
Was mathematics invented or discovered?
You could say that unless humans existed, things such as numbers, equations, etc.etc. wouldn't exist. But you could also argue that shapes such as circles, squares, etc. and concepts such as area, volume, etc., always existed, and they just needed to be found.
I feel this is a slightly foolish question, but I've not been able to find satisfying arguments on either side, so I would love to hear your opinions.
I don't think that mathematics itself has been discovered or invented.
Things in mathematics themselves like "pi" or "e" or formulas always existed since they are constant.
EDIT:
from further down wrote:  Mathematics itself isn't an invention or a discovery, but a collection of facts or theories which are inventions or discoveries 
Aliens (with also ten fingers) would have the same rules.
i don''t know about mathematical systems based on other numbers (we have a ten numbered based system)(forgot the proper name)
It is a very old question and there are supporters of both viewpoints.
Here is an interesting essay on the matter;
http://www.people.ex.ac.uk/PErnest/pome12/article2.htm
maths were disvocered, not invented
OK, I've read some of the article Bikerman linked to, and my opinion at the moment is that Mathematics was discovered (hardly surprising given that I would seem to fall into the category of absolutist). We didn't invent for instance the fact that the sum of the squares of the two smallest sides of a right angled triangle equals the square of the other side (Pythagoras' Theorem), we simply discovered that this relationship existed.
A more simple explanation would be that if you have one item and add another it equals two. This always existed, we simply gave the variables names. We didn't invent the idea that 1+1=2, we simply named the variables involved.
Now i shall read the rest of the article, and probably completely change my opinion.....
Genesiz wrote:  OK, I've read some of the article Bikerman linked to, and my opinion at the moment is that Mathematics was discovered (hardly surprising given that I would seem to fall into the category of absolutist). We didn't invent for instance the fact that the sum of the squares of the two smallest sides of a right angled triangle equals the square of the other side (Pythagoras' Theorem), we simply discovered that this relationship existed. 
Unless I'm very mistaken, we invented concepts like degrees of angles, right? We could have, for example, consider the curved end of an arc of a circle and call it an angle, right?
Genesiz wrote: 
A more simple explanation would be that if you have one item and add another it equals two. This always existed, we simply gave the variables names. We didn't invent the idea that 1+1=2, we simply named the variables involved.
Now i shall read the rest of the article, and probably completely change my opinion..... 
We could have also called the variables 1 and 2 as 5 and 6, so it would come to 5+5=6, which, under present system of Mathematics, a false statement.
what I'm trying to say is that we took some fundamental truths (such as 1+1=2), and then framed (invented) the entire Mathematics model around it.
Thanks, Bikerman, for that link. It was quite an absorbing read.
@Klaw 2 : From your post, what you are trying to say is that Mathematics itself isn't an invention or a discovery, but a collection of facts or theories which are inventions or discoveries, right?
saratdear wrote:  @Klaw 2 : From your post, what you are trying to say is that Mathematics itself isn't an invention or a discovery, but a collection of facts or theories which are inventions or discoveries, right? 
Yes I did mean that.
@saratdear  we didn't develop the ideas of angles and squares solely for the purpose of rightangled triangles. They were developed for something else, and then we discovered that this pattern existed. We didn't invent this relationship, or made it work around an existing system, we used the system we already had and discovered that this pattern existed within this system.
saratdear wrote:  what I'm trying to say is that we took some fundamental truths (such as 1+1=2), and then framed (invented) the entire Mathematics model around it. 
Surely you have contradicted yourself here. If we invented mathematics around a fundamental mathematical truth that already existed, then mathematics must have been discovered. We didn't invent these fundamental truths, they already existed.
Genesiz wrote: 
saratdear wrote:  what I'm trying to say is that we took some fundamental truths (such as 1+1=2), and then framed (invented) the entire Mathematics model around it. 
Surely you have contradicted yourself here. If we invented mathematics around a fundamental mathematical truth that already existed, then mathematics must have been discovered. We didn't invent these fundamental truths, they already existed. 
What I was actually trying to say was :
OK, we know that if we take 1 thing here and 1 thing there, we have two things. So if we add three things and six things, we get 9 things. That is the truth. Now, we invented basic procedures such as adding things together, subtracting, multiplying, etc. What I'm trying to say is that we could have invente a completely new calculation where...say...56#3 (where # is the sign, just like +. . etc.) means adding 3, then multiplying with 3, etc. So, we invented these.
I know it sounds weak, but I am pretty bad at framing arguments, so bear with me.
So how do you explain that things such as squaring and pi and e etc. have patterns in other things that we never even thought they had, and the 'discoverers' probably never even knew existed.
"Math" existed before life. I'm going to have to go with discovered.
Genesiz wrote:  So how do you explain that things such as squaring and pi and e etc. have patterns in other things that we never even thought they had, and the 'discoverers' probably never even knew existed. 
Now that is something which I don't have a definite answer of. We'll just wait for others I guess.
Well given that mathematics is universal, I'll have to go with the concepts being discovered and the symbols being invented.
Neither discovered nor invented  Just Needed
It seems a foolish question, but actually it's very deep. Is it mathematics really the language of nature? Does nature calculate numbers when phenomenons occurs? Or things just happens, and it's lucky that our calculus get the same results?
Mathematics defines stuff, and once defined, a whole bunch of consequences results. You just have to discover them. If you define things different, different results apears. So, my vote goes for invention.
But how likely is it that all the mathematical theories and formula's that we have fit into the overall 'big picture'. Given the massive unlikelihood of this happening, this must surely suggest that mathematics was discovered.
first there is an interesting book in this area
http://www.amazon.com/InventionInfinityMathematicsArtRenaissance/dp/0198523947
it is about infinity invention
THe review of the book says
"Here is the fascinating history of the emergence of modern mathematics during the Renaissance, and its intimate relationship with the artisan and artistic traditions of the time. The book covers the period from 1300 to 1650, when craftsmen were educated in "practical mathematics," and when the field of mathematics was gradually taking up a more significant place on the intellectual landscape. Field traces the influence of the mathematics of perspective in the arts, and shows how this led to the invention of a new kind of geometry in the 17th centurythe new projective geometry of Desargues" source Editorial Reviews
in my next post I will tell you some sources about the discovery aspect of maths
Math was discovered and then had names invented for its concepts. Math is like time, it existed before we named it but we had little understanding of it before then.
Math is usually regarded as being one of the most perfectly accomplished branches of science, as it has the ability to put up definite rules and answers, that match each other perfectly. Though at the same time, it can be argued that it is, if any branch, removed totally from the empirical 'reality' that it describes, as it works with symbols that are linked, not to the surrounding world, but to themselves.
Following this last point, one could see math as being a constructed abstraction. Though this totally depends on which angle you lay on this I guess...
Klaw 2 wrote: 
Aliens (with also ten fingers) would have the same rules.
i don''t know about mathematical systems based on other numbers (we have a ten numbered based system)(forgot the proper name) 
We have many other systems than decimals.
It would be cool if we met some aliens that calculated exclusively in binary. It's sure a lot more possible than that they would have chosen such a random number as 10 to base their main system on. Unless, as you say, they had 10 fingers.
Quote: 
There are 10 kinds of people in the world. Those who understand binary and those who do not.

I think that mathematics is neither invented or discovered. It just grows all the time. In the beginning, mathematics was counting some things. Out of this there are some rules intuitively growed. Like how to divide some food over a few persons. This was the start of equations...
More specialized forms of mathematics are invented or discovered by accident...
since part of my course is history of computing, i will have to say that maths was invented, it is a branch of science that allows us to come up with solutions to real life problems. (and some other speculative fields)
I didn't read all the posts, but someone mentioned a triangle...where in nature have you seen a perfect triangle? it is a man made invention. we made it and we found the relationship beetween it's sides and angles.
Also, the two words have very close meaning in this case, making this question stupid.
We invented the basic language and then discovered new ways to use it.
Does that make sense?
Like we created the system of numbers to describe real things (ex. counting) and then expanded it to basic algebra. Then based on our new "language" of math, we discovered (relative to the invention) that pi was 3.14....
Don't know if anyone has said this already because I'm much to tired to read everything BUT my two cents is this: Personally i think math was neither invented nor discovered. Obvious to me it was not invented because math is everywhere and has been since the beginning of... everything. I think math being "discovered" is very close, but does not fully justify. Personally, I think that math was defined. We always knew that if you had an apple and someone gave you another apple, you had two apples. All we did as said above, was give them variables, hence defining them. Like in programing languages, you need to define a variable for it to be used, and that is what humans have done over the years, is define relationships between terms. We don't "invent" these terms or relationships, because that would be the equivalent to creating water out of nothing. If we went: one, two, three, four, five, six, core (new number added) seven, eight, nine, ten. Nothing would completely make sense. Everything in our world works because it follows, "rules" or "laws" of nature or whatever. All we did was give those rules and laws names and figures to help us understand more about what goes on around us.
I think mathematics are something to be discovered, but I don't think there is only one way to define it. In fact, there are so many inefficient ways of calculating that I'm sure there is probably a more simple way to define our world.
Because we have ten fingers, base ten mathematics are the basis for everything, but who is to say that base ten is the best way? Maybe base 33 is the best way, I don't know.
Base 10 or 33 they are still part of Mathematics.
Reading all the mail on this topic gives discovered an overwhelming victory over invented!!!
3 cheers for Discovering Mathematics  I wonder who started it?
Hey, think abot other thing... such as opera singing...... humans could always sing in the opera voice, but they discovered it pretty late....
Everything in this universe is actually discovered, and named something new, to make it look as an "invention". thats a good theory
sweet and simple!
I was gonna say that it was invented but then when I was in the middle of my post, i thought it as discovered instead. So, instead f saying it's invented or discovered, I'll just say why I wanted to call it invented and why I wanted to call it discovered.
NOTE: I ain't a philosopher or some other genius/nerd, so don't start challenging my views in order to spark a great debate!
Invented: Because the results of the calculation are "written in stone" but we needing a reason for these results...the "why". And I guess the "why" is the reason we decided to "create/invent" mathematics.
Discovered: 1+1=2, I guess that one is sort of set in stone. 1 finger plus another finger is 2 fingers...of course it's common sense. And i guess it's this common sense or realizing that it's common sense that makes me see it as discovered.
This question has made me rewrite this post a lot of times, but hell...it isn't my job to figure it out.
After writing all this, and thinking for a lot longer...I have come no closer on making up my mind. But thanks for giving me something to think about when I'm bored
TheNisk wrote:  since part of my course is history of computing, i will have to say that maths was invented, it is a branch of science that allows us to come up with solutions to real life problems. (and some other speculative fields)
I didn't read all the posts, but someone mentioned a triangle...where in nature have you seen a perfect triangle? it is a man made invention. we made it and we found the relationship beetween it's sides and angles.
Also, the two words have very close meaning in this case, making this question stupid. 
It still wasn't "invented." It has always existed (at least since the beginning of the universe), whether we were ignorant of the relationships or not. The relationships between the sides and angles of a triangle existed before we knew about them. A triangle in nature doesn't have to be perfect (I don't really know what you mean by this...equilateral??). It still has the same general relationships of one drawn on paper. We didn't make up those relationships, so how could they have been invented? It's like saying we invented electricity, which obviously isn't true. We discovered it. Just because we didn't understand its mechanisms or interrelationships before the discovery/research doesn't mean we invented it.
However, I think some people used a better word: Defined. We obviously gave names and labels to things that never had them.
The relationship of mathematics to explain certain phenomenon like the gold ratio in many objects is discovered but the symbols to represent such relationships is invented.
Math was invented. It just happens to be able to predict things in a very powerful way.
Maths was discovered. Why? Maths is the language of the universe. If the universe always existed then the mathematics underlying it must have always existed too.
After discovery and curiosity, their came rise to additions to mathematics. Modern algebra, calculus, zero, infinity, geometry are all additions or inventions to mathematics. Mathematics was always there...and I believe there is still a large part of it undiscovered...this part is vital if we are to find the theory of everything.
After recently reading one of Dan Brown's books (the most genius author ever), I was amazed to find out that nature itself obeys simple math, with a prime example being the number 1.372
To explain this, if you take the length of your arm from shoulder to wrist, and divide that by the length of your forearm, you will get 1.372 (or thereabouts as inaccuracy has to be taken into account). This is the same for leg/shin, body/legs and so on.
Therefore math was discovered.
Personally I think Brown is a very mediocre writer. As for 1.372...I bet you don't get that ratio very often.
Dan Brown (who I personally think is a pretty poor author) greatly exaggerated (and even made up) a lot of stuff about the golden ratio. Don't take anything that he says in his books seriously. He's a fiction author.
It already existed even before it had a name so it was discovered.
Then people invented terms for it such as one and zero and Calculus. And stuff like that.
That's why people insist it's invented.
Good topic for debate. But I'd say discovered since it was discovered first.
Hello All
I haven't yet read the articles suggested by Bikerman, but right now I pretty much agree with what has been said in the "Are theories invented or discovered?" thread.
I mean, mathematics is something that we have built in our minds.
Take numbers and counting, for instance. Obviously we invented numbers for counting, and probably and alien civilization also would.
I don't believe we discovered numbers, because I don't think they exist in the real world  they are a tool we invented to describe (and predict) phenomena in the real world.
Now, what about 1+1=2?
In this case, I would say that having invented numbers and defined the rules that they obey (e.g. Peano's Axioms), together with rules of deduction (i.e. logic), the truths (or falsities) about the number system (such as 1+1=2) become inevitable. The proofs are just waiting to be discovered.
Now what about Pythagoras' Theorem?
Once again, we (or rather, the ancient Greeks) made a mathematical model of plane geometry with basic rules (axioms) and rules of deduction. Once this is done, truths (such as Pythagoras' Theorem) become inevitable, and once again the proofs are just waiting to be discovered.
To summarise:
In Mathematics, we invent systems with basic rules (axioms) and have to discover their consequences.
Most, but not all mathematics was invented as a way to make sense of our world and to improve our lives.
Clearly, Algebra, logarithms and calculus were invented. The symbology of math was invented i.e the naming of the numbers and their written form. The base ten decimal system was invented.
Alternately, realizing the constant ratio of a circle's circumference to its diameter i.e. pi, regardless of a circle's size, was always there, but it remained for humans to discover it. The same is true for the natural logarithmic base 'e' and other universal constants of nature. Certain biological growth patterns have been discovered in nature that follow the Fibonacci series. Was this curious series discovered or invented?
Perhaps both.
i think maths was dicovered
it was always there, just our brain had to discover it
math is the only thing in the world that exists without any ambiguity
Mathematics was born compulsion of necessity, when people began to engage in trade with each other.
According to me, mathematics was discovered, but numbers and operations on them were invented. A triangle was always a triangle, but it came to be recognised as one when it got that name. Similarly, Earth always revolved around the Sun, but it got its period of revolution after the invention of numbers  365 days. Angles always existed, but they came to be known as angles only after mathematicians named them so.
I hope this was useful.
According to me, mathematics was discovered, but numbers and operations on them were invented. A triangle was always a triangle, but it came to be recognised as one when it got that name. Similarly, Earth always revolved around the Sun, but it got its period of revolution after the invention of numbers  365 days. Angles always existed, but they came to be known as angles only after mathematicians named them so.
I hope this was useful.
Sorry. I accidentally pressed the "submit" button twice.
the mathematics that you and i learn, have developed gradually rather than being invented or discovered..
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Ankhanu
The laws of reality were discovered.
Our system of noting them was invented.
So I would use the word 'discovered' if I had to answer the question simply. Saying 'invented' gives us way too much credit.
