bluntwcrackrap said:Semantics.
homeomorphic said:I'm not sure that example is representative of all axioms. Non-euclidean geometry is not constrained by Euclid's axioms. Instead, Gauss ... asked what would happen if you dropped one of the axioms, the parallel postulate. In doing so, I would say he invented non-Euclidean geometry
... but you could also say he discovered it, in the sense that he seems to have hit on the fact that it could be consistent or should be taken seriously.
DrPinceton said:Math was not 'invented', math is merely expression of what one observes in an agreed upon symbolic language...
Or a point with no dimensions whatever. A Euclidian point is an invention.mal4mac said:You can't observe a line of zero width.
Pythagorean said:I think we're suffering from the lack of a third option that I briefly brought up early with the analog to natural language. Some of mathematics, like Euclidian geometry, is emergent...
You wouldn't call natural language completely invented...
Euclidean geometry is the formalization of something built into our perception about the universe.
homeomorphic said:Invention suggests freedom to choose. If that's what invented means, no, it's not completely invented.
Take fractions. If we consider some magnitude, we can set its value to 1 and mentally divide it into some whole number of equal parts. We could do 10 equal parts or 637 equal parts, or whatever. So, which of the whole number of fractions we could divide it into is "out there" waiting to be discovered? The answer is none of them. Fractions are an invention. The fictional whole number of equal parts is imposed upon the magnitude by the human mind. It's an awesome, versatile tool invented by man.Rocket50 said:I'd say it was discovered. We didn't really "invent" it as it was already out there, we just didn't know about it.
zoobyshoe said:Take fractions. If we consider some magnitude, we can set its value to 1 and mentally divide it into some whole number of equal parts. We could do 10 equal parts or 637 equal parts, or whatever. So, which of the whole number of fractions we could divide it into is "out there" waiting to be discovered? The answer is none of them. Fractions are an invention. The fictional whole number of equal parts is imposed upon the magnitude by the human mind. It's an awesome, versatile tool invented by man.
mal4mac said:I think this just adds confusion. Would you say a steam engine is emergent?
You could say the same about a steam engine. The steel & steam is discovered.
Yes - a formalization that we invented.
Pi and other irrational numbers, and the fact of incommensurate magnitudes, prove math is an invention, rather than a discovery about something in nature. The term ##\pi## is, in fact, an invention created to exactly name that which cannot be exactly designated with numbers, only approximated.PeroK said:I guess we invented ##\pi## and ##e##?
zoobyshoe said:Pi and other irrational numbers, and the fact of incommensurate magnitudes, prove math is an invention, rather than a discovery about something in nature. The term ##\pi## is, in fact, an invention created to exactly name that which cannot be exactly designated with numbers, only approximated.
PeroK said:And prime numbers, likewise, invented? Would it not sound odd to say that a new prime has been invented?
You're missing the point that irrational numbers and incommensurate magnitudes prove that fractions were not "out there" waiting to be discovered. They are an invention that breaks down in certain situations. This was a huge disappointment to the Pythagoreans who believed the universe was built on numbers, that they had discovered something amazing and eternal about nature.Pythagorean said:But again, the symbol isn't the math. We're actually talking about "that which cannot be designated". And it was certainly discovered if you're taking the the invented-discovered dichotomy ontology. Yes, the greek symbol pi was invented but it wasn't "created to exactly name" the number, it was a greek letter first: it's a cultural thing that pi is ONLY used for pi, many constants have symbols often used as variables, like permitivity uses mu Newton's gravitational constant is G (also used to represent Fourier transforms of arbitrary function, g.)
Anyway, the symbol isn't the math, as previously commented by another poster.
The circle was the thing invented (likely as an idealization of the observation of circle-like shapes and dynamics) and the ratio between it's diameter and it's circumference is the thing that was discovered about it... but the discovery goes beyond basic geometry, it has implications for a Euclidian universe in which Gauss's law (and it's generalization) is valid. Pi tends to crop up in lots of other places in physics, too.
No, I wholeheartedly endorse prime numbers as a discovery. Inventions often lead to discoveries. Once you invent something you discover it has properties that weren't part of the original intention and which could be useful in further inventions. Prime numbers are the basis of an invented scheme of encryption.PeroK said:And prime numbers, likewise, invented? Would it not sound odd to say that a new prime has been invented?
zoobyshoe said:[...]irrational numbers and incommensurate magnitudes prove that fractions were not "out there" waiting to be discovered[...]
It's immaterial where they got Pi from, and they could have called it "stargbast" for all it matters, the point is a term had to be invented to refer to the exact ratio of diameter to circumference because they could not do it with numbers. The exact ratio is: 1:Pi. Anything else is an approximation.
I don't believe the circle is an invention. That is one thing that I will wholeheartedly call a discovery. Good approximations of circles exist in nature: the sun and moon, esp, and a person playing with any kind of string, or even a bent stick can easily produce a circle by accident.
You were trying to correct me for having said Pi was invented, as if I had said the irrational ratio, 1:Pi was an invention. I said the TERM used in that ratio was invented to exactly describe what numbers couldn't describe. We had to invent a non-numerical term to express what is not possible to exactly express in numbers. My overall point was that the existence of irrational numbers proved math was an invention.Pythagorean said:I'm not sure what your point is. Or if the existence of irrational numbers prove anything about discovery vs. invention. Anyway, I've never argued for the Platonist stance in the first place, that fractions are "out there" and I don't see any parallel between it any my points. The point is that pi was found as a result of calculation (in many independent cases in physics and mathematics), not designed or constructed like other mathematical objects such as the Schrodinger equation.
The person I was addressing seemed to believe number systems are discovered. I repeated to you what I said to him, not because I thought you believed what he believed, but because you seem not to have understood what I was saying to him.I would say those are both immaterial with respect to whether pi was invented or discovered. I'm not sure why you're treating rational numbers as a righteous thing in this discussion, particularly since number systems are invented.
It sounds like you're saying "Ceci n'est pas une pipe."An approximation of a circle is not a circle. The circle is the idealization; the simplification. If you paint a picture of a mountain you don't say the painting was discovered because the mountain was, the painting was invented (created for appropriate connotation).
The representation of the object is invented to represent the real thing that's discovered. In this case, circles are invented to represent circular things (which are discovered).
Formally, your use of the premise:
"Good approximations of circles exist in nature: the sun and moon, esp, and a person playing with any kind of string, or even a bent stick can easily produce a circle by accident."
to conclude:
"circles are discovered"
is not sound.
We had to invent a non-numerical term to express what is not possible to exactly express in numbers. My overall point was that the existence of irrational numbers proved math was an invention.
It sounds like you're saying "Ceci n'est pas une pipe."
If the dichotomy in someone's mind is, "Math was either invented or it was out there in Nature waiting to be discovered. I think it's the latter," the counter argument is, "If it was out there waiting to be discovered, why does it break down so often? It must be an invention." If you contest that, you must be asserting it was out there waiting to be discovered. And I don't think that's what you are saying.Pythagorean said:I realize that; it's essentially what I'm contesting. I don't think the existence of irrational numbers demonstrates that mathematics is an invention, just that we can't interpret it to apply to reality in the ways we expect it to. Of course, the Pythagoreans had some incentive for denying irrational numbers, rooted in harmonic theory, where integer ratios were everywhere; not to justify their claims, but to explain them. Anyway... it's an example of a meaningful difference that emerges in reality between rational and irrational numbers.
Discovering the approximate circle leads right away to the concept of a circle in the same way seeing a pipe evokes the concept of a pipe. Once we experience a thing, we don't have to invent its concept.I guess that would be the artist's instance of the same thing, but in science it draws on a different analogy: "the map is not the territory".
http://en.wikipedia.org/wiki/Map–territory_relation#.22The_map_is_not_the_territory.22
zoobyshoe said:If the dichotomy in someone's mind is, "Math was either invented or it was out there in Nature waiting to be discovered. I think it's the latter," the counter argument is, "If it was out there waiting to be discovered, why does it break down so often? It must be an invention." If you contest that, you must be asserting it was out there waiting to be discovered. And I don't think that's what you are saying.
Discovering the approximate circle leads right away to the concept of a circle in the same way seeing a pipe evokes the concept of a pipe. Once we experience a thing, we don't have to invent its concept.
zoobyshoe said:You're missing the point that irrational numbers and incommensurate magnitudes prove that fractions were not "out there" waiting to be discovered.
Domenico94 said:I don't think so...If it was invented, why would it have so relevance in physics and in modern life? why would so many mathematicians agree, with time and space differences, over the same subject. I think the answer is just no.
Domenico94 said:I don't think so...If it was invented, why would it have so relevance in physics and in modern life? why would so many mathematicians agree, with time and space differences, over the same subject. I think the answer is just no.
That's fine, but when our interpretation of it breaks down, so must our interpretation of it as "out there in Nature waiting to be discovered" break down. A lot of great minds did, in fact, insistently interpret it that way; Newton and all his contemporaries, for example, and before them, the Pythagoreans.Pythagorean said:The issues is with the word "break down". Math doesn't break down, per say, our interpretations of it with respect to it's implications for reality is what breaks down.
What I'm asserting is that the Euclidian definition of a circle, which is a concept, pretty much drops in your lap during the act of drawing a physical approximation of a circle. You can see with your own eyes that the sine qua non of the circle is that the distance between the ends of the compass (or bent stick, or string on a peg) remains constant as you rotate it, and that fact is what gives the circle its circularity. Without it being an ideal circle, the idea of an ideal circle is immediately suggested.The "concept" (idealization) is an emergent product of our perception and the actual thing. Our formalization of it is invented with willful intention.
So, circular-like things are discovered, circles emerge from mind-reality interaction, and the geometric tools that build the equation of the circle is invented. The equation itself, you could argue, is discovered, but it's discovered within the invented geometry system.