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Is Mathematics Discovered Or Invented?
An anonymous reader points out an article up at Science News on a question that, remarkably, is still being debated after a few thousand years: is mathematics discovered, or is it invented? Those who answer "discovered" are the intellectual descendants of Plato; their number includes Roger Penrose. The article notes that one difficulty with the Platonic view: if mathematical ideas exist in some way independent of humans or minds, then human minds engaged in doing mathematics must somehow be able to connect with this non-physical state. The European Mathematical Society recently devoted space to the debate. One of the papers, Let Platonism die, can be found on page 24 of this PDF. The author believes that Platonism "has more in common with mystical religions than with modern science."
When faced with an awkward question, logical positivism asks: what would the answer tell me about the future?
Suppose you had a definitive, 100% guaranteed answer to the "discovered vs invented" question. What would it allow you to do that you couldn't do before? What could you predict? What would you gain?
Nothing, nothing and nothing.
It's meaningless; merely a matter of perception, wordplay and people having too much time on their hands.
Oh, and the correct answer is "discovered".
I much prefer the Kantian approach, which, simplified, is that space and time are the forms of human intuition, and it is these forms of intuition that lead to us understanding things the way we do (spacially and temporally, whose relationships are mathematical). "Things in themselves" are unknowable, and can only be approached through some set of references, whether it be through the space and time we perceive, other possible ways time and space could work (non-Euclidian geometries?), or ways we can't even imagine. Unlike Plato's idea, which is that mathematics involves universal truths we discover, Kant's "Copernican turn" puts the subject as the one who projects mathematics onto everything it experiences. Arguably, this is the idea that has lead to the "modern era".
This makes mathematics the study of these forms of intuition, so unlike Plato's approach, we're not "discovering" universal ideas, but rather coming to understand the way we interpret the world (and by "we", I mean me, the beings who do science that makes sense to me, and probably most beings on earth whose methods of sensation resemble that of humans).
To answer the question of discovery or invention from this perspective, we can invent ways to do mathematics, but the relationships themselves are a discovery of the way we intuit anything we can sense.
Of course the answer could lead to further locking up knowledge... You can't read my theorem until you pay the license type deal.
Shh.
The concept was invented.
What can be done with it is then discovered.
Paying taxes to buy civilization is like paying a hooker to buy love.
I'd say that one "invents" a set of axioms and "discovers" the inevitable logical consequences of those axioms. For example, one might invent a negation of Euclid's 5th postulate and discover non-Euclidean geometry. In the process, one might "invent" a proof which is a path that leads from axioms to theorems.
The point is that the axioms don't exist until we create them. But once we create a set of axioms, then the results are an inevitable (if arduous) journey of discovery which might require clever inventions to reach the destination of mathematical knowledge.
Two wrongs don't make a right, but three lefts do.
The question itself, as you pointed out but in a different way, is a false dichotomy (is it this or that??). There are a number of explanations that might be found in a mix of the two camps, or somewhere else altogether. As such, the question is pretty much meaningless, really.
You can go a lot more basic than 1+1=2. Go back to the Peano axioms and you'll find that all you have to assume is the existance of "0", a "successor" function, induction, and a few trivial things like the properties of equality and addition, and you get the whole of arithmetic -- including 1+1=2.
So you invent/assume your choice of axioms, and everything else follows from them and can be discovered at leisure.
What's purple and commutes? An Abelian grape.
"Yes".
To be more specific, Mathematical rules are discovered, Mathematical techniques are invented; "Mathematics" consists of both.
How would you suggest I measure an object in its rest frame?
This may seem like a nitpicking question, but it brings us to the point that I really want to make:
Mathematics is interesting because there are no ambiguities in a well described mathematical problem. There are many problems that have a finite set of solutions. However, every mathematical model we develop to describe our surroundings is only an approximation of our observations. With time, we can create more and more accurate models, but there will always be something about that model that is derived experimentally, and is therefor imperfect.
This does, in fact, tell us something about the underlying nature of the universe. Either it was created with some arbitrary parameters, or it exists in a way such that there is no way to perfectly describe it. Or maybe there are other possibilities I have not considered. What philosophical meaning you derive from all this is up to your own reasoning.