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

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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).

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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).

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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 :-)

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January 4, 2007 at 3:44 pm

MarcinZapowiada sie ciekawie, mam nadzieje, ze starczy zapalu :)

Zacheciles mnie do kupienia tej ksiazki, czytalem inne Penrosa i podobaly mi sie, ale pozniej moj zapal do popularnonaukowych zmalal (czytalem ich wiele i zbyt czesto powtarza sie to samo, w dodatku zwykle na bardzo ogolnym poziomie). Jezeli ta jest inna, bardziej scisla, chetnie po nia siegne (chociaz straszna cegla, przegladalem kiedys w ksiegarni :).

Chetnie poczytam okolomatematyczne wpisy na rozne tematy. Skonczylem MIMUW jakis czas temu i brakuje mi kontaktu z matematyka. Szukalem kiedys blogow matematycznych, ale zbyt wielu nie znalazlem. Jezeli Tobie sie uda, chetnie zobacze linki na Twojej stronie (a moze sa jakies matematyczne blogi po polsku? Tez byloby milo).

Hmmm, teraz pomyslalem, ze moze ten wpis tez powinien byc po angielsku :). No, trudno, nie chce mi sie przepisywac.

January 5, 2007 at 5:15 am

sirixAs to Penrose’s book: As I wrote earlier, it’s distinctive in that it’s definetely more than a popular book, but not academic one – it’s easier to read then academic one (but, of course, in comparison with academic ones many details are lacking).

I enjoy reading this book very much partially because I know most of the math Penrose refers to, so, at least for now, everything is very precise for me, even if Penrose skips some detail.

However, I have serious doubts whether to buy a copy for my brother who started to study physics this year (this book is rather expensive). On the one hand it could allow him to acquaint with some interesting, modern ideas of maths/physics at a very early stage of his studies (I regret that I didn’t have such possibility), but at the other hand, he doesn’t know as much math as I do, so this book could seem to him as just another popular science book and he wouldn’t get anything really deep from it.

However, I suppose that if You, Marcin, have already finished your studies in mathematics/computer science, than the book is worth Your money (btw, I read polish edition in a bookstore for about 15 minutes and I found polish translation very nice).

January 5, 2007 at 2:12 pm

AnonymousAh, so there is a Polish edition too? Probably even chaper… I am still abroad now I could get an original one easily, but as I’m comming back soon, I will probably wait a bit.

Thanks for your remarks, I think I will go for it (I graduated in math).

On my side I would recommend the book I bought quite recently: Visual Complex Analysis (http://www.usfca.edu/vca/) It’s highly recommended by Penrose himself, so I guess you may like it. It may be a nice gift to your brother, if he likes math (I hope he does). I regret I did not have it when I was his age.

January 13, 2007 at 3:28 am

Ponder StibbonsI’m an advanced undergraduate physics major, and I have to say that even though I have taken a year of real analysis and a term of complex analysis, I could not understand most of R2R (I bailed out halfway when it became clear that I was not getting much out of it). It’s definitely not a popular science book, even though it is branded as one. Penrose is not very good at explaining things to non-specialists.

January 15, 2007 at 4:42 pm

sirixI assure You that academic books that deal with some concepts R2R deals with are 10 times harder. Give it another chance :-)

I haven’t ever seen in any other popular book definition of what (for example) fibre bundle is. And so one has to spend some time if one sees it for a first time. But I prefer to have it this way than read an easy popular book and after don’t really know anything more I knew before.

January 15, 2007 at 4:49 pm

sirixGenerally, my experience says (unfortunately) that “to understand”

alwaysinvolves “to struggle”, “to be depressed because of one’s own weaknesses and so on.January 16, 2007 at 5:05 pm

rafalbI definitely agree that understanding is always associated with struggle, fight. I was always curious how it is in the case of geniuses… Do they struggle also?

Best regards, I keep my finger crossed for the development of your blog…

rafalb

January 17, 2007 at 3:40 pm

sirix“Do they struggle also?”

My general point of view on this comes from observation that all people dream. As I wrote in a comment to

another article, if I could imagine and recollect things as precisely as it is done in dreams, I would probably be a great mathematician.

And so I think that every of us has a body (in particular – brain) equal to this of Einstein. What differs common people and geniuses is perhaps abbility of the latter ones to use these “hidden resources” more effectively (and I believe that this abbility is an outcome of random coincidents). However, I don’t believe even geniuses can use these resources consciously. And so, in a sense, they at least “struggle less” than we have to:-)

There’s a great book about related issues:

Psychology of Inventionby great mathematician Jaques Hadamard (there’s a polish translation – it was published in a popular “Omega” series, one can often find it on allegro.pl for 10zl). He was undoubtedly a genius, and he tries to describe a process of invention. It was partially inspired by Henri Poincare, who also wrote few things on a subject, but I haven’t read them (yet).Also, Thanks for kind words Rafal :-)