I would be interested to hear your opinions on this question.

Galileo had written, "Nature's great book is written in mathematical language" Discovered obviously.

Discovered, of course. How can something be invented that is not already capable of existing?

This in no way detracts from the ingenuity and cleverness of those involved in discovery or "invention". I simply believe that ideas, inventions and formulas are latent - awaiting the right combination of serendipity and ingenuity to bring them to light.

Perhaps it is even more important to discover and communicate such an idea to others in a comprehensible fashion. This is the way mankind advances, whether it be Newton's methodology of denoting calculus in compact form or the engineering feat of assembling a working steam engine.

In any event, it must be quite the thrill to "discover" something of lasting magnitude - immortality in a bottle... :)

Some previous threads:
Is Mathematics invented or discovered?
Article on Mathematical Realism
See also the June 2008 newsletter of the European Mathematical Society (http://www.ems-ph.org/journals/all_issues.php?issn=1027-488X), page 19.

Invented...Where in nature can you find the golden ratio = 1.61803399?

Mathematicians discover natural patterns and formulate them. Not only that, they also invent some new patterns which do not exist in nature.

Yet when we use either one of these words, we may ignore the evolutionary construction of this discipline: Without previous knowledge, it was impossible to develop the new ones. "Discovery" or "Invention" almost equally focus on "suddenness". It is true that certain individuals come up with brand new theories and patterns, but none of them could be possible without depending upon other theories and patterns.

Invented...Where in nature can you find the golden ratio = 1.61803399?
http://www.google.co.uk/search?client=firefox-a&rls=org.mozilla%3Aen-US%3Aofficial&channel=s&hl=en&source=hp&q=phi+in+nature&meta=&btnG=Google+Search

http://www.google.co.uk/search?client=firefox-a&rls=org.mozilla%3Aen-US%3Aofficial&channel=s&hl=en&source=hp&q=phi+in+nature&meta=&btnG=Google+Search

My point was the number more than the actual design for you need numbers to get the design and interpret it into the golden number. Numbers in math are invented by humans.

My point was the number more than the actual design for you need numbers to get the design and interpret it into the golden number.
The ratio itself occurs all over nature.


Numbers in math are invented by humans.
Not quite. What we call a "number" is simply a human way of expressing a facet of reality.

The ratio itself occurs all over nature.


Not quite. What we call a "number" is simply a human way of expressing a facet of reality.

I do not disagree however I just think, IMO, that math was invented by humans to show what was already there in nature. We can pick this to death I'm sure but this is what I believe. I do see your point of view as well and respect your viewpoint.

Consider a circle, and the number pi.

The word "circle" is a designation, a human invention. It corresponds to a concept, the concept of a round shape, and the concept of the set of points in a plane that are a particular distance from a fixed point.
But the designation and the concept also correspond to real things, actual shapes. The shape of a circle is unambiguous. It has been realized independently many times, by different cultures, by other organisms consciously (e.g. dolphins making bubble rings) instinctively (spider webs, bird's nests) accidentally (ant hills, worm holes, fairy rings) and inherently (your eye, a daisy), and by non-living things such as bubbles, craters, weather patterns, stars, galaxies, and planets. Clearly, the concept of a circle is not a human invention. It is a discovery that has been given an invented designation.

The properties of a circle (radius and circumference) are part of the discovery. These properties are properties of actual circles that really exist. They may be given arbitrary designations, but the concepts themselves are not arbitrary, and nor is the relationship between them.

If pi were a human invention, if pi depended solely on its inventor, then it would be unlikely to be given the same value by two independent inventors. The fact that it has been discovered independently with the same value indicates that it relies on something beyond those inventors, something with its own independent reality.

If pi were simply an invention, why not just give it an easy value, like 3?

Similaryly, if the golden ratio is a human invention, why not invent it to be 1.5?

I do not disagree however I just think, IMO, that math was invented by humans to show what was already yhere in nature. We can pick this to death I'm sure but this is what I believe. I do see your point of view as well and respect your viewpoint.

The language and tools of mathematics are certainly inventions, but not the underlying concepts they describe.

Anyone can invent a new word and validly choose an arbtrary meaning to go with it, but can you do the same with a number?

The language and tools of mathematics are certainly inventions, but not the underlying concepts they describe.

That's my point.


Anyone can invent a new word and validly choose an arbtrary meaning to go with it, but can you do the same with a number?

Possibly if everyone else agrees with your hypothesis.

I agree it's useful, but usefulness and correctness are not always the same.

Precisely speaking, math is incorrect. Since the universe is reality and doesn't forget things, the reality that math and the universe are not compatible is evidence to the fact that math is a man made invention that works reasonably well for man, but nowhere close to being correct in nature.

No one claimed that Math is found somewhere in the nature as an element or force, of course it is man-made. Yet your "incompleteness" claim for math is equally true for human discoveries as well. That's why I asked you in the beginning if you could show us some complete or precise human discovery.

As far as I understood we are discussing because of the meaning we give to words. Correct me if I am wrong: You think that we can "discover" realities and "invent" unnatural things. What I say is this: What we think that we "discovered" is also a partial interpretation of nature. Or, if you want to give a different meaning to the word, we can get reality out of "inventions": Planes, computers, cars, bridges are all human inventions, yet they are as real as other agents of nature, they are working, they are affecting the environment, they are transforming the reality.

No one claimed that Math is found somewhere in the nature as an element or force, of course it is man-made. Yet your "incompleteness" claim for math is equally true for human discoveries as well. That's why I asked you in the beginning if you could show us some complete or precise human discovery.

As far as I understood we are discussing because of the meaning we give to words. Correct me if I am wrong: You think that we can "discover" realities and "invent" unnatural things. What I say is this: What we think that we "discovered" is also a partial interpretation of nature. Or, if you want to give a different meaning to the word, we can get reality out of "inventions": Planes, computers, cars, bridges are all human inventions, yet they are as real as other agents of nature, they are working, they are affecting the environment, they are transforming the reality.

Discoveries are found by man, inventions are created by man.

Man discovered that the Earth orbits the Sun.
Man discovered that the Moon orbits the Earth.
Man discovered life in the ocean.
Man discovered gold in the mountains, and oil below the ocean.

Man invented the airplane.
Man invented the light bulb.
Man invented language.
Man invented mathematics.

Discoveries are found by man, inventions are created by man.
Man invented mathematics.
So, still no actual evidence to support your contention then?
Read Pete's posts and then get back to us.
http://www.sciforums.com/showpost.php?p=2574077&postcount=19
http://www.sciforums.com/showpost.php?p=2574088&postcount=20

I've always been on the "discovered" side of the fence but, in any case, this is more a question of philosophy than mathematics. And as such the demands for "evidence" are absurd: this is not a question that can be settled by performing a science experiment.

Man invented language. Not your best claim, since bees and gorillas have language to varying extent.

And as such the demands for "evidence" are absurd: this is not a question that can be settled by performing a science experiment.
Yeah, evidence was perhaps the wrong word. Supporting data would have been better.

Not your best claim, since bees and gorillas have language to varying extent.

They do. I guess it's best to say language is invented, and leave it at that.

I could say living things create language, but then I would have to define "living thing."

Yeah, evidence was perhaps the wrong word. Supporting data would have been better.

Well, I think the most one can ask for is "supporting argumentation." This kind of purely philosophical position is always going to rest on a foundation of suppositions and beliefs, and not on anything that rises to the level of hard, objective "data" (which term, to me, implies stuff that lies firmly within the realm of the (natural and social) sciences).

I've always been on the "discovered" side of the fence but, in any case, this is more a question of philosophy than mathematics. And as such the demands for "evidence" are absurd: this is not a question that can be settled by performing a science experiment.

I agree that there is a large amount of philosophy embedded in my original question. I was tempted to post this topic in the philosophy section of this forum, but I wanted to get feedback from the math people especially, so I posted here.

Mathematics are a human invention; proportion is a property of the universe, or more than that, a property of existence itself.

Mathematics were invented to deal with the universe.
A universe that was proportional to begin with.

If not, mathematics would not sync with the universe and enable us to understand it.

I've always been on the "discovered" side of the fence but, in any case, this is more a question of philosophy than mathematics. And as such the demands for "evidence" are absurd: this is not a question that can be settled by performing a science experiment.

I think it may be more of a debate over semantics. Mathamatics is an outgrowth of philosophy, of logic in particular. It seems (subject to a variety of epistemological qualifications) likely that logic exists independently of mankind, and in that sense "mathematics" can be thought to exist independently in the abstract.

That said, the universe does not really "use" math, the universe is merely governed by consistent and stable rules that can be described, often very precisely, by mathematics.

Mathematics seems to be an area where neither the term is wholly adequate

That said, the universe does not really "use" math, the universe is merely governed by consistent and stable rules that can be described, often very precisely, by mathematics.

At this point, what you're talking about isn't math as such, but rather physics phrased in terms of math.

You could replace the word "math" in the above quote with "human language" and not change its meaning whatsoever, note. That aspects of nature can be described with math doesn't imply any separate, objective existence for mathematics, outside of the human mind. It just means that math is a sufficiently powerful descriptive system to convey ideas about physics (and other natural sciences).

Moderator note: 60 off-topic posts have been moved to a new thread, here:

Motor Daddy learns fractions

Please stay on-topic.

This might be more related to semantics than to math or philosophy.

BTW: Those who argue for invented rather than discovered point to sculpture. Id est: Did Michaelangelo merely chip away all the marble that was not David & discover that wonderful piece of art? I think this anaolgy is weak but interesting.

This might be more related to semantics than to math or philosophy.

BTW: Those who argue for invented rather than discovered point to sculpture. Id est: Did Michaelangelo merely chip away all the marble that was not David & discover that wonderful piece of art? I think this anaolgy is weak but interesting.Interesting that you should mention this. I'm sure you must be aware of the legend surrounding this sculpture (http://donnaferber.com/2010/02/michelangelo%E2%80%99s-david/):

One of the most renowned pieces of sculpture in the world is Michelangelo’s statue of David. Supposedly, when Michelangelo was asked how he created the magnificent statue David from a block of stone, he replied that he did not create David from the stone; rather he saw David in the stone and merely chipped away at the unneeded pieces until David emerged.

Perhaps not the most reputable of sources, but I'm too lazy to hunt an "intellectual" citation - the story persists however, I first heard it over 25 years ago. Here is another inane link. (http://www.streetdirectory.com/travel_guide/33279/self_improvement_and_motivation/chip_away_to_create_your_destiny.html) Search for yourself, you will find numerous citations.

Point being, if you believe the sculptor's words, things are "discovered", not invented - thanks for the extra point.

I wonder if Michaelangelo actually said that he chipped away all that was not David. If he did make such a remark, I wonder if he was serious.

It is my impression that Michaelangelo was a bit egotistic. Is such a remark consistent with his personality?

The language and tools of mathematics are certainly inventions, but not the underlying concepts they describe.

Anyone can invent a new word and validly choose an arbtrary meaning to go with it, but can you do the same with a number?

Invention IMO
Mathematics is a language.
The language of the natural world.

What the language describes, Nature, is "Discovered"
But the communication of the ideas via language in concise, elegant form is what Mathematics does so well.

Also, many topics and fields of Math can be invented.
Just as one can invent a word that has no correlation to any part of present reality,
so too Mathematicians [Theoretical Physicists] can be far ahead of the reality of experimenters and invent ideas that may or may not be useful some time in the future.

Perhaps English (& other natural languages) need a better word than discovery for referring to advances in fields like mathematics. "Discovery" does not imply the considerable intellectual effort & ability required to advance knowledge in the field of mathematics.

Many "discoveries" in the field of mathematics are the result of as much or more intellectual effort than that taken by Michaelangleo when he "discovered" David in the block of marble.

Until a better word is found, my vote is for invention rather than discovery when referring to advnaces in the field of mathemetics. of

discovered! duh!

Perhaps English (& other natural languages) need a better word than discovery for referring to advances in fields like mathematics. "Discovery" does not imply the considerable intellectual effort & ability required to advance knowledge in the field of mathematics.

Many "discoveries" in the field of mathematics are the result of as much or more intellectual effort than that taken by Michaelangleo when he "discovered" David in the block of marble.

Until a better word is found, my vote is for invention rather than discovery when referring to advnaces in the field of mathemetics. of

Do not forget discovering new species, new chemicals, new law of physics, new stars, even new islands require lots of intellectual and physical efforts! It does not make sense to compare the efforts involved in different kinds of discoveries. To make it explicit I disagree with that mathematics discoveries require more efforts than other kinds of discoveries therefore deserve a new name.

I think math is an inherent trait in all life. Animals have a superlative understanding of math, but in a different level from humans: a cat knows space, volume and distance, and can deduce a human cannot enter a hole - whereby it will cease running in that hole and stare at you unafraid. Speech is a unique triat with humans, but this is not thought to us - a parent only clicks a switch and we talk.

Mathematicians discover natural patterns and formulate them. Not only that, they also invent some new patterns which do not exist in nature.


I agree this. Zero was discovered or invented? I'd like to say it was discovered. However, Cartesian coordinate system was discovered or invented? For this question, I'd like to answer it was invented by Rene Descartes. This, I think, reflects where we put more weight when seeing things.

Invented.

It is then discovered that some aspects of nature correspond to some mathematics. If gravity worked according to a ratio to the cube of the distance instead of the square of the distance then we would simply have selected different equations.

If some aliens on the other side of the galaxy use base 8 instead of base 10 will their laws of physics be different?

psik

Sorry to jump in so late but this is a fascinating argument. It depends on what you mean by "mathematics".

Broadly speaking, we have two big high-level branches, pure and applied mathematics. Applied mathematics attempts to describe relationships between things in the physical universe. As such, the language, forms and methods are invented, but the relationships we try to describe are discovered, and of course new ones are discovered all the time.

Pure mathematics is the study of formal structures and systems, all of which were invented. There is no such thing as "two". We created this; now, if it so happens that you have a pencil and another pencil, and you agree we can use "two" to describe something about your pencil collection, well then, wonderful and God bless you.

So I think the passionate discussion about this is quite legitimate; it is not a simple question to answer.

Regards,
Scott Anderson
Instructor in Mathematics (whatever that is)
University of Detroit Mercy
Detroit MI

Absolutely invented.

Language was invented to describe and communicate; things that already existed.
Mathematics was invented to describe, quantify,etc; things that already existed.

I would be interested to hear your opinions on this question.Technologies are invented. Sciences are discovered. Mathematics is a tool used in both disciplines, as well as in commerce, art and elsewhere.

Our notational system, e.g., 1+1=2, was invented. It is a technology, a subset of written language. But the fact that one plus one equals two was discovered.
I could say living things create language, but then I would have to define "living thing."We have a pretty decent definition. A living thing must have most of the following:Homeostatis. Regulation of an internal environment to maintain a constant state. Organization. On our planet that means a structure composed of cells. Metabolism. Transformation of external energy into internal energy used to power the processes of life. On our planet that includes solar energy (photosynthesis), chemical energy (feeding) and ambient energy (sleeping on a warm rock). Growth. We're not sure how this would manifest in non-Earth life. Adaptation. Changing in response to the environment. This is fundamental to evolution. Response to stimuli. Living things interact with their surroundings. Reproduction. There are two primary means of reproduction on earth but there may be others elsewhere.Certainly when and if we discover life somewhere else we will have to update this list.

Nonetheless, I don't see a problem with a computer of the future being able to invent language. That would not make it alive.

Absolutely invented.

Language was invented to describe and communicate; things that already existed.
Mathematics was invented to describe, quantify,etc; things that already existed.

agreed. it's a language.

one apple plus one apple equals two apples. without the apples, the number two does not exist. it's just a concept.

if the earth disappears, the word 'earth' has no meaning except as a concept.

any number of nothings still equals nothing. pretty much makes it blatant that mathematics is an invention.

now 'formulas' and equations are a discovery but humans using mathematics to explain them.

MATHEMATICS IS A SCIENCE AS WELL. No doubt about it. It can be viewed as an art, a language or recreational, or many things. But it is a science as far as we're trying to discover the mysteries of nature. Nature uses the Natural numbers, and prime number theory is just the study to try to understand the "atoms" of the integers.

See this example:

This guy just draws consecutive circles of diameter 1 over consecutive circles of diameter 2 and so forth, and just studies the image created by analyzing how these Natural numbers fit into each other. There is no artistic control here, no language analysis, this is just how "God created the integers". The image is truely remarkable.

http://www.youtube.com/watch?v=UYkLz8BIS8k