GMontag

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?

Halx

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.

__________________
You have found this post informative.
-The Administrator
[Don't Feed The Animals]

adysav

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?

d*d

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)

borg7171

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

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

GMontag

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.

seep

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

d*d

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

Master_Shake

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.

__________________
-------------
You know something, I don't think the sun even... exists... in this place. 'Cause I've been up for hours, and hours, and hours, and the night never ends here.

asaris

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.

__________________
"Die Deutschen meinen, daß die Kraft sich in Härte und Grausamkeit offenbaren müsse, sie unterwerfen sich dann gerne und mit Bewunderung:[...]. Daß es Kraft giebt in der Milde und Stille, das glauben sie nicht leicht."

"The Germans believe that power must reveal itself in hardness and cruelty and then submit themselves gladly and with admiration[...]. They do not believe readily that there is power in meekness and calm."

-- Friedrich Nietzsche

Stompy

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 love lamp.

ARTelevision

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.

__________________
create evolution

Stompy

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.

__________________
I love lamp.

shakran

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.

bingle

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

Master_Shake

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.

__________________
-------------
You know something, I don't think the sun even... exists... in this place. 'Cause I've been up for hours, and hours, and hours, and the night never ends here.

Suave

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.

Stompy

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?

__________________
I love lamp.

asaris

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.

__________________
"Die Deutschen meinen, daß die Kraft sich in Härte und Grausamkeit offenbaren müsse, sie unterwerfen sich dann gerne und mit Bewunderung:[...]. Daß es Kraft giebt in der Milde und Stille, das glauben sie nicht leicht."

"The Germans believe that power must reveal itself in hardness and cruelty and then submit themselves gladly and with admiration[...]. They do not believe readily that there is power in meekness and calm."

-- Friedrich Nietzsche

roachboy

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.

__________________
a gramophone its corrugated trumpet silver handle
spinning dog. such faithfulness it hear

it make you sick.

-kamau brathwaite

Master_Shake

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.

__________________
-------------
You know something, I don't think the sun even... exists... in this place. 'Cause I've been up for hours, and hours, and hours, and the night never ends here.

Suave

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.

Stompy

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.

__________________
I love lamp.

Master_Shake

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.

__________________
-------------
You know something, I don't think the sun even... exists... in this place. 'Cause I've been up for hours, and hours, and hours, and the night never ends here.

Stompy

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.

__________________
I love lamp.

Master_Shake

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.

__________________
-------------
You know something, I don't think the sun even... exists... in this place. 'Cause I've been up for hours, and hours, and hours, and the night never ends here.

seep

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.

Suave

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

shakran

Why?

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

Halx

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.

__________________
You have found this post informative.
-The Administrator
[Don't Feed The Animals]

roachboy

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.

__________________
a gramophone its corrugated trumpet silver handle
spinning dog. such faithfulness it hear

it make you sick.

-kamau brathwaite

shakran

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.

GMontag

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.

noodles

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.

Suave

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.

CatharticWeek

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

asaris

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.

__________________
"Die Deutschen meinen, daß die Kraft sich in Härte und Grausamkeit offenbaren müsse, sie unterwerfen sich dann gerne und mit Bewunderung:[...]. Daß es Kraft giebt in der Milde und Stille, das glauben sie nicht leicht."

"The Germans believe that power must reveal itself in hardness and cruelty and then submit themselves gladly and with admiration[...]. They do not believe readily that there is power in meekness and calm."

-- Friedrich Nietzsche

antisuck

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.

Master_Shake

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.

__________________
-------------
You know something, I don't think the sun even... exists... in this place. 'Cause I've been up for hours, and hours, and hours, and the night never ends here.

anti fishstick

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?

__________________
And the day came when the risk to remain tight in a bud was more painful than the risk it took to blossom.
~Anais Nin