Loading...

## Differences in geometric assumptions

Risenhoover, James Alton

Risenhoover, James Alton

##### Abstract

Scope of Study: Several books on Foundations of Geometry and Non-Euclidean Geometry were examined. Those that gave a complete set of assumptions were given special attention. The sets appeared to be different . They were examined to see why they were different and if one could be said to be better than another. A composite set of assumptions with a discussion of each one was presented.

Findings and Conclusions: Although the assumptions appear different, and some cannot be checked directly against others, they must be assumed to be equivalent because they lead to the same theorems. No basis was found for saying that any particular set was better than the others. The different sets seems to be the results each author's attempt to improve some existing set. That is each author had corrected an error or changed something that he did not like in an existing set to get his set. Some authors stated that their set was a combination of two or more sets. The biggest difference is in the assumption that assign a metric property to geometry. That is most of the sets contain the same or equivalent assumptions except for the metric assumptions. The whole argument seems to be the result of a search for elegance instead of simplicity. Until a better set is written students must continue to use and study the existing sets. The set of assumptions presented in this paper would be an aid to studying the nature of assumptions.

Findings and Conclusions: Although the assumptions appear different, and some cannot be checked directly against others, they must be assumed to be equivalent because they lead to the same theorems. No basis was found for saying that any particular set was better than the others. The different sets seems to be the results each author's attempt to improve some existing set. That is each author had corrected an error or changed something that he did not like in an existing set to get his set. Some authors stated that their set was a combination of two or more sets. The biggest difference is in the assumption that assign a metric property to geometry. That is most of the sets contain the same or equivalent assumptions except for the metric assumptions. The whole argument seems to be the result of a search for elegance instead of simplicity. Until a better set is written students must continue to use and study the existing sets. The set of assumptions presented in this paper would be an aid to studying the nature of assumptions.

##### Date

1959-05