Was math invented or discovered?

[pincushionman]

The "is discovered/invented" question is difficult because "mathematics" is not exactly A Thing - it is a catch-all concept of both phenomena we intend to describe and the methods we use to describe them. One set of which we discover, and the other set we invent or develop.

The volume of an object is a phenomenon, and very much exists, no matter how we intend to describe it. But there are several different methods that have been developed ("invented") do describe it. Comparison, algebra, integration, differential equations, all of these things are valid approaches (methods), and are more or less appropriate depending on the context.

And sometimes, mathematics itself ceases to be a useful tool for describing a phenomenon. I'll give you the following story problem as an example:

There are three birds sitting on a fence. I shoot one. How many birds are left?

I'll let you ponder the answer, but it is most definitely NOT two (2). The simple math approach has failed us. You could add complexity to your solution by considering biology, acoustics, and statistics, all of which can be pressed back into mathematics with enough effort, but by doing so it makes clear that mathematics itself is the wrong approach to describe the phenomenon.

[Nikolai]

Notches are still in use today, as they are useful for slowly counting a small to medium number of objects when the total number is going to be unknown, especially on a substrate or with a medium that doesn't support erasing.

Once you have the concept of object permanence -- which isn't even exclusively a human concept on this planet -- it's a relatively short leap to this kind of test for equality. It's interesting to note that once you embrace it as a test for equality, though, it becomes useful for counting things even when the number of things is not small. Notably, Cantor used it to count an infinite number of things, and in so doing demonstrated that not all infinities are equal in size.

[rkman]

And sometimes, mathematics itself ceases to be a useful tool for describing a phenomenon. I'll give you the following story problem as an example:

There are three birds sitting on a fence. I shoot one. How many birds are left?

I'll let you ponder the answer, but it is most definitely NOT two (2).

That is not a mathematical description. It is language, which is notoriously ambiguous.

The correct answer is not 2 only if the context that is introduced at the beginning is excluded later in the story;

- There are three birds sitting on a fence.

- I shoot one (of the birds sitting on a fence).

- How many birds are left (of the birds sitting on a fence)?

And why would one ignore context? That is not how understanding in human communication usually works.

If the context has changed half-way through the story as it apparently has (if "2" is the predetermined wrong answer) - not telling us that is withholding of relevant information. If that is deliberate then it is deceptive, if it is accidental then it is poor story telling.

[LN400]

About the evolution of separate symbols for values, it is interesting to notice that some of the earliest notations we know of, were 'notches' if you like, on clay tablets dating back to some of the earliest forms of writing we know of.. Later, during the haydays of Greek maths, the Greeks used their alphabethical letters where their alpha would be the modern 1, beta would be 2 and so on. The Romans used letters as well. I, V, L, C, D and M. Symbols for distinct values seem to have appeared (and it's reasonable to think they did appear) around the same time humans thought of symbols with sound values.

Someone might correct me but it seems likely to me that numerals and sound symbols were part of the same process of concept evolution. We can only imagine how they would connect the value of sound or numerical values before that. One thing is certain though. The biological evolution hasn't changed humans all that much over the last, dunno, say 200,000 years. There hasn't been much time for evolution to make us much smarter (plenty of time for reality shows to make us dumber though) so people before recorded history I will say certainly had the same mental capacity to express both sound and numerical values in written form as we had 190,000-200,000 years later.

Could be plenty of reasons we haven't found explicit evidence of earlier writings, or maths notation. Could be as simple as some of them wrote it in the dirt using a stick and that's not well suited for long term storage. Could also be as simple as noone had thought of writing down anything before about 5-6000 years ago, other than pictures that weren't expressing concepts of values but rather just someone wanting to tell they saw a LOT of deer that day.

One has to be a bit cautious though. It was 'a proven fact' for hundreds of years that the Egyptian hieroglyphs were expressions of ideas and abstract concepts and not sound values. It was later proven however that the ancient hieroglyphs were, in fact 'modern' in the sense that each symbol represented sound values just like our modern alphabets. We can't easily rule out that markings in a cave were carved by a cavedwelling maths teacher, so to speak.

Problem is, we only know about genuine mathematical symbols that are about 5-6000 years old. Older than that, and it's theory.

[Jonboy]

Numbers and math are tools invented by humans. Just like any other tool, we use it to try to manipulate or describe the world around us. Humans began to see patterns in reality and created a system to describe and make predictions about those patterns.

The universe doesn't care about math, just like the universe doesn't care about scientific laws. The universe just "is", and math and the scientific method are systems of tools we invented to make sense of the patterns we see.

[LN400]

I have one main issue with the idea that maths as a set of concepts is invented. An invention does not demand consistency and abscence of contradictions, as such. Maths does. The evolution of maths is also an exploration of logic. Did we invent logic? Or did our sense of logic come through a chain of realizations? I don't know but I suspect strongly we did not invent logic. On the other hand, they say Newton invented calculus. I can not reject that but I am still curious if 'invent' is the right word, or if he happened to be the guy at the right time who concluded first that calculus would be a pretty neat thing to do.

But now I'm also questioning if I have got the right idea of what 'invention' means This is going to be a long journey for me.

[theend3r]

Numbers and mathematical axioms were invented, the rest was discovered assuming they work as defined.

[LABHOUSE]

It is a method that was invented to describe how things work. It governs nothing. You could find other ways to describe things, say, use an octadecimal system instead of a decimal system, and the mechanisms it covers are still the same. In fact, computers readily switch between decimal and binary. It is just a tool.

The different bases were invented, but numbers themselves were discoveredd.

[JedTech]

You "Discover" the Truth

You "Invent" a Tool

Consider the mathematical equation that you might use to describe an orbit or define how much energy is in an atom: These mathematical expressions are our tools to describe part of the universe that we have discovered. We invented these expressions to help us define and quantify what we have discovered. Did orbits exist before we described them with equations? Yes. Are our mathematical tools perfect at describing orbits? No, we know that they are not. There is still so much to be discovered about the nature of time, space, and gravity. Fortunately, in the future we will continue to use our inventions of math to further discover more of what already exists in the universe. In doing so, we may invent more mathematical tools to define the discoveries that we are new to.

To Summarize:

Math is not the truth of the universe. Math is a tool people use to describe the universe they perceive. There is much to discover about the universe. Our inventions of math and language help us to categorize, quantify, and further discover the truth of the universe.

[arkie87]

You "Discover" the Truth

You "Invent" a Tool

Consider the mathematical equation that you might use to describe an orbit or define how much energy is in an atom: These mathematical expressions are our tools to describe part of the universe that we have discovered. We invented these expressions to help us define and quantify what we have discovered. Did orbits exist before we described them with equations? Yes. Are our mathematical tools perfect at describing orbits? No, we know that they are not. There is still so much to be discovered about the nature of time, space, and gravity. Fortunately, in the future we will continue to use our inventions of math to further discover more of what already exists in the universe. In doing so, we may invent more mathematical tools to define the discoveries that we are new to.

To Summarize:

Math is not the truth of the universe. Math is a tool people use to describe the universe they perceive. There is much to discover about the universe. Our inventions of math and language help us to categorize, quantify, and further discover the truth of the universe.

I agree with your first two statements:

You "Discover" the Truth

You "Invent" a Tool

But i come to the opposite conclusion: the fact that math works (you cannot prove 1=2) shows that it is a universal truth; therefore, it was discovered.

[Superluminaut]

lol

You say you think math was discovered,

yet math is a language,

and your poll says that languages were invented.

[theend3r]

I agree with your first two statements:

But i come to the opposite conclusion: the fact that math works (you cannot prove 1=2) shows that it is a universal truth; therefore, it was discovered.

You can actually prove 1=2. It depends on how you define "1" and "2". It took Russell and Whitehead 100s of pages to prove 1 + 1 = 2.

Edited May 16, 2015 by theend3r

[Voidryder]

I'm going with discovered. Just imagine one of our ancient male ancestors and one of our ancient female ancestors on a cold winter night sitting in a dark cave, nice fire going, sharing a nice mammoth skin blanket, sipping a little bit of fermented berry juice. Nine months later they discover 1+1 = 3.

[hugix]

Math is a invention used to make physical things abstract. So we can work with them better.

Thinks of this example. A school has a yearly outing. Students get to choose between three different trips. One survival, one cultural and one sailing. A teacher in the school keeps track of how many students choose for each camp and keeps the numbers in a chart.

This chart is a conceptual representation of the choices and do not necessarily represent the students, you can't pick a student out of the chart. It is a clever way to represent something unrepresentative.

[Hunting.Targ]

Math was invented. Equations to describe a rule. Like 2+2=4. We invented that. E=mc squared was invented as well. The rules of the universe were there, and we discovered them. Math was invented to explain them.

Because I am a devout Christian, I view Math as God's invention but Man's discovery. I think that God is using man made symbols to describe God's handiwork.

A number is not a real thing. It is an abstract. We say something like two chairs, but one chair is not exactly like the other. They are just collections of somewhat, or not even, similar groups of atoms and forces. Yet numbers, or when converted to numbers, they are the same in the purest sense.

It really goes back to Plato's allegory of the cave.

You "Discover" the Truth

You "Invent" a Tool

Consider the mathematical equation that you might use to describe an orbit or define how much energy is in an atom: These mathematical expressions are our tools to describe part of the universe that we have discovered. We invented these expressions to help us define and quantify what we have discovered. Did orbits exist before we described them with equations? Yes. Are our mathematical tools perfect at describing orbits? No, we know that they are not. There is still so much to be discovered about the nature of time, space, and gravity. Fortunately, in the future we will continue to use our inventions of math to further discover more of what already exists in the universe. In doing so, we may invent more mathematical tools to define the discoveries that we are new to.

To Summarize:

Math is not the truth of the universe. Math is a tool people use to describe the universe they perceive. There is much to discover about the universe. Our inventions of math and language help us to categorize, quantify, and further discover the truth of the universe.

First of all, the system of mathematics and logic is a language. And like all language, it is bourne out of not just facts and concepts, but out of a certain way of thinking and perceiving. I once had an engineering teacher who came to my home country as a student not knowing the language, but he learned algebra, calculus, and chemistry just by reading and working with the formulae and symbols in textbooks. Any language has rules, and may be innovated and modified over time, but the purpose of modifications is to express thoughts, ideas, (and emotions, in the case of linguistic languages), and perceptions with some degree of clarity.

Mathematics is different in two respects from the linguistic languages of the world. First, it is universally recognized and understood, as I illustrated above. Because scientists are trying to understand the same phenomena in the same universe using the same methodology, they have developed, without negotiation or overt conflict, a largely universal system of representing their ideas. Second, mathematics allows little leeway in the way of context or subjectivity in how a concept is expressed; classical logic as the foundation of mathematical expression and reasoning requires certain conditions to be met when working with figures and quantities; the principles of identity, non-contradiction, and commutation of operation, albeit in slightly different forms, have appeared in multiple cultures in multiple eras throughout recorded history, just as have other ideas like currency, domestication of animals, art, and division of labor. From this, it is therefore likely, but not by any means certain, that it stems from fundamental properties of the the universe in which we live.

Beyond this, there are certain inescapable concepts, such as Pi and e, the trigonometric functions and the conic sections, that cannot be altered no matter what you do to the expressive or numerical systems themselves. No numerical system can accurately express the value of Pi, but our mathematical system can define and expess the idea of Pi in a finite expression. We can show the ideas, and can see that what we are representing in our system of expression are really fundamental properties of our reality, things that would not be different even if we contrived to make them so.

So I do think that everyone quoted above is as close to the truth as we can get without getting together and writing a paper about it (admittedly, it would be a philosophical, not a scientific publication).

"Mathematics is the language with which God has written the universe."

-Albert Einstein

[LN400]

That "maths" is universally understood I will dispute. Think about it. For a long time, zero was not accepted as of having anything to do with mathematics. Still to this day there are those who reject that zero has any mathematical meaning. Negative numbers, same thing. Complex numbers, those had an even tougher fight for acceptance. Sets of numbers. You will find people who believe all numbers are rational. There are societies where they simply have no concept of defined values greater than say 3, or 5. In these societies it makes absolutely no sense to talk about the value of 11. Either it's 1, 2 or 3, or it's 'many'. Still, they can tell that 20 is greater than 10, only they don't have the conceptual idea of defined values that great, at least not in their spoken or written language. Clearly the universal-ness of 'universally understood' isn't all that universal.

[Slam_Jones]

This is how I see it:

Numbers are just a representation of an idea, ergo they are a human concept. The shape and arrangement of them is arbitrary, as is the 10-based numbering system we use.

Math, on the other hand, can be proven. They are formulas that describe the nature of what is happening around us. The symbols we use to represent them are human constructs - the formulas themselves are not. You can plug any numbers (or symbols) into the formulas, and, given that the number system is logical, you will still get logical results.

[pendant]

Numbers are pure invention: they are tools used to describe what we perceive. For instance: for a long time, there was neither a concept of zero -- nor one of infinity. You can describe a sunset using numbers, but such a description is just a representation of reality (and using numbers in this way actually detracts from the reality because there is a huge difference between a colour palette and the beauty of a sunset). I offer as proof imaginary numbers.

Mathematics, on the other hand, describes reality directly; although it, too, is a tool, it is one used to describe what we discover about reality. Our interpretation of mathematics may be (is) incomplete: but, over time, we refine the tool to more accurately describe what we discover about the magical universe in which we live.

[Flymetothemun]

I say we invented it, but use it to describe the natural world. We invented languages, but we use them to describe things that have been around for ages. Plants have been around for a long time, before the invention of languages, but nowadays we use words like "Green", "Short", "Tall", "Delicate", "Strong" to describe them. I say it's the same with mathematics. Collections of natural things, like forests, have been around for a long time before the invention numeral systems, but when those were invented they could say that "There's 5 trees here" and everybody would know what they're talking about.

It's almost like a photograph with a camera. The camera will take photos of whatever you expose the film to, but the lens that you use changes what's being put into the film. Words are one lens to view the subject through. Numbers are another. Colors are yet another, and even sounds, to an extent, are another lens to view a subject through. The subject could be the natural world, machinery in a factory, cars on a road, computers on a desk, tools in a workshop, or anything you can think of. The film could be described as us.

[RainDreamer]

Math is a language. We use it to represent ideas, concepts, observation, imagination, into a form that we can teach others to understand. It is something we invented. However, all the things that it represents is discovered.

Makes me wonder, if we are to encounter aliens, how different their concept of math, if they have any, would be to us? Would they use a much more obscure system to us, or would there be similarity? Truly something to think about.

[BlueCosmology]

Everyone here seems to have some weird belief that maths is just used to describe the natural world. Many people seem to think that a proof of a mathematical idea is that it can predict some natural phenomena.

It isn't. A lot of maths ofcourse is used to describe the universe, but a hell of a lot of maths (certainly the majority of modern maths) is not at all and does not in any way describe anything in the universe. No mathematician would ever try to justify that their mathematics is correct through experimental evidence (much less claim that it is a proof).

Edited May 17, 2015 by BlueCosmology

[Steel]

BlueCosmology makes a good point, those who justify that their mathematics is correct through experimental evidence are physicists, not mathematicians.

On the point of maths being able to describe natural phenomena, is this truly a reason to believe that maths plays a role in our universe? If we think about it, a lot of physics uses completely arbitrary factors (by this I mean number that aren't just simple integers) to make the numbers agree with observed values (i.e. the gravitational constant). From this surely it's possible to argue that all we have are a system of rules and factors that give us the numbers that we expect to be associated with an object, and not any deeper understanding into the true nature of how the universe operates.

Just some food for thought.

- - - Updated - - -

A point made by one of my lecturers is a good example. It is something along the lines of:

Consider an electric or magnetic field. Is this field an actual physical object/construct/whatever you want to call it, or is it a human invention that allows our number to agree with what we see?

[rkman]

A point made by one of my lecturers is a good example. It is something along the lines of:

Consider an electric or magnetic field. Is this field an actual physical object/construct/whatever you want to call it, or is it a human invention that allows our number to agree with what we see?

Isn't there something really there to which our idea of an electric or magnetic field applies?

That idea is just an approximation - but then still it is an approximation of something real. That 'something' is not a human invention.

[BlueCosmology]

On the point of maths being able to describe natural phenomena, is this truly a reason to believe that maths plays a role in our universe? If we think about it, a lot of physics uses completely arbitrary factors (by this I mean number that aren't just simple integers) to make the numbers agree with observed values (i.e. the gravitational constant). From this surely it's possible to argue that all we have are a system of rules and factors that give us the numbers that we expect to be associated with an object, and not any deeper understanding into the true nature of how the universe operates.

That's a lot more just the fact that in everyday life we use unit systems that are convenient for our scale of life (metres, seconds, etc). In fundamental physics things like the gravitational constant are integers because that is the unit system that is most convenient in that case.

[Fel]

Why don't people realize that before they adequately define (not the smart-alec "dictionary/wiki/google" nonsense) what "invented" and "discovered" actually mean, your argument has no merit? To do this, you have to abandon the original argument; else you'll end up simply changing what needs to be defined. Saying that math is "x" and that as "x" it cannot be invented/discovered only means you now need to fully and utterly define what "x" is.

So, to the point, what does it mean to "invent" something, did humanity invent the steam engine?

Here we can place a few points

A) Normally, we attribute the original creator of a device the title of "inventor"

The steam engine itself is a tangible physical object that relies on physical processes to work

Now, using B we would say that, without a doubt, humans "invented" the steam engine as this "Numbers and Math" argument is centered around intangibles... but there is a problem:

We have historical precedence where two people "invented" something without interaction between each other and later the earliest one was acknowledged as the original inventor (only instance I am familiar with would be the Television, and that deals with copyright law and fun stuff)... so let's say that 4000 years ago on the planet Faux, a young creature also created "the steam engine". Because this creature on this different planet now is the FIRST to have created the steam engine, humanity couldn't have "invented" it.

Or, in plainspeak, humanity can only lay claim as the "original creator" to our devices so long as we do not know of any beings that existed prior to us that also made the same creation.

Hmm, this definition of "invented" now is starting to unravel.

But what does it mean to DISCOVER something? Provided the universe contains life other than ourselves and said life can exist in an infinite number of possibilities, we can say that any thought that is thought now will again be thought later on in the history of the universe. So if we "discovered" the steam engine, due to it simply being a product of natural phenomena, can we also say that the thought process that arrived to its discovery is also a product of natural phenomena?

Did we "discover" thought?

Certainly, the arguments that the CONCEPT of the steam engine is only information would support that the thought processes of how to arrive to it are also only information, and hence able to be discovered.

This topic is a very deep and philosophical topic; one that you all don't really do well with. It does it no justice if all you talk about is "math." "Math" is just a mechanism to the argument, a vessel to bring the topic to light; but the underlying topics of what thought we can actually call our own is where the argument originates from.

Edited May 18, 2015 by Fel