Physicist Eugene Wigner wrote a famous paper, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," beloved by my physicist friends. Newton, calculus, celestial mechanics, Hooke's law, Maxwell's laws of electromagnetism, matrix algebra and its use in quantum mechanics.
The dream dates at least to Pythagoras, the entire world captured in a small set of axioms — fundamental laws — that logically entail the entire becoming of the world. The Theory of, truly, Everything.
A brilliant physicist friend, who has made major contributions in biology, quipped that biologists, were they Galileo, would have dropped stones of different shapes from the Tower of Pisa to note minute changes in arrival time at its base, and missed the main point.
But my friend's quip also misses the main point about the biosphere, econosphere, culture and history: In their fundamental detailed becoming, these are largely beyond mathematics.
Darwin changed our world view with not a single equation in The Origin of Species.
Why? I think in biology part of the answer lies in the very notion of a biological "function." Ask Darwin what the function of your heart is and "pump blood" would be the prompt answer.
Imagine you remarked, as I've written, that your heart also makes heart sounds and jiggles water in the pericardial sac, why are these not the function of your heart? Darwin would answer that the heart exists in the universe as a complex organ, when most complex things will never exist at all, because it was of selective advantage to your ancestors to have hearts that pumped blood. So the function of the heart is to pump blood. The heart's function is a subset of its causal consequences.
Now the central question becomes: IS THERE A FINITE PRESTATABLE SET OF BIOLOGICAL "FUNCTIONS?" That is, is there a finite prestatable list of features of organisms that MIGHT serve a selective function in some selective environment?
I think the deep answer is NO.
Try it with screwdrivers. Is there a finite list you can come up with for all possible uses of screwdrivers for untold purposes? You can screw in a screw. You can use the screwdriver to open a can of paint. You can use the screwdriver to wedge a door closed, or open, or scrape putty from a window, or stab an assailant, or a friend, commit suicide, or as an object of art, or tie it to a bamboo pole and spear fish, or rent the spear to others and collect 5 percent of the catch, or with a rock, chop (slowly) down a small tree, or carve a stick or ...
We trust 007 or MacGyver to use a screwdriver in wondrous ways.
Is there a MATHEMATICS from which one can derive that screwdrivers work to screw in screws, open paint cans, or spear fish? How would we list all the possible uses to derive this mathematically? What is the problem? The famous frame problem of computer science at least. How many relational features do you see. We cannot list them. We do not know ahead of time what relational features will be useful for what unknown and new purpose, or function! But we USE just such relational features all the time to find new uses for screwdrivers.
Now recall my blog post with the lovely story of the Japanese man in Tokyo in a tiny apartment with too many books. Recall he scanned the books into his iPad, sold the books and had more room. Then he realized he had a NEW BUSINESS MODEL and went into business scanning books for others in tiny Japanese apartments with too many books into their iPads, charging for his service.
Note four things about this example: 1) The new business is the technological complement of iPads (or other scanning devices, but say iPads). 2) Until iPads came into existence, the very possibility of the new book scanning business did not exist. A new Adjacent Possible was really created with the invention of the iPad and its wide sale. We literally create the non-random possibilities we become. 3) The new USE, or NEW FUNCTION of the iPad was unexpected, and in general, not finitely prestatable, and new — i.e. "making space in crowded tiny Japanese apartments by scanning then selling book." Could you have predicted this new function and business after the iPad's invention? Before its invention? 4) The new business exhibits the FRAME problem, ever unsolved. Given a finite list of affordances, "is a," "does a," "uses a," "needs a," there is no guarantee that this list allows deduction of "scanning books and selling them to make space in tiny Japanese apartments." We cannot, in general, be guaranteed to derive mathematically all possible context dependent new uses of iPads from a finite list of affordances as "axioms" in untold "selective" environments for arbitrary purposes made possible by the very becoming of the biosphere, econosphere or history. Thus, this becoming of the econosphere in its detailed becoming is not mathematizable.
The unexpected uses of features of organisms, or technologies, are precisely what happens in the evolution of the biosphere and econosphere, and the analog happens in cultural evolution with the uses of mores, cultural forms, regulations, traditions, in novel ways. In general, these possibles are novel functionalities, in an unbounded space of functionalities, and so are not mathematizable and derivable from a finite set of axioms. I believe this to be true, do not know how to "prove it, and do not think is is just a version of Godel's theorem which is based on the axioms of arithmetic, or other axiomatized formal systems. What would be the basic axiom system from which we would try to derive all possible functionalities of objects and organismic feature in all possible selective environments, themselves emergent from the becoming of the biosphere, econosphere or history? We cannot mathematize this becoming. The set of possible functionalities in arbitrary selective environments that become in the adjacent possible of the evolution of the biosphere do not seem to be recursively enumerable.
In the past I've blogged about this in the form of Darwinian preadaptations. Preadaptations are unused causal features of an organ or part of an organism that come to be useful in a new selective environment, typically for a new function. Recall the swim bladder - a sac in some fish partly filled with air and partly with water, that adjusts neutral buoyancy in the water column. As I've blogged before, paleontologists think swim bladders derived from the lungs of lung fish. Water got into some lung(s), now sacs partly filled with air and water, and they are poised to become swim bladders.
But this is a new function, neutral buoyancy in the water column. Could we prestate this new function? In general, No. Could you prestate all possible preadaptations just for humans in the next million years? No. Not only can we not prestate the new function, there is no mathematics I can conceive of that would allow us to derive the emergence of the swim bladder as a theorem. Further, we cannot simulate the becoming of our biosphere: too many throws of the quantum dice.
More the swim bladder is a new empty niche in the growing Adjacent Possible of the biosphere, for a bacterium could evolve able only to live in swim bladders. (There is a bacterium only able to live in the lungs of sheep.) Like the iPad and the Japanese new business, where the invention and sale of the iPad literally makes possible a new Adjacent Possible business in the econosphere that could, in many cases, not have been foretold, the swim bladder makes possible a new niche for our bacterium using it as its sole home. The biosphere's evolution literally makes possible the new ways to make a living, the new niches, that it then fills! I find this enchanting. Evolution generates often unprestatable novel Adjacent Possible empty niches opportunities the biosphere will become.
I see no mathematics that derives these facts and becomings as theorems.
Thus my brilliant physicist friend, poking fun at biologists crowded atop the tower of Pisa, misses the biological, economic, and historical point: These becomings are beyond sufficient natural law and not mathematizable as axioms and theorems.
The world, Horatio, is richer than all our dreams.