The evidence that the Universe as a whole is flat - Euclidian, no curvature - is very recent and still not definitive. It could easily have been otherwise, as far as anyone knows. We know - at least, until someone finds something better than Relativity to describe things - that within our solar system, which would include all the applied geometry any human has done so far, the most accurate geometry is not Euclidian. Euclidian geometry does not describe the space of a gravitational field, according to Einstein, and we live in such a space. The geometry of this space is very well developed, being necessary for launching satellites to Mars and so forth. http://en.wikipedia.org/wiki/Riemannian_geometry
The eleven-dimensional universe theory of the 1980s already did that. The uncertainty principle and other problems of modern physics suddenly make sense if what we observe as subatomic particles are merely the manifestations of rapidly vibrating multidimensional objects during the instants when they pass through the three-dimensional "hyperplane" to which our senses are limited, in alternating directions. Once again from a linguistic standpoint I insist that this is more of a "model" than a "theory." Like imaginary numbers, those additional seven dimensions may represent nothing more than intermediate results necessary for calculations to be performed which explain and predict conditions in the one temporal and three spacial dimensions we can sense. What little I've seen of string theory that I can understand seems to be sidling into the same no-man's land between theory and model; between physics and mathematics; or perhaps even between science and philosophy. But this is really only the next iteration of the Ancients' discovery that the two-dimensional "world" they thought they were living on, which appeared to be a plane from their limited vantage, was actually the surface of a three-dimensional object, specifically a sphere. The earth's surface is in fact a Riemannian space when regarded as two-dimensional, yet when seen in the context of a three-dimensional universe any conflicts with Euclidean geometry simply evaporate. Similary, the curvature that we observe in our three-dimensional universe may resolve completely into textbook Euclidean calculations if and when we achieve a fuller grasp of a putative four-dimensional universe of which the part that we observe is only a "hyper-surface."
If mathematics is discovered instead of being invented by brilliant minds, why do morons not discover some of the advanced theorems? It seems to me that a brilliant mathematician is denied full credit for his efforts when you call his work discoveries instead of inventions. For this reason alone, I vote for invention. BTW: Did Michelangelo merely discover the sculpture of David? Obviously David was always in that block of marble. Did Michelangelo merely chip away all the stone that was not David? In many ways, I am arrogant relating to my various abilities, both physical & mental. However, I am in awe of the true geniuses. There is no way I could chip away all that was not David. There is no way I could have discovered (if you prefer) the mathematics found by Galois, Von Neuman, Gauss, Leibniz, Newton, Cantor, and all the others great mathematicians.
Discovery is not a passive activity. Brilliant people are more likely to discover things because they know how to look for them. Stupid people have to wait until they trip over them or get eaten by one. Relativity is one of the natural laws of the universe and clearly was not invented. Do you suppose even an average mind, much less below average, could have discovered it?
Math is invented, not discovered. It can't be discovered because it doesn't exist until someone makes it up. As a matter of fact, numbers themselves don't exist. They are a convenient, but abstract, idea that were invened like mathematics. But no, math was not discovered.
Numbers have to start somewhere. Until we got into agriculture and started living in static communities, numbers wouldn't have been more than concepts like "more" and "less", "together", "apart", and so on. When we needed to start counting things to keep track of them, and who owned what, we started counting things in discrete groups - the Babylonians had a complex (ac)counting system that was base 60 or something, which survived for a fair while. The Romans had a fairly inept math formalism, but it functioned well as a counting system - it must have or they would have abandoned or changed it instead of using it for all those centuries. It wasn't until we needed to start thinking about geometry and so on that math got past arithmetic.
Numbers and math may be abstractions, but they were abstracted from empirical observations of the natural universe. Two plus three equals five in reality, not just in the abstract. You can try it with your fingers, with sticks, or with the dead polecats you're bringing home for the tribe to make for dinner, and it will always come out the same. This is something that someone discovered. So is the ratio of the circumference of a circle, and π, the constant that keeps showing up in the natural universe every time we turn around. So are the formulas for the areas of squares and triangles, the volumes of various objects, the sines and tangents of angles, the Golden Mean, and e, the base of natural logarithms. For that matter, so is i, the square root of -1, which threw itself in our faces as soon as we discovered electricity and the relationships between its various measures. It's just as easy to argue that we didn't invent any of these things, we discovered them. But math is a technology and we can have the same intellectual argument over many of the earliest technologies. Did we invent agriculture or discover it? We discovered how animal and plant metabolism and reproduction work (science), then we invented ways of making them work for us (engineering). Metallurgy? We discovered that really hot rocks will melt and separate into different component materials (science), then we experimented with them and discovered that blending the tin from one city's rocks with the copper from another city's rocks created bronze, an alloy that was stronger than either (more science), then we invented tools made of that alloy (engineering). Primitive people typically had only the numbers 1-5 or 1-10. The Indo-European root phenqwe, "five," (cf. Greek penta, Latin quinque, German fünf, Sanskrit pancha, Russian pyat) also shows up as the Germanic word "finger." Dekm, "ten," was the largest number the Indo-European tribes had a root word for. Kmtom (cf. English "hundred," Latin centum, Russian sto, Sanskrit satem) is a compound word formed on dekm with the D eventually elided, with the basic sense "big ten." Just as the Germanic tribe went on to form thus-hund, "swollen hundred." The Meaning of Tingo delves into the number words of a few of the world's last remaining Neolithic peoples and the pattern is common. In one language the word for "eleven" is "have to go down to my toes." Sixty shows up a lot in early civilizations. We still have 60-second minutes, 60-minute hours and degrees, and 360-degree (60x6) circles. People become very comfortable with their numbers. When Fibonacci brought the Hindu-Arabic positional decimal number system to Europe in the 1200s, absolutely no one except mathematicians adopted it for several hundred years, not even shopkeepers. Look at my country's uncompromising rejection of the metric system. 128 ounces in a gallon, 63,360 inches in a mile. Although that happened rather early in all six of the world's civilizations. The Olmec, Inca, Egyptians, Chinese, Indians and Mespotamians all figured out the orbits of the heavenly bodies because they were key to religious rituals. And IIRC they all had a decent value for π.
There is a very finer line between invention and discovery, many ideas are a result of constant fine tuning or just plain accidental luck(ala Viagra), especially intangible ideas. Math like other languages was probably developed over time. I think, was it not the East Indians that invented zero and such?
Just think of an alien races math, they will have made observations with an entirely different neural system, yet would probably be strikingly similar to what we have (plus or minus a few oversights). We are just re-discovering something that was already there and giving it a name, Math.
I worked in civil service for most of my career. I wrote that book Please Register or Log in to view the hidden image! Still for the purpose of this discussion I think we can agree to define "truth" in a more scientific way. Nonetheless, some truths are more intractable than others and it would be quite difficult for a despotic government to convince people that 1+1=3 since even a child can test that hypothesis with nothing more than pebbles. This is why it's important to distinguish between mathematical theories and scientific theories. Mathematical theories can be proven true because they are based on pure logic. Scientific theories are based on empirical observation of the behavior of the universe and as such can only be proven false. Our canonical scientific theories are those which have withstood so much testing and peer review that we judge them "true beyond a reasonable doubt," to borrow the language of the law. But this is not the same as "true" with no qualification, which is the status of mathematical theories. The first documented evidence of a proper positional symbol for zero is credited to the Mayans in 36BCE. Since the evidence is outside the Maya homeland it's assumed that this invention is much older and may predate the Olmecs, a thousand years earlier. The first text with a proper positional symbol for zero in Eurasia is in India in 876CE. (I say "proper" because earlier attempts had been made to simply write the words for the numerals, and to invent a symbol but use it inconsistently.) The Uzbek mathematician Al-Khwarizmi brought the Indian positional decimal notational system to Islamic civilization and used it in his groundbreaking ninth-century book Arithmetic. It was a translation of this Arabic book that Fibonacci brought to Europe in the 12th century, and because of this we have always called these "Arabic numerals."
