Mathematics - Invented or Discovered

[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 consistancy 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 consistancy 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]

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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..
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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 quantuum 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 occurance. 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 excercise 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.

 

[TeV]

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. ..

IMHO,there are SOME differences between abstract math that never gets applied to physics and abstract maths that gets.
My impression:simple reason might be that nature uses the most simple wayouts,paths and mathrules in the base of its' modus operandi,while mathematician sometimes gets astray creating his own rules.Some of these rules might not be too efficient in the basis for the big machine of the universe and Nature rejects them.Example:transfite numbers are rejected by Nature in advance.
That's the freedom of math.Physicist seeks and explores laws of Nature while mathematician creates his *own* rules of game.Note the difference between used words :laws and rules.
Interestingly and quite obvious :mathematicians are also part of nature,beings made of flesh ,water and bones,bulks of matter that are organized in functional system ,and evolutionary quite efficient from the standpoint of mother Nature.Therefore...

 

[matt grime]

"transfinite numbers are rejected by nature in advance"
what does that mean? what does it mean to be accepted, and why are transfinite numbers not accepted. admittedly there is not an infinite number of anyone object in the universe, but even so, what's that got to to with it?

 

[p-brane]

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.

Here's where definitions of "discovery" and "invention" may come in handy.

When X-rays were discovered they were discovered, not invented. The various uses of X-rays was invented but not the X-ray itself.

When it was discovered that 1+1 objects equaled a group of 2 objects this discovery became one of the basises for several inventions in the realm of mathematical equations.

The difference between discovering X-rays and discovering a pattern in grouped objects is that the objects and their "mathmatical" relationships are judged to be mathmatical by our interpretation, from our perspective. The X-rays are "X-rays" regardless of what we think of that particular electromagnetic spectrum.

In a way it is beginning to look as though math is purely an invention, manifest of the imagination of over-cerebral humans, much like any other language. Thank you.

 

[TeV]

"transfinite numbers are rejected by nature in advance"
what does that mean? what does it mean to be accepted, and why are transfinite numbers not accepted. admittedly there is not an infinite number of anyone object in the universe

There isn't infinite number of any object in the universe I agree,but there's potential infinity in EM and gravitational force reach for instance (according to currently accepted models of these force-field interactions).Also,this is just a possibility and interpretations depend on model of the universe and destiny of the same.Potential infinity isn't the same thing as actual.Hence,no actual infinity-no transfite numbers , ordinals etc.
In sense I hold the universe is sort of "constructivistic machine".
Of course,this is my opinion.You may agree or not.

 

[matt grime]

but TeV, why must numbers only be things which count physical objects? why can they not be used to enumerate the state(s) of a system? and as such there are an infinite number of possible states of a system.

 

[p-brane]

My guess is that there are as many numbers, states, systems and purposes as we can construe out of the material we have available to our five or more senses. This variety and magnatude of states includes our actual senses as well. Its a never ending fractal... well, its never-ending until you stop looking into it. Then it ends.

Imagine that math goes away for the summer. There is sand but no one's counting the grains. There's an infinite party. Try to carry that thought into next fall. Cheers!

 

[TeV]

why can they not be used to enumerate the state(s) of a system? and as such there are an infinite number of possible states of a system.

With finite number of physical objects,recognizing finite number of fundamental force interactions in system,there "is" only possibility* for one infinity-trough the time unlimiting operation procedure:the potential infinity.This is also supported by spacetime quantization requirement in modern era of the science.One can make yourself easier by introducing continuum description of space in aproximation ,but this is just a cheating (a very good one since growth and number of states of the evolving system is astronomical).For * see 3 posts up:we are not sure about the destiny of the universe.Therefore,since the ideas of continiuum and actual infinities are completely human that leaves the debate of the meaning of actual realization in nature fruitless.
But even,from the standpoint of potential infinity sign of singularities in any physical theory are signs of the *sickness* in theory.That was what I wanted to emphasize.