Math: Discovery or Invention?
 

I was just curious what others' views on this subject are. When someone comes up with a new theorem or result in math, has he or she invented it? Or was it there all along and he or she simply discovered it?

It's quite definately an invention. This concept was proven by the discovery of a tribe in South America for which numbers have absolutely no meaning. They do not live with quantifiables. Also, as a child, you have to be taught how to count. It simply lends to the notion that math is an inventive use of our higher thinking processes.

Lot's of people will create philosophies based around various numbers that we can derrive out of our base 10 numbering scheme (pi, phi). How do you think it would be if humans only had 8 fingers, and thus used a base 8 numbering system instead?

Math is nifty, but it's not the truth.

Just because they don't understand mathematics, doesn't mean that 3 + 2 doesn't equal 5 if you see what I mean. It's an abstract concept.
Could 3 + 2 = 6 be correct if someone else invented addition?

Maths IS universal the figures and signs we have used to represent it are an invention. This tribe if shown two sets of pebbles one with 3 pebbles and one with ten would not know which set had the most (that is dealing with quantifiables). a base 8 numbering system is just as valid as base 10 and used in mathemetics regularly (i think binary is base8)

There are 10 types of people in the world....

Those who understand binary,
And those who don't.

Personally I feel its a bit of both. There is the invention portion where you come up with a new concept or area of mathematics (i.e. the invention of sets, algebras, etc.) and define everything and how it interacts, and then there's discovery portion where you figure out everything that logically follows from your starting axioms and definitions.

"Math" is an invention that we use to describe abstract concepts that we discover.

what??? abstract concept - whats abstract about addition, Maths itself is the abstraction

I think that asking if 3+2 could = 6 if someone else invented math misses the point. In another version of such math there might not be a 3 or a 2. Mathematics is just our way of dealing with abstractions or forms from the real things that our perceptions tell us exist.

For example, if you were to count all the books in a library you might come to the conclusion that there were 6,232 books in the library. But that's only because you identify each separate book as belonging to the general abstract group of "books."

What if you were able to perceive each individual book for what it was? Then you would have "The Hardy Boys", "A Brief History of Time", and "The Coming of Conan" etc... You would not have to lump all of these distinct objects into the group "books" and then deal with them in a like manner. Numbers of things, and the consequent addition or subtraction, would be meaningless because every individual thing would be unique.

I don't know that I really have a solid opinion on this issue, other than that a bald constructivism like Halx's is false. On the other hand, I do tend towards something of a constructivist view. On this view, mathematics is universal, but not objective. It's a construction of the human mind, but a very basic one, one that all human minds share. (and I want to say all non-human minds share as well, but we really don't have much evidence either way here.) It has to be taught, and there are societies which don't recognize it, and there are different ways of counting, but they all come down to the same thing. Whether you want to say 2+2=4 or 10+10=100, you're using the same numbers, just different numerals. Mathematics is, you could say, the form of our intuition.

Besides which, the things mathematics claims to be true can be proven, and can be proven much more rigorously than most of what philosophy claims. So the denial of the universal validity of mathematics is a denial of the universal validity of logic, and that's just a denial of the validity of philosophy.

I'd have to say it's a discovery.

Flat out, it allows us to describe objects in any way at any level.

For example, a volume of a sphere IS 4/3*pi*r^3. If you didn't have math, how would you know that? Eventually if someone wanted to find out and math didn't exist, they'd end up with... math.

Even if we were born with 8 or 29 fingers... I dunno, I'd have to think the numbers would still be the same. 10 is easy to work with mentally and visually. Probably just coincidence that we happen to have 10 fingers/toes. I couldn't imagine, if, say, we had 8 toes that things would be in octal.

I see it as an invention.

I have no idea what is really "out there" and I have no idea whether our descriptions of it - such as language and mathematics - actually describe anything other than internally consistent relationships.

Although, hm.. I'd say it's a mixture of both.

A tool used to relate the description of an object to our terms.

Let's just say we had a definite theory or equation for how blackholes work... if humans didn't exist, would black holes still operate in that manner? Yes, just... not in the units that we established.

Maybe if there was another being, it would be a completely different way to describe it, but in the end it would end up describing the same thing.

it is a discovery. Look at how kids are taught math:

"take one apple. Put another apple next to it. How many apples do you have?"

some time long ago Grog and Ug sat in a cave and noticed that they had a dead antelope and a dead antelope in front of them. They discovered that if you put one antelope next to another antelope, there are two antelopes. The concept of quantity was discovered, and from there stemmed all mathematics.

I would lean towards discovery, because it seems that all our concepts of numbers and logic are a priori. However, I'm not firm in that belief, because of course it's very difficult to tell what is actually a priori.

Bingle

shakran, we only count two antelope because we see those two objects as antelopes. Using our senses and the schemas in our brains we identify those two objects as having some characteristics that are the same in a very general way. But why do we have to think of things in such a general way? Certainly if you were to examine the objects closely enough you might be able to discern some differences. You might name them individually as Mr. T. and T.J. Hooker. Then, although they belong to our idea of what is an antelope, they are also very different things in that they are Mr. T and T.J. Hooker. If another person were to come along, he or she would recognize two antelopes, but you would see distinct objects, Mr. T and T.J. Hooker.

Which is right?

Are either of you more "correct" than the other?

Isn't it possible then that perhaps some alien civilization with more powerful sensory organs than us, or more developed schemas might see things in individual terms, rather than as parts of groups?

And Stompy, yes mathematics works very well to describe things that exist in mathematics, like spheres. But do such ideas really exist in reality? Outside of our minds do spheres really exist? Euclidean geometry works very well for simple objects that we interact with most of the time, but it falls apart on cosmic or atomic scales. If another civilization were to see things on a different scale they might come to very different conclusions than us.

It's a matter of perspective and understanding. Mathematics is useful to us as a method of describing the things we perceive around us. It doesn't actually tell us much about those things.

It's most assuredly an invention. Math is a language created to describe physical phenomena. By your logic, the English language is a discovery, because, if we didn't have English, we wouldn't know that "red" was a colour.

No, by my logic, that would mean communication was a discovery, because "red" can be described in all other languages - in the end you're describing the same thing no matter what method.

Likewise, if there was no communication and you wanted to convey an idea, eventually it would lead to communication.. because how else would you convey that idea?

Get it now?

Of course, it has to be an invention more like the invention of English than like the invention of, say, a religion. Both our native language and the language of mathematics affect how we think to such extent that it is virtually impossible to think otherwise without falling into contradiction. And while it is correct that whether or not we have two things before us depends on what we count as things, all the same, if we agree that we have one antelope and another antelope in front of us, we have to agree that we have two antelope in front of us. The point of my saying that it is a form of our intuition (and synthetic a priori) is that, while it may or may not map on to what is really 'out there', it does describe something accurately and universally -- the way we think about what's out there.

what asaris and art said.
to think otherwise would be to presuppose a natural order already extant in the world that human beings have simply discovered.
which is absurd.

But Math is just like the English language in that it is just another language or method of communication. Communication may be some kind of discovery, but the particular method of communication is entirely dependent on the person/thing creating it.

Yes and no. I don't understand why the lack of another language paralleling mathematics automatically makes it a universal truth rather than a language.

It's a universal truth to us.

We don't have any other thing to compare math to for our practical purposes.

Like I said, you'd be describing the same thing. If you said "red" in english, you could say the same thing in german, chinese, whatever.

If you use the method, for example, to describe the volume of a sphere and I say 4/3*pi*r^3, if you have another method, great, but in the end we're saying the same thing.

Even if we discovered this alternate point of view, dimension, whatever, there would still exist the method to describe that object to how WE see it for OUR practical purposes.

That's why I think it's a discovery, but at the same time an invention.. because usually inventions are created out of need for something, yet .. we needed something to help us describe everything around us. What we came up with so far has matched what is. The symbols, numbers, etc... yes, we made up, but they are used to convey the same basic idea (to other humans).

If it wasn't a truth, then half the things we have or use today that are heavily dependant on mathematics couldn't exist. Could there big a bigger picture? Sure, but for now, it works.

But isn't it possible to see things differently? What you see as antelopes I might see as two more specific things, Mr. T and T.J. Hooker. I only get to your level of perceptive analysis if I dull my senses, perhaps my sight, so that they look the same from a distance. But there are other ways of looking at those objects without labelling or counting them the same way as you. I could see them as two organisms, or as two million carbon atoms, or as zero uranium atoms, or as a subpart of the earth's ecological system, or as a heat emitting source, or as a light reflecting source, or any number of other ways that don't recognize them as antelope and don't recognize that there are two of them.

Through communication perhaps we can arrive at a common understanding that you are referring to two antelope, but there's no reason to assume that it's any more true than an opposing view of them as different objects, or as the same object.
But this assumes that there really is such a thing as a sphere, and not just our perception of a sphere.
I think you mean if it wasn't accurate then half the things... That's partially true. But remember it's only really accurate on our scale. Larger or smaller scales don't respect our cognitive distinctions.

Meaning, to us, there sphere is there. We see a sphere and that is how we measure it.

Sure, if you zoom out to outside the universe, things might seem different, just like if you zoom down to the sub-atomic level things appear different.

I'm sure a unified theory would elaborate more on this.

That's right, we see a sphere and measure it as such. Such an observation is accurate for our perceptions, fills our sphere-measuring needs, and allows us to use the thing as a sphere to construct a globe (or whatever it is one does with spheres)

But to say that the existence of such a sphere is truth, I think that's going too far.

I'd consider even something as simple as numbers ("1", "2") to be an abstract concept, and I'm calling "math" the concrete language we use to describe these concepts.

This is probably all semantic.

It is all semantic. The majority of philosophically based discussions are semantic.

Why?

(extra text added in parentheses to get around the crazy 10 character minimum post length :P )

I think the discovery of that tribe that has no concept of math proves that it is NOT a universal truth to us. Oh well, if you guys wanna make that horrible misstep of logic, by all means, go right ahead.

i referred only to asaris' s second post--it is an invention.
i am not a platonist--i do not believe there are forms in the universe somewhere that determine/condition human inventions.

It's not universal knowledge. Just because a group doesn't know about something doesn't mean it's an invention rather than a preexisting thing that was discovered.

People didn't know about Pluto for most of human history, but that doesn't mean that it's "discoverer" actually invented it.

No one invented gravity - it's a law of nature that is expressed with language.

By the same token, no one invented mathematics - it's a bunch of laws of nature expressed with numbers.

I think its fair to say that parts of math are invention. For example, Newton invented calculus. He came up with the concept of a limit, and of infintesimals (even though we don't use that concept anymore), and of a derivative. But he discovered the Fundamental Theorem of Calculus (the integral of a function is equal to its antiderivative). It was a discovery because it followed logically from the concepts he came up with, but he didn't know it when he came up with those concepts. It was all formed and waiting for him to find.

the tribe analogy is not valid. do you think they used electricity or knew what oxygen was? we discovered those last two. math is, to me, neither an invention or discovery. it is a description.

Gravity is not a law of nature, it is a law of human science ascribed to nature. Gravity is, in all forms we know it, a description of physical phenomena that may or may not be fully correct.

Similarly with mathematics; it is a construction of human intent. It is used by us to ascribe characteristics to aspects of physical phenomena, as well as theoretical phenomena.

The mere fact that we can describe something, or give attributes to something, does not automatically make it real, true, or correct.

I hate to cite this as an argument, but take the case of animals that have been trained to count (dogs e.t.c.), their minds have been conditioned to understand things mathamatically rather than discovering them themselves. Things will always be able to be compared as long as two things exist, it's the manner in which you do this, being able to determine the size of half an object, the size of two objects of the same size. Only certain trained minds can comprehend these things. Just as some say Grog and Ug discovered their latant mathamatical ability, I say that by mistake or otherwise, they re-invented their perception of reality to be able to understand it.

just my 0.02

The difficulty, Halx, is that you're operating under a false dichotomy. Either math is invented and is not universally valid, or it is discovered and is universally valid. But my point is that it is 'invented', but is universally valid. I would also say time and space are 'invented', but would you say that that means they are universally valid?

I'm putting invented in quotes here, because I don't mean, and I assume you don't mean, that some guy sat down some day and decided that 2+2=4. Sure geometry and calculus were invented in this way, but arithmetic? That's why I used the analogy of language. No one invented language, it simply arose out of changes in the human condition.

Science is, I think, a different sort of thing from mathematics. Math really does describe the way the world is. Science may or may not. There are lots of disputes about this, and I'm no philosopher of science. But the position that science is nothing more than a useful predictive heuristic is a reasonable one to hold; I don't think that, at the end of the day, such a position can be maintained with respect to math.

I've tended toward "invention" ever since reading Hoffstadter's Godel Escher Bach and trying to wrap my brain around Godel, who essentially proves that mathematics by definition cannot be both consistent and complete; any mathematical system which attains a certain level of "completeness", i.e., it is a powerful enough tool to describe pretty much anything, becomes inconsistent in that perfectly valid expressions can be formed which are paradoxes, both true and not true. And conversely, any system not plagued by this problem is simply too limited (not powerful enough) to be comprehensive.

To me, this sounds like mathematics is fatally flawed in much the same way as Newtonian physics, which turned out to be an over-simplification once we attained the ability to measure things which are very small or very fast. It seems to me if mathematics were an inherent part of "the way things are", it should work perfectly without paradoxes and inconsistencies. The fact that it doesn't work that way suggests to me that it is an invention of the human mind, and like many inventions is not so much perfect as it is a convenient tool.

I disagree, I think people did invent language, it just took a long time and no one person can be described as the singular inventor. Inventions don't have to be confined to physical or technological things. When scientists use gene splicing or similar techniques to change the physical attributes of an organism, for example the sturdiness of a tomato, is that a discovery or an invention. I would call it an invention.

From Dictionary.com

in·ven·tion ( P ) Pronunciation Key (n-vnshn) n.
The act or process of inventing: used a technique of her own invention.
A new device, method, or process developed from study and experimentation: the phonograph, an invention attributed to Thomas Edison.
A mental fabrication, especially a falsehood.
Skill in inventing; inventiveness: “the invention and sweep of the staging” (John Simon).
Music. A short composition developing a single theme contrapuntally.
A discovery; a finding.

dis·cov·er·y ( P ) Pronunciation Key (d-skv-r)
n. pl. dis·cov·er·ies
The act or an instance of discovering.
Something discovered.
Law. The compulsory disclosure of pertinent facts or documents to the opposing party in a civil action, usually before a trial begins.

dis·cov·er ( P ) Pronunciation Key (d-skvr)
tr.v. dis·cov·ered, dis·cov·er·ing, dis·cov·ers
To notice or learn, especially by making an effort: got home and discovered that the furnace wasn't working.
To be the first, or the first of one's group or kind, to find, learn of, or observe.
To learn about for the first time in one's experience: discovered a new restaurant on the west side.
To learn something about: discovered him to be an impostor; discovered the brake to be defective.
To identify (a person) as a potentially prominent performer: a movie star who was discovered in a drugstore by a producer.
Archaic. To reveal or expose.

Clearly the two words have some overlap, so that's probably where the confusion is coming from. But I also disagree that math really does describe the way the world is. Math describes the way we see the world. We see a sphere and treat it as such. But spheres don't exist as individual things in the real world, the only reason it's a sphere is because we call it such. Math is a subjective approach to reality. A single sphere to you might be a collection of one thousand carbon atoms to another. There are many ways of looking at things, not just ours.

And I don't think science is a description of the world, or a predictive method. I think it is a method for determining knowledge, a process exemplified by the Scientific Method (hypothesis, experimentation, observation...) Science is always changing as more accurate information becomes available.

And so what if this discussion is semantic? Does that somehow make it unworthy? The determination of meaning and changes in meaning is important if we want to communicate with each other. If we never addressed semantic issues and instead operated with our own individual ideas about language we'd have great difficulty communicating. Perhaps if you tried to order a cheeseburger you'd get crucified upside-down instead if we all ignored semantics.

What about the golden ratio? It's inherent in nature, the human form, even typography is based off of it! If there are things in nature that suggest mathematics and geometry, then wouldn't math be a discovery?

I would say that maths is the only objective approach to reality.
This wouldn't be a sphere.
You can mathematically describe any object, but simplifcations are often used because.. they're simpler.

anti fishstick, my response would be that to have an a priori notion of what is nature contradicts my statement "I have no idea what is really "out there"...

The acts of "seeing" or "observing" or "measuring" such things as golden rectangles or the golden ratio must involve the potentially erroneous practice of overlaying our own mental constructions on what is "out there."

That's the epistemological problem. The very act of interpreting golden-ratio-type relationships as preexisting in nature is a mathematical operation. What we do, it seems to me, is impose our internally consistent relationships upon whatever may be "out there." I can't see a way we could have any necessarily true idea of what nature is.

OK, what about dogs, such as mine, that have not been trained to count, but if you mess up and give them the wrong number of treats, they bitch at you until you add in the missing ones.

They've not been trained to count, yet they know about quantity.

Sorry for jumping back, but I thought this needed addressing.
"Special" numbers like Pi and Phi are usually determined by the ratio of two quantities. The units you use to describe these quantities is irrelevant, as the result is always the same.
Same with the description of a sphere that someone mentioned.

Your statement is a bit like saying "the distance between New York and Tokyo is greater if you measure it in km rather than miles."

Of course it would be a sphere! As a three-dimensional object the sphere would have to be constructed of something. Carbon atoms are just as good as anything else.

And yes, we use mathematics to describe objects, but the mathematics are irrelevant to the object. These simplifications are simple for us, not intrinsically. From a distance the Earth resembles a sphere (or an Oblate Spheroid if you prefer). For its inhabitants, most of the time, this macro-shape of the Earth is irrelevant. To us it is a collection of valleys, mountains, composite parts and different elements. For us to treat it as a sphere would often be useless. It is simpler to treat it as a bunch of component parts.

Consider, where do you live?

Apt. 2B, 1313 Mockingbird Lane, Scranton, Pennsylvania, United States of America, North America, Northern Hemisphere, Planet Earth, Sol System, Milky Way, etc... All of these answers are accurate, but only one is a meaningful answer to the question depending on who is asking and for what purpose. None of them are a universal truth, they all depend on the relationship of the person asking the question.

Try using mathematics to answer that question. You will still have to come up with arbitrary marks for a person to understand.

12 feet from the front door, or 600 Kilometers from the Washington Monument, or 42 degrees North Latitude, whatever. It's all dependent on the parties involved. There is no universal truth there, only a descriptive method for use between parties that have some frame of reference.

There's no reason dogs can't learn to count. But of course, we taught it to count, didn't we? It didn't discover it on its own. On its own, your dog probably wouldn't know it wants 5 milkbones, rather it would want enough milkbones to fill its stomach. It's the total mass of milkbone that matters to the dog, not the number of treats. Consider predatory animals in nature. They don't care if they take down 1 or 2 gazelle to feed. If the first gazelle is plump and its stomach is full, then its done for the day. But if the total amount of gazelle-age the animal has consumed is not enough to satisfy itself then it will take down a second gazelle. Not because it cares about the number it eats, but rather because it needs more stuff.

A sphere is an abstraction though, an ideal representation of a shape to which many things are similar, but nothing really is. A collection of carbon atoms in the approximate shape of a sphere is just that, a collection of things in an approximate shape.

A clearer example might be a Line, which clearly cannot exist having zero width, zero depth and infinite length. It's a useful abstraction though, a helpful model to have around. A sphere is a similar type of creature.

I think saying nothing really <i>is</i> may be going to far. There probably is something, even if I don't know what it is. I can perceive parts of it and I label these perceptions.

I dub thee: sphere!

That would be a good point, except that it's not right.

My dogs get all their treats at once. If I put down 4 (they get five) they won't touch 'em till I put the 5th one down there. Doesn't matter if I put 4 down fanned out, stacked up, or any other way, so it's not that they're seeing the wrong pattern on the floor either. They're counting 'em, and noticing that they're being shorted.


Also, we're talking about 3 basset hounds here. Basset hounds NEVER consider their stomach to be full, so if your argument were true for my dogs, they'd never stop bitching at me no matter how many treats I stuck down there.

Dude, there's no reason dogs can't learn to count. When you're dealing with domesticated animals that are fed sufficiently similar items on a regular basis they become conditioned to eating that many milkbones. I'm sure the dogs would love more milkbones, but they've become conditioned to accept the fact that you're only going to give them 5 milkbones. Try giving them 6 milkbones for a month, then try going back to 5 (or whatever the number is).

Without your training, the dogs would almost certainly eat until they were full (or some other cause stopped them). Where the animal behaviourists at? Can anybody back me up that animals don't much care for numbers on their own?

asaris, math IS a science, and is also the basis of all the natural sciences.

Everyone seems stuck on very archaic concepts of mathematics. As a whole, it is not based on the concept of numbers, and is quite separate from requiring any type of physical reality.

I think it would be useful to describe how a modern mathematician works. They basically do two things. The first of these is to set out definitions. Assume the (abstract) existence of an object X satisfying logical properties (1), (2), ... Any object with these properties is given some arbitrary name. This process seems quite clearly to be invention.

Once everything has been defined, the mathematician then tries to find interesting (meaning nontrivial) relations between them. These properties are logically deduced using only the definitions already given.

It is a little harder to say whether this step should be called discovery or invention. One is given a set system, and is finding rules for it that weren't known before. This is very closely analogous to the physicist finding rules to describe the universe. Both systems obeyed those rules independently of the discoverer(/inventor)'s understanding. I would therefore say that this part of mathematics is discovery.

There is an obvious counter-argument though. In the case of mathematics, all theorems are logically contained in the definitions (which are invented). It may therefore be said that the theorems were invented at the same time as the definitions. The statement is then that the inventor was simply not smart enough to understand the depth of his creations. Although I can see the point of this argument, it seems to run against the most common usage of the words we're discussing.

So I think math is fundamentally about discovery, but not the same sort of discovery that the traditional sciences strive for. Scientists have but one universe to study. Mathematicians create their own, and produce a new one whenever they get bored. In this sense, math contains elements both of the arts and sciences.

More practically though (to those of us who would like to actually use math for something practical), it is best thought of as a highly developed form of logical argument.

A close reading of this thread indicates that not everyone is stuck on very archaic concepts of mathematics.

Thanks for your views.

The fact is that every single concept in mathematics is an abstraction, from unity upwards.

It is a frame-work that we put around the world in which we live in the same way that physics is. It is however a human construction and a product of how we percieve the world.

It would be extremely interesting to see what another intelligent race would come up with in its place. I see it as hubris, however, to assume that it would be much alike.

Math is discovered. The relationship between numbers is there. Someone just discovers those relationships, he/she doesn't invent them.

language/tool

Mind you, I tend to be a constructiveness about nature, too. I've read too much Kant and Heidegger, I suppose. In a nutshell (and it'll sound loonier than it really is), all things are created by the human mind. There are no things "out there". There's no "there" out there.

axioms are invention.
proofs are invention.
theorems are discovery.

I guess apart from pythagoras's theorem as its equivalent to Eucilds 5th postulate. So that would make it an invention if viewed in that perspective:).

I wouldn't want to say that Math is a science, any more than I would want to say Philosophy is a science, and philosophy underlies science as much as math does. The experimental method seems to be fairly important to a natural science (though maybe I'm just being archaic again), and I've never known Math to use the experimental method. Of course, given what I've written above, it wouldn't hurt my point if math were a natural science.

And who's using an archaic notion of math? I'd be willing to bet that I've taken math classes at least as advanced as anyone else on this board; in fact, since I've taken a graduate level math course, it'd be hard for anyone to have taken a more advanced course. So you might want a little more content in that accusation.

I agree, math is a convenient construction used to describe the world in which we live. I think math is an invented means to describe and make use of discovered things.

Simply because something is not known does not make it untrue. Simply because something cannot be understood does not make it untrue. The discovery of 'that tribe' (the piranhas, I think) demonstrates the inherent limitation of all languages. But it has little to do with math or its efficacy.

Now, whether math can aid us in understanding natural phenomena is indisputable. It can, and does. Whether I understand it, have words for it, or not, it will do the same. Naming something one, un, uno, or whatever has no bearing on the matter.

yes

our numerical system is all made up. We as a society have decided that the symbol 1 represents one of said item and so forth. This includes language, math is an invention.

Well that depends, a system can be inherently inconsistent. At which point it gets rejected by those who might use it.

For instance a number system where somehow 3+2=5 and 3+2=6 would be break down pretty quickly. For it to be consistent a whole new layer of construction and method would need to be created. Some kind of modulus arithmetic.

But kd4, could the quantities represented by those numbers be such that 3 + 2 = 6?

Math does use the experimental method in a particular sense. Imagine that you are learning some type of math, and have just been presented with a set of definitions. These will generally seem to be too abstract to grasp on any sort of intuitive level, so you generally want to construct some examples to get a feel for how things work. Say that you now have several different examples, but are surprised to notice that they all share one particular property. You might then guess that all possible examples share this propery. So your examples have now given you an idea for a possible theorem. Examples are even more powerful in showing that a particular property is not correct when intuition would imply the opposite.

A well-known example would be Fermat's Last Theorem. Whatever progress Fermat may have made in proving it (nobody knows), the rest of the world didn't have a proof until a few years ago. It was, however, thought to be true because it seemed to work for every example that was tried. People kept showing that more and more special cases of it were correct. This gave people enough motivation to continue to try finding a complete proof even after 300 years.

So "experimentation" with mathematical definitions is an important part of mathematical progress. Unlike in the natural sciences, however, it is not absolutely required. Also, unlike math, science can never provide absolute proofs of anything.

I spend my days doing mathematical physics, so I also have plenty of experience with math. Its not terribly relevant though. I overspoke by saying "everyone," and I apologize.

Sure. As long as other additions keep the system consistent. And you can define consistency however you like, so you can really do anything at all.

Modern math considers numbers as we've all learned about them to be a particular example of a much more general type of object. For example, the integers form a ring, and the real numbers are a type of field. If you follow those two links and some of the pages linked from them, you'll start to get an idea of the types of things that you could come up with if you wanted to. Of course, you could say that those definitions aren't general enough either, but then you're really better off not using + and * signs anymore if you want anybody to read what you're doing.

All of these concepts are supposed to be completely abstract. The fact that we commonly use numerical quantities (along with the standard addition and multiplication operations) in describing the physical world is a completely separate philosophical problem.

yea, would have said the exact same thing http://www.lemonizer.com/upload/uplo...ober/dodgy.gif

I say basic math is discovery, because they have real-world meanings, like the number of objects and so on. Beyond calculus, I would say invention, because you start to define more and more concepts.

Math is a construct we have invented to represent things that we see. It may, however, be able to fundamentally represent everything in nature: if energy and matter are quantum, then can't they be expressed fully in numbers (it is my understanding that we can't measure them, but if we could..)?

Here's a mind bender: can god (assume god exists) change the value of pi? or phi? or e? or 1:1?

1st post :D

in·ven·tion ( P ) Pronunciation Key (n-vnshn)
n.
The act or process of inventing: used a technique of her own invention.
A new device, method, or process developed from study and experimentation: the phonograph, an invention attributed to Thomas Edison.
A discovery; a finding.

i vew math as recognition of relationships and patterns in and between quantities. we can create the symbols and methods for interpreting and dealing with what we discover, and ways to apply what we find, but they were related before we found them. though, i could say 5x=12y² and invent a relationship that doesn't necessarily have any real counterpart. depends whether an idea is considered an invention, i guess.

apparently, its all the same anyway.

I have so rarely seen math proven wrong so therefore it cannot be man made.

Math is an ascription of terms and methodologies used to process information found in the human psyche to the universe. I view it as being an intellectual construct used for processing information that exists soley inside the human mind. Does math exist outside the human in the world, or in the human as a means of processing that world in understood terms? I believe that the universe is bigger than the human mind, therefore math is something found within the human mind that serves as a looking glass, as opposed to being something fundamental to the the world or universe that humans can percieve suffeciently well to have, or at least claim, profeciency with.

That's my "I haven't had good sleep for 164+ hours and haven't had a good dose of caffeine for 40+" answer.