Do numbers have an objective existence? If life had not evolved on planet earth would there be numbers or are numbers an invention of human minds?
Are there any relevant works that discuss this? (I know of Husserl's Über der Begriff der Zahl and Frege's Grundlagen der Arithmetik already; are there any other notable discussions of this theme?)
The literature on these questions is immense, starting from Plato all the way to the modern mathematical logicians. Since your question is about the existence of numbers, you are concerned with the ontological status of numbers. So, with ontology in mind, you can distinguish the following schools of thought, according to the answer they give to your question.
AMBIGUOUS: Nominalism, Formalism and Logicism. There are several variations, reconstructions and weakenings of these positions that can be taken to occupy either side of the debate.
In the case of logicism, the answer depends on how you regard the ontological status of logic. Fregean and Russellian logicism as well as the early Wittgenstein undoubtedly thought that logic is in some sense what is given to us by the world and that, therefore, numbers do have some objective existence.
Formalism is ambiguous in the sense that although it is prima facie anti-realist because the naive formalist is taken to hold that mathematics is nothing other than systematic manipulation of symbols (which of course can only exist if human do) Hilbert himself (the originator of the position) held no such naive view. For Hilbert, there was a real core of mathematics (he called it 'real mathematics') that he believed was directly accessible by out intuition - this included basic arithmetic (1+1=2) as well as single-quantifier generalizations (For all x, x+1=1+x). And that sounds more realist.
Nominalism can similarly be seen as denying that set theory exists, but affirming the existence of numbers, or can be read as denying the independent existence of numbers altogether.
This is a broad outline of schools that have arisen out of convincing arguments proposed in answer to your question. Many of the arguments are compelling, all of them are interesting. I should also note that people like Alain Badiou (who says: 'Mathematics is Ontology') have also tried to answer the question from non-theoretical and less analytic perspectives. I am not familiar enough with such work to assess it, but it certainly sounds interesting.
Kronecker famously said:
„Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.“
(something like: "God made the integers; all else is the work of man")
...but I think even this is not true (besides there is no god ;-)
Even for the integers, e.g. the concept of one-ness, two-ness asf, you need some kind of abstraction, an intelligence and therefore an observer (not even a human one, but an observer nevertheless).
So the answer to your question in short is: No.
A number is an abstract thing. It doesn't exist independently of a thought about it.
For example a cat... . A cat can exist without a human observing it - ask the mouse! Or think about Brontosaurus. But a number is abstract - you can compare a basket with 3 bananas and a box with 3 letters, and the common thing among them is, that there are 3 elements.
But if you imagine a monkey, deciding between a basket of 3 bananas and another one with 4 - does he have an idea of the number 3, and does he see a connection to 3 strawberries? Or can he divide 3 bananas to 3 monkey childs?
I'm not sure, but most scientists assume today, that in the great universe, there is more live - not only on earth, and more intelligent species have been evolved, and they will need numbers as well to reason about their world.
See how numbers exist in different cultures, and how they were early used to document ownership. There are different number systems, and not everybody invented the zero, but see how fluently the most potent system was adopted elsewhere. The names of the numbers are convention, but not the numbers themselves.
Think about, how the numbers of electrons and protons determine the atoms.
Does an Oxygen atom observe the 2 Hydrogen atoms, or is water only existent, if there is an observer? It's a funny thing to do the brain experiment about observers, but it leads to nothing. If you apply the idea of the necessity of an observer, the whole idea of an objective reality breaks apart, but it takes everything with it and leads to nowhere.
Most things aren't observed non stop but only occasionally or in part or the surface or effects from them or have been observed or might get observed in the future. There is no useful application for the idea, that things stop existing, while you're not looking, listening, smelling, feeling, …
The question of existence is meaningless in the sense of Carnap and the logical positivists, and this question is precisely one which does not have meaning. The question of "existence" of one abstract thing in relation to the "existence" of another abstract thing has no bearing on any observation of the senses, and this question is just your brain fooling you into seeing a question from a collection of nonsense words.
Humans and Numbers exist independent of each other in separate layers of reality. The question assumes some type of interaction between humans and numbers as if they have any effect on each other. Do numbers independent of baboon obsorvers? the answser is yes, do they exist independent of baboon oborvers and also human obsorvers? we need another obsorver to report on that.
Numbers do not exist in this universe in the same manner that humans do and depend upon, so even after a complete catastorphe when all the livingthings have ceased to exist, and the next cycle of life starts all over again, the numbers will be there, ready to be used by any number recognising life form.
Whatever the different schools of thought there are, very roughly divided into the intuitionist/formalist/social constructivist side which says 'no' and the platonist/realist side that says 'yes', there is a great distinction between numbers (and mathematical objects) and physical objects like trees or air.
Just because we have thoughts about the number 2 as well as the tree outside my window, and the sentences 'the tree outside my window objectively exists' is just as parsable and understandable as 'the number 2 objectively exists' (without yet judging if the answers are the same or make sense beyond the superficial), just because we use 'exists' in both sentences, doesn't mean the uses of 'exists' are the same.
We can tell that a particular tree exists though our senses (with whatever classical doubts via dreaming, illusions, mistakes there might be) as well as a pair of physical objects, and we might have philosophical judgments about the objectivity of those kinds of sensations, but that is a different thing from 'tree-ness' and 'two-ness' (the latter is really what 'two' means: if you have two trees in your yard, you don't have 'two-ness' in your front yard).
So asking if numbers have objective existence is more like asking if concepts have objective reality (in distinction to objects that can be sensed directly).
There is the side issue of how can one 'sense' particular numbers. we all accept that we have sense organs that can 'capture' a tree (by sight or touch), but there is no immediate such sense for particular numbers. But even this issues shows that the concept of existence is different for numbers.
Number do exist as long as the countable objects exist, irrespective of the observer, but if there is no observer what loses its meaning are not the numbers but the "meaning" (understanding of reality) concept instead!
That is to say, numbers as adjectives can exist without observer (like in "two apples"), but as an imaginary abstraction no as the abstraction itself is a work of the observer's mind.
Also counting needs some order whose definition very much depends on the observer (which apple in the basket to be assigned 1, which 2, and etc.), but that a basket has 9 apples need no observer and will keep its meaning even in the absent of the observers.
However, if you ask about zero, infinity, cardinals larger than infinity, negative numbers, non-rational numbers, then one can continue discussing things further. All I said above was about the rational numbers, although I believe everything that we discover has or potentially can have a place in the real out universe.
And as a last point, the physical rules of nature exist irrespective of being witnessed by an observer or not. Such rules have always numbers with themselves (like Pi, e, and maybe many more) even though there be no observer writing down the discovered rules on a piece of paper. That Einstein said Nature is numerically integrating somewhat clarifies this idea better.
If to exist is to be the value of a bound variable, and if 1728 is the smallest number that is the sum of two cubes in two different ways, then the number 1728 exists. If moreover, in the absence of human (or other) minds, 1728 would still be the smallest number that is the sum of two cubes in two different ways, then the existence of the number 1728 does not depend on the existence of human (or other) minds.
It seems to me be to be self-evident both that 1728 is the smallest number that is the sum of two cubes in two different ways, and that this would remain true in the absence of human (or other) minds.
As somebody pointed out above - Circles exist regardless of observers, and the ratio between any circle and its radius is constant. For that matter, the ratio between the distances between the center of a sphere and any point on its surface is constant too (1), so these "values" must exist as well, independant of thought about them. In fact, a sphere is defined as a shape where every point on the surface is equi-distant from the center. If numbers don't exist, then that concept can't exist either...
I hear the arguments against the independant existence of numbers, but I cannot wrap my head around their non-existence, where I can easily understand other such arguments. I find it easier to understand the concept of something concrete ("a tree" for example) not existing than numbers not existing. I tend to trust my understanding in issues like this, if something feels like its bordering on the nonsensical like this does, I cannot accept it as true.
I welcome an explanation of a number-less world that I can wrap my head around - I just haven't heard one yet...
Here's an argument I presented that the number "6" must exist independently of the mind. The number "6" is the first in the sequence of numbers that are sum of their divisors since 6=1+2+3 and 6 is also divisible by 1,2 and 3. Therefore the number "6" is the answer to what's first in the sequence of an algorithm and this algorithm have existed even if human minds didn't exist.
And the number pi appears when calculating how a river flows etc and that river exists independently of the human mind, therefore the number pi also exists indepedent of the human mind and is not just a mental construction.
Mathematics is a science that studies common properties of physical objects which do not depend on the actual physical substance of which the objects are constructed. For example it studies what is in common with all round things, with all triangular things etc.
Certain objects can be said of to consist of other constituent objects. There is a property of being composed of only one object, of two objects, of three objects etc.
All objects that are composed of six other objects have something in common. For example, they can be divided in two objects that have three parts or into three objects that have two parts. This property of being composed of six parts is named "number 6" so the mathematicians could study this common property and then make conclusions about other composed objects (a bunch of flowers, a set of cards etc) without studying them separately.
Thus the numbers as properties of real-word objects exist independently, like any other properties, such as being big, or being round, or being heavy (the common properties of all heavy things for example studies mechanics, thus mass is also something that exists independently).
Not really, they exist in a relation and commitment between God and mankind. To this degree they have a practical status as being ontologically existant.
The fact that this topic has been debated inconclusively for more that two millennia should be a massive clue that the question you are asking is not the sort that has an absolutely right answer. It boils down to a disagreement about the meaning of the word 'exist'. Clearly numbers exist in certain senses of the word and not in others. They exist as ideas, as printed symbols etc. They don't exist in the way that planets do, as big objects subject to gravity. If you consider only those two aspects, you would conclude that numbers are a human invention. The question then is whether it is meaningful to say that numbers exist is other ways between those two extremes, which allows one to say that they exist independently of human minds.
Suppose I have a field with six apple trees, and that overnight tonight all humans die. Does my field still contain the six trees? It seems to me that while there would be nobody to count the trees, or even to say what was or wasn't an apple tree or a field, there is a sense in which there are six apple trees regardless of whether humans exist to contemplate them. You can chose either to extent the word 'exist' to cover this instance of the number six, or you can chose not to. Either way, the question is really about how you prefer to use the word 'exist' as a label- it doesn't change the underlying reality (assuming there is an underlying reality).
What we typically mean by "number" is abstracted quantity. We might say we have 5 apples or that marriage is between two people, forming 1 union. In all of these cases, the quantity is always of something. It is an accident, something that cannot exist apart from some object that can. But we can entertain in our minds, through the intellectual operation of abstraction, these quantities apart from those objects, analogously to the way we can entertain the color red apart from anything that is red, even though in the real, redness cannot exist apart from things that bear that quality. We can take 5 on its own, and examine that quantity as such, on its own, in itself, in our minds. We can see that it is not divisible by any other number save for itself and 1, for example, as a number theorist might.
So, in that sense, numbers as abstracta do not exist apart from minds that can abstract them from the real, just as colors cannot, but quantities of things do exist in the real as accidents in a manner analogous to color. One exception would be the so-called mind of God, which may be identified here with the Platonic realm of ideal forms. But this, too, at least understood analogously, is like a mind, except that instead of abstracting from some existing reality, it can impart concrete reality to those forms.
Numbers are representations, e.g. 3 and (Roman) III are understood to represent abstract concepts having the exact same properties. Numbers themselves are mere labels or identifiers of truth which exists independent of observation.
So, do numbers exist independently from observers? No, but what they identify does.
