It is classic, but I would like to know what you all think.
invented. but its not as invented as chemistry - damn thats made up! (yes i am a student)
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.
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.
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.
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.
... based on the identification with a body.
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.
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.
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.
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".
Both!! Mathematical concepts are invented when the axioms of a mathematical system are given, discovered when they are later conjectured of proved.
I don't think basic mathematics is invented. Even monkey's can count and add.
Going alongside Platonism in math and saying discovered.
I'm with Godel, Hardy, and Penrose, i.e. I am a Platonist.
"... 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
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.
Realizing that one and one banana is two bananas is not the same as saying "1+ 1= 2".
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
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..."
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?
I feel Mentors have some obligation to reply to a reasonable request.
This is what the thread author asked:
This is how I responded:
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.
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.)
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.
Yes, I see that now. Thanks Matt.
Separate names with a comma.