Informally the role of teacher may be taken on by anyone (e.g. when showing a colleague how to perform a specific task). In some countries, teaching young people of school age may be carried out in an informal setting, such as within the family, (homeschooling) rather than in a formal setting such as a school or college. Some other professions may involve a significant amount of teaching (e.g. youth worker, pastor).
In most countries, formal teaching is usually carried out by paid professional teachers. This article focuses on those who are employed, as their main role, to teach others in a formal education context, such as at a school or other place of initial formal education or training.
Mathematicians can exemplify cumulative mental experience in the approaches and skills with which they work. Mathematical realism, like realism in general, holds that mathematical entities exist independently of the human mind. Thus humans do not invent mathematics, but rather discover and experience it, and any other intelligent beings in the universe would presumably do the same. This point of view regards only one sort of mathematics as discoverable; it sees triangles, right angles, and curves, for example, as real entities, not just the creations of the human mind. Some working mathematicians have espoused mathematical realism as they see themselves experiencing naturally occurring objects. Examples include Paul Erdős and Kurt Gödel. Gödel believed in an objective mathematical reality that could be perceived in a manner analogous to sense perception. Certain principles (for example: for any two objects, there is a collection of objects consisting of precisely those two objects) could be directly seen to be true, but some conjectures, like the continuum hypothesis, might prove undecidable just on the basis of such principles. Gödel suggested that quasi-empirical methodology such as experience could provide sufficient evidence to be able to reasonably assume such a conjecture. With experience, there are distinctions depending on what sort of existence one takes mathematical entities to have, and how we know about them
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