After demolishing erroneous ideas some 25 centuries old, some brand new, I explain why Mathematics Can Be Made To Correspond To A Subset Of Neurology. And Why Probably Neurology Is A Consequence Of Not-Yet Imagined Physics.
SOCRATISM, PLATONISM ARE WRONG:
Einstein famously declared that: “How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?”
Well, either it is an unfathomable miracle, or something in the premises has to give. Einstein was not at all original here, he was behaving rather like a very old parrot.
That the brain is independent of experience is a very old idea. It is Socrates’ style “knowledge”, a “knowledge” given a priori. From there, naturally enough aroses what one should call the “Platonist Delusion”, the belief that mathematics can only be independent of experience.
Einstein had no proof whatsoever that”thought is independent of experience”. All what a brain does is to experience and deduct. It starts in the womb. It happens even in an isolated brain. Even a mini brain growing in a vat, experiences (some) aspects of the world (gravity, vibrations). Even a network of three neurons experiences a sort of inner world unpredictable to an observer: https://patriceayme.wordpress.com/2015/03/15/three-neurons-free-will/
Latest Silliness: Smolin’s Triumph of the Will:
The physicist Lee Smolin has ideas about the nature of mathematics:
“the main effectiveness of mathematics in physics consists of these kinds of correspondences between records of past observations or, more precisely, patterns inherent in such records, and properties of mathematical objects that are constructed as representations of models of the evolution of such systems … Both the records and the mathematical objects are human constructions which are brought into existence by exercises of human will; neither has any transcendental existence. Both are static, not in the sense of existing outside of time, but in the weak sense that, once they come to exist, they don’t change”
Patrice Ayme: Smolin implies that “records and mathematical objects are human constructions which are brought into existence by exercises of HUMAN WILL; neither has any transcendental existence”. That’s trivially true: anything human has to do with human will.
However, the real question of “Platonism” is: why are mathematical theorems true?
Or am I underestimating Smolin, and Smolin is saying that right and wrong in mathematics is just a matter of WILL? (That’s reminiscent of Nietzsche, and Hitler’s subsequent obsession with the “will”.)
As I have known Smolin, let me not laugh out loud. (“Triumph of the Will” was a famous Nazi flick.)
I have a completely different perspective. “Human will” cannot possibly determine mathematical right and wrong, as many students who are poor at mathematics find out, to their dismay!
So what determines right and wrong in mathematics? How come enormously complex and subtle mathematical objects, which are very far from arbitrary, exist out there?
I sketched an answer in “Why Mathematics Is Natural”. It does not have to do with transcendence of the will.
AXONAL LOGIC IS MATHEMATICAL LOGIC, NEUROLOGY IS MORE:
Neurology, the logic of neurons, contains what one ought to call axonal logic, a sub-category.
Axonal logic is made of the simplest causal units: neuron (or another subset of the brain) A acts on neuron (or brain subset) B, through an axon. This axonal category, a sub-category, corresponds through a functor, from neurology to mathematical logic. To A, and B are associated a and b, which are propositions in mathematical logic, and to the axon, corresponds a logical implication.
Thus one sees that mathematics corresponds to a part of neurology (it’s a subcategory).
Yet, neurology is vastly more complicated than mathematical logic. We know this in many ways. The very latest research proposes experimental evidence that memories are stored in neurons (rather than synapses). Thus a neuron A is not a simple proposition.
Neurons also respond to at least 50 hormones, neurohormones, dendrites, glial cells. Thus neurons need to be described, they live, into a “phase space” (Quantum concept) a universe with a vast number of dimensions, the calculus of which we cannot even guess. As some of this logic is topological (the logic of place), it is well beyond the logic used in mathematics (because the latter is relatively simplistic, being digital, a logic written in numbers).
The conclusion, an informed guess, is that axons, thus the implications of mathematical logic, are not disposed haphazardly, but according to the laws of a physics which we cannot imagine, let alone describe.
And out of that axonal calculus springs human mathematics.
HOW TO PROVE THAT MATHEMATICS IS NEURONAL PHYSICS?
If my hypothesis is true, mathematics reduces to physics, albeit a neuronal physics we cannot even imagine. Could we test the hypothesis?
It is natural to search for guidance in the way the discovery, and invention, of Celestial Mechanics proceeded.
The Ancient Greeks had made a gigantic scientific mistake, by preferring Plato’s geocentric hypothesis, to the more natural hypothesis of heliocentrism proposed later by Aristarchus of Samos.
The discovery of impetus and the heliocentric system by Buridan and his followers provides guidance. Buridan admitted that, experimentally heliocentrism and “scripture” could not be distinguished.
However, Buridan pointed out that the heliocentric theory was simpler, and more natural (the “tiny” Earth rotated around the huge Sun).
So the reason to choose heliocentrism was theoretical: heliocentrism’s axiomatic was leaner, meaner, natural.
In the end, the enormous mathematical arsenal to embody the impetus theory provided Kepler with enough mathematics to compute the orbit of Mars, which three century later, definitively proved heliocentrism (and buried epicycles).
Here we have a similar situation: it is simpler to consider that mathematics arises from physics we cannot yet guess, rather than the Platonic alternative of supposing that mathematics belong to its own universe out there.
My axiomatic system is simpler: there is just physics out there. Much of it we call by another name, mathematics, because we are so ignorant about the ways our mind thinks.
Another proof? One can make a little experiment. It requires a willing dog, a beach, and a stick. First tell the dog to sit. Then grab the stick, and throw it in the water, at 40 degree angle relative to the beach. Then tell the dog to go fetch the stick. Dogs who have practiced this activity a bit will not throw themselves in the water immediately. Instead they will run on the beach a bit, and then go into the water at an angle that is less than 90 degrees.
A computer analysis reveals that dogs follow exactly the curve of least time given by calculus. Dogs know calculus, but they did not study it culturally! Dogs arrived at correct calculus solutions by something their neurology did. They did not consult with Plato, they did not create calculus with their will as Smolin does.
It’s neurology which invents, constructs the mathematics. It is not in a world out there life forms consult with.