I read somewhere that the term for a concept that we don't try to

[Fredrik]

I read somewhere that the term for a concept that we don't try to define is "primitive". Primitives are supposed to be to definitions what axioms are to theorems. I also read that when one of these deliberately undefined concepts is explained (but not really defined) by a description in plain English, such an explanation is called an "elucidation".

My problem is that I tried to find out if this terminology is standard, and I was only able to find a few sources who used the word "primitive" in that sense, and none that used the word "elucidation". (I don't remember where I learned those terms). Now I'm wondering if these are not the standard terms, and if there are other terms that are used more often by mathematicians and philosophers.

 

[SW VandeCarr]

Primitives are undefined object names in an axiomatic system. From such a system, models can be built when the primitives are given an interpretation.

http://www.math.uiuc.edu/~gfrancis/M302/handouts/postulates.pdf

 

[Fredrik]

Thanks. That's a nice article. This guy seems to know what he's talking about and he's at least using one of the two terms I mentioned in the way I expected.

It seems like a lot of other authors choose to avoid the term "primitive". For example, I found a searchable pdf version of "A concise introduction to mathematical logic" by Wolfgang Rautenberg, which seems like a very good book. It doesn't use the term primitive at all.

Now I'm also confused about how differently Francis and Rautenberg are using the word "model". Francis say that a model is an interpretation that assigns meaning to the primitives, so that theorems can be considered true or false. The example he uses in the article is to take the "points" mentioned in Birkhoff's axioms for Euclidean geometry to be points in $$\mathbb R^2$$. But Rautenberg says that a model is a mathematical structure together with a function that assigns a value to all the symbols representing variable names. A "value" is just a member of the underlying set of the structure. A model in the sense of Rautenberg gives a truth value to arbitrary formulas involving variable symbols, like x=y.

Hm, these two notions of "model" are quite similar. Maybe they are the same, and I just don't get it.

Edit: I found my original source for the terms "primitive" and "elucidation". Page 7 of "Set theory and its philosophy" by Michael Potter. He says that the term "elucidation" was used by Gottlob Frege.

 

[SW VandeCarr]

Thanks. That's a nice article.

You're welcome.

Now I'm also confused about how differently Francis and Rautenberg are using the word "model". Francis say that a model is an interpretation that assigns meaning to the primitives, so that theorems can be considered true or false. The example he uses in the article is to take the "points" mentioned in Birkhoff's axioms for Euclidean geometry to be points in $$\mathbb R^2$$. But Rautenberg says that a model is a mathematical structure together with a function that assigns a value to all the symbols representing variable names. A "value" is just a member of the underlying set of the structure. A model in the sense of Rautenberg gives a truth value to arbitrary formulas involving variable symbols, like x=y.

I would say Rautenberg's (R) view is essentially the same. R says a function assigns a value. Given the broad modern concept of a function as a mapping, an interpretation (I would think) could be thought of as a function that assigns a meaning to a primitive. In R's example, that "meaning" remains abstract.

Perhaps others will post. I'm not a mathematician; just a consumer of mathematical and statistical products, como usted.