Are mathematical objects invented or are they discovered?

Discovered of course.

I would say discovered, though I don’t know very much about these things. The universe does what it does, it behaves the way it behaves according to it’s own laws. When we look hard enough and smart enough we notice it, discover it. But the universe does what it does, it doesn’t need us to understand it.

People have been asking this for a long time. On one hand, you have Plato’s perfect forms, and on the other, Aristotle’s ideals.

Plato said that there was a realm of perfect things, and the things we see here are mere shadows of that perfection. For instance, we can’t draw a perfect line or a perfect circle, we can only approximate the perfect shape that exists in that other realm.

Aristotle didn’t believe in any such perfect forms. He said they didn’t exist, that nothing was perfect. However, in our heads, we know what a perfect circle or line ought to be – we can conceive of it, even if it doesn’t exist. The idea of the thing exists in our minds.

Now, if you believe in something like Plato’s perfect forms (and one could say that the idea of Heaven borrows heavily from this concept), we might say that we “discover” the mathematical object – or, at least, the shadow of it that manifests in this world. And if you believe that mathematical concepts mostly exist in the mind of the thinker (because, after all, if there’s no mind to think a theorem, does it exist?), you might say that we’ve invented the object to explain a concept.

If you figure out a way to beat Plato or Aristotle at this game, let me know. ;)

no forgiveness granted

@laureth I think Hilbert’s formalism could be construed as a way to claim agnosticism with respect to this question and go on. A dodge rather than beating them at the game, maybe.

I think they are invented. Sometimes out of necessity, sometimes out of curiosity, but always invented.

Math is a tool that we created to help us manipulate the world. Not in a physical sense like a hammer, but in a mental sense.

@Sueanne_Tremendous Oh well, I don’t know how many times I’ve been told it’s “better” to ask for forgiveness than for permission. ;)

I once had a dream of “something-like-the-afterlife”. Every person who had ever lived was enshrined as a number (though no explicit “encoding” method was at hand in this dream, imagine having your neural stream-of-consciousness encoded somehow into a digital stream, from conception to death it would form an incredibly long number in base two) and they were hanging like paintings in a vast gallery, but with no floors or ceilings. I suspect this dream was a result of my reading about set theory.

I would say both, math is so important so universal that you wouldn’t be able to function without it. It’s hard to say exactly which one for some situations…like.. is 1+1 = 2 discovered or invented? It’s easy to say that d = vi(t) + ½at^2 was invented as it’s a formula for physics, but before all this was allowed to be formulated other areas of mathematics must have been discovered.

I know you asked this for new answers but for new jellies who want to read some other answers there are some great ones here

@shrubbery good catch. That one predates my time here.

I don’t think that invention is necessarily different from discovery. Many inventors claim that they’ve discovered their inventions, like patterns floating in the ether.

I think a better way to think about it is that human brains can filter mathematical objects.

Numbers are a way to explain what we observe.

It is an accepted fact that the acceleration of gravity is 32’ per second per second, but gravity does not follow this law, the law was written to comply with what gravity was already doing.

If there was no math, everything would still happen the same way.

@Qingu I think that’s pretty close to how it is. Our brains capture and categorize patterns. An interesting thing about the mind is that it can abstract some information from these patterns and check the predictions of it with reality. In the case of mathematical objects we can abstract them in our minds and when we see that their predictions are borne out in real life we believe that they exist outside of our minds. When many people abstract these objects in the same way we see them as universal.

Poincaré quotes!

“Mathematicians do not study objects, but relations between objects. Thus, they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: they are interested in form only.”

“If geometry were an experimental science, it would not be an exact science. it would be subject to continual revision… the geometrical axioms are therefore neither synthetic a priori intuitions nor experimental facts. They are conventions. Our choice among all possible conventions is guided by experimental facts; but it remains free, and is only limited by the necessity of avoiding every contradiction, and thus it is that postulates may remain rigorously true even when the experimental laws which have determined their adoption are only approximate. In other words the axioms of geometry (I do not speak of those of arithmetic) are only definitions in disguise. What then are we to think of the question: Is Euclidean geometry true? It has no meaning. We might as well ask if the metric system is true and if the old weights and measures are false; if Cartesian coordinates are true and polar coordinates are false. One geometry cannot be more true than another; it can only be more convenient.”

@Qingu…

“There is no scientific discoverer, no poet, no painter, no musician, who will not tell you that he found ready made his discovery or poem or picture – that it came to him from outside, and that he did not consciously create it from within.”
William Kingdon Clifford

…which then puts me in mind of…

“Nothing is more important than to see the sources of invention which are, in my opinion more interesting than the inventions themselves.”
G.W. von Leibniz

Mathematical knowledge strikes us as objective and necessary; the mathematical community is global and there is almost universal agreement on which mathematical statements are true, or at least, proven. Platonism, the view that there are abstract mathematical entities with properties that we discover through some faculty of reason, explains the objectivity of mathematical knowledge.
The problem with platonism is its conflict with physicalism, which, for the problem at hand, is the doctrine that nothing in the mind changes unless something in the brain changes first. If there are abstract entities, how would we ever know? Such entities are not situated in spacetime and, thus, do not participate in causal relationships. If such entities cannot affect the brain, then they cannot affect the mind. As most of mathematics deals with the infinite, of which we have no direct experience, mathematics must be something we invent.

Intelligent creatures from another world may use mathematics but it would have to be the same mathematics that we know. One plus one can never equal three. I think that mathematics, unlike ordinary language is discovered and not invented.