I think there exists a way to describe all known physics without using any mathematical formula, a way to describe what a "wave equation" really is without using mathematically defined fields of any kind, just using algorithmics, in fact using just a single, small and compact algorithm.
OK, it sounds crazy, but I have dealing with this idea back when I coded the "Fractal quantum simulator", well, a naive sketch of it. I realized I could use a fractal growth algorithm, applied over a big amount of simple particles following a simple loop, that made physical properties into the forming "clouds" of those particles. I thougth it was curious and that I should come back to this code some day.
Watch a couples of videos to have a taste on the idea: Particles are in the middle of each cloud (indeed they are ion the center of masses of the clouds), while sparks evolve around them randomly interacting, but without using forces of any kind, just entanglements, teleportation, random moves, etc (like in a vitamined Conway's game of life).
But the universe as we know is really doing exactly that! And it is not using mathematics at any moment, by the way...
The thermodynamic naive example
To understand the real idea, let me first use a simpler and more daily example: thermodynamic laws doesn't exists, and the maths used are useles in the underlaying reality.
Presure is a thermodynamic main concept, but we discovered later that it really were the millions of molecules in the cloud of gas that, individually, cuased small forces by just colliding in unpredictable -to our scale of time and distances- trajectories that used just simple Newtown laws, but in such a big number that it was more practical to stick to fake laws that worked well, again at our sclae of time and distances.
The quantum way
Think on quantum computers. Make the smallest one you can that still works, or just think this should exists. You have a "spark". It would be smaller that a photon, or not even have a size at all. This spark would be the smallest particle possible, the one that makes all the other things just by running the code they... are? in a big enough group of sparks.
The code it should run would be the simpliest program you could code for that quantum computer the spark is. I always think on Mandelbrot set to imagine that code: a simple line of code that generates a increible complex figure you can zoom in and out and explore ad infinitum, full of similar but not repeating structures. This is the kind of code that, when made the quantum way could generate a fractal universe like ours.
Now place a big amount of those "computer" together and switch them on (actually they are always on and consume no energy). Their programs will make them interact here and there and form connections, building a complex network of such computers. Some sparks would become "switch" of the net, more connected that the average. Those switches could represent photons, gluons, and so on. Those "switches" or fat sparks, by still running the same original code, woud form "hubs" of hiper-connected nodes, a even more connected spark that mostly connect "switches" spark. They could be fermions, electrons, etc.
But here you don't have a CPU to run the code, RAM to store values and screen dots to show the results: those sparks are all those things at the same time. Connections made by spark in the network are the RAM, the code they individually run, when used at the same time inside this net of inter-connections is the CPU, and the hubs big enough for our measurement abilities are the dots on the screen, the particles.
Those new "hubs" continue on building even more complex structures as same program runs over "hub" nodes, creating atoms, then simple molecules, complex molecules, life, intelligence, consciousness and, who knows, something even bigger in some more CPU time.
More than a philosofic idea?And the question I would like you to think about and comment is: is there any known and used physical concept that could never be described by maths, any maths, except in a probabilistic averaging form (like in the thermodynamic example)?
My bet is: wave equation. What is really happening down there that, from our scale can only by understood probabilistically?
And more interesting: can the algorithm used by those hypothetical sparks been described, simulated, and used to fill the gaps that the math based laws of physics will never fill, like the nature of this new "quatum presure", the wave equation?
I have dreamt many nights with that algorithm, draw plenty of versions of it, and played near of it with some fractal algorithms, and I feel some one will find it and say: WTF, only this short code?
No, I don't have this algorithm nor a clear idea of it, nor I know for sure it does exists, but it could be such a definitive and elegant solution, such a doable thing even if it sounds crazy today, and it is so inexpensive to think about it, that I bet this algorithm will be found some glorious day.
What I can tell you is that, the deeper I get into the my fractal algorithms, the easier it seems to me that some day some one will find this golden algorithm, the smallest one generating the most complex structures ever: The algorithmic seed of our actual fractal universe.
And that was all, folks, I started to write this post being 48 years old and now I am 49. Happy birthday to me!