Is Mathematics Discovered or Invented?

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.

Comments

[Stephen]

Just because Plato had the attitude that "it's out there", and it sounds like religeon doesn't mean math isn't there to be discovered. He might have had some good ideas with some bad ones. That must be true for everyone. When you set up some rules, you end up with a search space. Some of the bits out there to be discovered may come in handy, like bisecting angles. You might have invented the search space, but you discover some really cool theorem.

But you could say that the Universe presents a similar search space for gadgets, and some are handier than others. The search space was there, so there is no invention, you discover the rules. And that's for gadgets. Hey, if i get a long stick, i can hang a heavy load off it, put one end of the stick over my shoulder and let the other end drag on the gound. Handy. It's trivial, so it's discovery. Or was it trivial when it was discovered before fire?

My son noticed something cool about dandelions, looking out the window. There seemed to be more of them in the bright sunshine than at dawn. That's clearly discovery. But maybe someone will use that and we'll have Velcro, or something that handy.

I've done some invention. I documented my process most carefully. It lead to nonobvious new functions, but the process was strict search using textbook techniques. It felt like discovery.

The real question is, What are the implications for discovery vs invention? If it's antibiotics, for example, it doesn't matter legally, and the question is moot. You have to demonstrate that a 'compound' has some effect.

[ganeshbrhills]

Or perhaps it was sheer necessity?

[Jan-Michael Guzman]

It would seem to me that 1+1 would always equal 2 regardless if we were here to witness it or not. I feel mathematics is discovered but the method to its discovery is always created to fit the situation.

[Richard@Home]

"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?”"

If you turn a painting round and look at the back, does the picture still exist?

[Winnipeg Theme]

Was math or properly mathematics "discovered or invented" ?
It is often the idea that great minds think alike
Often the same "discoveries" are invented ( for good reasons and purposes) at different places and at different times - to even disappear or reappear yet again
It all helps the scientific and application discoveries , comprehension and ultimately applications
You mind find one source and time of invention and yet some time later you or another researcher poking around may find yet another source of all
This is the nature of mankind and the development as well as growth of mankind and the scientific knowledge attributed to manking

[Jason]

Great Digg. Maths is entirely invented and taught, just like language.

We use decimal and count to ten because we have 10 digits on each hand. Our measurement systems and number base only have meaning to us. Ask a rabbit if he knows, or cares - how many hours are in a day.

Thanks to maths (specifcally, our measurement of time!) we can look back in history and clearly identify significant inventions : the length of metre, the weight of kilogramme, the number of days in a week etc......and my personal favourite, the relationship between the the radius of a circle and it's circumference....yep....guy named Pythagoras "discovered" it. The actual discovery is that our numbering system (which works so well in many cases) - is insuffiently sophisticated to describe it !

[Mark]

Surely maths is just a framework that we have placed around the universe, a tool we use to help explain these 'universal truths'.

So does this make it a combination of both a discovery and an invention? The fundementals have always been there, we have just invented a system which we can use to investigate/discover them!

[David]

Who is to say that math is not something that has been both invented and discovered. Take for example the advent of numbers. The base 10 numeric system was something invented thousands of years ago to count something as simple as cattle or live stock. With out the invention of numbers how would we have a system to portray how much of something we have. I believe that then using "invented math" things about the world and what not were able to be explained. If you look into just about any higher level mathematical proof you will find that many things will be assumed or "invented" along the way. It seems that things have been discovered about the world based on the foundation of mathematics that has been poured by the greats such as Newton, Liebtniz, and Pascal.

[KG]

So we discover the world around us, now how do we use and manipulate this world. Invent Mathematics.

[Adam]

What is the practical benefit of having such a debate other than it keeping some writers in a job at the European Mathematical Society Newsletter division?

[1...2...5]

2+2=5 is correct when you simply assume that 5 comes after 3 on the scale of 1-10. Numbers and mathematics are simply symbolic of the observed/implied relationship between objects that are grouped together. There is no such thing as "the number 5" in the nature of the universe.

[Toby Lethby]

Surely Maths is discovered. Tomorrows mathematical equation is still there today, it just hasn't been found, as opposed to it being created from scratch tomorrow? If you catch my drift....

[Mark]

1 + 1 could never equal anything else but 2, but isn't it possible that a mathematical system could be devised where '1' doesn't actually exist? Where there is an entirely different concept of 'numbers' and 'values' than what we have? Maybe many different kinds of alternative mathematical systems as different from each other as they are from ours.

Admittedly something like that is difficult to imagine simply because the math system that we have is so ingrained into our brains.

[Francis Caestecker]

Some ignorant replies here. The notational views of math are ofcourse invented. The symbol for an integral isn't the correct one, the language it is portrayed in and written down in is invented, but so are the names of atoms. But chemistry isn't invented now is it?

First of all, you've got two type's of math. The maths that interact in our universe, and the maths that dont interact in our universe. The first is physics. If math is invented, so must be physics. Everything we see is math, everything we percieve is math. Every science is math. Because math is simply the logic on which everything is based on, and therefor plato is right. It is there to be discovered. Now ofcourse we also have the different type of math, the math that has no application whatsoever on the universe. This is more modern math, because most traditional math actually stems out of physics. This math might not have an immediate link to the universe, but that doesn't mean that link isn't there. It might still be undiscovered. Either that, or it is an extension of laws that hasn't been reached in the universe yet. But that doesn't mean its unreachable. Who knows what effects can come to exist, if only the logic exists to hold it.

Therefor, math is the science of logic. The laws of our universe, and the laws of our possible universe. It is uninventable, but it is possibly non-consistant. Who knows if 2 + 3 = 4 (in literal modern day math language), in a different paralel universe with a unique logic of its own.

But in this universe. Math is consistent.

[Cowboy]

The bigger questions remains: How long did you spend on the title of this post?

[J]

Has everyone forgot Kurt Godel's work.
Just sit and read On formally undecidable propositions of principia mathematica and look at the proof of why a complete system can not be shown to be complete. Then you will see that only a complete system must have a design set up from day one. By the way, 1+1, has nothing to do with math. It is a definition of numbers and its relationship. Math is systems of structures that relate truths. Numbers are only a way to express this.

[LZ]

Hate to be a diletante on a subject a lot of serious people give serious thought but I find myself inclined to believe that math is NOT a property of the universe but of our (human) conscience - thus it should be discovered.

[HJ]

Isn't math as we understand it more of a framework by which we categorize the universe, much like written language or musical notation? So, when we get that sense of discovery by taking an uncategorized physical event or process and putting numbers/words on it, are we doing anything more than bringing it into a framework where we can understand it?

Clearly there's something to be discovered (physics, etc.), but thinking about this question always brings me back to this:

1+1 = 2
1+1 = 1+1
anything = itself

Which says to me that math is more of an invention for the purposes of understanding.

[SC]

HJ hit the nail on the head. Math is a framework, a model, or an abstraction, if you will, to sometimes understand reality. While it follows the laws of logic, it should never be mistaken for reality.

[supercharged love]

There is water. A bucket can hold it, or a waterproof, flexible bag. A bucket can exist without anyone agreeing upon a name. Many ideas can apply to the bucket and describe it very well, in a useful way, even if the bucket has many names, too. Sometimes the order of the universe shapes the nature of the descriptive language here or there in scientific and mathematical pursuits, and sometimes that order of nature is brought to light in a useful way by the creation of a system and terminology. The terminology of quark types is a good example. Color, spin, etc. It serves a purpose, yet quarks do not reflect light in a macroscopic way. It is just utility. The terminology, roughly, is use of labels for common understanding. 2 + 2 = 4 seems far more concrete, and it is so basic, there is no need to change the names of numerals. The names are not creative to us, though. Two (2) is integral to our understanding as a name, but the change of the name will not change the equation. At least, according to most. A better name alone can sometimes yield better understanding, however. I understand this because my talent is language, and a simple name is not as useful as a name that reflects the nature of a thing. For a rather philosophical non-expert, to me it seems this issue is like many others. Sometimes there is black and white between many shades of gray.

[CC]

There are certainly some aspects of Math which are invented, such as notation. But these aspects are not what we would typically understand as the core activity of "Mathematics," any more than you would say that Chemistry is merely the symbols we use to represent different elements.

At its core, Mathematics is the exploration of what a given set of axioms imply. The axioms themselves may be invented (and are usually assigned based on what would likely be of use), but the implications of those axioms - the resultant properties of the universe they define - are discovered. And this process of discovery is what we'd commonly refer to as the pursuit of mathematics.

To demonstrate that these properties are discovered, all you need to do is demonstrate that given the same axioms - the same starting point - people can reliably reconstruct the same properties of the universe they imply. In other words, the properties of integers, say, remain static no matter who studies them.

For example, give ten school children a stack of six pennies, and ask them to arrange them nicely. Many will stumble upon the fact that they can make a nice 2x3 rectangle, a consequence of the fact that 6 factors into 2 and 3. This is not an invented trait of the number six - it's a trait that exists independent of our awareness of the concept of factoring. It happens whether we understand the concept of factoring or not.

Other kids may make a triangle, with one penny in the top row, two in the middle row, and three in the bottom. Again, that you can make a triangle out of six coins is not an invented trait of the number six, but is a consequence of the fact that it satisfies the formula for the summation over a sequence. The number five does not have this property, nor does seven. These properties of integers exist, regardless of our understanding of them.

The above examples shed some light into the discoverable nature of mathematics. The abstract mathematical concepts of factoring or summation of a sequence are overtly suggested by real-world incidents of the mathematics at play. It's not a stretch to wonder "how many pennies would be in a triangle with ten rows? Twenty? A hundred?" You don't invent the answer to this question - you discover it. And anyone who tries to tackle these questions will arrive at equivalent answers (or answers that do not stand up under experimentation). That the situation of triangles of pennies is contrived and perhaps not naturally occurring in nature is irrelevant - the mathematics lies in the extrapolation of the known to the unknown. But more importantly, the knowledge gained can be abstracted so that it pertains to all similar cases, not just pennies being arranged into triangles. You can use it to compute the number of nickels in a triangle, or how many total gifts your true love gave you on the twelve days leading up to Christmas - any case where you're summing a similar sequence. Once you can abstract the concept away from concrete things like pennies, you can apply it to any situation that matches the axioms it is based on - which is where its usefulness in the sciences becomes clear.

[Alex]

Mathematics deals with the logical relations between quantities, independent of arbitrary notation. Thanks to observation, the universe is known to consist of the logical interaction of physical quantities. Some theoretical physicists, like Max Tegmark, argue that the universe is an abstract mathematical structure. The fact that the laws of physics are invariant under a particular symmetry group and other such symmetries underpin the most fundamental physics is suggestive of this.

It is clear, however, that other axiomatic systems could exist without logical contradiction. Axioms are invented, and from these, theorems are discovered. While these theorems may not describe our universe, they are underpinned by the same logical structure, which leads me to believe Russel was right when he said, "Mathematics takes us still further from what is human, into the region of absolute necessity, to which not only the world, but every possible world, must conform."

It is perhaps that mathematics and all universes conform to a higher logical structure than either, which is not yet well understood.

[Math Quiz]

Mathematics was discovered, probably about the same time that water was. It is something that was so necessary that you couldn't keep it far from creation.

[Fastbullit]

Translated is the correct term, I believe.
Math is a language, the language of the world, and all its physics and functions.
For us humans to be able to precisely understand the nature of our world and apply it in our societies.
You guessed it, we use mathematics.

[phoenixbyrd]

Math is invented, just as language was invented and time was invented.

There is no underlying universal principle of mathematics in our universe.

Time does not exist, no matter how many clocks you wanna point at.

An alien species will/should not come up with the same exact measurements of mathematics, language, or time. Yet they can still accurately describe a rock being thrown in a wide arc and measure the angle of the arc and the estimated time it will take to reach the ground.

[Bruce LaDuke]

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

http://www.anti-knowledge.com/book.html

[twiart]

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.

[Ryan]

This is a good read

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

http://perso.unifr.ch/rafael.nunez/welcome.html

[James]

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.

[lotusandbob]

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.

[5-15-9]

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.

[phoenixbyrd]

@lotusandbob

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.