Is Euclid's Fourth Postulate Redundant?

[lugita15]

Euclid's Elements start with five Postulates, including the fifth one, the famous Parallel Postulate. Less well known, however, is the Postulate that forms the basis for the fifth: the fourth one, which states that "all right angles are equal." Students who see this for the first time might find this puzzling, because obviously two angles which are equal to a 90 degree angle are equal to each other, since Common Notion 1 says that "things which are equal to the same thing are are also equal to one another". But then they realize that the matter is so straightforward: the definition of a right angle is an angle produced when two lines intersect each other and produce equal adjacent angles, and it's not clear why an angle produced by one such pair of lines should bear any relation to an angle produced by another such pair of lines.

So Euclid's fourth Postulate is not redundant for the reason that beginning students might think. But my question is, is it nevertheless a redundant postulate, although for far less trivial reasons? David Hilbert, in his Foundations of Geometry (Grundlagen der Geometrie in German), claims to prove Euclid's fourth Postulate in theorem 15 (on page 19 of the PDF or page 13 according to the book's internal page numbering), prefacing the proof by saying "it is possible to deduce the following simple theorem, which Euclid held - although it seems to me wrongly - to be an axiom."

Now it's fair to say that Hilbert was working in a different (and more rigorous) system of axioms than Euclid was, but I think Hilbert's proof should be seriously considered for two reasons. First of all, why would he dub Euclid's decision to call "all right angles are equal" a Postulate as "wrong" if it merely reflected a stylistic difference concerning what you choose as starting assumptions and what you consider theorems? But more importantly, by tracing back all the assumptions used in the proof of theorem 15, it seems to me that only four of Hilbert's axioms are ultimately used: IV 3, IV 4, IV 5, and IV 6. And I don't think Euclid would have objected to any of these statements:

1. IV 3 follows directly from Euclid's Common Notion 2.

2. IV 4 is partly stated in Euclid's Book I Proposition 23, which doesn't depend on the fourth postulate, and the part of IV 4 which (I think) is not stated is easily provable in Euclid's system.

3. IV 5 follows from Euclid's Common Notion 1.

4. IV 6 is just part of Euclid's Book I Proposition 4, which doesn't depend on the fourth postulate at all.

So could Euclid have proven his fourth Postulate as a theorem instead of just assuming it?

Any help would be greatly appreciated.

Thank You in Advance.

 

[Simon Bridge]

I think you need to show that the fourth postulate follows from the other four alone to consider that it is redundant to Euclid.

 

[lugita15]

I think you need to show that the fourth postulate follows from the other four alone to consider that it is redundant to Euclid.

Well, I showed that all of Hilbert's assumptions in the proof were either assumed or proven by Euclid independent of the fourth postulate, so what more is needed?

 

[Simon Bridge]

I'm not so sure that you did - you certainly asserted that this was the case.
You should be able to work it backwards so that you can start from Euclid to get his fourth without assuming it.
It should be straight forward enough to show step-by-step, and provides a way to confirm what you've done.
Which is what you need - or did I misunderstand your question?

 

[CellCoree]

I would say that this is a fascinating question that has been debated by mathematicians for centuries. Euclid's Elements is a foundational work in the history of mathematics, and the question of whether or not his fourth postulate is redundant is an important one.

On one hand, Euclid's postulates were meant to be self-evident truths that did not need to be proven. The first three postulates are relatively straightforward and can be easily understood and accepted. However, the fourth postulate, stating that all right angles are equal, does seem to require some explanation and justification.

On the other hand, as you have pointed out, Hilbert was able to prove this postulate using only four of his own axioms. This raises the question of whether Euclid could have done the same, and if so, why did he choose to include it as a postulate rather than a theorem?

One possible explanation is that Euclid's system of geometry was not as rigorous as Hilbert's. In Euclid's time, the concept of proof and axiomatic systems were still developing, and it is possible that he did not have the same level of rigor that Hilbert had.

Another explanation could be that Euclid's postulates were not meant to be absolute truths, but rather starting assumptions that were necessary for his system to work. In this case, the fourth postulate may have been included as a way to ensure the consistency and coherence of the rest of the postulates and theorems.

In conclusion, whether or not Euclid's fourth postulate is redundant is still a matter of debate among mathematicians. While Hilbert was able to prove it using only four of his own axioms, it is possible that Euclid's intentions for including it as a postulate were different. Further research and analysis may shed more light on this question, but for now, it remains an intriguing and open topic for discussion.