Mathematics: Invented or Discovered?

I admire your certainty. It takes stones to state this as a truth that will stand forever.

My apologies for misinterpreting your earlier comments on abstraction.

I was responding to statements like:
Math is certainly a “natural” force - sort of 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.
Thinking of it in that sense, what room is there for “invention”?
Put me in the discovery camp.
I think the analogy to language is a poor one. We can ascribe different attributes to objects and actions and use different words for them, but I don’t see any room to do that with mathematics.*


What, then, is invented? Does the word have any meaning in your lexicon?

All independent thinkers did not come up with the same mathematics.


Well, I mainain that many (if not most) of the “quantities and relationships” are dependent upon the invented form not upon observed natural phenomena. I agree, though, that observed phenomena were the impetus for the invention of maths in the first place.

math geek

You said: we invented a language to describe things that we discovered. That part of your position requires no clarification.

Do you apply the same rules when speaking of other language? Does English include the referent rock as well as the label “rock”? If not, why is mathematics special in this regard. All languages are invented to describe phenomenologies, many of which have independent existence in the non-solipsisteic world.

I fail to see the analogous elements of zero and inertia. Can you please draw this existence of zero explicitly. The laws of physics have been aproximated from increasingly detailed observations of physical phenomena. Are you saying that “absence” falls into that category? Surely the distinction between something observed and something not observed is relevant when speaking of a “natural” property of reality. The leap between “I don’t have a rock” and “I have zero rocks” was certainly not intuitively obvious to the majority of early civilizations.

Yes, mathematics contains many elements that are useful models of natural phenomena. English also contains many elements that are useful models of natural phenomena. Languages are invented to meet need. Sometimes that need is to describe an observed natural phenomenon. Other times (unicorn, high order matrices) it is not.

“Trees” is an element of English. Trees are not.
“fibonacci sequence” is an element of mathematics. Petal, petal, petal petal, petal petal petal, petal petal petal petal and petal are not.

I’ll try and clarify my stand on some of the questions just raised; if I accidentally omit any, just ask and I’ll clarify further.

(At least) two things are common to any science (and for this post I’ll classify mathematics as a science, but it’s not necessarily clear to me that that’s entirely appropriate). One, observation/discovery of the properties of whatever is being studied; two, invention of a model to accurately describe those observations. I was about to make the distinction that, in this thread, it seems that many of us on the invention side claim “mathematics”, by definition, to be entirely contained in the latter category–simply that mathematics, by definition, is the model/language. I was also going to argue, as I have been doing, that there is also the observation aspect to mathematics, common to all sciences. We observe, as I’ve mentioned before, the properties of quantity, for example. I was going to say that, in this sense, it is more appropriate to define “mathematics” as not only the language used, but also the inherent, observed properties of number that we use that language to study.

However, I just read panamajack’s post, and it occurred to me that that might be an even better way to describe my opinion on the matter–that mathematics is nothing more than a language, but the fact that it is the only such possible language implies that the language itself is a discovery. (It should be noted here that in this interpretation I’m still claiming there exist the observational “roots” of mathematics (again, observations of the properties of quantity, for example), the distinction I’m now making is that I’m simply not including the observational aspect as part of the definition of mathematics). Let me try and clarify my take on this further.

Take physics (or any science), for example. Physics certainly includes both discovery/invention aspects common to all sciences, we can observe the phenomenon of gravity, then build a model to describe it. What if our particular model of gravity comes from the mind of “God”, so to speak? What if our model not only accurately describes gravity, but is, in fact, the actual way gravity works, in some sense? Certainly, by one definition our model is still an invention, but is it not also correct to describe it as a discovery? If there was some omniscient being watching over this, could it not correctly exclaim, “My God, they actually discovered the Law of Gravity!”?

Of course, such an event would be unknown to us–it is epistemologically impossible for us to know the true nature of the universe, it is only possible to build better and better models scientifically, ignorant of how close or how far we actually are from the truth. At this point I’m reminded of a passage from G.H. Hardy’s A Mathematician’s Apology; unfortunately I don’t have a copy at hand, but I will try and paraphrase the hell out if it, while keeping the point I’m trying to make. He said: I was speaking somewhat provocatively, perhaps, since I was addressing a group of physicists, when I mentioned that if there was a fundamental difference between what mathematicians and physicists study, it would be that the mathematicians are in much closer contact with reality than the physicists. This will probably surprise many people, since it is the “real world”, after all, which physicists study, so allow me to explain. Take a chair, for instance. Depending on your perspective, a chair can be many things–it can be a place to sit, it can be mass of solid matter, or it can simply be a collection of swirling elementary particles. On the other hand, for example, the number 29 is prime–not because the universe is constructed in some particular way, not because of some peculiarity in the way our minds think, it is prime simply because it is prime.

Anyway, the point I’m trying to make here is that while one may think of mathematics as an invented language, it is also the way things must necessarily be (and we demonstrate this through mathematical proof), and can therefore also be truthfully called a discovery. Sure, idealized circles do not physically exist in the universe, but why do we see so many close approximations to circles throughout the universe? Because of the mathematical properties of the circle, which are unchangeable.

I’m gonna have to cut myself off here, this is long enough as it is, but I hope that was clear and that, at the very least, clarified some of the details on my position.

Well, the logic is interesting, but the assertion that any mathematic is the only possible such language is false. The history of math shows us several algebras. Newton and Leibnitz developed distinct bases for calculus. Which of these languages do you now claim was not possible?

x[sup]2[/sup] + y[sup]2[/sup] = k
{The points in a plane distance k removed from the origin}

Which of those is the only possible mathematical expression of a circle?

It is also odd, less than 30, whole, and a factor of 87. How does selecting one mathematical property from among many grant 29 a special distinction that selecting one physical property from among many would not grant to a chair?


gravity discovered?? we have “discovered” that things fall toward each other and have mathematical models to describe that behavior. as we gain knowledge we update those models. we have not a clue as to why things fall towards each other. we have hypothesized gravitons, but afaik, never seen one. i wouldn’t get too excited about this discovering gravity, my cat has discovered that on her own.

joe cool:

not even close. the mathematical definitions of all these concepts is precise. each planet has wobbles outside it’s elliptical orbit and outside it’s plane or revolution. also, the planet defining that ellipse in space is hardly a point, which is a very important part of the definition of an ellipse.

irises and pupils?? to quote you: very nearly circular. sorry, we are talking mathematics. close enough is not good enough.

the sun is not elliptical, it is a solid, therefore an ellipsoid [but only roughly].

there are no circles, ellipses, ellipsoids, spheres, lines or planes. except in our minds. and that is where mathematics resides. we try to descibe nature as well as we can so that we can predict it. but we are always approximating.

none of the geometries says it only applies in some part of the universe and not others. they each start with basic unprovable assumptions [axioms], and go from there. the axioms are unprovable. the only “proof” lies in whether they produce a useful model for the universe. if they are useful for part of the universe but not other parts that seems to me to prove they are not natural, but human-made.

Comparing math to “other languages” would seem to presuppose that math is nothing but a language, which you already know I don’t agree with. But if you’re asking “why do I consider math more than a language”, it’s because of phenomena like cardinality which are pre-existing in nature (and hence no more part of any language than rocks are part of English), and which I consider to be part of mathematics.

Inertia and absence are both phenomena whose existence has been confirmed by repeated observation. The words “increasingly detailed” don’t seem to apply in the case of absence, but other than that I would say yes, “absence” falls into that category.

Ah, but “absence” is being observed. The property of reality that we’re talking about is being observed. The rock or hammer or whatever isn’t there isn’t the property of reality that we’re talking about at the moment.

So? Neither was the leap from phlogiston to atomic theory.
Just because our understanding of something is occasionally refined doesn’t mean we invented that something.

Once more, my position is as follows: mathematics is more than a language (and hence is more than just something we invented). There are phenomena, like cardinality, which are inherent in the real world and which were not invented by us. We merely invented a language to describe them. (A fibonacci sequence is a more complicated example of the same type of phenomenon.) The reason why I feel that these phenomenon are not just what Nen called a “perceived property” is because of their impact on other physical phenomena, like whether a particle of antimatter exists after an experiment (my example) or the chemical properties of an atom (Cabbage’s example).

Yes. Exactly. And we’ve also discovered other phenomena, mathematical phenomena, like cardinality. That is discovered, and is a part of mathematics.

I will attempt to contain my excitement. Give your cat a scratch behind the ears for me.

No, the people using the formal systems of geometry to describe the mathematical properties of the universe say that. The formal system of Euclidean geometry describes the shape of space…not perfectly, but that’s not the point. The more general formal systems of Riemannian geometry describe phase spaces, which are an aspect of the behaviour of other physical systems.

Exactly. The models, which are one part of mathematics, describe certain aspects of the universe which are another part of mathematics.

Our models of the phenomenon of gravity (don’t worry, I’m containing my excitement) are not useful when it comes to describing nuclear forces. This does not imply that we invented gravity. We invented the models that we use to describe gravity, just as we invented the models which describe the natural numbers. But that’s not the same thing.

I’m going out of town tomorrow, so don’t expect replies until Sunday evening.

Mathematics is a language. Is language invented or discovered? To claim language is discovered is absurd; it would mean that language exists “out there” somewhere and we stumble upon it. Therefore, one jumps to the claim that language, and by extension math, is invented. But look upon the stunning lack of success of all of the invented languages, the most famous of which is Esperanto. Language cannot be invented, either.

Math is not a language, or not a full language that you can woo someone with ask them to please lower the toilet seat tell jokes in express gratitude for a superbly cooked meal followed by a mediocre movie and a mind blowing blow job in. Math is a specialized, ‘laboratory’ language, which, rather spookily in my opinion, mimics and predicts… physical attributes in the universe.

It ain’t all made up to get published in the journals. Is it Kempler’s Law that predicted the orbits of the planets? Doesn’t that indicate that the dipoles of the OP – discovery versus invention – don’t cover the whole interaction between our tools to understand the universe (like math, like language) and the universe itself?


Originally posted by Alessan
Plus, if you put a wall up at any angle other than 90 degrees from the ground, it’ll fall over.

Originally posted by Nen
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
I’m building a geodesic dome. May Bucky Fuller haunt you to the end of your days!

have done so. she appreciates it.

but that is exactly the point!! if mathematics is the actual rules that govern the universe, then no variation from those rules is possible.

we may discover a relationship between two things. but as you say, the people using the geometries to describe mathematical properties say that the geometries do not perfectly describe the shape of space. [you are supporting my argument.] since the mathematics describe the relationships rather than “are” the realtionships, mathematics is merely a language. languages are created. you have become so accustomed to using mathematics to describe the phenomena that you have come to believe it is the phenomena.

an example to illistrate what i mean: chinese people learn to read chinese characters from an early age. they so identify these characters with the meanings, that they will insist that the characters resemble what they stand for. the characters for the sun, the moon, and river are often cited. if one is an outsider to chinese writing, and sees these three characters, they will not guess their meaning from their depiction. [the character for the sun is essentially a square with a horizontal line bisecting it.]

For the latter, those are simply two different ways of describing the same object, the circle; I didn’t intend to imply that there is one and only one “True” description of any mathematical object, what I’m trying to say is that the mathematical object was discovered, not invented. Of course our mathematical notation is invented, that much is obvious, perhaps I should clarify my definition of mathematics to include the ideas expressed by the notation.

Apparently what led you to ask that question was my previous post, where I defined “mathematics” as the language, and claimed it was the only possible language. I had not thought that out fully enough before I posted, and I apologize for that. (I’ve never given this debate as much thought before, so I guess I’m still working on the best way to verbalize my thoughts on the subject, but I thank ya’ll for the opportunity :D). I was confusing some of the words in an effort to describe my position, certainly any language is invented, including the language of mathematics, it would have been more appropriate in my previous post to have replaced “language” with “ideas” when I said, “the fact that it is the only such possible language/idea implies that the language/idea itself is a discovery.” I do claim that the ideas are discovered, and that they are universal–regardless of how the universe is constructed, or how our minds function, the ideas of mathematics remain unchanged. In fact, I now feel it’s more appropriate to describe mathematics as the ideas represented by the language itself, not the language. Again, I apologize for my confusion and lack of clarity on this before. Let me try to clarify the difference, for me, between the language of mathematics, and mathematics itself, by giving a paricular example of each.

First, here’s a property: Take any four digit number (not all digits the same). Write the digits in ascending order, then write the digits in descending order. Subtract the smaller number from the larger number. Repeat the process with the number. Then again. And again. After at most seven times of doing this, you’ll end up at Kaprekar’s constant, 6174, regardless of your original starting number. An interesting curiosity, but I wouldn’t truly call this a mathematical property. It’s simply an artifact of working in the decimal system, which is a completely arbitrary choice; if I switch to binary, octal, hexadecimal, or whatever, this property may either change or even disappear completely. It’s a consequence of the language we use for mathematics, and I’m more inclined to label it as a consequence of the invention of our decimal notation.

On the other hand, take for example (and here there seem to be many, many choices on what to use as an example) the prime numbers. The prime numbers are not dependent on the language used, and as I’ve mentioned before, since I consider the counting numbers and their multiplicative properties to be discoveries, the prime numbers must also be a discovery, since they are completely determined by the multiplicative properties of the counting numbers.

I believe the same argument applies to Newton and Leibnitz. To be honest, I’m not at all familiar with much of calculus as they knew it, only what’s been carried over into the modern language and notation, but as I understand it, while their language and notations were different, the underlying ideas were the same, which, again, seems to give strength to the discovery argument. And that’s not to imply that anything was incorrect or invalid about their language and notation; there is nothing invalid about Roman numerals, but it’s simply a bitch to do any calculations with them. Over the course of history, the notation simplifies, but the underlying ideas are constant, and more discoveries can be made with simpler notation.

The way I read it, the point is not so much the varying ways a chair or a number can be described, but the nature of those descriptions (this may have been lost in my paraphrasing, I don’t remember how the original reads). Our observations of the physical world are inherently subjective, to an extent. It’s conceivable to me that, if we had a different set of senses other than our given five, our models of the universe could be significantly different. Or perhaps our bodies could have been made of a different type of matter, allowing us to walk right through a chair without even recognizing it as a solid object as we do now. Additionally, it’s conceivable that the physical properties of the universe could be, to an extent, arbitrary. I believe it was Einstein that said if he could ask “God” one question, it would be, “Are the physical laws of the universe necessitated, or is it possible that the universe could have been constructed differently?” (again paraphrased). In that sense, it’s questionable whether we discover any “Ultimate Truth” through physics; we can have some success at discovering some of the properties of this universe, but those properties are not universal {multiversal?) throughout the realm of “all possibilities”. Again, however, I argue that mathematical properties are “universal/multiversal” throughout the realm of all possibilities. It’s not some quirk of this universe, or some quirk of our minds, that makes 29 prime (or, indeed, has any of the properties you mentioned); it is an absolute property of 29, it is independent of all other things–it is because it must be.

The axioms are invented, the theorems are discovered.

Not sure about the logic. The discoveries are made with respect to an invented analytic system.

Math as a language Not sure about this. Typical languages have an invented vocabulary. AFAIK, some linguists believe that the rules of grammar are an artifact of the study of language and that in some sense a typical sentence by native speaker can not be ungrammatical. So the rules of grammar are empirical generalizations which must be discovered. But the underlying structure of the language is invented.

Now, I suppose mathematics has an invented vocabulary (eg 1,2,3, right angle, hypotenuse) or at least invented labels for these concepts, but as far as I can tell the scope for invention within the grammar of mathematics is limited (again, the theorems must be discovered). Of course, I’d have more confidence if I knew what the heck “the grammar of mathematics” was.

And deciding whether the concepts “2” or “right angle” are more like theorems or axioms is a question that I’ll have to leave to those with a better background in set theory. Apologies if the above is hopelessly confused or misses the point.

If you knew what they grammar of math was, you’d be more famous than Cecil, so don’t sweat it tonight.

Languages don’t typically have an invented vocabulary; languages generally have a stolen vocabulary. English is approximately 30% anglo-saxon, 30% latin, 30% greek, & 30% other. Sheriff. Boondocks. Robot. Adobe.

Yeah, that’s over 100%, that’s only ONE of the unruly behaviors of a REAL, evolving language.

Math can attempt to explain simple things like gravity, in a theoretical sanatized universe.

Can it delineate the break-up of a three year affair, & how miserable you feel after?

Can Math stroke the absense of her fiery Red hair?

Math is not more than a language, it’s less than a language. It is very similar to a language; hence the poetical metaphor. Back in high school, a current tagline was “Give me x+1 fucking breaks.” The humor of this derived in part from the fact that the connotative aspects of existence aren’t quantitative.

We’re not “varying from the rules”. We just don’t yet have a perfect description of the rules.

No, I’m not. Once more: the models, which are part of mathematics, describe certain aspects of the universe which are another part of mathematics. The fact that the descriptions aren’t perfect does not support the position that the things being described were invented. The fact that the descriptions aren’t perfect has no bearing on whether or not the things being described were invented or discovered, which is what I’ve been trying to tell you.

Please do not tell me why I believe something. (That’s rather patronizing.) Instead, either explain to me why cardinality (for example) is not part of mathematics, or explain to me why both my arguments and Cabbage’s as to why cardinality is a discovered phenomenon are wrong.

i did not patronize you. i’ll try not to tell you why you believe, and you can not tell me how to make my argument.

discovery: finding something, then trying to figure out what it is.

invention: making something that didn’t exist before [usually with object or tools that did exist previously], normally needing to make adjustments and refinments to improve it.

those seem like reasonable descriptions of those processes to me. it seems to me that mathematics fits invention rather than discovery. we use it to describe discoveries, and refine it and improve it as time goes on.

I’m in the ‘Math is discovered’ camp, although I don’t think I can state why as eloquently as Cabbage, Math Geek & others here.

I remember a definition of math to the effect that it is a kind of language about which the subjects & predicates are complete unknowns. (It was a little clearer than that; perhaps someone else knows what I’m referring to & can state it?)

Anyway, this definition always struck me as a rather impoverished view of math. It focuses on the mathematical proof (clearly the backbone of math) by denying the ‘content’ of math; e.g., numbers & geometrical concepts.

Also, this definition is a retrofit; it reflects a revisionist view of the history of math.

No. mathematics IS. We humans have created many languages to describe and explore it, but the languages aren’t math, they are our LANGUAGE of math.

You can only reach the conclusion that math is a language a-priori. By DEFININING the word mathmatics as ‘the language whereby we do math’, which is pretty unsatisfactory, since that leaves us without a word for the fundamental truths that we discover when we use that language.

Put me in the ‘math is discovered, but the languages are invented’ camp. But really, you guys are only arguing semantics.


So far no-one in this interesting thread has questioned the validity of the invented/ discovered dichotomy. Why not?

It is not obvious to me either that evolved brains would be able to see Truth, nor that they would have the capacity for creative mathematical thought without there being a use for it. Rather than a sharp distinction between invention and discovery, how about a fuzzy one between embellishment and unfolding?

I think you want to be a bit careful with this line, since it is you arguing that some strings of symbols have an inherent meaning and relationship with what “is”.

I had something to say, but it has already been covered in this thread, so I will just cast a vote for discovered

Evolved brains are not able to introspect Truth… see the history of Western philosophy.

Why should we have evolved the capacity for creative mathematical thought at all? Pragmatically… you can’t ward off a sabre-tooth tiger attack with it, it don’t cause babes to swoon at your feet (if we’re just meat sacks designed for the tranfer of DNA, being a math geek (in the generic and not specific sense, MG) it would have been bred out of us long ago.

The real question is, if math is made-up, why would it be predictive of the physical universe? In some sense, it has to be discovered.

On the other hand, since, like religions, there’s a plethora of maths out there, we can clearly see that they’re made up.

picmr, you are outside the box. You have the soul of a poet. We’re obviously asking the wrong question here.

GEB was a big influence on me growing up. Mapping concepts, Bach’s fuges, the lack and importance of a frame of reference in the universe… it was a great eye opener.

I spent a couple of years as a linguist, thinking I wasn’t fit for anything other than Academentia in America.

Eventually I got a job that covers the rent. Whippie.

I’d rather get off the gerbil wheel & re-invent the universe with the tools at haand.