Is Mathematics Discovered or invented?

[waht]

All notation, rules, axioms, were invented to aid our brain to describe something that our brain perceives is out there.

And we have applied these constructs to describe the physical world ideally. When some of these constructs are intermingled in such a way that they produce a meaningful result, then we call it discovery.

 

[matt grime]

Mathematics goes far further than that, waht. I'm not sure I'd feel confident to state that *all* rules, notation and axioms are invented to describe things that are out there, or even idealized versions of what is out there. That sounds suspiciously like physics to me.

 

[waht]

As a physcial world I meant geometry, not physics. By noticing basic patterns in euclidian geometry, mathemaicians have layed out simple axioms and theorems such that our brains won't have to refer to geometry directly to describe what happens.

And these relationships and rules describe a phenomenon that is discovered. For instance, the pythagorean theorem wasn't invented, it was discovered, because it always was and will be, we just invented a language to describe it and prove it by means of logical deduction which is ultimetaley derived from our physical world.

But, when we mix together these axioms, some produce a consistent result which is still analogous to geometry, but other combinations give rise to an abstract entity which is still consistent, and even beautiful, but feels completely alien and unintuitive. And for it to understand, mathematicians keep inventing notations, and more and more axioms to better describe it.

As far as I know, we could be just exercisesing neuron pathways in the brain.

My 2 cents.

 

[matt grime]

So, in you opinion, all mathematics consists of is Euclidean geometry...

I'm going to take a stab in the dark and say you're not a mathematician.

And please feel free to produce for me a 3-4-5 triangle in the real world that is and always be, though I won't hold my breath.

 

[Cincinnatus]

Rade and Waht are both making the mistake of treating mathematics like other sciences. There is an essential difference between them in the sense that science is empirical but mathematics is decidedly not.

Any scientific theory can be disproved by observation, but this is certainly not the case in mathematics, there is absolutely no observation that could be made that would invalidate the pythagorean theorem (or any other mathematical theorem).

In fact, as Matt Grime said, the objects that are studied in mathematics don't even exist in the world.

 

[waht]

I have all respect for you guys, just trying to explain this in a different perspective. My math background goes all the way up to topology if you want to know.

Euclidian geometry, like all branches of mathematics, is ideal. The real physical world is not. Our brains definatetly prefers to deal with the ideal world hence it's studied more closely. When applied to physics, we get fair approximations as compared to the ideal.

What I'm trying to explain, (i'm bad explaining) is that the basis of logic that has been embedded in our subconscious mind since childhood, and has manifested itself to produce many possible combinations which spawned algebra, geometry, calculus etc.

Can you define a point or a line without ever experiencing the real world?

 

[sameandnot]

So, in you opinion, all mathematics consists of is Euclidean geometry...
I'm going to take a stab in the dark and say you're not a mathematician.
And please feel free to produce for me a 3-4-5 triangle in the real world that is and always be, though I won't hold my breath.

of course, euclidean geometry was meant to symbolize the world, idealized or not, but it turned out to not even be the ideal geometry. so we invented non-euclidean geometry. of course, one can not make a 3-4-5 triangle in euclid's geometry that holds up in reality, but the "new" geometries and mathematics, also do not hold up in reality.
me must have surely recognized that no matter what mathematics we intend to apply to the "real world" indefinitely and absolutely, can never "hold up". nature always seems it to invalidate mathematics as a proper representation and knowledge of it.
as a "mathematician" you must surely know of godel's theorem, that any mathematical system is always doomed to have some statement that cannot be proved.
this surely negates the idea that any mathematical model can be made to truly reflect reality. so, we must concede that we are attempting to resolve mathematics to Reality... meaning that we are using math to symbolize the world, whether it's physics, economics, or any other kind of rational examination.
word do not exist anywhere either, but they are surely symbols. numbers and theorems are not different from words and statements, in this respect.
if i am mistaken, please enlighten me, guru.

 

[Cincinnatus]

Can you define a point or a line without ever experiencing the real world?

I thought you might go this direction waht.

In my opinion, the answer is yes, I think you can. Or rather that mathematics does not require me to define these terms in order to do geometry. In the sense that any mathematical statement can be represented as a statement about a formal system mathematics is completely a priori.

An example of what I mean by formal system can be found here:
http://www.math.uncc.edu/~droyster/math3181/notes/hyprgeom/node27.html"

Using such a system, we don't need to define what is meant by terms like point and line in order to derive all the relevant mathematics about them. Any other information we would add by defining these terms is "extra-mathematical".

 

[Rade]

And please feel free to produce for me a 3-4-5 triangle in the real world that is and always be, though I won't hold my breath.

But why should there be such an entity in the real world ? In fact, many physical phenomenon of nature are exactly as predicted by mathematics, many are not, does nature care what mathematics predicts ?--I think not.
I offer the following from this link:
http://users.powernet.co.uk/bearsoft/Maths.html
It is important to realize that nature is not able to cross multiply equations, make algebraic substitutions or perform feats of differentiation and integration. These are mathematical devises by which we seek to construct mathematical models which are isomorphic to the way in which nature works. The problem is that we do not know when our model is isomorphic and when is is homomorphic. If it is isomorphic, then anything we do in our mathematical model has a direct parallel in nature and vice versa. If on the other hand, it is homomorphic, then we will find an exact parallel in our mathematics to anything which nature can do, but there will be things which we can do in our mathematical model which nature will not be able to parallel. The consequences of this discussion are that we must at all times in our mathematics be aware of the limitations of nature's power to do mathematics. Our mathematics has great limitations because unlike nature we are not able to perform the almost infinite number of calculations corresponding to the action of every charge in the universe on every other charge. Nature performs a summation. We perform an integration to get the same result. The validity lies in understanding the correspondence between the two and not allowing our mathematics to stray into further deductions without establishing the continued correspondence between our calculations and the way in which nature performs her own.

 

[matt grime]

as a "mathematician" you must surely know of godel's theorem, that any mathematical system is always doomed to have some statement that cannot be proved.

If you're going to cite goedel's theorem at least get it right. (and look up the works of tarski to see that the conditions that you've failed to mention are both sufficient and necessary.)

I have no idea what you post was about in regards to mine. Indeed I've no idea what you even men by resolving mathematics to reality or whatever. If you are saying that whatever mathematical objects are then they do not exist in this universe then i heartily agree. if however you think mathematics is purely limited to modelling things that do exist in this universe then i would tend to disagree; i feel mathematics has gone way beyond that. As Conway once facetiously said of some large number, 'if that even exists'

 

[matt grime]

Can you define a point or a line without ever experiencing the real world?

Yes, you can, though the invention of such things would appear esoteric. Lots of maths is invented before a use is found for it, as is inevitable it is inspired by some maths that was inspired by some maths that probably was invented for a real world application.

An example that is interesting me at the moment would be topological quantum field theory.

Firstly there is no observational data to imply that string theory is correct, and secondly a 2-d TQFT is a functor from a category whose objects are (finite) collections of circles and whose morphisms (think of this as evolution in time) are given by riemann surfaces with certain openings. None of those objects was invented to describe physical phenomena directly, and arguably they still aren't being used to describe physical phenomena.

 

[matt grime]

But why should there be such an entity in the real world ?

! that was the point I was making.

http://users.powernet.co.uk/bearsoft/Maths.html

ooh, a powernet user, that's bound to be mathematically sound, and not at all an uniformed ramble by someone who doesn't know their arse from their elbow. How about some links to papers in peer reviewd journals? One that isn't written entirely in the style of a 'pathetic fallacy', for instance?

 

[veij0]

However, I would take the point of view that "one apple plus one apple makes two apples" does NOT mean that "1+ 1= 2" divorced of any specific objects- that's a completely different definition!

You certainly have a point. perhaps I tried it too simple.

 

[Rade]

And please feel free to produce for me a 3-4-5 triangle in the real world that is and always be, though I won't hold my breath.

OK, here goes. Consider the stars of the night sky. Are not stars in the real world that is and always be? Take many pictures, say with the Hubble telescope. Get your ruler, and the 3-4-5 triangle you seek will be found in the positions of those three stars whose shape yields these measurments. Note, you asked for a 3-4-5 triangle, not 3.000000-4.000000-5.000000, so please no argument on effect of measurment error. Is this perhaps not what Pythagaras did on a clear night many years ago, looked to the stars, and in their shapes as expressed by numbers, found the ultimate elements of the universe ?

 

[sameandnot]

there is no point in doing mathematics nor could it have arisen at all, without its being associated directly and drawn from reality. there is no meaning to mathematics without application to reality, in any way. there may not even be such mathematics existing... i would like to see one show me such a math.

thank god for "margins of error"...

 

[matt grime]

OK, here goes. Consider the stars of the night sky. Are not stars in the real world that is and always be? Take many pictures, say with the Hubble telescope. Get your ruler, and the 3-4-5 triangle you seek will be found in the positions of those three stars whose shape yields these measurments. Note, you asked for a 3-4-5 triangle, not 3.000000-4.000000-5.000000, so please no argument on effect of measurment error. Is this perhaps not what Pythagaras did on a clear night many years ago, looked to the stars, and in their shapes as expressed by numbers, found the ultimate elements of the universe ?

And now you're claiming that the fabric of space time is euclidean are you? That might come as a small surprise to physicists, who generally consider it to be curved and as we all know the pythagorean theorem fails to be true in hyperbolic geometry. Then there is the small problem that the stars are moving, and there is the problem of measurement: how can there be no issue of measurement error when you first line is: get a ruler?
(Pythagoras's theorem actually fails spectacularly in spherical geometry as is easy to see. Consider a right angled triangle the base lying on the equator and the apex at the north pole. the two sides from the apex to the base always meet the base at right angles, and they always have the same length irrespective of the length of the base, hence the pythagorean theorem fails to hold *on the surface of the earth*)

And where are these three stars that form a perfect 3-4-5 right angled triangle?

 

[matt grime]

there is no point in doing mathematics nor could it have arisen at all, without its being associated directly and drawn from reality. there is no meaning to mathematics without application to reality, in any way. there may not even be such mathematics existing... i would like to see one show me such a math.
thank god for "margins of error"...

I already gave you an example of theoretical physics that has no basis in the observed data of the real world.

As for other things: category theory was not developed with the intent of doing anything for the real world.

Arguably non-euclidean geometry was developed without recourse to the real world, it was an attempt to see if the parallel postulate was independent of the other axioms, and its models were a long time in being invented. Oddly, one is of course spherical geometry, the geometry that is most natural ro describe the Earth's surface.

Just because something has now got a use modelling the real world doesn't mean that ti started off with that intention.

It would of course be disingenuous to deny that if you go back in the evolution of ideas far enough that you won't find some real world impetus for a lot of mathematics, but at which point does the chicken in the chicken and egg 'paradox' actually stop being a chicken?

Here's one: is there anything in the real world that is actually a continuum? We pass to the continuous because that makes our life easier.

Groups were invented to study the roots of polynomials. Does that make them motivated by the real world? They also describe the symmetries of objects (not necesarily realizable in 3-d real space) so are they discovered or invented?

What about schemes? Can you clarify what is necessary for something to be considered applicable to the real world?
This is now no longer a question about mathematics' philosophy but its inherent merits. There are whole swathes of research out there that were done with no application in the real world, that was even Hardy's defence. If you want to start another thread about 'is there any merit in mathematics for its own sake' please feel free but it is not really part of the debate out formalism v platonism, invention v. discovery.

Most (all?) people who make such claims as mathematics is only worthwhile if it is based in the real world usually annoy the hell out of me so I won't bother to participate, which will probably come as good news to you should you start such a debate. The reason being that the person with that thesis is almost never a mathematician, knows little about mathematics, and doesn't ever state what is necessary to validate mathematics as a worthwhile cause. The notion of necessarily directed research generally is indefensible since most distinguished discoveries have come about by undirect research (penecillin, polio vaccine etc) of course one the discovery is made then it is necessary to direct it to its natural conclusion.

In mathematical terms, how about the categories of sheaves over (ringed) topological spaces (eg smooth projective varieties). Is that esoterically abstract subject worth studying? Such things were developed independently by mathematicians and now turn out to be of interest to mirror symmetry physicists. Fermat's Little theorem, that for all primes p and for all a in Z, a^p=a mod p is a little bit of abstract pure maths, something that had no real life usage for centuries, but is now the essence of RSA.

Surely we could make physics defend itself against the charge that its research must only be done with practical applications in mind. That would undo most physics research grants since they work on scales that are usually unapplicable to any real life situation.

 

[sameandnot]

I already gave you an example of theoretical physics that has no basis in the observed data of the real world.

As for other things: category theory was not developed with the intent of doing anything for the real world.

Arguably non-euclidean geometry was developed without recourse to the real world, it was an attempt to see if the parallel postulate was independent of the other axioms, and its models were a long time in being invented. Oddly, one is of course spherical geometry, the geometry that is most natural ro describe the Earth's surface.

Just because something has now got a use modelling the real world doesn't mean that ti started off with that intention.

please try not to be short-cited.

first, theoretical physics has, as its foundation, the intention of really representing reality, no matter how complex and abstracted the math has become. it is fundamentally the same as when it was in classical mechanics and also the same as it was when it was developed, in general.
just because it is presently abstracted so greatly, and distantly from its origins, does not mean that it is not fundamentally intended to reflect reality, correctly, in some way. this goes for all mathematics.

the fact that modern mathematics is extracted and developed from the simple mathematics of observation/association, means that it is, in fact, no different in its nature, it is just manifest in more complicated abstractions/forms.
it is important not to be decieved by the vast complexity of forms, now present, in mathematics and recognize its nature as being that which is meant to correctly reflect reality, in some way.

math, no matter how complex and distantly abstracted from its original form it is, is still inherently the same as it always was. it has just been developed to greater and greater complexities.

it has as its purpose to give an account of reality; founded on the belief that reality is divisible and logically consistent.

math has, now, become so developed and complex that it is often perceived to be an entirely separate entity, in it own. math can be developed by math, alone, but it is developed in this way, from the essential seed (philosophy and perception of reality) from whence it grew.

pardon me if i seem to be saying the same thing repeatedly, but the point must be understood.

"the world of mathematics" is, because it grew from a distinct perception of reality... and therefore, from a distinct, single, philosophy of reality. it is always trying to fulfill that basic perception, though the recognition of that perception has been lost in the ensuing world of numbers, equations and theorems from whence it issued.

 

[Sir_Deenicus]

Im going to go with Disconted or Invovered. The same thing can be said about orange juice. I discover orange juice but I must invent the concept for the proper description and handling of my new perceptual experiences.

I will also note that Gauss was on a team to measure the magnetic field of the Earth or some such and was so motivated to consider a spherical goemetry. As for Bolyai, I do not think it appropriate to say he completely disconnected from reality since hiw work led from observations and thoughts on the parallel postulate whose form was the formaliztion and abstraction of centuries of applied work.

Also, one can arguably trace the foundations necessary for Galois work back to the babylonians and their need to encode certain problems to do with the marking of land for the deciding of inheritances (and other such) in terms of polynomials. While far removed to today's modern methods, their per problem and heuristic methods served and more importantly they laid a set of problems and general ideas that would serve to be imeasurably important for time to come.

btw, I find Applied Maths to be horridly boring.

 

[matt grime]

first, theoretical physics has, as its foundation, the intention of really representing reality, no matter how complex and abstracted the math has become. it is fundamentally the same as when it was in classical mechanics and also the same as it was when it was developed, in general. just because it is presently abstracted so greatly, and distantly from its origins, does not mean that it is not fundamentally intended to reflect reality, correctly, in some way. this goes for all mathematics.
the fact that modern mathematics is extracted and developed from the simple mathematics of observation/association, means that it is, in fact, no different in its nature, it is just manifest in more complicated abstractions/forms.

That is because you are choosing your definitions to fit your opinion. no harm in that but you should state them first.

Thus anything that has at any point had any connection with modelling reality or has derived from such is in your view an attempt to describe it. Thus of course you are correct. You could not be wrong.

As someone employed to do mathematics research I feel that what I do has no basis in reality and am perfectly happy with that position, as are a great many other people who are in maths. You might care to take into account their views before telling them what they do.

 

[matt grime]

As for Bolyai, I do not think it appropriate to say he completely disconnected from reality since hiw work led from observations and thoughts on the parallel postulate whose form was the formaliztion and abstraction of centuries of applied work.

But the point was no one could find a geometry in which the parallel postulate failed and there was no natural model for hyperbolic geometry until after it was given some abstract ones, thus despite being the most naturally occurring geometry, it was purely invented before it was found in physical form. The invention of hyperbolic geometry was a purely theoretical exercise, founded from a desire to see if the parallel postulate was intrinsic to geometry.

 

[arildno]

it is important not to be decieved by the vast complexity of forms, now present, in mathematics and recognize its nature as being that which is meant to correctly reflect reality, in some way.
math, no matter how complex and distantly abstracted from its original form it is, is still inherently the same as it always was. it has just been developed to greater and greater complexities.
it has as its purpose to give an account of reality; founded on the belief that reality is divisible and logically consistent.
math has, now, become so developed and complex that it is often perceived to be an entirely separate entity, in it own. math can be developed by math, alone, but it is developed in this way, from the essential seed (philosophy and perception of reality) from whence it grew.
pardon me if i seem to be saying the same thing repeatedly, but the point must be understood.
"the world of mathematics" is, because it grew from a distinct perception of reality... and therefore, from a distinct, single, philosophy of reality. it is always trying to fulfill that basic perception, though the recognition of that perception has been lost in the ensuing world of numbers, equations and theorems from whence it issued.

Worthless philosophical crap emanating from your woefully inadequate and simplistic "definitions" of what math is supposedly to concern itself about.

If you are in desperate need for a definition of what math "is", then you might as well regard the nature of math as to be that of a game.

 

[sameandnot]

Worthless philosophical crap emanating from your woefully inadequate and simplistic "definitions" of what math is supposedly to concern itself about.
If you are in desperate need for a definition of what math "is", then you might as well regard the nature of math as to be that of a game.

why not regard mathematics as a game?
it has its winners and losers, does it not? it has its "hall of fame".

i am not concerned with the concerns of mathematicians. only, am i concerned with, the facts of its being. essentially.
surely it is invented and discovered.

we are continually discovering (really unfolding) the possibilities of the invention's unfolding, logically, by way of the logical rules.

math is a conceptualization, in numerical form and the consideration of the relationships between said numerical formal concepts. how could a concept not be invented?

we are really unfolding (discovering) the nature of logic, which we created, based on a basic perception of reality; the perception of reality from whence math is founded is the idea that "objects" are the "building blocks" of Reality; the perception of "objectivity" as being Real. not to say that it's not, but we can say that the subject has invented "objectivity" (the idea of a world of distinct, individually existing objects) in the same manner that math was created.
so, math and objectivity are really the same. especially when considered that they both originated from the intention of defining reality in "knowable" parts, and knowing it, by way of examining its parts.

 

[waht]

we are really unfolding (discovering) the nature of logic, which we created, based on a basic perception of reality; the perception of reality from whence math is founded is the idea that "objects" are the "building blocks" of Reality; the perception of "objectivity" as being Real. not to say that it's not, but we can say that the subject has invented "objectivity" (the idea of a world of distinct, individually existing objects) in the same manner that math was created.
so, math and objectivity are really the same. especially when considered that they both originated from the intention of defining reality in "knowable" parts, and knowing it, by way of examining its parts.

Well said, this is what I was trying to explain.

By building new concepts on previous one's, you go ad infinitum. And the more sophisticated the concept, the more possiblities present itself.

By looking at this from more of a psychological point of view, the conciousness is the root of the problem, which roughly speaking is a huge association machine, whose basis is derived from early childhood experience of the world. Taking into account emotions like inspiration or awe which define our drive of curiousity; mathematics itself fails to exist as an independent entity.

 

[Rade]

A question. If "all" mathematics is "invented" by the human mind, then it seems a reasonable hypothesis open to falsification that the relative number of blind mathematicians (e.g., #/1000 indivduals selected randomly) should be the same as those with sight. The reason being that, if all mathematics is invented, then what need to discover any spatial relationships between objects via evidence of the sense of sight--such mechanism would be of no value.
As to the comment about hyperbolic geometry and that it must be "invented" because it was not predicted a priori from reality--I find this to be false reasoning because the concept derives ultimately from sense of sight dealing with reality of parrallel "lines", and of course parrallel lines exist as a concept because they are discovered via our sense of sight. No mathematician "invented" concept of parrallel lines, where in history of mathematics do we find this as fact ? Thus, since parrallel lines can only be discovered not invented, any concept built on investigation of parrallel lines (such as hyperbolic geometry) is by definition discovered via evidence provided by reality, not invented by human mind outside connection with reality. And please, quantum mechanics does not predict that "reality out there does not exist"--nonsense, if there is no reality there cannot be "quanta". Do not confuse this with discussion of Bell Theorem, which deals only with entangled entities, not entities bound by the strong force.
One easy way to provide answer to this thread--bring forth a peer reviewed mathematical paper by a person 100% blind from birth where they "invented" a new concept of mathematics that could never be derived from evidence provided by the sense of sight. Until I see this paper, I will hold that mathematics is "ultimately" discovered via evidence provided by the senses, never invented by the blind transcendential mind.

 

[Cincinnatus]

Forget geometry for a moment then, do you agree at least, that number theory could be a priori?

It certainly doesn't make sense to say that rade's blind mathematician could plausibly be at a disadvantage in number theory since no one can "see" numbers anyway.

 

[sameandnot]

do you need to see to know space? or even to have a concept of "obects"/objectivity.
surely it is foolish to think that is true. one feels many things with the hands, expecially when blind. one still has to manage one's way through space and time. rade, you are, in this instance, not thinking clearly about what you are saying. space is known, and "parallel lines" are known, not through sight, but through perception, in general.

even helen keller eventually was able to conceptualize the world, and thereby learn enough about conceptual reality to speak and write.

can number theory be known without appeal to experience?
what is an example of something that can be known without appeal to experience? or prior to experience?

this is an incredibly difficult question to answer.
if there is a subject who is experiencing, at all, how can it be said that anything can be known prior to experiencing? or without appeal to experience? isn't using the rationality an experience?
can number theory exist prior to the experience of existing?

no. it appears that experience, in whatever form, is the base. there must be the experience before anything can be known to exist. experience permeates the entire fabric of one's knowing. it is the foundation of knowing anything.

i will need an example of something known a priori. even if the thing is not experienced, directly, the inference of its existence is drawn from experience. all knowledge refers to experience, essentially.

again, give examples and we can explore it together.

 

[Rade]

Forget geometry for a moment then, do you agree at least, that number theory could be a priori? It certainly doesn't make sense to say that rade's blind mathematician could plausibly be at a disadvantage in number theory since no one can "see" numbers anyway.

No, we can "feel" numbers, thus one apple, two, etc. Numbers are not a priori to the evidence of the senses. Now, perhaps you will argue that the set of all numbers [ - infinity number <---> + infinity number] is a priori to reality, but I hold that even this must fail because it is reasonable to conclude that this concept derives from existence itself in spacetime, thus [past time existence <----> future time existence]. Thus, when a mathamatician says, I can always add another number to either end of the scale, the philosopher responds, I can always add another thing that exists to reality, both past and future.
Thus I hold that number theory is not a priori to existence, numbers (... -2, -1, 0, +1 +2 ...) have direct association with spacetime units of existence, which has no limit to ultimate alpha and omega. As for 0.0 and its relationship with reality, it is that which exists within the concept of the "present".

 

[Cincinnatus]

i will need an example of something known a priori. even if the thing is not experienced, directly, the inference of its existence is drawn from experience. all knowledge refers to experience, essentially.
again, give examples and we can explore it together.

Most people agree that things like "I think therefore I am" are a priori knowledge.

So if we hypothesize a mind that has experienced nothing whatsoever this mind would still be able to think presumably. Then there is nothing stopping this mind from inventing a formal system on its own.

You must agree that this mind would certainly be capable of arriving at all the theorems about various mathematical objects and thus derive mathematics if it started with the appropriate definitions of the mathematical objects.

So, the question then becomes is there any way in which it could be natural for a mind with no experiences to define mathematical objects.

Looking at mathematics in a purely formalist way provides the answer. This comes from a recognition that mathematics doesn't actually need all the facts about its objects that are commonly assumed to be true of them. An example being the fact that we all have an idea of what a line "looks like". However, there are no theorems in mathematics that depend on our "vision" of a line, that makes this vision extra-mathematical. (not math!)

In fact, according to formalists, mathematical objects can be defined purely syntactically. That is, it doesn't matter to mathematics what a line IS but only how it relates to points, planes, other lines and whatnot.

Viewing mathematics in this way it is easy to see that a mind devoid of experience could make up such a system and do mathematics with it.

This link discusses David Hilbert's formal axiomatization of geometry in the purely syntactic way I mentioned.
http://www.math.uncc.edu/~droyster/math3181/notes/hyprgeom/node27.html"

 

[AKG]

It comes down to whether or not you think mathematical objects have existence independent of the axioms that specify them.

If you think mathematical objects are "real" things and our axioms only serve to describe them in a mathematically useful way, then you would answer that mathematics is discovered.

Whereas, if you think mathematical objects are "created" by the axioms that uniquely specify them then you would (probably) say mathematics is invented.

In the case of number theory, no (decidable) set of axioms uniquely specifies or characterizes number theory. That is, if number theory is created by some set of axioms, it is no set of axioms that any person has created.

I think the question, "Mathematics, invented or discovered?" is just a bad question. All of mathematics? Including all mathematical methods, theorems, facts, objects, etc.? Can some parts not be both invented and discovered? Can some things not have parts invented and parts dicsovered?

I think one thing we can say is that when people prove entirely new results, they aren't inventing anything. Fermat did not invent his little theorem, he discovered that it is true. But that's not the whole picture. Because his discovery differs, at least in some respects, to discovering gold buried under a mountain. There is no question that gold, mountains, and the fact that the gold was under the mountain were not invented. "There is gold under the mountain" is a proposition whose truth is discovered, and not invented. And it is about things that are not invented. But what of mathematical theorems? They can't be said to be invented, and they should be said to be discovered, but are they about things that are invented?

If I invent a set of rules for manipulating symbols, I don't know, a priori, what symbols I will get if I apply this rules to some initial symbols. I am discovering the consequences of my rules. So is mathematics like this? Do mathematicians invent their subject matter, and then discover the consequences of this invention? Or do they discover their subject matter, and also discover the deeper consequences and properties of these subjects?

I think mathematical methods, just like other kinds of methods, are probably both invented and discovered. They are discovered in the sense that we say things like, "I discovered a way to eat food with my feet." If you discover a way to eat with your feet, then there must have been a way to do so all along, i.e. it is not that it was impossible for one to eat with his feet before you happened to think of a way. Certainly, the way you've found to eat with your feet was always a possible way to eat with your feet, and now you've thought of it, so you could certainly be said to have discovered it. At the same time, you thought about it yourself and tried to come up and invent a way to eat with your feet.

I think the most interesting question is in regards to mathematical objects. Are they invented, discovered, or both in some sense? If they are invented, can they still have independent existence in some sense? It seems to me that when someone found the Monster group, it is not as though they invented it, it was there all along. Even when someone says that {0,1} with addition mod 2 forms a group, it seems this fact was true even before anyone talked about groups, i.e. the associativity of addition modulo 2 seems to have nothing to do with whether or not humans talk about it being associative. But does the set {0,1} or the operation of addition require human invention?