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.