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## Is Mathematics Discovered or invented?

 Quote by sameandnot 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 seperate 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.

 Quote by arildno 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.

 Quote by sameandnot 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.

 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.
 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.
 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.

 Quote by 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.
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".

 Quote by sameandnot 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/m...om/node27.html

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 Quote by 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.
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?

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 Quote by Cincinnatus 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.
Easy in what sense? Surely, only the most theoretical, hypothetical sense. The ability to picture things and get an intuitive sense of what's going on would be entirely lost on this person. Perhaps someone could theoretically come up with the same definitions that we have, but most of the mathematical things we investigate can be traced back to something having inspiration from the physical world. We are probably inspired to investigate quantity because we perceive objects as distinct, i.e. we can see a number of distinct objects sitting on a table, we don't just see one mass of visual data. We live in space, so we have natural ideas about length, area, volume, etc. All the things we study about Rn, what reason would someone with no experience have to think of such a thing?

 Quote by AKG 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.
However, Cincinnatus only said that mathematical objects can be defined syntactically, not that number theoretic truth could be so defined. So the second-order Peano axioms, which are categorical (they have the natural numbers as their unique model up to isomorphism), could be said to syntactically define the natural numbers, even though they fail to prove all second-order arithmetic truths about the natural numbers.

 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.
And that it was a sufficiently interesting to be considered a genuine mathematical theorem. There is an infinity of mathematical theorems that Fermat could have come up with, but most of them would have been trivial and uninteresting (the sum of the first three primes being 10 is a theorem, for instance). How does the special appeal to mathematicians of Fermat's Little Theorem (and generally, most theorems we prove) factor into this?

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 Quote by 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.
That isn't a remotely reasonable assumption. Indeed it implies that being unsighted would be a positive bonus in doing mathematics.

Even assuming that sight or lack of were independent of mathematical ability then at best the proportion of unsighted should be exactly as it is in the rest of the world.

However, I wouldn't even bother going as far as pondering that as the hypothesis that 'mathematics is an invention of the mind hence blindness should be no bar in doing mathematics' should be examined carefully. It assumes that universities and education in general does not in anyway discriminate against the blind. Nice as that thought is and as much as I wish it were true I seriously doubt that that is the case.

Mathematics is principally a printed medium too and (I would suggest that) no books and certainly no papers have had print runs in Braille.

Blindness is certainly less of a bar to appreciating or composing poetry if it is any at all, and it might lead to greater appreciation of poetry as an audible object; do you suppose that there are as many blind as sighted poets, as your hypothesis would seem to imply there ought to be?

so, if one has the experience of existence, primary to any investigation into the existences of "things", how can anything be said to be a piori?

 Quote by cincinnatus 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."
but, where does syntax derive its existence from? at the very least, it comes from the experience of exisiting; one must have the experience of existing, in order to even create syntax, or know syntax. how can one know anything, without first appealing to the experience of their being?

things are known, because one has the experience of being able to know. I have the ability to know, because i have the experience of existing. if there was no experience of being existent, how could there even be the question of knowing?... let alone the ability to know?

the concept of a priori knowledge, may be short-sighted. nothing can be known without appealng to the experience of being able to know, initially.

 Quote by akg 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?
yes. i said this in a post at 9:07 on thursday the 15th.

 Quote by akg Originally Posted by Cincinnatus 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.
the idea of having a mind, and concurrently, that that mind is devoid of experience, is a contradictory statement. to be in a state of non-experiencing... there must be no being/existence.
at the very least, there is the experience of reasoning. but this example is lost, as well... it only serves to elucidate the idea of experience; to extend it beyond the perceptions of the sense-organs, and to show that experience is founded in the subject's very existing, and not in a perception of something "exterior". sense-perceptions merely combine with the basic experience of being existent, and thereby become interwoven in the essential experience of being, accentuating and coloring the basic experience... it appears. because we are, we can not claim to be able to know things without referring to any experience at all. this is self-contradictory.

 Quote by akg Easy in what sense? Surely, only the most theoretical, hypothetical sense. The ability to picture things and get an intuitive sense of what's going on would be entirely lost on this person.
who would be such a devoid being? a nothing? a non-existing? are we asking an inert (dead) body, to tell us what's up? i don't know, but i know that we need to re-think the concept of "a priori" knowledge.

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 Quote by VazScep However, Cincinnatus only said that mathematical objects can be defined syntactically, not that number theoretic truth could be so defined. So the second-order Peano axioms, which are categorical (they have the natural numbers as their unique model up to isomorphism), could be said to syntactically define the natural numbers, even though they fail to prove all second-order arithmetic truths about the natural numbers.
Okay, I see.
 And that it was a sufficiently interesting to be considered a genuine mathematical theorem. There is an infinity of mathematical theorems that Fermat could have come up with, but most of them would have been trivial and uninteresting (the sum of the first three primes being 10 is a theorem, for instance). How does the special appeal to mathematicians of Fermat's Little Theorem (and generally, most theorems we prove) factor into this?
I don't understand the relevance of this question. Fermat did not invent the fact that $a^p\equiv a\ \left({{\rm mod\ } {p}}\right)$, it was discovered to be a consequence of other things.

 Quote by AKG I don't understand the relevance of this question. Fermat did not invent the fact that $a^p\equiv a\ \left({{\rm mod\ } {p}}\right)$, it was discovered to be a consequence of other things.
I'm not suggesting Fermat invented his theorem. But why is the theorem that the first three primes sum to ten not listed as one of his theorems, or as the theorem of any other mathematician? Very few of the infinity of possible theorems in number theory are ever mentioned. What makes us single out Fermat's Little Theorem as special? I believe this sort of question needs to be considered when determining how mathematics proceeds.

Did Shakespeare just discover a particular sequence of English sentences when he wrote Macbeth?

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 Quote by VazScep I'm not suggesting Fermat invented his theorem. But why is the theorem that the first three primes sum to ten not listed as one of his theorems, or as the theorem of any other mathematician?

surely you jest?

 Very few of the infinity of possible theorems in number theory are ever mentioned. What makes us single out Fermat's Little Theorem as special?
because we are not so stupid as to be unable to appreciate what is genuinely hard and original.

 I believe this sort of question needs to be considered when determining how mathematics proceeds.

not really, or rather not unless you know nothing about mathematics.

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