Does mathematics exist independently of intellects, or is mathematics an invention of the intellect?

What a layman calls “math” is a combination of both.

The words “multiplication” and “exponent” are human constructs, along with the symbols 0123456789 and !^*-+=<> etc. If an alien civilization existed, it would not use those words, will not use those symbols, and would represent math in a completely different notation.

While the notation is a human construct, I believe that the underlying concepts are not. While other civilizations might not share the same notation, they almost certainly have something that means “first number added with second number.”

If math were purely a description of the known universe, then I don’t think that it could be as powerful of a prediction tool as it is. Take the Higgs Boson for example. Nobody had ever observed it before, but the math demanded that it exist. And it does. The same thing happened with black holes. And even the atomic bomb.

Are mathematical objects discovered or invented? – an oldie but goodie, lol. In the philosophy of Plato they have a reality in some realm independent of the world including us. Platonists say math is discovered. Non-Platonists say that mathematical objects are products of human mind and language. For them a triangle or the number 7 doesn’t really exist; mathematics is invented.

This is in contrast to science, where there’s (presumably, as I am not a solipsist) a pre-existing reality “out there” independent of our minds, which we may come to know through physical observations. These observations, in turn, form patterns and can be organized by various laws for which mathematics seems remarkably perfectly suited for describing a great deal of physical law. Moreover newly created objects of pure mathematics are eventually found to be perfectly suited for describing scientific theory not yet discovered at the time they were created! That’s curious no matter how you view mathematical objects.

Personally I’m a non-Platonist. I suspect that mathematics is invented in our minds, which have been shaped from birth to recognize concepts corresponding to the physical world in which we live, so in the end the applicability of math to science isn’t such a big coincidence after all.

Independently. Mathematics abounds in the natural world.

This is a wonderful classical tautological question. I’m firmly in the camp that math exists and we discover them. But it’s a fun thing to think about.

In one sense it is a meaningless question. From a pragmatic point of view, how we answer the question has no impact on what we do. That being said, I am a Platonist with regard to math. There is more to math than numbers, but just consider numbers for a moment. Can you even conceive of a universe where numbers do not apply?

I am willing to concede that some of mathematics is invented. Consider the idea of continuity. The definition of continuity, with its deltas and epsilons, was a great achievement of 19th century mathematics, but does continuity really exist? Motion is quantized. It has been speculated that time and space are also discrete, and perhaps time as well. In that case, the mathematics of continuity would turn out to be a beautiful and useful metaphor for an illusion.

Platonist here as well with regard to most of math. I say most, because @LostInParadise‘s exception is valid, and not the only exception. Still, the bulk of maths are discoveries of things that rule this particular universe. They could be different in some other universe, but that doesn’t seem to be the question.

Our hypothetical aliens, regardless of how different or how distant their homeworld, would never get here is they did not have an understanding of what we mean when we say 1 + 1 = 2. Our aliens would realize that the square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the other two sides. They would have derived π, and so on.

They might say these things in a very different way—almost certainly they would do so. But if you learned their system of description you’d find they were right in sync with Pythagoras.

What would mathematics be without a mind to understand it?

@flutherother Still there, running the Universe, waiting for a mind capable of understanding it to come along.

There is no clear answer. Math can both be viewed as invented or discovered.

When intelligent aliens contact humanity at some point in the future, it’ll be interesting to learn how they do math. This could help us find an answer to this open question.

Nothing + what you had before =0, 0 + the first 1 given by the everything = 1, add 1 to what you had before 0+1=1 Creation? now add what you got now to what you had before 1+0 = 1 add what you got now to what you had before 1+1 =2 add what you got now to what you had before 2+1=3 add what you got now to what you had before 3+2 =5 add what you got now to what you had before 5+3 = 8 dar dae dar add what you got now to what you had before 8+5=13 the spiral in a snail shell or the milky way the pattern in a sunflower seeds the breading habits of rabbits maths has been part of the begining.

@graynett Interesting set of non-sequiturs proving nothing.

What @greynet wrote reminds me of Peano’s axioms for natural numbers. The function S referred to may be thought of as meaning successor and is just a formalization of the notion of counting. From these simple axioms it is possible to define addition and multiplication and to prove the commuative, associative and distributive rules and all the properties of natural numbers.

The set N of natural numbers has the following properties:
1. The set contains an element designated as 1
2. x =x for all x in N
3. x = y implies y = x for all x and y in N
4. x = y and y = z implies x = z for all x,y and z in N
5. There is a function S defined for all elements of N all of whose values are in N
6. If S(x) = S(y) then x = y for all x and y in N
7. There is no element x of N such that S(x) = 1
8. Any set X that contains 1 and is such that if n is in X then so is S(n), is equal to N

Definition of addition:
1. x + 1 = S(x)
2. x + S(y) = S(x+y)

For example, if we define 2 as S(1) then
x + 2 = x + S(1) = S(x+1) = S(S(x))

@graynett Now that @LostInParadise has explained what you wrote, and I have a chance to review it with a clear head, not falling asleep at the switch, I regret my hasty and off-base answer. Please accept my apology. Indeed, maths are all around us in the structure of the Universe, waiting to be discovered and understood.