reflection of what we can discern about the natural world around us. And this is the essence of mathematics, not the numbers, or theorems, or formulas, but the actual idea that is being represented. Everything else stems from this basic foundation.

So as an example just to demonstrate this point a bit (well at least the best I can): Take any object on your desk at the moment, let's just say it's a cup. Now you can describe this cup through the English language. You can talk about its size, its shape, its color, its density, its material, the sound it makes when you hit it, etc... Will you ever adequately represent exactly what's sitting on your desk? Probably not, but you can get pretty darn close to representing as fully as you can this object through the use of language. So the language itself will always be inadequate, but the cup will not. The cup is the focus, not the language. An idea of mathematics is the cup. The language of mathematics is English. (Hopefully that makes some sense :/)

So mathematics, the language itself, was "invented." But the concepts it represent (which is the crux of mathematics really) were discovered. You cannot invent something that has always existed in the natural world.

That's why this all just turns into semantics of what is "mathematics," what is "invent," what is "discover," and is really meaningless to talk about. What's important is understanding what mathematics is conceptually, and everything else will just follow naturally.
 
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So mathematics, the language itself, was "invented." But the concepts it represent (which is the crux of mathematics really) were discovered. You cannot invent something that has always existed in the natural world.

That's why this all just turns into semantics of what is "mathematics," what is "invent," what is "discover," and is really meaningless to talk about. What's important is understanding what mathematics is conceptually, and everything else will just follow naturally.

I think this is what I was trying to say, in so many words. In order to discuss and manipulate mathematics, humankind had to come up with some sort of system...but the basis for that system was observation of natural phenomena.

The Fundamental Theorem of Calculus, for example...someone had to sit down and write it and its parts down. However, does that mean that the ideas behind the formulas wouldn't exist had they not been written down? It seems to me the debate just dissolves down into semantics and not mathematics after a certain point.
 
mathematics as we know it is the language used to explain organization, invented by man. said organization exists in nature, has always been and always will be, long before and long after humanity ceases to exist. we can explain it, but we didn't create it - it was always there.

basically, i'm saying mathematics is the study, not the thing being studied. in that sense, we did create math. but we didn't create the concepts it teaches.
 
We've invented a system for interpreting mathematical concepts, which are based on the world around us. If there are two dogs, there are two dogs, no matter what word you use for "two" or how you convey that concept. Extraterrestrial civilizations thousands of star systems away might not use the same numerical systems that we use--they're probably not going to use base ten, for example--but that doesn't mean that the underlying concepts based upon the world around us aren't universal.

So, in the end, asking whether we "invented" or "discovered" math is a meaningless word game. The fact of the matter is that we discovered the way the world works and we invented a system to measure it (Which is probaby still very, very flawed; we still have much more to learn about how our universe works).



I agree with everything this man said. In fact its what I was planning to say until I read this post. We could call the number three banana and four apple and we would still have the same result.....pinapple ^^
 
Is it not true that Einstein himself (I'm going to paraphrase, here) realised that he could not get particular equations to work the way he wanted, so invented differentiation and integration in order to manipulate them the way he wanted?

Incredibly complicated thing to do, I imagine. Fantastically useful...but man-made, wasn't it?
 
Yes, but that would also imply that there was a point in time in which differentiation and integration don't apply, which is untrue; we're just unaware of it.

It's the same with non-Euclidean geometry; Gauss was aware of its existence way before the people who published their results on it. He just never considered his theories about it "perfect" enough to be published, so it remained unknown to the rest of the world. If our world is non-Euclidean, than it was so before people published papers on it, and it was so before Gauss discovered it, and it's probably always been like that, unless there's evidence to suggest otherwise.
 
Is it not true that Einstein himself (I'm going to paraphrase, here) realised that he could not get particular equations to work the way he wanted, so invented differentiation and integration in order to manipulate them the way he wanted?

Incredibly complicated thing to do, I imagine. Fantastically useful...but man-made, wasn't it?


Actually it was Isaac Newton.

Newton was insanely smart, in chronological order:
Discovers law of optics (white light is composed of other colors).
Discovers 3 laws of Motion.
Discovers universal law of gravitation.
Then he figures out why planets orbits are ellipses- and proves this by inventing/discovering calculus.
AND THEN HE TURNED 26.

Think about that.
 
Math is really just another way of explaining how nature works. So, again: man invented math, a language used to explain the laws of the universe. Man did NOT create said laws, they have always been in place.

Actually it was Isaac Newton.

Newton was insanely smart, in chronological order:
Discovers law of optics (white light is composed of other colors).
Discovers 3 laws of Motion.
Discovers universal law of gravitation.
Then he figures out why planets orbits are ellipses- and proves this by inventing/discovering calculus.
AND THEN HE TURNED 26.

Think about that.
I'm thinking about it, and it's making me feel like a total failure in comparison. At least I have my video games.
 
I think it's a mixture of discovery and invention. Man has always seen a numerical significance to everything, and through that, they made a mathematical system.
 
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