Yeah, but that wouldn't be mathematics.

But let us just take the integers with the common definitions for addition, subtraction and multiplication, but then redefine them so that every operation first performs the usual operation and then increments the result by one.

  1 + 1 = 3
  1 * 1 = 2
  (1 + 2) * 3 = 13
  (a + b) * c = a * c + b * c + c - 4

You will discover them, not invent. Albeit subtle, there is a difference between these notions, and you will indeed discover the integers in (almost) exactly the same sense as electron was discovered - what you do is, you put forward an idea, play with it, test it, correct mistakes, and in the end you find what you have been looking for. What you can invent in mathematics, is a proof; but even in this case the path may lead to a series of discoveries. Whether it does, depends on the attitude towards it (Grothendieck's analogy between building a proof and opening a nut).

Before you write down the axioms for integers, they do not exist, you invent them. That there are infinitely many primes is a consequence of the axioms, you just have to notice those special numbers and give them a name. Here I find it actually quite tempting to say you invent the primes, or maybe better the idea of primes, but I still think it is more correct to say you discover them, given the axioms they were always there even if you have not yet noticed, named and described them. The same for the proof that there are infinitely many of them, given the axioms and the laws of logic, the truth of that statement and the possible proofs for it are fixed, you just have to find one.

Maybe back to the integers. Yes, you may play around with different ideas, tweak definitions and so on. But what you are actually doing is inventing a whole bunch of similar structures and then you pick the one that you like the most, that works the best or whatever. Some of your attempts might be inconsistent, some might not do what you want, but you all invented them. And among them you might discover one that works just like wanted it to work.

You say that no sane person will deny the existence of three, I am actually willing to go even further, I am willing to deny the natual existence of your apples and horses. The idea of cutting the universe into pieces is arguably an invention. It is of course an extremely useful idea for comprehending the universe, but if you think about it, you are somewhat arbitrarily drawing borders around collections of atoms.

There is one forest. And a hundred trees. And millions of cells. Is the water in a tree part of the tree? What about the water inside of the cells? When does it become part of the tree? What about a water molecule just evaporating from a leave, when does it become part of the atmosphere?

But even if there are things that can be counted, or if quarks and gluons behave in a way that can be described using SU(3), does that really imply the existence of the mathematical structure used to describe those things? Where do you get multiplication from? Or tetration? Are they invented while the basic structure of the integers is discovered? What exactly does existence even imply or require in this case?

And another thing just came to my mind. In which way do the apples and horses tell us anything about three? I can see that you can probably demonstrate that you can pair your apples and your horses, but that is quite far from telling us anything about three as you can do the same with any number of objects.

You also say the common thing is three, but that does not really help to pick out what three is, therevare many common things. Also note that you introduced the idea of a set, where does that come from?

Three is not something that applies to one of the apples or horses, it applies to the sets of them. Does the set of three horses exist or is that not something you made up, this, this and that, this are my three horses?

And now that we are talking about sets, because that is where three comes into play, why did we talk about apples and horses to begin with, it totally doesn't matter what you put into your sets, just how many things. So did we just lose the relationship to the real world or at least conclude that it never mattered?

I would at least say that it is certainly not as easy as you want it to be, there is much more nuance to this than three apples, three horses, therefore natural numbers. It actually sound circular. Let me pick three apples and three horse, now look, here are two sets of three things, so three exists.

Maybe it would be a nice challenge to tell me how to find three in nature. You obviously can not say take three horses. It seems tempting to say start without a horse, then add one, then another and yet another. But did you then not just invent that, did you not just come up with the axioms of the natural numbers, just expressed with horses instead of symbols?

A blind person would definitely disagree, having “invented” things he keeps bumping into, one after another… See, knowledge is never invented, it’s based on discovery; and mathematics is a form of knowledge.

> anything about three

Well, the discussion wasn’t about any particular number or concept, but if you want to define 3, you should look no further than the special property of a three-legged stool, the smallest polygon, or, if you look close enough, the proton. The general notion of the number, then, may come from a comparison (trying to see if there is a one-to-one correspondence) between the legs of a stool that is safe to sit on and one you risk breaking your neck if you try. If that’s full of “nuance,” I don’t know what isn’t.

I was not talking about inventing the things themselves but logically dividing them up. For the blind person it makes not difference whether he bumps into a tree because you decided to divide the forest into several individual trees or whether he bumps into the entire forest. But if you want to count things, then it of course makes a huge difference whether there is just the forest or whether there is a collection of trees. This subdivision of the universe - or the forest - into several individual object - or trees - is arguably invented, not the universe or the forest itself.

Knowledge is a thing you have, your awareness of some fact. Mathematics is not that, it is some form of fact you can be aware of. You can have knowledge about mathematics but it is not knowledge itself. If I name my dog Beethoven, that establishes the fact that the name of my dog is Beethoven, that is a kind of invention. If I tell you about this, you gain knowledge about the name of my dog. No discovery involved, neither when establishing the fact nor when you learn about it.

[...] look no further than the special property of a three-legged stool, the smallest polygon [...]

Okay, I found a stool that does not wobble on uneven ground and a triangular rock. Nothing about them on their own is related to three. You can of course explain to me what a leg is or the corner of a triangle and then ask me to form the sets of legs and corners and point out to me that the sets have the same number of elements, but there is lot of stuff going on here. You made me find things related to three, and I never doubted that there are such things, the number of dimensions of space would be another good candidate. But you mostly gloss over the hard part of actually extracting the natural numbers in general or three in particular.

It is of course trivial in everyday language as we learn about pairing and counting things relatively early in our life and humanity has made use of those ideas for a long time. Look, the number of legs equals the number of corners. And there are as many of them as I have horses and apples. But notice that those sentences are full of ideas and words related to numbers - number of legs, equals, as many.

But just as with the horses and apples, nothing about the non-wobbly stool or the smallest polygon - actually you mean the polygon with the fewest number of sides and note that there the idea of numbers already sneaked in again - is intrinsically related to three. It is the set of legs and the set of sides that are related to three, it is the carnality of the sets that behaves like the naturals. The process of forming those sets does a lot of heavy lifting to get you towards finding numbers in nature. And I do not think you can just brush that under the rug, you will have to justify that forming sets and looking at their cardinality is not something that humans invented.

Exactly. It makes no difference for him that you want to see a proof of individual trees' objective existence - because he already knows this for a fact! That's what that pesky objective reality does, sometimes forcing knowledge about itself upon us, whether we like it or not, or whether we would prefer some other "proof." The proof is in the pudding, as they say.

> Mathematics is not that.

Sure it is. One who knows about numbers knows more about the objective reality than those who don't. One who knows about Lie groups knows even more.

> you gain knowledge about the name of my dog. No discovery involved

That's not quite true: I discover that you have a dog (as long as you did not "invent" it).

> Nothing about them on their own is related to three.

It does, if you look at it from the right angle. There's a different thing at play here. While the number of legs could be easily matched with the corresponding number of apples, by itself this correspondence does not necessarily make any particular number stand out (although in some cases, like, say, in the case of a non-wobbly stool, a triangle, or the number of eyes and hands, it would - simply because there are many pairs of eyes, etc.); what's also at play here is different ways to look at numbers, which includes seeing them not only as "cardinals" (which is what you are still limiting yourself to) but also as "ordinals": the number 1 "stands out" as the smallest ordinal (greater than "nothing"), the number 2 is what follows it, etc. It is all these aspects combined that form the true content of the notion of the number.

You are missing my point here. The blind guys knows he bumped into something, so something exists. He could just say he bumped into a part of the universe, he is not forced to say he bumped into a forest or a tree. He could even consider himself part of the universe and say one part of the universe bumped into another part of the universe, just as one part of you bumps into another part of you when you clap your hands. The consequence of that is that there are no distinct objects to count, it is just one really complex object, the universe, interacting with itself.

Sure it is. One who knows about numbers knows more about the objective reality than those who don't. One who knows about Lie groups knows even more.

No, that is a kind of map territory thing. You can have knowledge of mathematics but mathematics is not knowledge. You can have knowledge of my dos's name but my dog's name is not knowledge.

It does, if you look at it from the right angle.

Switching from the cardinals to the ordinals will not really make a difference, you are glossing over a lot of heavy lifting. I have to repeat myself, the non-wobbly stool is not related to three, it is the set of its legs that is related to three. On to get there, you have to single out parts of the stool and combine the parts into a set and the take about the cardinality of it. There are a lot of steps and concepts on that way which makes it at least very non-obvious how the number three was always there and is not just the result of that process.

I am open to an example how I would find the ordinals in nature, I am not sure it will be any easier than with the cardinals. Non of the trees in the forest is the first one, you will have to impose an order an them. Maybe something with time, sunrises or days, they are at least already ordered.

Not at all! Talk about losing the trees for the forest... Even the blind guy knows, viscerally, that what he is hugging is not the "entire universe," or that stepping off a cliff would result in a dramatic experience; that a sighted person (and a philosopher) can see "a bigger picture" does not change the facts; indeed, a focus on "a bigger picture" can be deceiving (think of quantum vs. classical mechanics).

> my dog's name is not knowledge

We are going in circles. Mathematics is not just "names."

> the non-wobbly stool is not related to three

It is. You have a class of non-wobbly stools with a matching number of legs. You have a class of girls with a matching number of eyes that boys like more than others. Examples abound.

> how I would find the ordinals in nature

The early bird may get the worm, but it's the second mouse who gets the cheese.

See also https://byjus.com/maths/ordinal-numbers/

Also the non-wobbly stool is just that, one non-wobbly stool. [1] It again takes your mind to divide it up into components, including the legs, collect the legs back into a set and count them. Take a monocrystalline non-wobbly stool, it is just a weirdly shaped single crystal, nothing indicates that one should consider three specific areas of it as legs. Look, I am not arguing that it makes no sense for humans to think of this thing as a stool with three legs, but if you want to argue that numbers exist in nature independent of humans, then you have to keep human concepts out of the argument. Because we see the stool as a distinct part of the universe with three subcomponents that we can group together in a set to count them, does not mean that there was a stool or legs or a set of legs or a set of cardinality three before a human mind had a look at the situation.

You have to argue that objects are not a human invention, that cutting a forest up into trees instead of considering it one whole thing makes sense in absence of humans. If you have that, you have to argue that collecting objects into sets makes sense in the absence of humans. And then finally you can argue that the cardinality of those sets is not an human invention but was discovered.

[1] Or maybe not even that, maybe just a specifically shaped part of the universe.

I think this would be a wrong way to look at things; in the end, though, all this comes down to people taking different philosophical positions which cannot be reconciled in principle, like science and religion, for example.

It is all physically there in the universe, but not logically split into pieces and labeled with words, but this is what you need to count things. The entire universe physically existing is not enough, you also have to logically break it into countable pieces. This is really the core of my objection against three horses and apples, the atoms [1] are there even without humans, but the atoms [1] being there is not enough to logically have apples and horses to be counted.

[1] I really do not want to use the idea of atoms here but I can not think of a good way to this without making this even more confusing.