Is Math an Invention or a Natural Phenomenon?

[Pythagorean]

Semantics.

Ultimately, yeah. What's the tools and techniques vs. what's the thing being studied? However you define that probably predicts your answer to some degree.

 

[mal4mac]

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

Good point. I did say 'axioms were mostly discovered' :) I'd agree that non-Euclidean geometry was invented.

... 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.

But a steam engine is consistent with the law of thermodynamics, and that was certainly invented! Strictly speaking *all* axioms are invented. Euclid's first axiom comes from raising an everyday observation to the status of universal truth, together with inventing the ideas that lines have no width and points no dimensions.

 

[mal4mac]

Math was not 'invented', math is merely expression of what one observes in an agreed upon symbolic language...

You can't observe a line of zero width.

 

[zoobyshoe]

You can't observe a line of zero width.

Or a point with no dimensions whatever. A Euclidian point is an invention.

 

[Pythagorean]

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. It was a mixture of invention and discovery, but some things (like our senses and sensory processing) lead to a particular perspective (the human perspective) that we take for granted.

You wouldn't call natural language completely invented, but you wouldn't really call it discovered either. It emerges as a way to describe things, but it's not just semiotics, it's syntactic too: english has cause/effect operators (verb) and properties (adjectives) and coarse quantification arises from adjectives: small/medium/big; north/south/east/west. And these can even have finer quantifications with some applied logic: small-medium, medium-large, northeast, southwest, etc. We know that southeast must be between south and east because the logic is built into the language.

Euclidian geometry is the formalization of something built into our perception about the universe. In that sense, it's both discovered and invented, as well as a naturally emerging aspect of human consciousness.

 

[mal4mac]

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...

I think this just adds confusion. Would you say a steam engine is emergent?

You wouldn't call natural language completely invented...

You could say the same about a steam engine. The steel & steam is discovered.

Euclidean geometry is the formalization of something built into our perception about the universe.

Yes - a formalization that we invented.

 

[MadAtom]

Invention suggests freedom to choose. If that's what invented means, no, it's not completely invented.

I agree with that. I believe (forgive me if I'm wrong) that maths is all about the numbers and the relations with those numbers (axioms, theorems and all that) which are ultimately mechanisms to produce even more numbers. so let's imagine each integer as fixed points, equally spaced, on an infinite surface (let's call it the Number Grid). then derivation, integration, and all human ''tools'' are just ways to go from a point A to a point B of the Grid. Since the grid is already established (a matter in which we didn't have any freedom to make it fancier) can we say that we are inventing these paths from A to B or we are just discovering like Columbus?

So I think this all problem boils down to whether integers are invention or discovery, and I'd answer can 1 not be 1? But that answer wouldn't even be wrong...

 

[iDimension]

I prefer to think that Math was discovered and not invented. Math would still exist regardless of if humans are around to discover the equations and assign symbols to them.

2 + 2 would still equal 4 but there just wouldn't be any value or symbols defined. Just as E=mc² would still be true regardless is humans existed, or any life at all for that matter.

 

[Rocket50]

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]

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.

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.

 

[PeroK]

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.

I guess we invented $\pi$ and $e$?

 

[Pythagorean]

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.

Apples and oranges. A steam engine has a defined plan and purpose and is carefully engineered. Disciplines like mathematics are social entities; they grows organically based on personal interests and perceptions of the universe.

It is more complicated, but that doesn't make it wrong. It's an oversimplification to say that all of maths was either invented or discovered.

Yes - a formalization that we invented.

You've revealed your conclusion and stance, but you haven't made an argument that actually confronts my point.

 

[zoobyshoe]

I guess we invented $\pi$ and $e$?

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.

 

[Pythagorean]

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.

 

[PeroK]

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.

And prime numbers, likewise, invented? Would it not sound odd to say that a new prime has been invented?

 

[Pythagorean]

And prime numbers, likewise, invented? Would it not sound odd to say that a new prime has been invented?

Prime numbers are discovered, but one could argue that the classification "prime numbers" was invented. So all the prime numbers we discover are the result of the invented classification scheme.

 

[zoobyshoe]

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.

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.

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.

 

[zoobyshoe]

And prime numbers, likewise, invented? Would it not sound odd to say that a new prime has been invented?

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.

 

[Pythagorean]

[...]irrational numbers and incommensurate magnitudes prove that fractions were not "out there" waiting to be discovered[...]

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 Schrödinger equation.

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 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.

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.

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.

 

[zoobyshoe]

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 Schrödinger equation.

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.

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.

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.

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.

It sounds like you're saying "Ceci n'est pas une pipe."

http://en.wikipedia.org/wiki/The_Treachery_of_Images

 

[Pythagorean]

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.

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.

It sounds like you're saying "Ceci n'est pas une pipe."

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]

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.

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.

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

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.

 

[mal4mac]

Math is a product of the human mind and we make mathematics up as we go along to suit our purposes. If the universe disappeared, there would be no mathematics in the same way that there would be no football, tennis, chess or any other set of rules with relational structures that we contrived. Mathematics is not discovered, it is invented. This is the non-Platonist position.

"If we accept that mathematics is invented, rather than discovered, we can be more daring, ask deeper questions, and be motivated to create further change." Derek Abbott, http://www.huffingtonpost.com/derek-abbott/is-mathematics-invented-o_b_3895622.html

 

[Pythagorean]

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.

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.

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.

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.

 

[Domenico94]

Why should it be?

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.

 

[PeroK]

You're missing the point that irrational numbers and incommensurate magnitudes prove that fractions were not "out there" waiting to be discovered.

What about the number of days in a year? That's not rational.

 

[mal4mac]

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.

The reason mathematics is admirably suited to describing the physical world is that we invented it to do just that. We make mathematics up as we go along to suit our purposes. Mathematicians agree on the same subject partly because there is continuity of knowledge across time and space. The Arabs learned from the Greeks, and the West learned form the Arabs.

In cases of similar developments, but no contact, there is "one obvious invention". For example, integers up to about seven are gifted to us by evolution, and it seems a very obvious step to make up more integers by "adding 1", just as we make 2 from 1 by adding 1. One can imagine many cultures inventing this "adding 1" process to "invent new integers" without learning from another culture. Of course, this process would be useless if "adding 1 object" led to to arbitrary number of objects. If that had been the case the invention of integers would have died at birth, like a tissue paper steam engine.

 

[Pythagorean]

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.

"Relevance in physics and modern life" doesn't seem to be the aporopriate condition on determining invention vs. discovery, primarily because inventions can be designed to be relevant to physics and life.

 

[Windadct]

IMO - "Math" is just a language - the beauty and science of the ocean existed before we invented language to describe it. Just because we can describe a physical system mathematically - does not mean the "math" was there first. The systems we encounter are relatively simple - and relatively simple math can be used to describe those systems.

 

[zoobyshoe]

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.

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.

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.

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 equation of a circle, (x-a)2 + (y-b)2 = r2 is a different kind of definition than Euclid's and is surely the product of deliberate invention. (At least, it couldn't be defined that way until the invention of integers and Cartesian coordinates.) So, yes, that definition is the product of invention.

Withall, I am hearing what you're saying, but am continuing to discuss it as if the dichotomy were the only way of looking at it because I feel the best way to prepare people for the introduction of the concepts you want to introduce is to first get people to appreciate the very much 'invented' aspects of math.