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Is Mathematics Discovered or Invented?

Stella_2 For centuries people have debated whether – like scientific truths – mathematics is discoverable, or if it is simply invented by the minds of our great mathematicians. But two questions are raised, one for each side of the coin. For those who believe these mathematical truths are purely discoverable, where, exactly, are you looking? And for those on the other side of the court, why cannot a mathematician simply announce to the world that he has invented 2 + 2 to equal 5.

This question pops to the surface of the math world every so often, like a whale surfacing for air. Most mathematicians will simply set aside this quandary for those from the philosophical realm, and get on with proving theorems.

However, the mathematical whale has surfaced this year, thanks to the European Mathematical Society Newsletter’s June edition, where the question will once again be raised.

If you’re looking for a side to join, then maybe the Platonic theory is your cup of tea. The Classical Greek philosopher Plato was of the view that math was discoverable, and that it is what underlies the very structure of our universe. He believed that by following the intransient inbuilt logic of math, a person would discover the truths independent of human observation and free of the transient nature of physical reality.

“The abstract realm in which a mathematician works is by dint of prolonged intimacy more concrete to him than the chair he happens to sit on,” says Ulf Persson of Chalmers University of Technology in Sweden, a self-described Platonist.

And while Barry Mazur, a mathematician at Harvard University, doesn’t count himself as a Platonist, he does note that the Platonic view of mathematical discovery fits well with the experience of doing mathematics. The sensation of working on a theorem, he says, can be like being “a hunter and gatherer of mathematical concepts.”

Mazur provides the opposing view as well, asking just where these mathematical hunting grounds are. For if math is out there waiting to be discovered, what once was a purely abstract notion then has to develop an existence unconceived of by humans. Subsequently, Mazur describes the Platonic view as “a full-fledged theistic position.”

Brian Davies, a mathematician at King's College London, writes in his article entitled “Let Platonism Die” that Platonism “has more in common with mystical religions than with modern science.” And modern science, he believes, provides evidence to show that the Platonic view is just plain wrong.

So the question remains; if a mathematical theory goes undiscovered, does it truly exist? Maybe this will be the next “does a tree falling in the forest make any sound if no one is there to hear it?”

Posted by Josh Hill.


I wrote this book when I was younger and it isn't very well written, but it does answer your question:

We didn't invent language,evolution did.

Math is a framework that consists of mathematical equations that are invented by us TO fit the evidence,facts observed in the Universe which in turn makes the 2+2=4 that me mathematically valued makes sense.Most importantly,math like evolution theory,at first it was just another theory or an invention if you may..but now it is already a FACT,because it is provable,observable,consistent with the changes of variables.

This is a good read

George Lakoff & Rafael E. Núñez - Where Mathematics Comes From

Basic mathematics is discoverable, but specific to humans our math system is how it is purely because of our physical attribute of having 10 fingers and toes. If we ever meet another intelligent being figuring out their math system may be as easy as counting their fingers or toes and discovering what their number for 1-6 are. More complex mathematics such at geometry is required for any complex building or vessel, so it is doubtful we will ever meet a intelligent race without a understanding of mathematics.

We humans will never be able to precisly understand Nature completely i.e., the complete workings of the Universe just as we will never understand ourselves completely. Godels's theory of Incompleteness can be applied to Physics because Physics is based on Mathematics. We will not understand the mind of God .The Universe may be an abstract Mathematical structure as Max Tegmark suggests but, if our number theory is built on axioms ,Kurt Godel proved we must go to a higher level of more abstract Mathematics to prove some true theorems which can't be proven within the original number theory with its axioms. That higher level of Mtahematics will also be incomplete.
We can't use finite integer equations to develop a formula for pi because it is transcendental but we can go to infinite series and do it using integers.
Pi/4 = 1 - 1/3 + 1/4 - 1/5 - 1/7 ....(-1)^(n+1}/(2n-1) .......
We get closer and closer to pi as n increases.
In Physics we get closer and closer to calculating the true orbit of a satellite by including more and more parameters/terms in the partial differential equations describing the orbit , e.g. the oblateness of the earth and more exact gravitational constant. Newton knew that d(M X V)/dt = SUM(applied forces) and solved that to get an approximate parabolic trajectory for a canon ball and Kepler knew applied Mathematics and came up with an ellipse for the orbit of the planets but he did not have abstract Mathematics like Group Theory or Topology to prove theorems in Complex Variables which are higher level subjects.
I think humans who are part of the universe are only capable enough to understand the tip of an iceberg when it comes to the ultimate bedrock of reality and unknown science.
Godel proved that there are an infinite set of true conjectures/theorems which are definitely true some of which are not provable and only a small number of these true theorems can be possibly proven within the system of numbers and we can not know which have a possible proof and those which we don't. For awhile it was thought that Fermat's Theorem was a true one but one that had no possible proof.
I think this applies to Physics also and we must go out of our universe to prove some things within our universe . We are limited by our brain power to really see the mind of God.
An insane person can not prove he is insane or that he is sane because his brain is limited by being insane , so he must go outside his own system to evaluate himself.
We are in the same boat and can not understand the universe because our brains are limited (but maybe not insane - although many of our leaders seem like they are beyond insane)
I believe the Universe has a built in underlying mathematical foundation and we can only discover some of that mathematics , at least enough to approximate the goings on in the universe and , like an infinite series , come a little bit closer to understanding true reality but , like I said before , it is and can only be the tip of an Iceberg.

Math invented? Hmmm. Sorry, I can't buy that for a second. Granted there are new ways of mathematics that we think we invent but this is akin to saying that because we didn't know what lightning was, it must be Zeus creating lightning bolts and hurling them down from the clouds. What we "invent" are more perfect methods of manipulating and understanding numbers, which of course were already there waiting to be discovered. It's not magic or creation on the part of humans.
If you have 1 man and add another, you have two. It can be nothing else at all. There is no other number for two. That's unchangeable to the utmost.
In short, math=concrete truth, universal. The invention? Our understanding of it.


What physical laws/properties of the universe can you proudly point at and say see, that there says math exist's and is discoverable!

There is no such thing as apple(a) and apple(b) equaling two apples, not without an intelligent species to denote a value system to those apples.

And yes, it can be a different number other then two apples. It depends on the invented value system we both agree upon to describe the amount of apples. Hence the reasoning behind math being an invented construct.

wait. you didn't answer the question. Is mathematics really invented or what?? If ever it is invented, who invented math?? don't forget to add me on myspace!!

if a mathmaticaltheory exists but we havent discovered it does it exist. that is why i leave math alone the basics are good enough to get me throught my life but some of my teachers in high school have killed my dream of being smart as i formally was.Thanks to the price is right and Mrs bird son...


I also love him but i wouldnt let him go to the bathroom.

Mathematics is a tool - an invention - that is used to discover properties of the universe (which can be mathematical). Mathematics (along with other modern knowledge)is now advanced enough to discover new properties about itself (with Godel limitations in mind). The discovery that the golden ratio is used in nature (part of the universe) can be defined (invented) mathematically.

On other things mentioned, like to point out that:
a) 1+1 = 0 with a carry (base 2) - there is no '2'.
b) Base 12 and 60 where probably more commonly used in the earlier use of mathematics. Base 10 seems natural because we learn it early in life. It is convenient because our modern times have better machines and procedures to handle base 10.

The idea of '1' - however it is written - is used because it correlates (a discovery) to whatever is being counted (an invention). We then invent a more compact way to represent higher counts or other numbers:'i' for the square root of negative 1 for instance.

Off subject: I am against the idea that time does not exist. How would you measure entropy? Or, for that matter, define speed without using a definition of time? At least, in our macro world. You never grow old and never go anywhere. You look older today. So, I refute thee modern George Berkeley!

Mathematics was invented

me and my partner are writing a report, (highschool kids) and we think that maybe math was discovered and how to use it was invented. ummm get what i mean? not really sure , just throwing it out there....

Descriptions of mathematics are created; mathematical truths are discovered.

The decimal system, or a mathematician's formal proof, are both creative works; the underlying truths, however the fact that no whole number solutions exist for x^n + y^n = z^n for n > 2 exists regardless of whether or not anyone knows it; the mere fact that Andrew Wiles *proved* that, did not magically cause all the whole number solutions to that equation to magically disappear; they never existed, and they never will.

I meant to say, "however, such as the fact..."; I also meant to add a comma after "n > 2" -- pardon my typos.

Math ? Invented or discovered. First, there is language and definition that must be agreed upon and interpreted. Now, lets just talk about Pythagoras. There is a physical relationship between the diameter and circumference of a circle that cannot be changed. When you find what that is, that is discovery. When Pythagoras decided to number the degrees or angles in a circle, that was invention. He chose 360 because that number was sufficiently large enough to include many evenly divisible numbers. Example 360/2=180, 360/3=120, 360/4=90, 360/5=72, 360/6=60,360/8=45, etc.

Whether someone is the first to discover it or is its inventor, what is the difference for all intents and purposes? Is there any perceptible difference from that moment on any any human perception on activity in which this distinction makes an actionable difference?

I think most people are just overstating the problem. If anyone can explain to me how we "invented" pi = circumference/diameter then I will accept that math is invented. I'm not arguing that we didn't invent the terms, we did, but the relationship in pi is not something anyone could "make up." It has always been true. Someone just found that it was.

i hate math, and whoever invented/discoverd it.

a helecopter invented math so came along the bird to ruin it an the moon exploded and then the moon made dinosaurs.then i got very very tall like tall and crushed the city.

i invented a math that 10-3 equals 12. or 48-10=92 or 48-(10+x)=23 and y=Nfq's. Nfq's = number of frequencies

Math is obviously discovered, because if a caveman had one rock and found another rock, he will have two rocks and there is no arguing against that. And if those rocks were to go missing, the caveman will not have any rocks; therefore, 1+1 will always equal 2 (Name changing is negligible), and 2 - 2 will always equal 0.

Mathematics has always been discovered,is being discovered and will always be discovered.
Surely the measurement systems,decimal systems,time clock are invented but mathematics not.Those are just tools and methods for using mathematics.
Take for instance,lets assume i have two apples and i have to distribute it equally among two persons,surely whatever measurement system i follow,whether i knew mathematics or not ,I will give one apple to each one. Agreed?

The basic principle underlying behind Mathematics is pure logic and nobody invented that.

Wherever in space you go to "THE LOCUS OF A POINT MOVING AT A FIXED DISTANCE FROM ANOTHER POINT" will always be a circle,the theoroms of geometry would still apply. Mathematics is discovered,tools and methods to use it are Invented.

this stuff about math is so ccccccooooolllll!!!!!

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