Was mathematics invented or discovered by humans?

Answer 1

Short answer. A bit of both.

AnswerHere are some of the arguments I came across. I've put them in order, ie. "Math Was Invented", "Math Was Discovered", and "Math is/was Both Invented and Discovered".

Math Was Invented

I see mathematics as a invented/developed

approximation by us to describe what we find in the

real world. I use the word developed because we

develop closer approximations to real life as time

progresses. We should realize, however, that it still

is only an approximation; life itself has not been

adequately described using mathematics to say that we

can predict with absolute certainty future events.

To some extent this is a matter of definition and semantics. My opinion:

Mathematics, as a formal system, is invented in the sense that one starts

with a certain set of elements (usually numbers, but not necessarily so) and

defines a set of rules regarding various relationships between, and

operations of, those elements. And you can invent any rules of the game you

desire.

Mathematics, as a discipline, may be quite a different process. One may

perceive a certain pattern, regularity, or whatever word you might want to

describe some insight (usually regarding numbers, but not necessarily so)

and try to "explain" that insight. This process may be quite circuitous, and

"illogical".

There are any number of observations (conjectures) in mathematics that are

put forward without any proof. Perhaps "Fermat's Last Theorem" is the most

famous, but by no means the only, example of something that was found to be

empirically true for a long time before it was proven in a formal sense. If

you read any of the popular histories of numbers like 'pi' , 'e' , or 'i' it

will be clear that there is much more discovery than invention.

Math itself is invented. What it can be used to model clearly is not.

If the mathematical ideas are out there, waiting to be found, then somehow a purely abstract notion has to have existence even when no human being has ever conceived of it. Because of this, Mazur describes the Platonic view as "a full-fledged theistic position." It doesn't require a God in any traditional sense, but it does require "structures of pure idea and pure being," he says. Defending such a position requires "abandoning the arsenal of rationality and relying on the resources of the prophets."

Mathematics is an abstraction. Because of this, the best part of it was invented.

I vote discovered.

It was well known for quite some time that there existed mathematical paradoxes that could yet not be explained (Zeno's Paradox) which eventually become solvable through limits and calculus. The solution to the problem did, of course, exist in Zeno's time ... but yet lie undiscovered, or thought of. Because developments in Mathematics require new and sometimes counter-intuitive thinking, one can be led to believe it was invented, but I consider it discovery in a way of thinking as opposed to the conventional definition of coming upon a physical thing.

Math Was Discovered

Did man create all math? If man created math, then all math can be manipulated, and it is whatever we want it to be, right?

Sometimes the axiom will be based on physical observation, such as if you were using maths to predict where an object is going. But in pure maths, axioms can be based on a random whim or tie in with the "real world" but are basically just decided upon.

Math - Both Invented and Discovered

Both. An axiom is invented and the conclusion drawn from it is discovered.

But that doesn't answer the question, if an axiom agrees with physical observation and prediction, we have still discovered the axiom. The axiom is only derived from our experiences, which are a product of our environment.

I'm with Sisyphus on this one; it frustrates me. I don't have a real answer (one I can fully rationally defend), but my suspicion is that it is discovered.

The 'reason' (if I can give it such a generous title) goes something along these lines (which was presented to me by someone else, not my own creation):

Imagine a nuclear Holocaust which killed all the humans, non-human animals, and anything even remotely sentient, say leaving only apple trees. If, in the course of events, the wind blows through the apple tree and knocks off two apples. Are there really two apples on the ground, or does someone need to reinvent math to deduce that?

I just can't wrap my mind around the idea that there aren't two apples on the ground without someone's having 'invented' math to tell me that. I may not exist to know that the trees and apples exist, and consequently cannot make the discover about the number of apples, but what I know and what is are distinct for me (I'm sure there is some philosophical term for this perspective, and it wouldn't surprise me to find out that there are many good arguments against it). And, if two more apples fall, that the invention of math is needed to tell me that 2+2=4 apples on the ground is too mind boggling for me.

I'm not saying I've proven anything, I just find the abstraction that there aren't four apples on the ground without an observer who has invented math to be frustrating.

Where, exactly, do these mathematical truths exist? Can a mathematical truth really exist before anyone has ever imagined it?

It's discovered, else physics wouldn't work.

Zah? Would physics stop working if we didn't have the words to describe it? Where do mathematical concepts lie for them to be discovered?

The question assumes that an invention is different from a discovery.

Is this true?

Good question. Could it not be said that an invention is nothing more than a discovery of some entity, concept or system.

In any case, regarding mathematics, the universe would behave mathematically with or without our discovery, invention or our existence, for that matter.

I would try to explain the success of physics by first agreeing that the universe most likely has some constant, underlying structure. As mathematics is the study of structure, it's then natural that math is successful in physics, and since we're always immersed in it, it's also natural that much of mathematics is heavily inspired by the physical world.

So my counterargument is basically thus: the underlying universal structure "could have been" different, and mathematics would still deal with it--it's a study (the abstraction) of any structure, not just the ones that actually obtain. Your above response seems to be that we aren't really stepping outside the boundaries of the structure when we imagine the alternatives. That's consistent, but then it also carries a bit more metaphysical baggage than necessary to explain the success of physics.

But you also said that it's free of any semantic content. But your statement seemingly ignores the observation that Wigner called "The unreasonable effectiveness of mathematics in the natural sciences". One could, in principle, develop a mathematics in complete isolation, and eventually come up with pi. So why is it that circles in the real world, approximately speaking, have a circumference pi times the diameter?

There is a very mysterious connection between math and the real world. I can't explain it except to posit that math is in some sense more real than the universe itself, and we are just here to discover it.

So, what you're saying is that it is in principle possible to randomly create abstract structures without any empirical input whatsoever, and then after the fact find out that one has stumbled upon something that has a correspondence to the real world? What's so strange about that, exactly?

What would be far stranger is that if it was the case than any mathematical structure one concocts in such isolation has some corresponding part in the real world. I suppose it could be true, but it's such a hefty claim to accept without very good reasons. And if it's not true, then we don't need to bother with the heavy metaphysics imposed on us by declaring that mathematics is discovered.

Maybe there is a universe where the truth of the axiom of choice matters. It doesn't seem to be this one, so we're able to invent math beyond what matters to physical law. Anyway, your situation would be very strange indeed, but I don't see how it makes the current one not strange.

* * * * *

Interesting but a bit too anthropocentric. Mathematics is not entirely a human construct. There are some aspects of mathematics, conservation of numbers, for example, that is grasped by several mammals.
I'm sure that half of math was invented and the other half was discovered..

I see it this way:

the "blueprints" for math was always there, but someone had to build it.

addition was discovered because animals had always took many objects and put them together

multiplication was discovered because it is multiple addition

subtraction was discovered just the opposite way addition was.

division was discovered because it is the opposite of multiplication.

numbers were discovered, but base ten (0,1,2,3,4,5,6,7,8,9) was invented. There might as well be base three (so numbers would look like this if they were base three: 0,1,2,3,10,11,12,13,20,21,22,23,30,31,32,33,100,101,102,103,111,112,113,121 . . . . . . . )

and so on . . .