Mathematics - Invented or Discovered

[Janitor]

To get science-fictional about it...

If the first twenty spacefaring alien species that we earthlings come into communicative contact with all tell us that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the sides--making allowances for differences in language and symbology of course--then it would have to be a pretty obstinate person to say that all of mathematics is the free invention of the human mind, and nothing more than that. Or so it seems to me.

 

[12345]

Why? How?
Why do you assume that it was invented?
How was it invented without discovering first the mathematical relationships in nature and then inventing a symbolic mathematical language to describe these natural relationships.
Mathematical relationships exist in nature and are observed by us. We invented the symbolic mathematical language so that we don't have to have oranges and/or apples on hand every time we want to perform a mathematical operation just like we invented algebra and the use of letters to represent numbers either as unknowns or as unspecified i.e. "x" ,A+B=C, 1,2,3...n etc.

The symbols and language of mathematics is beyond doubt an invention or convention but mathematics itself is a discovery of nature. Mathematical relationships exist in nature, the universe, with or without mankind. At least this is the position of Penrose and others including myself since reading his book.

exactly...i should be more specific about what i mean. i meant the number system, and the other mathematical terms have been invented by man. i believe that every thing in nature is relative to an equation.

 

[matt grime]

But Pythagoras's theorem is only true in Euclidean Geometry. It certainly isn't true in other geometries (which are more natural, despite what people think), and is a consequnce of the *definition* of geodesic.

 

[honestrosewater]

It seems that by some of these arguments, humans have never invented anything; since any invention (with the usual stipulations that an invention be intelligible and useful) is the result of observations. Surely the lightbulb existed in some form in nature, in however many fractions, but it required a kind of thought that differs from "mere" discovery. Discovery and invention may or may not require intent, yet that matters little here. Discovery entails only finding or happening upon something and does not include making modifications to or extractions from the thing discovered. The latter is invention.
Well, that's MHO anyway.
Happy thoughts
Rachel

 

[sol2]

Some like to watch the steering of boids:), and some like to watch the Dance of the Honey Bee:)

Heck, why not refer to John Nash again here, and the principals of negotiation? The consequence of conceptualization had its basis in mathematical interpretation. So given the gifted eye of the mathematicain, how watchful is s/he of the anomalies in nature?

 

[honestrosewater]

If you want to delve into a mathematician's brain, why not deny the invention of mathematical language? That is certainly more reasonable than denying the invention of mathematical concepts.

Language, along with it's cognitive structures, has a longer history and wider usage, even amongst nonhuman animals. The seemingly innate mathematical abilities lose their mathematical character when viewed as the accompaniments or by-products of already developed liguistic cognitive structures. How does simple counting differ from putting a name to a face? But there is more to math than putting two and two together. (And I have yet to see those monkeys on typewriters write Shakespeare BTW )

From denying mathematicians credit for their conscious intent and manipulation of ideas follows the elimination of the entire category of human invention, thus making the entire invention/discovery distinction pointless anyway. Perhaps the "real" question is then, "Conscious or mechanistic"?

Sorry, I'm tired and a bit grumpy.
Happy thoughts
Rachel

 

[hypnagogue]

It seems that by some of these arguments, humans have never invented anything; since any invention (with the usual stipulations that an invention be intelligible and useful) is the result of observations. Surely the lightbulb existed in some form in nature, in however many fractions, but it required a kind of thought that differs from "mere" discovery. Discovery and invention may or may not require intent, yet that matters little here. Discovery entails only finding or happening upon something and does not include making modifications to or extractions from the thing discovered. The latter is invention.
Well, that's MHO anyway.
Happy thoughts
Rachel

The way I take the phrasing of this problem, something that is invented is something that is completely novel (ie, had not yet existed until its invention), whereas something that is discovered is something that already existed.

In the case of math, it's obvious that the specific formal languages and so on that we use to describe mathematical relationships are just human invention. However, to say that math is a whole is an entirely human invention would mean that mathematical relationships do not really exist in nature. If this is the case, it's not clear that Newton would have ever been able to write his Principia.

 

[hypnagogue]

But Pythagoras's theorem is only true in Euclidean Geometry. It certainly isn't true in other geometries (which are more natural, despite what people think), and is a consequnce of the *definition* of geodesic.

If you draw two perpendicular lines measuring 3 and 4 inches respectively, then a third line connecting the two non-intersecting endpoints will always be 5 inches. Try it.

There is a question of domain of applicability-- from a QM perspective, on a small enough scale it would probably be impossible to construct a right triangle in the first place. But in our familiar human scale corner of nature, where continuity and Euclidean space are great approximations, the Pythagorean theorem will not fail to hold. Therefore the Pythagorean theorem describes a mathematical relationship that exists in our familiar niche of nature, even if it fails to hold in more exotic circumstances.

 

[honestrosewater]

The way I take the phrasing of this problem, something that is invented is something that is completely novel (ie, had not yet existed until its invention), whereas something that is discovered is something that already existed.

*Completely* novel? I think it is exactly this definition which excludes the possibility of invention. Existed in what form- in the same exact form as the invention? I think invention is higher up the totem pole. An invention being based in reality does not exclude it from being an invention.
Is there any thought in your head or any work of your hands that is *completely* your own and not based on anything else?

In the case of math, it's obvious that the specific formal languages and so on that we use to describe mathematical relationships are just human invention. However, to say that math is a whole is an entirely human invention would mean that mathematical relationships do not really exist in nature. If this is the case, it's not clear that Newton would have ever been able to write his Principia.

How so?! language is not *completely* novel. language evolved from cave paintings, pictographs, etc. which were certainly based on observations of nature. The language and the concepts can have similar bases. But the observation of a ladder leaning against a wall and the concept of the relative sides & angles of a perfect right triangle are two different things. They are as different as Chopsticks and Chopin, if not more so.

Happy thoughts
Rachel

 

[matt grime]

If you draw two perpendicular lines measuring 3 and 4 inches respectively, then a third line connecting the two non-intersecting endpoints will always be 5 inches. Try it.

No it isn't 5 inches exactly. Firstly real world measurements don't take that accuracy, and secondly you're ignoring the (possible) curvature of space-time. And by using the word measure you are eliding the issue of in which geometry your geodesics are defined.

 

[aqualine]

The argument has gone in several directions. i think the question is not about "math", but the substance that math asserts to delineate, and whether those motions are real-- and disregarding of human note-- and so "discovered", or completely dependent upon that human notice and non-existant without it: invented. But then again, "math" as a construct is really inextricable from the mechanics it denotes in as much as it denotes those mechanics and no others. So it is incomplete; but that does not make it necessarily unreal. You could say that if it is incomplete it is relatively finite and if finite, infinitely irrelevant. But again, an irrelevant mechanism is not necessarily a non-real mechanism. Crudely, were you to need to cross an ocean, you would have to take a path to get there. But is that path real, or would it die with the need to cross? Certainly mechanically there remains its possibility and physically there were even its effect. And there is the real question: Is the cause/ effect relationship real or does it hang on perception? Two particles in space could be seen to be on a path to collide, and we can induct that they will collide. But it is possible to imagine (in the same way that it is possible to imagine nothing or everything) a sort of omniscopic point of view whereat they are not approximating, but traveling apart. That would make the initial perspective, along with its perceived mechanism, not irrelevant, but have been all along unreal. That, of course, would appear to require more than a single "math"-- neither one ending with a dynamic or inequality, nor one ending with a static or equality, but both without regard for time-- and that is not possible as far as we are concerned. But that does not make it impossible. Our concern with its math does not make it impossible that our math with its mechanics is perfectly unreal.

 

[robert Ihnot]

Is this bull**** or what?

No it isn't 5 inches exactly. Firstly real world measurements don't take that accuracy, and secondly you're ignoring the (possible) curvature of space-time. And by using the word measure you are eliding the issue of in which geometry your geodesics are defined.

The Egyptians had to constantlyl survey the land as the Nile frequently overflowed. Try telling them your bull****!

Math was not well-developed in those days, but the Egyptians had practical abilities. That is how they survived.

 

[robert Ihnot]

Deep Mystery

If you want to delve into a mathematician's brain, why not deny the invention of mathematical language? That is certainly more reasonable than denying the invention of mathematical concepts.

Language, along with it's cognitive structures, has a longer history and wider usage, even amongst nonhuman animals. The seemingly innate mathematical abilities lose their mathematical character when viewed as the accompaniments or by-products of already developed liguistic cognitive structures. How does simple counting differ from putting a name to a face? But there is more to math than putting two and two together. (And I have yet to see those monkeys on typewriters write Shakespeare BTW )

From denying mathematicians credit for their conscious intent and manipulation of ideas follows the elimination of the entire category of human invention, thus making the entire invention/discovery distinction pointless anyway. Perhaps the "real" question is then, "Conscious or mechanistic"?

Sorry, I'm tired and a bit grumpy.
Happy thoughts
Rachel

It does seem in a way that language is more innate. Clearly the practicality of language is enormous. Math is a little different. Sure, there is reason to know how to count, but that ability like writing was often left up to scribes who recorded cattle sales, etc. Detailed math begain as the job of the specialist.

Clearly, people can see that a picture represents a thing, not itself. Animals, maybe can not. Without this abstract ability nothing much would have been posible as civilization progressed.

But, with math, such as tensor analysis or 4 dimensional space, how on Earth would such math ability help most people? Thus math ability, real abstract ability, which is rare, seems to be a special gift of the gods that to a primitative society would be meaningless.

Thus a new dimension of the question between invention and discovery is how are mathematicians able to preceive, to become cognative of sophisticated mathematical structure at all?

This seems to be really a more important question. After all, what society can not preceive or undersand can't hurt you, or anyway, no one wouldl know the difference!

To go over this question again: What use would a savage have of Relativity? What possible survival value would there be in such advanced thinking? Thus how could the ability to understand that develope at all?
roberteignot.

 

[p-brane]

I've been reading Roger Penrose's " The Emperor's New Mind." He brought up a question that I once thought I knew the answer. "Is Mathematics invented or discovered?" I.E. Is Mathematics a purely mental construct or does(do) Mathematics exist, as in Plato's forms, somewhere, as Truths of Reality that we discover rather than invent?

And, by extension is Logic invented or discovered truths of the universe?

Think about it. Then give us your thoughts on the matter. It is not really as simple or straight forward as it first appears.

Math certainly exists now.

I would say that math is a tool that was waiting to be discovered and used... much like a stick that was discovered and used to coax ants out of an ant hill by a bird or a chimpanzee.

 

[TeV]

Math certainly exists now.

I would say that math is a tool that was waiting to be discovered and used...

I certainly can't claim it exists since when you speaking about existence,it must be carefully clarified what it means for abstractive terms.But,wether we found that "tool" somehow implemented in world around us,or order in it had just inspired us to invent the "tool",something is interesting:There is tight connection between math development and physics development (the science describing world around us).In other words the language of physics uses language of math.Not just technically,that wouldn't be so weird.It is nearly parallel in History.Many abstract math theory "structures" become sooner or later part of some branch in physics.
Personally,the most amazing thing to me is we managed to mathematically formulate laws of quantum theories that work so well ,while not knowing/having clear idea why Nature behaves at quantum level in so strange way that there are many interpretations of QM driving people crazy.

 

[sol2]

I certainly can't claim it exists since when you speaking about existence,it must be carefully clarified what it means for abstractive terms.But,wether we found that "tool" somehow implemented in world around us,or order in it had just inspired us to invent the "tool",something is interesting:There is tight connection between math development and physics development (the science describing world around us).In other words the language of physics uses language of math.Not just technically,that wouldn't be so weird.It is nearly parallel in History.Many abstract math theory "structures" become sooner or later part of some branch in physics.
Personally,the most amazing thing to me is we managed to mathematically formulate laws of quantum theories that work so well ,while not knowing/having clear idea why Nature behaves at quantum level in so strange way that there are many interpretations of QM driving people crazy.

There is always a graduation in thinking from our predeccesors?

Reimann did it having consumed Gauss, and Saccheri.

There were save assumptions in a euclidean world, and now, the quantum of the math is evolving. Imagine, Topological considerations in a such an abtract world.

http://superstringtheory.com/forum/geomboard/messages4/18.html

I think this is Einsteins lesson about the origins of interpretation? That it must be geometrical defined. If this is the case then we must have some origins from which to begin?

 

[TeV]

There is always a graduation in thinking from our predeccesors?
...
Einsteins lesson about the origins of interpretation? That it must be geometrical defined. If this is the case then we must have some origins from which to begin?

Yes.What would we be without our predeccesors?Newton himself said he had seen further becouse he had been on the shoulders of the "giants".
As I have heard of Einstein's standpoints on the philosophy of physics ,it was sort of speak "geometrical".He even liked ,believe it or not,Platon's idea where universe was just empty space and bodies in it are chunks of empty space separated by geometrical surfaces.

 

[sol2]

Yes.What would we be without our predeccesors?Newton himself said he had seen further becouse he had been on the shoulders of the "giants".
As I have heard of Einstein's standpoints on the philosophy of physics ,it was sort of speak "geometrical".He even liked ,believe it or not,Platon's idea where universe was just empty space and bodies in it are chunks of empty space separated by geometrical surfaces.

Plato's solids? Where does it end? Bucky Ball's or crystallography? Fractals?

So we see this shift on Pythagorean harmonies as a extension of string theory and Lqg on the other hand? There still is this struggle to define on the issues of background and non background.


What do we mean when we say "continuum"? Here's a description Albert Einstein gave on p. 83 of his Relativity: The Special and the General Theory:

The surface of a marble table is spread out in front of me. I can get from anyone point on this table to any other point by passing continuously from one point to a "neighboring" one, and repeating this process a (large) number of times, or, in other words, by going from point to point without executing "jumps." I am sure the reader will appreciate with sufficient clearness what I mean here by "neighbouring" and by "jumps" (if he is not too pedantic). We express this property of the surface by describing the latter as a continuum.

http://superstringtheory.com/forum/geomboard/messages2/117.html

So you can see how this issue can materialize respective positions?

This directed my attention to how all maths arise, and the source from which they materialize. If it is inherent in observation that nature will supply us this definition, then how shall we describe it?

Platonic Solids and Plato's Theory of Everything

The Socratic tradition was not particularly congenial to mathematics
(as may be gathered from A More Immortal Atlas), but it seems that
Plato gained an appreciation for mathematics after a series of
conversations with his friend Archytas in 388 BC. One of the things
that most caught Plato's imagination was the existence and uniqueness
of what are now called the five "Platonic solids". It's uncertain who
first described all five of these shapes - it may have been the early
Pythagoreans - but some sources (including Euclid) indicate that
Theaetetus (another friend of Plato's) wrote the first complete account
of the five regular solids. Presumably this formed the basis of the
constructions of the Platonic solids that constitute the concluding
Book XIII of Euclid's Elements.

In any case, Plato was mightily impressed by these five definite shapes
that constitute the only perfectly symmetrical arrangements of a set
of (non-planar) points in space, and late in life he expounded a
complete "theory of everything" (in the treatise called Timaeus) based
explicitly on these five solids. Interestingly, almost 2000 years
later, Johannes Kepler was similarly fascinated by these five shapes,
and developed his own cosmology from them.

To achieve perfect symmetry between the vertices, it's clear that
each face of a regular polyhedron must be a regular polygon, and all
the faces must be identical. So, Theaetetus first considered what
solids could be constructed with only equilateral triangle faces. If
only two triangles meet at a vertex, they must obviously be co-planar,
so to make a solid we must have at least three triangles meeting at
each vertex. Obviously when we have arranged three equilateral
triangles in this way, their bases form another equilateral triangle,
so we have a completely symmetrical solid figure with four faces,
called the tetrahedron, illustrated below.

http://www.mathpages.com/home/kmath096.htm

You could see how useful the monte carlo method might be in describing quantum gravity in such a model of triangulation?

http://superstringtheory.com/basics/gifs/SinPlot.gif

Or Pythagoras as the first string theorist?


http://wc0.worldcrossing.com/WebX?14@194.h1WobsZmbR1.25@.1dde7b2e Do not forget to scroll down immediately.

 

[matt grime]

The Egyptians had to constantlyl survey the land as the Nile frequently overflowed. Try telling them your bull****!

Math was not well-developed in those days, but the Egyptians had practical abilities. That is how they survived.

you're confusing the practical with the theoretical. the idealized world in which pythagoras's theorem is true is eulcidean geometery, which is reasonably approximated by the Earth's surface locally. key word approximated. or are you a flat earther as well who hasn't noticed the natural geometry of the surface is spherical? mind you you also seem to think matter is infinitely divisible too, so who knows what crackpot theories you hold.

if another alien speices did not view the universe in the same manner as we (which was the original point in question), perhaps they would not have developed an axiomatic geometry which had euclidean geometry as a model.

 

[Royce]

you're confusing the practical with the theoretical. the idealized world in which Pythagoras's theorem is true is Euclidean geometry, which is reasonably approximated by the Earth's surface locally. key word approximated. or are you a flat earther as well who hasn't noticed the natural geometry of the surface is spherical? mind you you also seem to think matter is infinitely divisible too, so who knows what crackpot theories you hold.

if another alien species did not view the universe in the same manner as we (which was the original point in question), perhaps they would not have developed an axiomatic geometry which had euclidean geometry as a model.

Another term for euclidean geometry is plane geometry the geometry of a flat plane as opposed to spherical geometry or any other curved space or plane. On a flat plane all of Euclid's axioms hold true and cannot be denied. Any species who can imagine or experiences a flat plane will by necessity develop euclidean or plane geometry with all of the relationships being the exact same, thus supporting the position that mathematics is discovered rather than invented.
The same holds true for any curved space or surface. The relationships will remain exactly the same as they, the relationships are intrinsic properties of the surface at hand. There are no other possibilities!
It is then, as I said, axiomatic that mathematics are discovered properties of nature, reality or the universe, whichever you prefer, and not pure abstract constructions of our minds.
Try to describe the motion of a falling body in a gravitational field using any mathematics other than the one that we use now and learned in Physics 101.
Calculus is a natural and logical result of any such attempt. Newton and others did not invent calculus purely out of thin air but were led to it virtually by the hand out of necessity to describe such motion and other such phenomena.
While it is true that relativity and curved space can and does effect the results of such calculations it is only at the extremes that they have any significant effect at all. In a practical sense here on Earth they can be safely ignored except when traveling great distances of the surface of the earth.
Then spherical geometry is the applicable math to use. The old adage of using the right tool for the right job applies here as everywhere. I would not use a sledge hammer to attach a glass plane to a window frame any more than I would use pane geometry to describe the surface of a sphere

 

[sol2]

I liked your description Royce.

There are reasons why Enstein choose a positive curvature in regards to Reinmann's sphere?

I can't help but be enamoured with the relationship of people like Sachheri, Gauss, and Reinmann, but this by know means demonstrates the geneology of this developement, but reminds us of the issues with Bolyai, Minkowski and Lorentz.

Moving to hyperdimensional realities is as much the acceptance of the world of Gauss, as it is to see how Einstein evolved this move and was futher extended in the visions of Klein. Imagine a cylinder, but before this Mercuries orbits, Taylor and Hulse and then the Bose Nova?

Indeed the need for this consistency is extremely important as I believe it has to form the basis of understanding the move to quantum geometry. Einstein just didn't understand the geometry but raised the issue of gravity in GR. We are being lead along here I think when we consider Kaluza and Klein, and the unification of electromagnetism with Gravity. There seems to be no other avenue as far as I can tell where such consistency demonstrates it's uniquesness, not just with the geometrical evolution, but also demonstrates a call for a experimental justification.

That's just my point of view though

 

[TeV]

On a flat plane all of Euclid's axioms hold true and cannot be denied. Any species who can imagine or experiences a flat plane will by necessity develop euclidean or plane geometry with all of the relationships being the exact same...
The same holds true for any curved space or surface. The relationships will remain exactly the same as they, the relationships are intrinsic properties of the surface at hand. There are no other possibilities!

Well,Euclid's axioms hold true on flat plane becouse they are defined as axioms.Guided by everyday experience people (in world around them) picked some "obvious" truths and formed minimum set of axioms that satisfied logical requirement that no paradox can be derived from them.Axioms ,of course,cannot be proved or disproved.
On the other hand,Lobacevski showed that one of these axioms (axiom of paralels) can be altered on the very same flat plane to produce new logically valid geometry.Representations of these geometries in 3D Cartesian system generates "curved" sufaces (pseudospheres etc..) ,though.

 

[BobG]

Yes.What would we be without our predeccesors?Newton himself said he had seen further becouse he had been on the shoulders of the "giants".

That's a quote with a funny history. Newton said that as a way of closing one of his letters to Robert Hooke. Newton totally despised Hooke, so much so that it controlled quite a few of his actions and caused him a lot of problems. So much so, that in spite of Hooke's legendary obnoxiousness, you had to have some doubts about Newton's emotionaly stability, as well. Hooke also suffered from some physical handicaps, so much so that people compared him to a dwarf. Even if a veiled insult, it has some irony. Newton's Principia was at least partially motivated by a disagreement Newton had with Hooke.

Since I should say something to keep this on topic, how about a quote from Leo Kronecker, "God created the integers, all else is the work of man."

Contrary to what a previous post implied, only humans can 'count'. All animals, including humans, have some innate ability to descriminate between a certain amount of objects without having to resort to counting. Humans are supposed to be able to recognize up to 4 objects without counting (which would explain why, when tallying things with marks, we tend to use a slash through the previous 4 marks to indicate 5 and why 5 is such a recurring base in the world's numbering systems). Counting, the first mathematical development, is an invention of man that enables him to extend his natural abilities beyond just four. (I guess Leo wasn't quite right - maybe God just created the first four integers)

And while math was invented to describe relationships between different things in the physical world, the math isn't the things described. For example, what is the ratio between the circumference of a circle and the diameter of a circle? The relationship exists for every circle and the math we invented can approximately describe that ratio, but math isn't going to give you the 'real' ratio. There are many relationships in nature that can't be expressed, only approximated, by numbers or mathematical equations.

As such, math is a logical language invented by mankind. Like all other human languages, its is easily expandable and very adaptable. In the same manner one might 'discover' the similarities between the Sun and a red rubber ball and invent a new 'simile', mathematicians and scientists can 'discover' how adept previosly invented mathematical concepts are at describing newly discovered phenomenon.

 

[TeV]

****************************************************
Contrary to what a previous post implied, only humans can 'count'. All animals, including humans, have some innate ability to descriminate between a certain amount of objects without having to resort to counting. Humans are supposed to be able to recognize up to 4 objects without counting..
****************************************************
Hmm,I thought It was scientificly proven that some animals can "count" or better say recognize set of object up to about 4 too.
Crows and dolphins in particular .But don't take me for word,I'm not sure.

 

[sol2]

Maybe the Bee's can count considering their dance?

Oh I forgot to mention http://superstringtheory.com/forum/geomboard/messages/27.html and since TEV did, I too acknowledge his contribution.

 

[Gza]

Here's what I think mathematics is. I'm not sure of the philosophical sophistication of the people posting here, but there is a concept know in philosophy as "background" that is important to understand my idea. It is basically the unavoidable basis that you are required to base any opinion you have about the world upon.(an example of background would be in our visualization of more than three dimensions. The operational nature of our three dimensional processing visual cortex serves as the background to our vision of geometry. We cannot make a sensory based visualization of geometry in more than 3 dimensions, so our visual cortex limits us, and thus serves as the background in this analogy. We must base all visual opinions against this background, and can never overcome it as a consequence of our biological structure.)

I could also analogize it mathematically, bringing it closer within context, as a the basis vectors that span a vector space. You can make any judgement of the world, but all these judgements are nothing more than some simple or complex "linear combinations" of background "basis vectors". (I really hate to define mathematics in terms of itself, but bear with me.) Math to me is the actual study of background, abstracted away from the various ways it can be scaled and combined within its own background space. We are determining the rules for construction of the background basis, so that we are free to "build up" all the logical combinations possible. You could also say my entire definition is equivalent to saying nothing more than mathematics is an axiomatic, deductive brance of reasoning, but I think this definition is stronger than that. The deepest axioms we can come up with are the ones derived from background limitations.

 

[p-brane]

I certainly can't claim it exists since when you speaking about existence,it must be carefully clarified what it means for abstractive terms.But,wether we found that "tool" somehow implemented in world around us,or order in it had just inspired us to invent the "tool",something is interesting:There is tight connection between math development and physics development (the science describing world around us).In other words the language of physics uses language of math.Not just technically,that wouldn't be so weird.It is nearly parallel in History.Many abstract math theory "structures" become sooner or later part of some branch in physics.
Personally,the most amazing thing to me is we managed to mathematically formulate laws of quantum theories that work so well ,while not knowing/having clear idea why Nature behaves at quantum level in so strange way that there are many interpretations of QM driving people crazy.

Yes, you have several points there. I believe math to be a language developed to interpret our physical surroundings and our observations of the same. That would explain the tandem effect seen in the parallel evolution of math and physics, which you have pointed out.

Sometimes the language of math gets ahead of the observations we make so that we are witnessing a description of something that has not been actually observed. Not unlike predictive science fiction. Then that bit of language becomes validated by a physical quantum. Take Einstein's prediction of black holes for example.

But, in the end I would say that math is and always will be a simple description we have devised and/or discovered. We use math as an overlay on the functions we find in our natural surroundings. We take our observations and we conveniently package them in mathmatical formuli and they remain as records and testament to what we have seen in nature or to what we can calculate as being possible in nature.

The proof that math is a relative interpretation of nature is in the fact that it is a tool that can be used in a large variety of situations. Its usefulness always remains dependent, relatively, on the requirements and nature of those situations.

 

[selfAdjoint]

But what about all the abstract math that never gets applied to physics? You can assert that it all will be ultimately relevant somehow (I have heard mathematicians do this), but that is faith in things unseen. I think mathematics is the free exercise of some mental abilities we are born with. The more I think about Chomsky's new theory that recursion pure and simple underlies our language skills, the more I suspect that it underlies mathematics as well.

 

[matt grime]

Another term for euclidean geometry is plane geometry the geometry of a flat plane as opposed to spherical geometry or any other curved space or plane. On a flat plane all of Euclid's axioms hold true and cannot be denied. Any species who can imagine or experiences a flat plane will by necessity develop euclidean or plane geometry with all of the relationships being the exact same, thus supporting the position that mathematics is discovered rather than invented.
The same holds true for any curved space or surface. The relationships will remain exactly the same as they, the relationships are intrinsic properties of the surface at hand. There are no other possibilities!
It is then, as I said, axiomatic that mathematics are discovered properties of nature, reality or the universe, whichever you prefer, and not pure abstract constructions of our minds.
Try to describe the motion of a falling body in a gravitational field using any mathematics other than the one that we use now and learned in Physics 101.
Calculus is a natural and logical result of any such attempt. Newton and others did not invent calculus purely out of thin air but were led to it virtually by the hand out of necessity to describe such motion and other such phenomena.
While it is true that relativity and curved space can and does effect the results of such calculations it is only at the extremes that they have any significant effect at all. In a practical sense here on Earth they can be safely ignored except when traveling great distances of the surface of the earth.
Then spherical geometry is the applicable math to use. The old adage of using the right tool for the right job applies here as everywhere. I would not use a sledge hammer to attach a glass plane to a window frame any more than I would use pane geometry to describe the surface of a sphere

As you admit in this post plane geometry is only an attempt to model that which we see locally. That does not support the idea that it was discovered, and in fact surely it demonstrates that we invented it? The fact that we can take the parallel postulate as true or false without contradicting the other axioms also backs up this assertion.

 

[BobG]

Calculus is a natural and logical result of any such attempt. Newton and others did not invent calculus purely out of thin air but were led to it virtually by the hand out of necessity to describe such motion and other such phenomena.

Maybe it's almost just semantics. To be 'mathematics', it has to have some set laws and constraints to keep it internally consistent - as opposed to the real physical world, which sometimes yields some unexpected exceptions. Not all math is based on real observations, but sometimes just contrived so that 1+1 doesn't just usually equal 2, or usually equal some value around 2 - it always equals exactly 2, no exceptions. (In the exponential function, a^x, why does 'a' always have to be greater than zero and not equal to 1 - what real world phenomena does that relate to? You mean I can't make any boxes that are 1 meter in length, 1 meter in width, and 1 meter in heighth? Then I guess it's a good thing all my boxes are 39.37 inches by 39.37 inches by 39.37 inches!).

Newton discovered a relationship about the slope of the tangent to a curve plotted on a graph. But, even the math he invented based on this relationship wasn't exactly accepted. In fact, quite a few ridiculed it - you can't have instantaneous velocity, because, if time is zero, then there is no motion (poor Zeno was rolling around in his grave). Plus, you can't divide by zero - it's a mathematical law! It still needed quite a bit more work (add some more laws, constraints, etc - specifically, somebody had to invent 'limits') to seal up the 'holes' in it before it 'officially' met the standards of being a branch of 'mathematics'.

The final 'invented' product was not the same as the 'discovered' relationship that inspired it. No real difference from most other inventions, which are almost always inspired by something observed in the physical world.