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If we imagine the world and what surrounds us as concepts, if you then had to explain these ideas (i.e. What's a person? What's an animal? What's a living being? ...) you'd then need to recursively explain other concepts. What would be the simplest idea(s)?

Let's pick a person as an example. You could say a person is a thinking animal, you'd then need to explain what thinking and thought means, and what an animal is. From that you'd need to explain what their 'parent' ideas are etc.

What concepts can instead be explained without explaining other concepts?

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    Hi, welcome to Philosophy SE. The theory of constructing complex ideas from simple ones was developed by Locke, see simple ideas, but it is considered too naive in modern cognitive science and philosophy. According to modern studies, concepts are mastered in clusters, with interrelated "ideas" mastered simultaneously through their interaction in various contexts, and there are multiple ways of "building" some from others, which can then be declared "simple" for the purpose. Various formalizations of mathematics are a good illustration of that. – Conifold Oct 7 '19 at 3:06
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    Surely 1 is the 'simplest' number, especially if you're trying to explain the concept to other people. It took us many thousands of years of counting before we discovered 0 – JeffUK Oct 7 '19 at 10:54
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    You may wish to look up Natural Semantic Metalanguage (which I learnt of from here) – Rusi-packing-up Oct 8 '19 at 5:15
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    Wittgenstein, Tractatus, 2.0211 ff. on simple objects is relevant. – Geoffrey Thomas Oct 8 '19 at 16:47
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According to empiricism [1], everything we can think of is just a set of compound ideas. An apple cannot exist without a color, or a taste. The set of properties that our sensibility grants to a thing form that thing. So, all physical things have some degree of complexity.

In addition --as you mention in your question--, concepts we built upon other concepts finish following circular references. The philosopher Immanuel Kant suggested that reason is sustained upon a set of tautologies, and that's quite logic.

Therefore, perhaps the simple concept that one can think of is a thing, which is something that, essentially, we cannot explain. It is simple in the sense it is the building block of our reason (I think your question goes in such sense), and not in the sense that it is an object[2] that is easy to explain, without circular references. Things are mental features, built over sensory and reason-related information.

On one hand, physically, things are very difficult to define. If you try to define a home, you will have a lot of problems finding its boundaries, which finally end up being subjective: they depend on each person (e.g. does a home include a garden? its inhabitants? the water in its pipelines?). The same happens with an atom or a rock, the problem is identical, at different scales. Moreover, things depend on our ideas of time and space, which, according to Kant, are also subjective.

On the other hand, things are more difficult to define as logical entities. We form ideas in our brain (we don't know if they correspond to neurons, electrical or chemical contents!). We grant them of properties and we use them to interact with the physical world. But what they are is a long philosophical debate [3].

[1] https://plato.stanford.edu/entries/rationalism-empiricism/

[2] This contrasted concepts, common in philosophy, are useful to get: object: the target of reason, what a subject thinks of, instead of a physical or ideal entity; subject: the individual that uses reason over an object; thing: a physical manifestation, which can be an object to reason.

[3] https://plato.stanford.edu/entries/object/

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