h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

You're right to doubt me, Toy Box, as I'm talking through my hat. My brain really isn't functioning up to scratch these days as I am still recovering from sickness. My genuine apologies. The correct terminology is:

Postulates - things which are assumed because they are either self-evident or impossible to prove (also often known as axioms)

Common Notions - other things which are self evident

(why Euclid draws a distinction is not clear)

Propositions - statement which are then proved, or constructions

I'll amend accordingly.

I haven't had time to look through Icy's list (thanks, Icy), as I'm in the middle of my father-in-law's funeral. I might have time on Tuesday.

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From what I remember of old geometry courses, common notions more or less describe the objects you are working with (points, straight lines,...) and 'axioms' how they behave (given a straight line and a point out of it there exists a unique..., etc.)

It seems like a good addition to the guide.
One sentence in the summary sounded like it was phrased poorly:
<
...anyone with a bit of work can appreciate some of,although most people won't have the stamina to trawl all the way through it.>
I probably would have said
Anyone willing to devote the time and effort can appreciate some of it; it is unlikely ,however, that many would have the stamina to trawl all the way through this work.

>>Nobody has ever shown Euclid to be substantially wrong in his description of geometry, but about two centuries ago it was shown that two of his postulates are in fact far from obvious. Euclid assumed them to be true because he couldn't prove them. Assuming them to be false, however, leads to an alternate equally valid description of the world, known as non-Euclidean geometry, which makes strange non-intuitive predictions for large distances.<<

What does it *mean*?!
- He isn't wrong but sometimes he's far from obvious? How does one follow the other? Why can't something be un-obvious and not-wrong?

- Assumes them true because he couldn't prove them? Hold on a sec... I learned the opposite - something isn't true unless you *can* prove it. Until then it's a theory.

- Assuming them false leads to an equally valid description of the world? How can there be two types of geometry? Doesn't geometry deal with things you can touch and feel - things with dimensions and area? How many ways can there be to look at a pi?

- strange non-intuitive predictions: well are they accurate? How long are the distances anyway? Why are they strange? If this is valid, why don't we learn it in school?


Er, sorry. I got a bit hung up on that paragraph. I suspect you have a lot of background knowledge about this stuff that I don't, so you don't realize that us mathmetical slackers won't understand a word.

"[...] about two centuries ago it was shown that two of his postulates are in fact far from obvious."

Indeed this should probably be rephrased (and many things are both non-obvious and non-wrong). These were properties which looked obvious and which should (as in: moral duty ) be provable by using only the remaining axioms. Euclid didn't manage to do so however, and so after a while he decided to write them as axioms:

"Euclid assumed them to be true because he couldn't prove them."

For centuries afterwards, numerous people tried (as Euclid had) to prove them using only the remaining axioms, and they all failed. Until one day someone came up and said: "Oh, right then, let's assume that these axioms do not hold and let's see what geometry would looks like". This procedure turned out to lead to:

"an alternate equally valid description of the world, known as non-Euclidean geometry, which makes strange non-intuitive predictions for large distances."

These are fascinating worlds and you can make them actually very concrete, but just now it is not the point.

"something isn't true unless you *can* prove it. Until then it's a theory."

A hypothesis, to be precise (or a conjecture if you like). Actually, at the beginning of a Theory you have to declare some properties to be True without having to prove them - these are the Axioms. That's a little bit like fixing the rules of the game, describing what you are allowed to do and what you are not. Because in order to prove the first theorem ever, you cannot rely on a previous theorem but you cannot either rely on an absence of true property - so you need to impose it to be true.

"How can there be two types of geometry? Doesn't geometry deal with things you can touch and feel - things with dimensions and area?"

That's easy once you know the trick. Imagine that you are "doing geometry" on a big sphere or a globe. So "straight lines" are the lines you obtain if you go along one direction and never stop. On a sphere, if you do this you end up tracing big circles like those which join the two poles. In particular, "any two distinct straight lines meet at exactly 2 points": N and S poles. This is typically false in Euclidean geometry.

Also, imagine a big triangle drawn on a globe, with 2 vertices somewhere on the equator and one on the North pole. Then you can see that it has 2 right angles at the Equator vertices, and one non-zero angle at the North pole vertex. Thus, the sum of angles is more than 180 degrees - again, something which is not true in Euclidean geometry.

"If this is valid, why don't we learn it in school?"

Because you cannot learn everything. Better to focus on one easier (and more useful) theory than confusing people with the general theory.

...and, just because I like shameless self-promotion: A1003294

Aaaand, triple post .

Entry: Euclid - A13456082
Author: Gnomon - [ 3 stars - off to Budapest and Greece next week ] - U151503

Is there an edited version of that, too?

Of course there is: A1134091. Sorry about that.

I don't want to get too hung up on non-Euclidean geometry since it is all covered in A3251693 (Curved Space and the Fate of the Universe).

The point, Howl Leo, about assuming things to be true because you can't prove them is that since everything must be proved based on simpler and more basic things, you must eventually reach things that are so simple and basic that they can't be proved. So you have to assume them to be true, because you believe them to be true.

For example, we assume that given any two points, we can draw a straight line between them, and that we can extend this line at either end as far as we like. However, we can't prove it. The first part, about drawing the line between them, seems reasonable, so we are happy to assume it. The other part, about extending the line indefinitely is not quite so reasonable. Who knows what happens at the far side of the universe?

I get it now. Thanks. But um, how about in the entry?

I'll try and reword it to make it a little clearer. But not now. Later.

The best time to do anything.

I can see that this Entry needs some more work to get it just right. Thanks, everybody, for your very helpful suggestions. I'm going away on holidays tomorrow, so I don't have time now to make the necessary changes.

I would like to put this on hold, and ask scouts not to pick it. I'll sort it out in about two weeks' time when I return.

Note to self:

Need to bring out the fact that Elements isn't all about geometry. There's quite a lot of number theory too, including the celebrated "Proof of the Infinitude of the Primes".

I've reworked this somewhat. The biggest change is that I added the section on Number Theory, but there are some other small changes.

I'd be delighted if someone would look it over again.

OK then, let's have a look.

"one of the definitive works of geometry" - since number theory is also mentioned, maybe this statements becomes slightly unaccurate.

Oh, and shouldn't you mention Euclidean division too?

Oops, I have to for now, sorry about that. Will try to come back to it later

And one last question (beyond the scope of the entry but I just remembered, maybe you know): was it known to the Greeks that for a circle of radius r, the perimeter is 2*pi*r and the surface is pi*r^2 ? if so, how did they prove it was the same pi involved?

It looks good to me, Gnomon.

Thanks, Icy.

Toy Box,

I've changed "one of the definitive works of geometry" to "one of the definitive works of mathematics".

I've never heard of Euclidean division, so I don't know whether I should mention it or not.

The ancient Greeks certainly new about Pi, although they didn't call it by that name. One of them produced an approximate value for it by constructing polygons inside and outside a circle, and calculating the perimeter length of these. The circumference length had to be between these two lengths.

It's a pretty simple matter to show that area of a circle is the product of half the circumference times the radius. This means that the pi must be the same one. But I don't know whether the Greeks did this.

Oh, that may be the French terminology then. I'll write to the Science Academy to suggest we call it Bourbaki division then

Actually it is the division with rest (given n and m two natural numbers, there exist integers q and r such that n = qm + r and r < m).

Surface is radius * half-circumference: you mean it is fairly easy to prove with elementary methods (i.e. no integrals)? Oh, I'll think of that. Thanks