I do not know how I have missed Raymond Lull’s Ars Magna for so long. After all Raymond Lull (1232-1316) wrote at least 250 books and most of them are applications of Ars Magna. I learned of it just today from Martin Gardner’s Science good, bad and bogus (1981). As so many of these skeptics and protectors of science, Gardner did not have any scientific education. Why I usually dislike such skeptics is that they often have a too naive believe in science. One look at quantum field theories, say, to Feynman’s calculations of renormalization (I will write a post of this a bit later), give a considerably different view of hard and real science. (I also checked if Laurence Gardner maybe was Martin Gardner’s brother, but apparently not. Laurence Gardner did not have much education either, but a bit more than Martin Gardner.) Nevertheless, after reading the book I must admit that Martin Gardner is better than skeptics typically are and the book is quite fine.
Anyway, the book did have a chapter on Lull and it explained the method of Ars Magna, for sure picked there as an example of pseudo-science, but Gardner also found some reasonable applications to the method, it was not all negative. Of course I had to immediately try it and the method is quite good, as a medieval method. It is very simple: you take any field that you want to study and two sets of categories or basic truths or something of this type. Then you go through all possible combinations of the elements in these two categories and see if the combination gives you the solution you are looking for. Lull usually divided two round discs into sectors, one disc smaller than the other one, wrote one letter symbolizing each element of the categories in each sector, and then fixed the smaller disc on top of the larger one with origins in the same point. Thus, turning the smaller disc he could easily find all combinations of these two categories.
This does sound like a very simple method indeed and you may wonder what it possibly could help, but the help is in going systematically, and rather fast, through all combinations, not to omit any of them. Additionally, there is the very important observation of Lull that you can study any field, and even obtain insight to very difficult problems, by just looking at combinations of the main principles of the field. Let us make a very simple example. What is the name of the President of Finland? For me, as a Finn, this is not so difficult, but for a foreigner it may well be. Once you probably heard it but cannot remember. It may be difficult for a Finn also because many Finns call the President Joker, as while smiling he bears an uncanny resemblance to the Joker in a pack of cards. In order to solve this problem with Ars Magna, let us take two circles and write the letters of the alphabet on both circles. Obviously, in the field of names the most important principles are the letters. You turn the disc starting e.g. from A-A. After some time you get N-I and remember, it was Niinistö. Not remembering his given name, you repeat this procedure and after some time find S-A, yes, if was Sauli Niinistö. You can do this memory trick without the discs, just going through the alphabet in the head, and often it is enough to try the first letter, but absolutely, it is Lull’s Ars Magna method and it does work.
Now, let us apply this to quantum gravity. As you know, quantum gravity has not been solved yet, but Lull’s method gives immediately good ideas. We have two categories: gravitation and quantum field theories. Quantum field theories are today gauge theories. We pick up the most important principles of gauge theories. They include such issues as renormalizability, mass generation by spontaneous symmetry breaking and coupling of interactions to the free propagator. We find a set of such important principles, but let us include these three and give them letters R, M and C. The current theory of gravitation is General Relativity. Einstein formulated many principles, one of them being the equivalence principle (the gravitational mass equals the inertial mass) and another one is the stress tensor. We find a set of such principles and our set includes the equivalence principle and the stress tensor, which we give the letters E and S.
Writing the letters on the discs, dressing in medieval gown, and turning the disc all the time attentively searching if this combination might give some new ideas, we soon come to the combination R-S: we should renormalize Einstein’s field equations where there is the stress tensor in the right side. This cannot be done, so we cannot have the stress tensor. We have to drop the Einstein equations and take Nordström’s scalar gravitation theory. Indeed, a scalar field can be renormalized easily.
Then we come to the combination M-E and notice that there is a conflict: in quantum gauge field theories the masses of massive fermions and bosons are generated by Higgs fields via spontaneous symmetry breaking, while the equivalence principle says that the masses are very closely associated with the gravitation field. In order to have the gravitational mass equal to the inertial mass generated by the Higgs mechanism, we have to have gravitation coupled to Higgs fields and as gravitation is long range, we take a (massless) goldstone boson field as the quantum version of Nordström’s scalar gravitation field. But generation of masses to quarks requires another set of Higgs fields than generation of masses to electroweak bosons and leptons in the electroweak theory. Thus, we leave the electroweak theory as it is and do not couple it with gravitation. After all, practically all mass is in hadrons (protons and neutrons). It is enough for the equivalence principle (in the precision we can achieve) that only quarks have the gravitational mass equal to the inertial mass. Electrons do not need to feel gravitation, they are so light compared to hadrons. We may want to work in the Unified Field Theory of SU(5) and have 24 Higgs fields that generate the mass of quarks. From this system we take a single goldstone boson as the gravitation quantum field. As the goldstone bosons of the strong interaction QCD are gluons, we have to introduce a heavy gluon (as one goldstone boson was used for gravitation), and this hopefully changes QCD so much that we get quark confinement and avoid proton decay. (You have to work on this, Ars Magna cannot help in calculation.)
We still turn the discs and get C-S and notice that of course, the stress tensor is a coupling of gravitation to the electromagnetic field. There would not be any stress in the space if mass would not resist the gravitation force. Massive point particles would simply accelerate in the force field. The reason there is stress and why Einstein has the off-diagonal elements in the energy-stress tensor is that the General Relativity is coupled to electromagnetic fields. Once this coupling is removed, the field equation gives Nordström’s scalar gravitation field. We also see that QCD must be coupled to gravitation because of the masses of quarks, while QED and electroweak theories can be as they are.
I guess this is enough as a demonstration of the great usefulness of the simple method of Ars Magna. Though it is intended for solving theological problems, it may be especially good in the creation of fundamental theories in theoretical physics. Though Einstein stole ideas of other researchers, he was correct in one thing: a fundamental theory in theoretical physics must start from a small set of fundamental principles. This is a totally lullian view and Einstein probably stole it from Lull. Correspondingly, combining two fundamental theories in theoretical physics must start by investigating combinations of the fundamental principles of these theories, i.e., from Ars Magna. Only after this step we can continue to calculations.