Is Math an Invention or a Natural Phenomenon?

[Pythagorean]

Well certainly there is something there that math is representing and it appears independent of the observer (thus objective). But still, math is a social construct used to understand the thing that is and its construction and motivation often strays from reality. Other times, it can be used to make predictions about reality... but some additioblnal interpretations and assumptions are often required, justified not from the math, but from observation.

In this way, math is more of a logical clay. Scientific fields use interpretation and evidence along with the logical clay model nature, but the construction of the logical clay and the discovery of its properties is more a subject of math than science.

 

[zoobyshoe]

Well certainly there is something there that math is representing and it appears independent of the observer (thus objective). But still, math is a social construct used to understand the thing that is and its construction and motivation often strays from reality. Other times, it can be used to make predictions about reality... but some additioblnal interpretations and assumptions are often required, justified not from the math, but from observation.

In this way, math is more of a logical clay. Scientific fields use interpretation and evidence along with the logical clay model nature, but the construction of the logical clay and the discovery of its properties is more a subject of math than science.

I am happy to see you invoke logic with regard to math as opposed to language.

On a different tack: I'm surprised you, of all people, haven't suggested the whole dichotomy, invention vs discovery is false and the question improper. Without a neural substrate for math, it wouldn't exist. Math, number manipulation, should therefore be regarded as an evolved capacity, selected for the advantages it gave us. In other words, it makes no more sense to ask if math is discovered or invented than it does to ask if our ability to see color was discovered or invented.

However, maybe you don't see it that way.

 

[zoobyshoe]

Well certainly there is something there that math is representing and it appears independent of the observer (thus objective). But still, math is a social construct used to understand the thing that is and its construction and motivation often strays from reality. Other times, it can be used to make predictions about reality... but some additioblnal interpretations and assumptions are often required, justified not from the math, but from observation.

And: Yes, well put.

 

[Pythagorean]

On a different tack: I'm surprised you, of all people, haven't suggested the whole dichotomy, invention vs discovery is false and the question improper.

Well I did say math is both invented and discovered :)

I am happy to see you invoke logic with regard to math as opposed to language.

I think natural language is a logic system, too; just not as robust or accurate as the language of mathematics.

 

[zoobyshoe]

Well I did say math is both invented and discovered :)

Which is OK with me as long as you're not saying "invented" and "discovered" are just about synonyms. This discussion is all about "le mote juste" for me, and in my mental inertial frame it is an invention. I can easily see that a person can mentally enter the inertial frame of 'pure numbers' and in that frame it's really mostly discoveries. Mathematicians make a lot of discoveries about numbers as numbers. I don't like that frame, though. A person could turn into John Nash in there.

I think natural language is a logic system, too; just not as robust or accurate as the language of mathematics.

I think I'd say the usually slack natural language can be brought to attention when the subject of quantities is discussed. I can't accept math as a language separate from the language of the person speaking. If I write A=B, B=C, A=C, it is a statement in English. It's just written in shorthand. A equals B, B equals C, therefore, A equals C. It's relatively easy to achieve accuracy and be rigorous when you are limiting yourself exclusively to the subject of quantities.

 

[Pythagorean]

Which is OK with me as long as you're not saying "invented" and "discovered" are just about synonyms. This discussion is all about "le mote juste" for me, and in my mental inertial frame it is an invention. I can easily see that a person can mentally enter the inertial frame of 'pure numbers' and in that frame it's really mostly discoveries. Mathematicians make a lot of discoveries about numbers as numbers. I don't like that frame, though. A person could turn into John Nash in there.

I think I'd say the usually slack natural language can be brought to attention when the subject of quantities is discussed. I can't accept math as a language separate from the language of the person speaking. If I write A=B, B=C, A=C, it is a statement in English. It's just written in shorthand. A equals B, B equals C, therefore, A equals C. It's relatively easy to achieve accuracy and be rigorous when you are limiting yourself exclusively to the subject of quantities.

I guess it would be more accurate to say that math and language are both a class of semiotics and syntactic structures rather than say math is a language. In math, the semiotics are better defined and the syntactics are complex and rigorous. In language, the semiotics are diverse and not well defined (confounded by things like connotation).

 

[sonicelectron]

What is interesting is that this universe at least, does appear follow some sort of strict reasoning and logic (to put in human terms...). Regardless of what we invent, it all points back to the system as a whole that permitted it to happen. Sort of like a computer program. Conway's Game of Life for instance, you have a few simple rules and look what pops up in that program. You can't just say the "Glider Gun" made the "Gliders", it did in some sense, but ultimately the program is what did it. That's why I'm using the term "generated", instead of "invented". But with that in mind I could easily make the jump into the "mathematical forms" thing, which I will not, because it requires a huge leap of faith. All I can say is what I said above, appears to follow some sort of strict reasoning/logic/rules/laws as a whole, how and why? No idea, but when something that resides in/is a part of, can, from following these set of rules, describe them somehow and use that reasoning to build upon and "invent" new things, that's a very interesting thing to think about.

Pure Junk.

 

[DrPinceton]

Math is merely a agreed upon defined symbolic language for observation

Math was not 'invented', math is merely expression of what one observes in an agreed upon symbolic language, that being modern number theory using agreed upon symbols. So while pure math, observation of natural phenomena as per the Pythagorian system, where all things can be defined as 'number', math is purely observation of natural phenomena, but the symbols used in math, are 'invented', symblos are symbols, hwoever, 'math' is pure observation of natural phenomena, if you are of the vein of the Pythagoream system.

When a sentient being 'observes' natural phenomena, if they have a symbolic language such as what humans call math, they can then express the observation via mathematical symbols that are agreed upon, just like all 'languages' use agreed upon symbols (words/letters/etc) to express 'thought'.

The relating of that observed symbolisms to other natural phenomena that has mathematical composition within it, such as nature using the Fibonacci sequence, etc, math becomes just another 'language' to express observed phenomena.

 

[homeomorphic]

Euclid's axioms - aren't they discovered? "1. A straight line segment can be drawn joining any two points." This is something we discover in first grade drawing lines between two points with a ruler. Because rulers & humans are inaccurate, we can't be certain the axiom relates precisely to the real world, but we raise it to the status of absolute truth by calling it an axiom. So axioms are mostly discovered, and the only invention is raising an empirical observation to the level of absolute truth.

I'm not sure that example is representative of all axioms. Non-euclidean geometry is not constrained by Euclid's axioms. Instead, Gauss or someone like that 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. It's semantics. Only later on was it discovered in nature, in the form of Einstein's theory, and as Beltrami later pointed out for hyperbolic geometry, in the geometry of surfaces like the pseudosphere.

So, I'd say they are both invented and discovered, depending on the axioms in question and your semantics.

 

[DivergentSpectrum]

Semantics.

 

[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 Schrodinger 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 Schrodinger 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.

 

[zoobyshoe]

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

I think that fact supports my argument, doesn't it? It's another example of Nature not working out to perfectly dovetail into clean, whole numbers, or whole number fractions of 1, the way the Pythagoreans thought it would at first.

 

[Pythagorean]

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.

Only after you've invented the concepts of points, centers, and planes with which to construct the Euclidean definition.

Still, the circle was never "out there" waiting to be discovered. Circularity is a property we can discover in the natural world (only because it's something we've classified as "this property of things we find in nature that looks like this", but the noun, "circle", is rather abstract and meaningless (with respect to natural phenomena) when used alone.

 

[zoobyshoe]

Only after you've invented the concepts of points, centers, and planes with which to construct the Euclidean definition.

OK, I shouldn't have said "the Euclidian definition," but, "the essence of the Euclidian definition." The concept of an ideal flat figure distinct from other flat figures, and characterized by the equality of any measurement made from center to circumference drops into your lap from the act of drawing the circle, without you even needing to have a name for any of it. Given time you could work out how to articulate that in words. (The concept usually precedes the ability to articulate it.)

Still, the circle was never "out there" waiting to be discovered.

It was not invented, and if it wasn't invented and it wasn't "out there" waiting to be discovered, it must have been "in here" waiting to be discovered. By which I mean our ability to conceive of the 'mathematical object' (I think that's the term I want), the circle, emerged along with our ability to perform the varied logical processes concerning abstract quantities that constitute math. When we discover things in math, I think they're discovered in an interior conceptual arena constructed to try out model trains of logic.

 

[zoobyshoe]

Incidentally, it occurred to me that any math must, by necessity, be built on the assumption of conservation of quantity. If that assumption isn't made, there's pretty much no point to the math. It's possible those primitive tribes who don't count higher than 10 haven't made that assumption, and may unconsciously suppose quantities might spontaneously change.

 

[mal4mac]

It [the circle] was not invented ... and it wasn't "out there" waiting to be discovered, it must have been "in here" waiting to be discovered.

Are babies born knowing what a circle is?

This is open to empirical investigation. here's some background:

http://www.psychologytoday.com/blog/babies-do-the-math/201101/brainy-babies

"babies appear to be born knowing that objects cannot magically appear or disappear, that they cannot pass through each other, and that they cannot move unless contacted by another object. These expectations hold for objects, but not for non-object entities like substances (e.g., liquid, sand)."

So babies are born knowing quite a lot of physics .

"by their first birthday (and long before they can talk), babies exhibit quite sophisticated number knowledge. They can enumerate visual and auditory items, items presented sequentially and items presented simultaneously."

Nothing there about a circle being "in there". "Numbers" and "assuming continuing existence" are in there because they are useful to evolution. But is spotting a circle useful to evolution? Being able to choose two apples, rather than one, is obviously useful; but taking the more circular apple doesn't seem useful. My guess is that the circle was invented, and has to be explained to children - it's not "in there", and no perfect circle is "out there", so it had to be invented.

 

[zoobyshoe]

Are babies born knowing what a circle is?

This is open to empirical investigation. here's some background:

http://www.psychologytoday.com/blog/babies-do-the-math/201101/brainy-babies

"babies appear to be born knowing that objects cannot magically appear or disappear, that they cannot pass through each other, and that they cannot move unless contacted by another object. These expectations hold for objects, but not for non-object entities like substances (e.g., liquid, sand)."

So babies are born knowing quite a lot of physics .

"by their first birthday (and long before they can talk), babies exhibit quite sophisticated number knowledge. They can enumerate visual and auditory items, items presented sequentially and items presented simultaneously."

Nothing there about a circle being "in there". "Numbers" and "assuming continuing existence" are in there because they are useful to evolution. But is spotting a circle useful to evolution? Being able to choose two apples, rather than one, is obviously useful; but taking the more circular apple doesn't seem useful. My guess is that the circle was invented, and has to be explained to children - it's not "in there", and no perfect circle is "out there", so it had to be invented.

Due to (PF member) Pythagorean's earlier criteria about what constitutes a circle:

An approximation of a circle is not a circle. The circle is the idealization; the simplification.

the capacity to idealize a circle from experience of the approximation should be "in there" with numbers. We perform math on idealized internal models. We can account for a large herd of sheep by idealizing the sheep as all equal, when they're actually all different weights, and colors, etc.

 

[mal4mac]

the capacity to idealize a circle from experience of the approximation should be "in there" with numbers.

Idealizing a few sheep by making them "equal and simple" in the process of "number discrimination" seems an easy thing for the brain to do - they are obviously separate white bits on a field of green. But idealising objects as circles seems more difficult - is an apple circular, is an orange? Does the baby know a circle at birth? Or does the baby learn to distinguish circles later on? I suspect a long time "later on" - after learning about apples, oranges, balls... and why one is more circular than another. I might, of course, be wrong. There must be experiments on this - does a baby distinguish a circle from an oval "at birth"?

 

[Pythagorean]

OK, I shouldn't have said "the Euclidian definition," but, "the essence of the Euclidian definition." The concept of an ideal flat figure distinct from other flat figures, and characterized by the equality of any measurement made from center to circumference drops into your lap from the act of drawing the circle, without you even needing to have a name for any of it. Given time you could work out how to articulate that in words. (The concept usually precedes the ability to articulate it.)

It was not invented, and if it wasn't invented and it wasn't "out there" waiting to be discovered, it must have been "in here" waiting to be discovered. By which I mean our ability to conceive of the 'mathematical object' (I think that's the term I want), the circle, emerged along with our ability to perform the varied logical processes concerning abstract quantities that constitute math. When we discover things in math, I think they're discovered in an interior conceptual arena constructed to try out model trains of logic.

This is essentially what I've been saying about the third option, but to me it appears to emerge from the interaction of external and internal, not solely internal or external. We don't have a circle clump of neurons, per say, we have a more generalized visual system and circular things are common enough that we lump them into one category (like we do with so many other things).

 

[iDimension]

So can't we just say that some areas of mathematics are invented and some are discovered? The perfect circle doesn't exist in nature anywhere we know of which in turn would mean that Pi can't exist until humans or some other intelligent life draw the perfect circle.

It's a hard one but after reading some of the other posts I think it's fair to say that some math is invented and some math is discovered. The concept or mathematics must surely be discovered and not invented as any intelligent life form must surely use mathematics too.

The symbols and notation is invented but the idea and concept is discovered.

 

[mal4mac]

The concept or mathematics must surely be discovered and not invented as any intelligent life form must surely use mathematics too.

Maybe. But they might invent different mathematics, just as they would invent different steam engines (or different engines if they skip steam!) The Greeks got a long way without algebra. Calculus was developed using limits, but now calculus using infinitesimals is acceptable. Maybe the aliens would invent mathematics based on infinitesimals from the get go - or maybe just have a discrete mathematics rather than messing with all those nasty infinities.