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Topic: [HM] arithmoi poetoi
Replies: 1   Last Post: Sep 20, 2005 3:40 AM

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Jose Ferreiros

Posts: 57
Registered: 12/3/04
Re: [HM] arithmoi poetoi
Posted: Sep 20, 2005 3:40 AM
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Dear friends:

Roger Cooke and Panagiotis Stefanides have kindly helped with the
translation. For instance, Roger writes:

The phrase "poietos arithmos" means an invented or artificial number
(literally, "made") , "eidetikos" means shaped or formed ("eidos" means
shape, form, genus, and so forth, and is related to the simple past form
"eidon", which means "I saw"). Usually "-ikos" translates into English as
"-ics" as in "mathematikos" (adept at learning), "praktikos" (skilled in, or
ready for, action).

On this basis, my proposed translation of "arithmoi poetoi or eidetikoi" is:
"numeros creados o ideados", in English "created or ideal numbers". In the
1880s, Cantor adopted at several places the view that numbers are
"creations" of the human mind (a view characteristic of Dedekind). He also
emphasized that both the real numbers and the transfinite ordinals are
"ideal numbers" in the more or less the sense of Kummer. At some point he
even states that the modern definitions of the real numbers make explicit
their "ideal" nature. Now, in the paper I was quoting ("On the different
standpoints regarding actual infinity") he goes on to say that his novel
order types are nothing but an extension of the same train of thought. But
by phrasing this in Greek, he is also calling attention to the connection to
Plato's theory of ideas, "eidos".

Best wishes from Seville,


You know Cantor had a
tendency to "explain" his ideas with references to words in Latin or Greek.
E.g., in 1883 he explains his concept of set or "manifold" by saying that it
is similar to Plato's "eidos or idea" (Abhandlungen, 204). In a later
publication he explains that actual infinity is conceived or grasped by the
human mind under the form of his transfinite numbers, "or even under the
more general form of the transfinite order types (arithmoi poetoi or
eidetikoi)" (Abhandlungen, 372).

Jose Ferreiros
Departamento de Filosofia y Logica, Universidad de Sevilla
Camilo Jose Cela, s/n. E--41018 Sevilla, España
Tel: +34.954.557825
Fax: +34.954.551668

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