Is Mathematics Discovered or invented?

[ComputerGeek]

It is classic, but I would like to know what you all think.

 

[Serene_Chaos]

invented. but its not as invented as chemistry - damn that's made up! (yes i am a student)

 

[selfAdjoint]

It is classic, but I would like to know what you all think.

I b3elieve it's invented, but it FEELS like discovery. And I believe this is so (I have posted this before) because mathematical ideas are by definition WELL-DEFINED. That is they each have a limited and explicit definition agreed on by all, which gives them a sharp-edged character, just like our sensations of a rock or a chair in our environment. So the mind treats them that way and they feel like discoveries.

 

[Pengwuino]

I feel it is a discovery because any invention can be modified to do anything. We don't invent F=ma for example, we discover it... or else we would be perfectly capable of saying F=2ma. Who would be to say we're wrong if its just an invention? Inventions are never "wrong", some are just better then others.

 

[Rade]

I agree with Ayn Rand on this point..."a vast part of higher mathematics...is devoted to the task of discovering methods by which various shapes can be measured" (e.g., integral calculus used to measure area of circles as one example). In this way, the mental process of "concept formation" and "applied mathematics" have a similar goal--identfying relationships to perceptual data.

 

[saltydog]

It is classic, but I would like to know what you all think.

Mathematics is a product of the human brain. What better way to co-exist in a world than to become in some ways like that world. I think the human brain has done that by evolving a neural architecture that closely resembles the non-linear dynamics all about the world we live in. It is this synergy in dynamics I feel, that allows the emergence of a phenomenon called mathematics that works so well in describing nature. Mind, nature, and math. They are all cast from the same mold. It is not that math exists indepenently within nature to be discovered, but rather that nature has conspired to re-create itself within us in such a way that leads to its dynamic representation within our brain that we call mathematics.

 

[sameandnot]

I agree with Ayn Rand on this point..."a vast part of higher mathematics...is devoted to the task of discovering methods by which various shapes can be measured" (e.g., integral calculus used to measure area of circles as one example). In this way, the mental process of "concept formation" and "applied mathematics" have a similar goal--identfying relationships to perceptual data.

... based on the identification with a body.

 

[matt grime]

Boy does that Ayn Rand (appear to) know jack about mathematics, unless she has such a suitable vague notion of what a 'shape' and its 'measure' are as to make her statement vacuous.

 

[Cincinnatus]

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.

 

[Cincinnatus]

Inventions are never "wrong", some are just better then others.

This is interesting,
One can say that mathematical theorems are never wrong either.
If you view mathematical theorems as statements about what is provable from certain axioms then they will never be wrong. That is, if you regard all mathematical theorems as conditional statements of the form:

if (axioms) then (theorem).

Then no mathematical theorem will be false either.

 

[saltydog]

Boy does that Ayn Rand (appear to) know jack about mathematics, unless she has such a suitable vague notion of what a 'shape' and its 'measure' are as to make her statement vacuous.

I'm flat-out disappointed in this. Perhaps you've already addressed the question asked by the thread author in an eariler post and just don't want to be bothered again by what many would consider a premier philosophical question in mathematics. And please spare me any retaliation against my post as I've never claimed to be an ace in mathematics. Just expected more from one I think is.

 

[HallsofIvy]

I'm flat-out disappointed in this. Perhaps you've already addressed the question asked by the thread author in an eariler post and just don't want to be bothered again by what many would consider a premier philosophical question in mathematics. And please spare me any retaliation against my post as I've never claimed to be an ace in mathematics. Just expected more from one I think is.

You're disappointed that he responded (negatively) to your post rather than addressing the orginal post- which you also did not address? If you think you were addressing the orginal post then either you did not understand what it was asking or you did not understand what Ayn Rand was saying (I suspect the latter). The original post asked, as you said yourself, a "premier philosophical question in mathematics". The Ayn Rand quote did not address itself to that but simply spoke of mathematics as a search for formulae for "measuring shapes"- without specifying what she meant by either "measuring" or "shapes".

 

[HallsofIvy]

Is Mathematics Discovered or invented?

Both! Mathematical concepts are invented when the axioms of a mathematical system are given, discovered when they are later conjectured of proved.

 

[tribdog]

I don't think basic mathematics is invented. Even monkey's can count and add.

 

[PerennialII]

Going alongside Platonism in math and saying discovered.

 

[George Jones]

I'm with Godel, Hardy, and Penrose, i.e. I am a Platonist.

Regards,
George

"... and there is no sort of agreement about the nature of mathematical reality among either mathematicians or philosophers. Some hold that it is 'mental' and that in some sense we construct it, others that it is outside and independent of us ... I believe that mathematical reality lies outside of us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations', are simply our notes of our observations."

G. H. Hardy

 

[saltydog]

You're disappointed that he responded (negatively) to your post rather than addressing the orginal post- which you also did not address? If you think you were addressing the orginal post then either you did not understand what it was asking or you did not understand what Ayn Rand was saying (I suspect the latter). The original post asked, as you said yourself, a "premier philosophical question in mathematics". The Ayn Rand quote did not address itself to that but simply spoke of mathematics as a search for formulae for "measuring shapes"- without specifying what she meant by either "measuring" or "shapes".

Hall, I request you kindly explain your response above to me: Matt did not respond to my post; he responded to Rade's post. The original poster simply asked if math is invented or discovered. I believe I did address that question. Why do you think I did not?

Edit: Oh yea, I'm disappointed because I would liked to have read what Matt though about math being invented or discovered, one of our brightest members expousing elequoently about the matter, as opposed to what he actually said. I remain disappointed.

 

[HallsofIvy]

I don't think basic mathematics is invented. Even monkey's can count and add.

Realizing that one and one banana is two bananas is not the same as saying "1+ 1= 2".

 

[scott1]

Kind of both discoverd because you can count that there 3 calcutors even if we didn't anything about numbers there's still 3 calcutors invented because of stuff binary and we invedted the method we use to count the number of calcutors

 

[Rade]

Boy does that Ayn Rand (appear to) know jack about mathematics, unless she has such a suitable vague notion of what a 'shape' and its 'measure' are as to make her statement vacuous.

As defined by Rand, measurement "is the identification of a relationship--a quantitative relationship established by means of a standard that serves as a unit". "A shape is an attribute of an entity--differences of shapes, whether cubes, spheres, cones, etc. are a matter of differing measurments; any shape can be reduced to or expressed by a set of figures..."

 

[VazScep]

The question looks vague to me, even if it is a philosophical cliché. What is meant by mathematics here? If mathematics is a practice or academic tradition, say, then what would it mean to say it is invented or discovered? Does the question really ask what numbers are, or what some other set of mathematical objects are and what their origin is? Are discovery and invention mutually exclusive?

 

[saltydog]

Hall, I request you kindly explain your response above to me: Matt did not respond to my post; he responded to Rade's post. The original poster simply asked if math is invented or discovered. I believe I did address that question. Why do you think I did not?

I feel Mentors have some obligation to reply to a reasonable request.

This is what the thread author asked:

Is Mathematics Discovered or invented?

This is how I responded:

Mathematics is a product of the human brain. What better way to co-exist in a world than to become in some ways like that world. I think the human brain has done that by evolving a neural architecture that closely resembles the non-linear dynamics all about the world we live in. It is this synergy in dynamics I feel, that allows the emergence of a phenomenon called mathematics that works so well in describing nature. Mind, nature, and math. They are all cast from the same mold. It is not that math exists indepenently within nature to be discovered, but rather that nature has conspired to re-create itself within us in such a way that leads to its dynamic representation within our brain that we call mathematics.

Looseyourname, if Hall can't get off his high horse and explain his comment to me, I ask you as mentor of this forum to please explain to me why the above response is not "addressing" the question of the thread.

 

[HallsofIvy]

Sorry, Saltydog, I didn't read the name on the post. I assumed, incorrectly, that it was the person who first quoted Rand that was responding.


(It's not a "high horse" (well, more of a pony). I haven't been on line since early yesterday morning.)

 

[matt grime]

I was not at all addressing the original part of this thread since my opinions on it are fulsomely expressed in other threads on this topic in these forums. Shockingly this is n't the first time someone has brought up this topic.

I will summarize my posts on it:

I now think I don't really know what the exact definitions for each philosophical position are; they seem to change depending on whom you ask. I tend towards formalism, and not platonism. Moreover a complete ignorance of the philosophical issues is no barrier to doing maths showing just how unimportant *mathematically* this question is. Further, I do not think it is the premier philosophical question in mathematics. It might be the premier mathematical question in philosophy, or the premier question in the philosophy of mathematics; I do not regard them as being part of mathematics.

My contribution was initially only meant to point out that Ayn Rand is either uninformed about what mathematics at a higher level is, that the quote is out of context, or that is just plain being misused. Since the poster of that quote cited integrals as a means of evaluating areas and volumes as an example of higher mathematics who knows what we're supposed to think, since elementary calculus from some centuries ago hardly counts as cutting edge research.

Arguably all mathematics is about "shape", by which we mean "some set of things with structure" and if we regard "measuring" as "finding out things about these sets" then it is vacuously true. This can even include primes as geometric objects such as arithmetic curves, so schemes or something like it, can't say I know much about them.

If we think shape is soley an attribute of euclidean geometry, the platonic solids, things you can draw on paper, then it is obviously false. And Rade's explanation of her definitions just introduces the now undefined entity of 'entity', but that is failry typical of non-mathematicians trying to do maths. And no, that doesn't mean mathematicians have fool-proof definitions, but that they tend not to bother with what the philosophical nature of anything is since that has no bearing on actually doing mathematics.

 

[saltydog]

Moreover a complete ignorance of the philosophical issues is no barrier to doing maths showing just how unimportant *mathematically* this question is.

Further, I do not think it is the premier philosophical question in mathematics. It might be the premier mathematical question in philosophy, or the premier question in the philosophy of mathematics; I do not regard them as being part of mathematics.

Yes, I see that now. Thanks Matt.

 

[matt grime]

perhaps it is "the premier philosophical question ABOUT mathematics"? My interest in it, and in its posited solutions has no bearing on what i do when i sit at my desk trying to solve problems.

 

[Rade]

If we think shape is soley an attribute of euclidean geometry, the platonic solids, things you can draw on paper, then it is obviously false. And Rade's explanation of her definitions just introduces the now undefined entity of 'entity', but that is typical of non-mathematicians trying to do maths.

You are correct, Rand is not a mathematician, she is a philosopher. Let me continue with Rand definitions if it will help since you now ask about "entities". Rand defines "entity" as something that exists that has a specific nature and is made of specific attributes. Then she offers some examples. She claims that of the human senses, only two provide direct awareness of entities, sight and touch. The others give only awareness of "attributes" of entities (e.g., hearing, taste, smell). Then, she states: "attributes cannot exist by themselves, they are merely the characteristics of entities, motions are motions of entities, relationships are relationships among entities". Hopefully this helps explain why Rand views mathematics as discovered from shapes of entities gained via perception, and not from relationships of entities invented by the mind. I have no idea the answer to this thread question, I only posit what I understand Rand to be saying, since she is no longer with us to tell us directly. If Rand has something to offer to this thread, great, if not, so be it.

 

[matt grime]

And what does she define "exist" as? Don't answer for my sake, I don't actually care. In fact I actively care not to know the answer.The thread has no answer, by the way; it is in the philosophy section for a reason.

 

[veij0]

mathematical truths are analytic truths. Since they are actually true because of the definition of its particles. 1+1=2 because the definition of two ones and a plus is two. One apple plus one apple makes two apples. That is the definition of two. Basically the same pinciples flow through the whole concept of maths.

 

[HallsofIvy]

mathematical truths are analytic truths. Since they are actually true because of the definition of its particles. 1+1=2 because the definition of two ones and a plus is two. One apple plus one apple makes two apples. That is the definition of two. Basically the same pinciples flow through the whole concept of maths.

The first part of that, I agree with completely. That was Kant's view wasn't it?

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!

 

[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.