As I wrote earlier, I’m reading “The road to reality” (R2R) by Roger Penrose (RP). I want to share some thoughts about what I have already read:
1. The greatness of Euclid (and also Gauss, to some extent)
2. Uses of negative numbers in physics
1. R2R made me aware of how great Euclid’s genius was. Though I was lucky enough to hear about Euclid’s axioms as early as in a highschool (RP seems to suggest that this is common in UK; it is not in Poland), I didn’t think about them as something really worth of attention. RP tries to show them as Euclid’s foreseeings of very modern ideas:
- 2nd axiom: every straight line segment can be extended infinitely in both directions. In highschool I thought of it as a merely technical one, allowing for certain geometrical constructions. RP suggests to look at it as “completeness axiom”, as well as infinity (possibly non-compactness ?) axiom – it is amazing if Euclid could think in similar terms.
- 3rd axiom: existence of circles with arbitrary centers and radii. As above. Also, it can be perceived, together with 4th axiom, as axiom of homogeinity of space.
- 4th axiom: “equality of all right angles”, which is, according to RP, axiom of isotropy (of geometry being the same in every direction). Wow.
As to the famous 5th axiom – given a point and a line there exist exactly one parallel to this line going through given point – one hears about it from time to time and, unlike with axioms above, I occasionaly thought about it while in university. Still, I never understood why Euclid (and later generations of mathematicians) made so many efforts to understand it. RP says (more or less) that it’s because Euclid (and next generations) wasn’t actually sure that euclidean geometry is the geometry of universe. That is, Euclid wanted to describe the real space and he could possibly understand that it is not clear whether 5th axiom actually holds in this real space!
(Maybe I went to far – maybe Euclid didn’t doubt the 5th axiom in physical universe, but still, he’s great in understanding that 5th axiom is connected to a fundamental properties of reality (though, as we know, it doesn’t hold in a universe). However, there is a serious issue about whether Gauss didn’t ponder over idea of 5th axiom being false in a universe, as he was among the discoverers of noneuclidean geometries and also interested in a geodesy – PR gives reference to J. Fauvel, J. Gray, The History of Mathematics: A Reader, which I hopefuly will check after getting back to Warsaw next week.)
When I learned geometry I never used axioms (accordingly, I had some difficulties with proving most basic theorems), thinking that they are just curiosity – indeed, it’s cool that all theorems follow from few basic facts, but this is not really where all the fun in geometry is. RP’s interpretation of Euclid’s axioms says that the point of view of Euclid was 10 times deeper than wanting to prove all theorems from few facts.
After finishing highschool, for few semesters I conducted math classes for pupils who wanted to participate in mathematical competitions. As I was very fond of classical geometry I always started with it. One mistake I think I made is that I played to much with axioms – I didn’t want these kids to have problems similar to those of myself in their age.
Now I come to believe that it’s not that bad if young students don’t derive everything from axioms (but instead they “see” the truth of particularly basic statements), because perhaps axioms are not about it – they are more about deep insights of Euclid (&co.), which can be appreciated only after aquainting with moderately modern mathematical ideas (that is, after at least 1 or 2 years of mathematical studies – in my case after 3.5).
2. Nobody before turned my attention to the fact that it is difficult to find physical application for negative integers, where physical means “really physical”. That is, negative temperatures, velocities/distances and bank accounts are all cheating (because, respectively, there is absolute 0, velocities/distances aren’t really negative – it can only be convenient to say so, and bank accounts aren’t very physical).
However, RP gives an example: electric charge. This is very cool, but for some reason I prefer the same example, but with different name: electric field (possibly because I’m used to think that charge is (number of electrons – number of protons) which leads to rather formal than actually physical use of negative numbers). Nevertheless, this is only change of words.
Also, RP points to the possibility of having -1 cow by building the cow out of antiparticles, but this isn’t very convincing (as RP himself admits).
That’s all for now. I find reading R2R very enjoyable and will continue to write “reports from lecture”. Also, this week is a last week of my extended holidays – after getting back to Warsaw I’ll probably start to write more specific stuff. I suspect that it can take form similar to John Baez’s This Week’s Finds in Mathematical Physics (or actually, I would be happy if if took such a form). Of course, I understand that John Baez’s TWF is a Platonic Ideal I can only try to aproach :-)