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.
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.
> it makes not difference whether he bumps into a tree because you decided
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.
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!
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.
The blind guy is of curse not hugging the entire universe but a part of it. Where there trees before humans existed? No. It took our minds to look at the universe and say »Hey, it would be really helpful for talking about the universe if we would consider those wooden pieces of the universe that are roughly cylindrical and have the green stuff attached on the top individual things and give them a name, maybe something like tree.« We could have decided otherwise, never invented the concept of a tree and always just talk about forests. The subdivision of the universe into objects, at least on the macroscopic level, is not intrinsic to universe, it is an invention of the human mind.
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.
You are essentially saying that the huggable "part of it" did not exist until the blind person has bumped into it.
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.
Depends on what you mean with exist here. It exists in the sense that it is part of the universe, the atoms are there [1] and they can interact among each other and with the rest of the universe including the blind person part of the universe. What I think does not exist independent of a human is the idea of a tree or maybe even the idea of an object in general, grouping together some atoms [1] and labeling them as a tree, blind person or whatever.
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.
You are saying two things: what we call objects (perhaps, except for "atoms") did not exist - in a meaningful sense of the word - until humans appeared and found it convenient to perceive them; and the nature admits no actual boundaries between objects (which is why these objects do not, in fact, exist), rather itself being kind of a soup of atoms. There are terms for these philosophical delusions, one being (a mild form of) idealism, the other, reductionism. They are indeed delusions, because neither is useful or can be proven in any logical or scientific way. Why, for example, not take you reductionism a step further and say that atoms, elementary particles, etc. do not exist, either? You are left, then, with a rather featureless, uniform "matter" (you could call a "matter field" or something). Your philosophy ends there, as there is nothing else left to talk about; my point, on the other hand, has been that the objective reality is much more complex than that; that, in particular, there are emergent phenomena - which are no less real than what they came from; even the very convenience and practical usefulness of perceiving things the way humans do is, in fact, dictated to us by something that is outside our control. I call that something "objective reality."
I completely agree, talking about the macroscopic [1] universe as one thing is at least complicated and unwieldy and maybe even severely limiting. But that does not actually matter, it must be outright wrong to do so. If I can talk about the universe without splitting it up and counting things, than that undermines your position that the numbers are inherent in nature. And I agree with you that it is much easier to think about the world split up into many countable objects, but that does not make the idea of numbers inherent in nature. Maybe they are inherent in the human brain, maybe they are a complete invention, I do not really care for the sake of this discussion. Or maybe that is what you actually meant all the time, inherent in human perception and thinking, whereas I really mean inherent in the universe, predating humans.
With regard to the delusions I can only say think again. A non-wobbly stool for a human to sit on or a five-finger glove to keep a human hand warm, those things can not possibly have existed before humans. There might in principle have been a collection of atoms in the shape of such a stool or glove before humans, but they would not have been a stool or a glove. Objects as we humans use them are more than the sum of their atoms, they also capture things like intentions, usages and even their history. A tree trunk can just be a tree trunk or it can be a bench to sit on. An atom by atom copy of the Eiffel Tower is not the Eiffel Tower build by Gustave Eiffel. The human notion of an object is in general very rich, often fuzzy and inseparable [2] from the human mind.
I have no real objection to the claim that they are for all practical purposes part of nature to humans, those concepts could be hard wired into the human brain and that seems even very likely to me. So if we are limiting the discussion to the universe as perceived by humans, maybe there are numbers or at least precursors to them in our minds, encoded in the connections of our neurons controlled by our DNA. But if we step back further, if we try to look at the universe independent of the way it is perceived by us, there I do not think you said anything so far that convinces me that numbers are already exist there.
[1] As before I want to avoid also having to deal with elementary particles and given the examples we have discussed so far sticking to the macroscopic world does not seem to be a limitation.
[2] Inseparable is not the proper word here but for brevity I will gloss over that until it becomes relevant.
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.