Is Math an Invention or a Natural Phenomenon?

[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.

 

[zoobyshoe]

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).

I'm really not sure how much weight to put on the external input. But for anyone still in the "math is out there in the universe" camp, it is important to emphasize how much of math is actually "in here".

The brain seems to spontaneously generate geometric images under certain circumstances:

http://plus.maths.org/content/uncoiling-spiral-maths-and-hallucinations

Less formal blog:

http://disregardeverythingisay.com/post/9331287956/the-visual-components-of-a-psychedelic-experience.

The same thing happens to some people experiencing visual migraine "aura".

These experiences seem to "drop out" of the way neurons interact when under the influence of a variety of "toxins" or pathological situations, or they may "drop out" of the cytoarchitecture of the brain, itself, under those circumstances.

This should cast doubt on whether "invented or discovered" is a proper question, and get people thinking about the "third option."

edit:

There must be experiments on this - does a baby distinguish a circle from an oval "at birth"?

See the above. You can't find the answers to some questions by experimenting on babies.

Edit: Article from Scientific American:

http://blogs.scientificamerican.com...ts-nature-turing-patterns-and-form-constants/

Edit: Wikipedia article on Form Constants:

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

 

[Medicol]

I think Math is Expanded, not invented or discovered.

 

[BobG]

What does discovering a constant (Pi) have to do with whether math was discovered or invented?

Math is invented. Hopefully, the invention is useful for modeling things that occur naturally in the real world (vs the imaginary world of mathematicians ).

Take the invention of quaternions. Hamilton spent a long time trying to invent trinarians, but it was virtually impossible to develop a mathematical system that would work with one real and two imaginary components. He invented quaternions because it was actually possible, most importantly, and if you just set the real part to zero and used three imaginary components, his quaternion math could accomplish all the things Hamilton hoped his trinarian system would accomplish.

And, like a true mathematician, the new math he invented didn't actually solve any problems that were very pressing during his time (although over a few hundred years, other people did actually find some problems that quaternions were good for). Quaternions were just something that gave him cool things to carve into bridges while on moonlight strolls with his wife (See what I mean about mathematicians?)

 

[homeomorphic]

And, like a true mathematician, the new math he invented didn't actually solve any problems that were very pressing during his time (although over a few hundred years, other people did actually find some problems that quaternions were good for). Quaternions were just something that gave him cool things to carve into bridges while on moonlight strolls with his wife (See what I mean about mathematicians?)

Disagree. The main reason why quaternions are useful was already apparent back then. That is that they can be used to calculate rotations. That might not be all the applications, but the use in computer graphics today, for example, is basically not anything new, other than how to get a computer to do it and make the computer. A lot of the things mathematicians do today are much more removed from reality than quaternions are. What could be more real than 3D rotations? Of course, then Hamilton went and tried to do all sorts of things with quaternions that weren't so successful, but I think one of his main motivations was to describe 3D rotation.

Some of the semantic difficulties are also evident in your post. By math, do we mean math as a whole subject or do we mean specific math? Also, using one example is not really sufficient. If you look at a wide variety of examples, you see that it becomes awkward to always insist on either "invention" or "discovery", and that it is semantically much more natural to use both words, depending on the math in question.

 

[BobG]

My comments about the usefulness of quaternions were slightly tongue in cheek (but vector analysis enthusiasts sure spent a lot of time bashing quaternions, as this quote shows: "Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way" - Lord Kelvin, 1892).

They're actually pretty useful in satellite attitude control, as well.

But they are a good example of "discovered" vs "invented". I could easily understand a person saying the commutative property of multiplication was discovered, as any person looking at a group of chairs assembled in five rows of 10 could easily discover that property themselves. But that doesn't mean multiplication has to be commutative. A person could design a non-commutative version of multiplication - and Hamilton did with his quaternions.