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In mathematics, there are many things that can be rigorously proven to exist, but at the same time rigorously proven that no concrete examples of it can be given. This is a crazy idea for non-mathematicians.

Have any philosophers studied things that certainly exist but can never be seen? Descartes, based the existence of many things on our mind. (I think, therefore I am.) However, in the above situation in mathematics, the mind can tell us that (1) something must exist, or else there would be a contradiction, and (2) that thing does not exist in my mind since neither examples of it nor method to produce it are existent, which are two contradicting messages.

Examples of things that exist but do not exist:

  1. Non-Lebesque measurable sets: the existence of this is established by the axiom of choice, so no concrete examples can be given.
  2. Brouwer fixed point theorem: this is proved by contradiction, giving no methods of finding such a fixed point.

Just tell me some theories on this topic. (I am not professional in philosophy.)

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    These things are called non-constructive existence proofs, usually proofs by contradiction or those that use the axiom of choice. Some of such objects have counterintuitive "pathological" properties. Constructivists reject the existence of such things, and traditionalists often simply find it technically convenient to act like they exist, after all mathematical "existence" does not really amount to much. – Conifold Aug 12 at 8:47
  • @Conifold +1 for substantial comment but I do disagree with the last statement. – Bertrand Wittgenstein's Ghost Aug 12 at 21:15
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    @Conifold “does not really amount to much” What do you mean by that??? – Ma Joad Aug 12 at 22:41
  • Except to few platonists, it means that we can talk about X without running into known contradictions, not that X exists in any real sense. Similar to fictional characters, only obeying much stronger logical restrictions, and having practical application to things that do exist in reality, see What is existence and how far does it extend? – Conifold Aug 13 at 0:42
  • @MaJoad, since I see you're more active over on math.SE, could you give some examples of mathematical concepts which have no "concrete" examples? My first thought is a Klein bottle, but it's sort-of a narrow idea of non-existence in 3-space...maybe you have other things in mind? – John Hughes Aug 13 at 16:21

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