Expanding a little on what others have already said, there's a way to frame this with respect to a broader epistemological point that might be of interest: Mathematics is conceptual, and the answer to whether mathematics is invented or discovered is the same as the answer to whether any concept is invented or discovered -- and this answer of course depends on your theory of knowledge.
Consider Platonists (Extreme Realists), who hold that concepts/universals/essences exist as real entities in another realm: their answer would be that these things are obviously discovered, not invented -- the opposite would be incoherent to them.
And consider Nominalists in contrast, who hold that concepts/universals/essences are basically chosen by us: their answer would be that these things are inventions, not discovered -- the opposite would be incoherent to them.
In contrast to both, there is Objectivism, which explains that concepts/universals/essences are objective in that they are the man-made (i.e., product of a volitional process) form in which we grasp things/attributes/relationships in the world, be they man-made or metaphysically-given. This is the serious, technical root of why an Objectivist might cheekily answer that Yes, mathematics is both discovered and invented.
That's all pretty abstract, so maybe I should risk an example... Consider color concepts: Our culture has dozens if not hundreds of color terms in broad use, yet there have also been cultures which have used just two color terms, one for all the colors we might recognize as darker, and one for all the colors we would recognize as lighter.
Does this mean one is right and the other is wrong? (Yep, the Realist would say, at most one of them could have discovered The One True Set Of Color Concepts "out there".) Or is it just that color is subjective? (Yep, the Nominalist would say, this is just another illustration of how universals aren't "out there" and so they must be "in here" and simply invented.)
An Objectivist will explain that both cultures are indeed attending objectively to the facts out there: the metaphysically-given range of the electromagnetic spectrum that humans perceive, and the properties of things which cause them to reflect those frequencies -- and that these facts are discovered, not invented. But at the same time, it is not written on reality how many "buckets" Thou Shalt Use to categorize those metaphysically-given frequencies and the things which reflect them. How many color terms we use legitimately depends on us, our context, our purposes.
For a much better understanding, I suggest reading Rand's short monograph on concept formation, Introduction to Objectivist Epistemology, and/or the first three chapters of Peikoff's book, Objectivism: The Philosophy of Ayn Rand.
What is the method we arrived at this? Did we discover or invent it?
Objectivism, per se, does not address this issue directly. So one might ask: what are the methods that might lead to it indirectly?
Pat Corvini might offer some fodder for thought on this matter. (One might start with "Two, Three, Fours and All of That") Number is objective. The number line consists of an open ended set of concepts that, depending on the direction, persist as one more than, or one less than, the direction traveled, i.e., in accordance with a quantified application of Aristotle's view of the more or less.
The concepts of "two", "three", "four", are 'invented' to designate the specific quantities referred to. Metaphysically, "two" differs from "three" in the same respect as "three" differs from "four".
The relationship of any two numbers being one unit apart was the discovery while the concepts were invented to concretize this relationship in conceptual form.
In this sense, the relationships are discovered, while the specific concepts are "invented" or assigned to differentiate the discoveries one from another.
"Two", "three" and "four" take their place in the number line as "||", "|||" & "||||" differing - ultimately getting their identity from their relationship to the group with one of its members taken as a unit.
answered May 30 '14 at 00:07
The Objectivist answer is "yes" -- i.e., math is both invented and discovered. "Invention" isn't necessarily independent of "discovery." Indeed, most human inventions involve and depend heavily upon a process of discovery. Inventions such as the internal combustion engine and the electric light bulb were not concocted by man "out of the blue"; they depended on innumerable discoveries. Consider the historical development of math, from primitive counting (preceded by the use of bags of pebbles to represent herds of animals, with one pebble for each animal), followed by adding quantities together, or subtracting them, or multiplying or dividing; and the development of algebra, statistical analysis, geometry, trigonometry, calculus, and beyond. There were new discoveries as well as inventiveness at every step. All concept-formation and reasoning involve an interplay of discovery (observation) and human conceptual processes to essentialize and integrate man's knowledge.
answered May 30 '14 at 00:08
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