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As another comment said. What do you mean by exist? It requires a little more rigour here. Using your example: What does it mean for three horses and three appl

As another comment said. What do you mean by exist? It requires a little more rigour here. Using your example: What does it mean for three horses and three apples to exist? The better question: What does it mean for three (of anything) to exist? Why not one thing, one thing, one thing? The definition of existence you appear to be using is the physical proximity of those objects. But even that can get quite hairy. If there is two apples within inches of each other and another apple 100 feet away, are there three apples or two? The answer to this question depends on what you mean by the existence of three apples.

akomtu 4y ago
If an apple is cut in half, does the apple still exist? When the halves are brought together, at what distance between the two does the apple start existing?
- ffhhj 4y ago
  Tarski would like to cut that apple into thin slices: https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
- anthk 4y ago
  Is there an atomic bond? No. Then, there's no whole apple any more.
danielam 4y ago
> What does it mean for three horses and three apples to exist? You tell us. You seem perfectly comfortable speaking of a multiplicity of instances. (Personally, I would begin the discussion with substantial form. I can speak of many triangles that instantiate the same triangularity which allows me to assert the same properties of triangularity of them all. And yet triangularity is not to be identified with any particular triangle.)
- Koshkin 4y ago
  This is a valuable point. Our thinking is too much tied to the language. Of course when we say that the abstract triangle exists we mean that what exists is the property of triangularity. Properties (of things) do exist, whether we like it or not. Just like the number 3, say, also exists - in the form of “threeness” property (of a collection of objects). There is not much more to be said here, except maybe that most, if not all, properties manifest themselves in, or indeed are, relations between things, and also that they are not arbitrary products of the mind, being impressed into us by the objective reality - and how can something be impressed into us that is not itself part of reality, i.e. does not exist in the most obvious sense of the word.
  - TuringTest 4y ago
    > they are not arbitrary products of the mind, being impressed into us by the objective reality - and how can something be impressed into us that is not itself part of reality, i.e. does not exist in the most obvious sense of the word. They are not arbitrary products of the mind, for the reasons you explain; but they are products of the mind. The property of NP-completeness didn't exist until we invented algorithms and analysed their time complexity, though many problems do posses that property; and in the same way the property of triangularity didn't exist until Greeks started imagining geometry in terms of idealized regions of space, defined in terms of simple relations between points without size. (And then, the triangularity property changed a lot when people started questioning the parallel postulate and discovered non-euclidean geometry). It is not necessary that an entity exists for it to form impressions in the mind, it is enough that the mind can imagine it from actual existing perceptual elements, and our imagination fills in the details. Is it truly needed for a monster to exist under the bed for it to impress our juvenile minds?
    - Koshkin 4y ago
      > they are products of the mind Well, a fighter jet, too, is a product of the mind. > triangularity didn't exist until Tell this to triangular molecules. > a monster This says more about people than it does about the reality (though even fears and phantasms can indeed be rooted in reality).

akomtu 4y ago

If an apple is cut in half, does the apple still exist? When the halves are brought together, at what distance between the two does the apple start existing?

ffhhj 4y ago
Tarski would like to cut that apple into thin slices: https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
anthk 4y ago
Is there an atomic bond? No. Then, there's no whole apple any more.

danielam 4y ago

> What does it mean for three horses and three apples to exist? You tell us. You seem perfectly comfortable speaking of a multiplicity of instances. (Personally, I would begin the discussion with substantial form. I can speak of many triangles that instantiate the same triangularity which allows me to assert the same properties of triangularity of them all. And yet triangularity is not to be identified with any particular triangle.)

Koshkin 4y ago
This is a valuable point. Our thinking is too much tied to the language. Of course when we say that the abstract triangle exists we mean that what exists is the property of triangularity. Properties (of things) do exist, whether we like it or not. Just like the number 3, say, also exists - in the form of “threeness” property (of a collection of objects). There is not much more to be said here, except maybe that most, if not all, properties manifest themselves in, or indeed are, relations between things, and also that they are not arbitrary products of the mind, being impressed into us by the objective reality - and how can something be impressed into us that is not itself part of reality, i.e. does not exist in the most obvious sense of the word.
- TuringTest 4y ago
  > they are not arbitrary products of the mind, being impressed into us by the objective reality - and how can something be impressed into us that is not itself part of reality, i.e. does not exist in the most obvious sense of the word. They are not arbitrary products of the mind, for the reasons you explain; but they are products of the mind. The property of NP-completeness didn't exist until we invented algorithms and analysed their time complexity, though many problems do posses that property; and in the same way the property of triangularity didn't exist until Greeks started imagining geometry in terms of idealized regions of space, defined in terms of simple relations between points without size. (And then, the triangularity property changed a lot when people started questioning the parallel postulate and discovered non-euclidean geometry). It is not necessary that an entity exists for it to form impressions in the mind, it is enough that the mind can imagine it from actual existing perceptual elements, and our imagination fills in the details. Is it truly needed for a monster to exist under the bed for it to impress our juvenile minds?
  - Koshkin 4y ago
    > they are products of the mind Well, a fighter jet, too, is a product of the mind. > triangularity didn't exist until Tell this to triangular molecules. > a monster This says more about people than it does about the reality (though even fears and phantasms can indeed be rooted in reality).