Several days ago, I got into a debate with some friends about whether we invented mathematics or discovered it. Most of them seem to have gravitated towards the “discovered” side, while I am left alone with only a few allies to defend the “invented” side. I’ve used this board for referencing other subjects we’ve argued about, but I cannot turn up any related threads by searching for keywords like “math,” “discovered,” and “invented.” So I present this question: Was mathematics invented or discovered by humans?

Briefly, here are the main points of each side so far. I cite examples such as logarithms; the way it was once explained by a teacher was along the lines of “…and here, mathematicians were stuck on how to proceed. So they made up a new method.” I also use right triangle trig ratios. What’s special about a 90 degree angle in nature? Some of the other side’s arguments are based on the natural world having many patterns, such as the multiplication rate of bacteria. All of classical physics is also a sticking point – if we invented mathematics, did we also invent physics? Or just invent the way to interpret physics?

[My apologies if this appears twice, I’m getting error messages from the server when I try to submit this…]

That’s a tricky question to answer until you distinguish between math as “mathematical properties of the real world” and math as “formal systems of mathematics”.

As an example of the former, consider the phenomenon of “one”, i.e. unity. We certainly didn’t invent unity. But on the other hand we did invent the formal system known as arithmetic. Specifically, (if I may over-simplify for a moment here) the mathematician Peano set down the axioms and rules of arithmetic. Those axioms and rules together make up a formal system which appears to describe certain natural phenomena like “one”, “zero”, and “addition”, but the formal system is not the same thing as the natural phenomena.

In other words, you and your friends are each partly correct: we invented a language to describe things that we discovered. In the same way we invented the colour wheel as one way of describing colours, and Newtonian mechanics as one way of describing the motion of the planets.

I’m going to cast my vote for discovered. Although the development of mathematics could have gone slightly differently, the general principles are set and can’t be altered.

Take the specific example of the 90 degree angle. Right angles have a lot of important occurences in “the real world”, so I doubt that any mathematical system could exist without giving them special signifigance. In force fields such as gravity or magnetic fields, you do no work moving a particle in the field if and only if you move the particle at right angles to the field.

Well, I’ve been on hiatus of late and after having spoken with Gaudere the other night, I felt I ought to return to her domain once again. So…

The real world doesn’t posess any mathematical properties, it only has physical properties. Mathematics is a construct of human mental faculties (although I suppose it is possible that other species may have some simliar notion). This construct is utilized to understand our perceptions by processing the information and assimilating it into the construct. The earth has physical properties of magnetic field, mass, motion, etc., and the nature of these properties is expressed through a formal system. Motion is not a mathematical property, but velocity can be quantified in terms of other constructs, namely space and time. Space and time are constructs in and of themselves, but they are also understood in terms of formal systems.

Although I find your definition of the numeral one intriguing, I think it is misleading. Is the number one of mathematics really equivalent to the concept of unity? If I think of the universe as a whole, the concept of one and unity are synonymous. If I think of a number line, the concept of one and unity are not synonymous. Then again, I think unity is a concept, or construct, just as is the number one.

Would you provide some examples of how the items you listed are phenomena in nature? I fail to understand how zero is an occurence in nature or how addition is a phenomenon.

That is precisely my point: we invent mathematical constructs to understand and express information about a physical world. However, I infer from your post that you don’t distinguish between mathematical and physical properties.

Right angles are significant in electromagnetism. Pi is significant in Euclidean geometry. i is signficant in electrical engineering. Consider the equation (which I personally find to be blissfully aesthetic):

e[sup]i*pi [/sup]= -1

It reveals some wonderful mathematical properties…in the decimal system. If one tries to express that equation in binary, it ceases to have the same aesthetic. Any number can be quite significant pertaining to the right situation in the right formal system, but I doubt that makes mathematics predetermined.

Mathematics is continually being developed. At one time there was only Euclidean geometry, yet now there is Riemann geometry. At one time there wasn’t calculus. We continually invent constructs to assist us in understanding our world around us, much as we continually create words to describe things outside our lexicon.

I’m sorry to save this question for last, but I thought that after my little diatribe it would be clear that I am inclined toward the latter.

Oh yeah, two more things: 1) I don’t like you dissin’ my Inuit friends and their igloos, and 2) I used to be a carpenter–their definition of a right angle tends to be a bit less rigid than that of a mathematician.

Humans only invented the terms and definitions that make mathematics understandable. The same way they make a distinction between “granite” and “limestone”.

Math is certainly a “natural” force - sort of - in that you can change the terms or definitions, but “1” will still be “1”, “2” would still be “2”, and “69” would still be “69”.

An integer by any other name would smell as sweet.

Decimal vs. binary has nothing to do with it, the equation is precisely the same regardless of what numerical base you’re using.

As for the OP, consider some of the most elementary levels of math. Suppose there are other civilizations of intelligent beings in the universe. I find it incredibly difficult to believe that they would not also develop the notion of counting things, just as we have done. The language, notation, and so forth used would most definitely be different, but the ideas involved would necessarily be the same. Perhaps counting itself is an invention used to describe properties of the universe, but when that invention is invariant throughout the universe, I’m more inclined to describe it as a discovery.

From there, things such as addition and multiplication would be immediate (If I have n things and he has m things, how many things do we have together? If n of us each has m things, how many things do we have altogether?). The idea of prime numbers would not be far behind (Can I or can I not divide these objects up evenly among everyone?), and all of these notions would necessarily be identical to our own. Thinking of it in that sense, what room is there for “invention”?

One of the Socratic dialogues involves a discussion of whether we are BORN knowing mathematics. Socrates argues that mathematics training is really helping the student to remember what he already knoew. This position is not widely held today (to say the least).

One way to more precisely define the OP would be to ask whether a race of intelligent aliens would be expected to devleop the same mathematics. In fact, the Search for Extra-Terrestrial Intelligence is beaming some mathematcs into space. I think SETI is using something like the values of pi or e in base 2.

Their approach supports the idea that some mathematics is natural or discovered. However, having spent a long time as a math grad student, I believe that much mathematics is totally invented, and for a specific purpose – to get an article published.

Mathematics is a language (several languages, actually). The fact that the languages of math can usefully describe aspects of the world does not mean that the particular syntax invoved was “discovered”. Does the word “rock” mean that English was partly discovered?

Mathematics does not exist outside of the human mind. Much as I admire the Socratic Dialogues, mathematics also does not exist as an intrinsic property of the human mind. Thus, it cannot be discovered.

No. It supports the idea that any intelligent life advanced enough to allow interstellar communication would most likely have invented tools for describing the physical properties of the Universe. Binary mathematics is one such tool, and the extremely simple nature of its syntax makes it a good candidate for broad understanding.

What do the following things have in common: a pair of apples, a pair of motorcycles, a pair of hats, and a pair of birds? I’m sure there’s more than one answer, but one answer is that there’s a pair of each. That is what I call a “mathematical” property of the real world, the fact that some sets of objects share the same cardinality. The number “two” is the word we use to describe the property of having the same cardinality as a pair of apples.

And as Cabbage said, once you recognize things like cardinalities and ordinalities, other mathematical properties of the real world like addition start to become apparent.

(Perhaps you recognize those properties but refer to them as “physical” ones. In which case we’re just arguing over a definition…)

Nope. That’s why I put “one” in quotes. The number one, to me, belongs in the world of formal mathematical systems, whereas the phenomenon of being only one thing (instead of two things, or no things) is a observable feature of reality.

Originally the ancient Greeks (possibly the first to think about the philosophy of mathematics) thought about numbers this way: as a way of describing certain observed properties of things around them. Specifically, some things are numerous and others aren’t.

(The Greek philosophy of numbers goes farther than that, of course: Platonic forms and all that. But I’m not going there.)

It’s only in modern times that arithmetic has moved away from being a “physics of counting” to an exploration of a formal system. But the dichotomy between the formal system and the real world is still there. The difference between the two was cast into very stark relief by Godel’s Incompleteness Theorem, which among other things draws a sharp distinction between the truth of a number-theoretical statement (i.e. how well it models the behaviour of the real world) and its provability (whether or not it follows from the axioms of the formal system).

That summarizes our philosophical differences in a nutshell…or almost:

I believe that the universe has both physical and mathematical properties. (Although I’d be willing to bet that those categories overlap here and there.)

There are other systems of “arithmetic” out there. It’s just that the ones that differ from conventional number theory either don’t describe the real world very well (“One, two, three, many.”), or else they’re extensions of conventional number theory like the ones generated by Godel’s theorem, which differ only in some bizarre axioms which may or may not reflect real-world behaviour.

Its true that conventional number theory seems to be the theory to describe counting. We certainly don’t expect to make some new observation about the real world which forces us to update the theory, in the way that the orbit of Mercury forced us to improve upon Kepler’s and Newton’s laws. But perhaps somewhere some alien civilization (presumably with a reeeeally long lifespan) has sat down and counted some really big piles of frobniks, and discovered something which will someday force us to add a new Godel-like axiom to number theory.

Okay, so maybe even I don’t believe that. But it’s theoretically possible

These things are not pairs until I decide to see them as such. Until then, I see only apple, apple, motorcycle, motorcycle, hat, hat, bird, bird. Only when I seek to define and describe relationships between these things do I fire up the math. The relationships between physical things described with mathematics only exist when I employ mathematics to describe them. I could just as easily describe them as red, black, gray, yellow. They are only pairs when I need them to be. From this, I believe that mathematics, like poetry and song, is a language invented for the specific purpose of describing the observed world in a way made useful only through the developed understanding of that language.

Of course, I’m a bit out of my element here, and you are all more than welcome to point and laugh.

I agree with MathGeek: “we invented a language to describe things that we discovered.”
e.g.
I am writing a computer program using the most efficient algorithms to find digits of pi. Am I “inventing” the digits of pi or “discovering” the digits of pi? I say that I am “discovering” them, since these are values that exist independently of the method I use for my calculation (and even exist in the absence of anyone attempting that calculation).

When I prove a theorem that states that there is no largest prime number, I am demonstrating a fact that already existed even before the first human had imagined the “concept” of a prime number. How can one say that I am “inventing” this statement? I am “inventing” the proof, but I am “discovering” the fact that there is no largest prime number.

Spiritus Mundi and Tymp, I wholeheartedly concur. But since you asked, Tymp, I must now point and laugh at you.

But mathematics is not an entity in itself. As Spiritus Mundi, Tymp and I have attempted to point out in various ways, mathematics is a language or construct. Granite is a mineral consisting of such-and-such elements in such-and-such lattice. Granite exists physically independent of our attribution of nomenclature. The rock itself does not inherently possess the nature of the label “granite.” Nor do two rocks inherently possess the labels of “one” and “two.” Concepts of “oneness,” “twoness,” and “graniteness” are attributes we apply to objects—they are not attributes of the object itself. There exists a significant distinction between naming something and the actual properties of that something.

I concur, but you’re missing my point. This appeal to discovery seems to be partially based on aesthetics. Referring to the numerous examples of right angles in nature does not make right angles inherent properties of an object. It is our perceptions, and the constructs through which we describe our perception, which possess this aesthetic. My point was that that particular equation is aesthetic only in a particular system or language. When one changes the language, the aesthetic changes.

Again, mathematics is a language. One (human or alien) may enumerate, add or divide objects. Those actions are descriptions of our perceptions. Stating that the formal systems of numbers and arithmetic are discoveries is like stating that the formal systems of phonetics and syntax are discoveries. The capacity to wield language is based on physiology. The capacity to understand language is based on physiology. The innate-ness of language in the environment is absurd. As I stated above, “Concepts of ‘oneness,’ ‘twoness,’ and ‘graniteness’ are attributes we apply to objects—they are not attributes of the object itself.” Mathematical properties are not properties of the physical world—they are properties of the construct we employ to understand the physical world.

Yet that cardinality of “pair-ness” is a construct. It is a result of our perception; it is not inherent in the objects themselves.

It is precisely that definition which is at the root of this debate. Coming to an acceptable definition is essential to arriving at a resolution.

How does a number, an element of a language, exist prior to the conception of the language? In discovering the digits of pi or discovering that there is no largest prime number, you are discovering properties of the language which has been invented. You may be discovering properties of the language which has been invented (much as linguistics is an analysis of the properties of languages), but those properties aren’t inherent physical properties of the world–they are properties of a construct. Constructs, models, languages, et cetera aren’t discovered.

Yes, invented - even proofs in number theory are only true within a particular mathematical system. I used to subscribe to the idea that mathematics existed “out there” and all we did was discover its eternal truths. Then I read The Mathematical Experience by Davis and Hersch. You can get different geometries by affirming or denying the “parallel postulate”. You can get different ideas about the real numbers by affirming or denying the “continuum hypothesis”. Then you look at the world and ask which of these systems can be mapped onto it so that the correspondence is meaningful. Goedel’s theorem tells us that any mathematical system has undecidable propositions- this includes systems that deal only with integers. In other words, there is some statement like “All prime numbers have the property that …” which is true for one definition of integers and false for some other definition, but where both definitions include the Peano postulates.

But Arnold, those are statements that we make with mathematics. We invent the ships that allows us to discover the new lands. “No largest prime” is not mathematics. This sentence is not the english language.

Nen: How does a number, an element of a language, exist prior to the conception of the language? In discovering the digits of pi or discovering that there is no largest prime number, you are discovering properties of the language which has been invented.

But the digits of pi, or the fact that there is no largest prime number, is independent of the properties of the language I am using. (Disregarding of course the semantic differences between pi expressed in base 10 and pi expressed in base 16). If I tie a string to a stake planted in the sand, stretch the string as far as it can go and draw a line, I end up with a circle. The ratio between the line drawn in the sand and the length in the string will exist whether or not I invent a language to describe it.

FriendRob: Goedel’s theorem tells us that any mathematical system has undecidable propositions- this includes systems that deal only with integers. In other words, there is some statement like “All prime numbers have the property that …” which is true for one definition of integers and false for some other definition, but where both definitions include the Peano postulates.
But some statements are true regardless of your definition of the integers. Or are you saying that one could come up with a definition of the integers in which there would be a largest prime number? I thought Goedel’s theorem said that some statements will be undecidable, but it did not follow that “for any statement X, one can invent a system in which statement X will be demonstrably false.”

Spiritus Mundi, we seem to have a different definition of mathematics. I use the term mathematics to include the methods or language we use to find facts about (e.g.) the natural numbers, but also the facts themselves, whereas you restrict mathematics to the first clause of my definition.
If I say “my observations show that there is a nova at position X in the sky”, what would be the proper relationship of that sentence with the word “astronomy”. Would you say “that is a fact that I proved using the methods of astronomy”? i.e. astronomy describes the constructs I use to find out facts about the universe?

That’s not the case. If one constructs a language to enumerate objects, the elements of that language, i.e., numbers, are not independent of the properties of the language. Consider the English language. The term “tree” is a name applied to vegetation meeting specific requirements. A linguistic analysis shows that the term in question is a noun. One can speak English without knowing what a noun is, much as one can be unaware of all of the digits of pi, but the elements of a language are defined by the language–they are not independent of it.

Again, I disagree. The ratio of which you speak is not physical property. Physical properties of the scenario you describe displacement and mass. To describe how it appears one applies the term “circle.” The name “circle” does not exist without language, nor does the term “ratio” or it’s value. These terms are elements of the construct. An element of the construct cannot exist without the invention of the construct. If you disagree, can show me a ratio without language, mathematical or otherwise?