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:

Smolin:

*“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.

Patrice Ayme’

Tags: Axonal Logic, Buridan, Celestial Mechanics, Lee Smolin, Mathematical Logic, mathematical objects, Mathematics, Neurology, Physics

April 23, 2015 at 12:17 am |

Like Eugen, I think it’s an awesome, astonishing idea, and like him I will study it.

Did you try to expose that strike of genius at Sci Sal? Or did they block you? That’s where you found out about Smolin’s completely ridiculous idea?

April 23, 2015 at 1:27 am |

They blocked all comments of mine that described my own theory, calling it “irrelevant” to the Original Post or the discussion. I must admit that my own idea lacked the drastic “triumph of the will” aspect of Smolin’s…

September 16, 2015 at 4:01 pm |

“However, the real question of “Platonism” is: why are mathematical theorems true?”

Is there any mathematical theorem that you know is true for nature? Mathematics is completely false and is never valid for anything in nature. Can you perform 1+1=2 for any two objects of nature? Can you add one apple and one orange and say two fruits? You cannot. Both 1s must be real numbers, after adding, the result must also be a real number. Two fruits is not a single real number or a single object. Therefore you cannot add apples and oranges.

Can you add two apples? No you cannot. No two apples are identical and therefore you cannot add them. All objects of nature are different. Math requires everything must be real numbers. Real numbers are not objects of nature. Therefore real numbers are false. How can you create something true using something false like mathematics? You cannot. There is no math that is true for nature.

For more details take a look at chapter one on Truth at the blog site https://theoryofsouls.wordpress.com/

September 16, 2015 at 9:20 pm |

Welcome idpnsd! I do not believe in real numbers either! Not because they “lie”, but because infinity is an impossibility.

1 + 1 =2 is a physical object. A neural network IS a real object.

Of course, one can “say two fruits”. Just try it, it works.

What is going on?

One (neural network, “NW”), orange (NW), plus (NW), one (NW), apple (NW), equal (NW), two (NW), fruits (NW). Then tie up all these NW together by other NW.

In other news, you do not seem to be aware of the cardinal theory of Georg Cantor, as modified by Naïve Set Theory. “2” is the set of all sets which can be put in bijection with the set {0, {0}}… Where “0” is the empty set.

Mathematics live as all bijections between neural networks.

PA

September 21, 2015 at 7:36 pm |

Real numbers are closed under addition operations – that means – the result must be a real number. All real numbers are single points on real line, including cardinality. The addition operation is defined only over real numbers. Objects of natural world cannot be added.

Engineers have designed many kinds of guns that generate real numbers for any object of nature. These guns have a display. You point the gun to a dog, and shoot, it will show a number on the display. Similarly you point the same gun at a cat and it will show you another digital number on the display. It will be meaningless to add such numbers and then do math on them. Cardinality is one such a gun. Internally these guns use analog to digital converters of many different kinds.

September 21, 2015 at 9:36 pm |

Any digital display is… finite. Any digital computation is… finite.

Telling me real numbers are points on the real line is a tautology. I want to give you bananas, and add them, day after day… Hoping you will understand bananas can be added. A three year old human tends to understand that.

September 21, 2015 at 10:28 pm |

But what is the point adding a number from a dog and a number from a cat? What it will give us about their characteristics? How do you compare a cat and a dog?

Are two bananas same? Are two humans same? Isn’t everything in nature like apples and oranges?

September 21, 2015 at 10:45 pm

As you yourself said, a dog plus a cat make two animals. That’s kindergarten level exactly (I have a five year old, she understands this very well). Your problem is with CATEGORIES (a branch of math sort of inaugurated by Aristotle, and which can be used foundationally in math).

September 22, 2015 at 5:07 am

Give me any peer reviewed article in any professional journal that uses math to describe nature – and I will show that it must be false. As an example take a look at the chapter on quantum mechanics at the blog site https://theoryofsouls.wordpress.com/category/f-ch6-quantum-mechanics/ which shows that the Uncertainty Principle is wrong.

You do not have to go to such level of details. Real numbers are not objects of nature – because they neither grow on trees nor can be mined from earth. Therefore real numbers are false. How can you create something true using something false like real numbers?

September 22, 2015 at 9:51 pm

Your neural networks are real. They build approximations of real numbers. This is how computers work, BTW. Inasmuch as real numbers require infinity, I agree, and have long argued, that they are, indeed, not real.

Mathematics as used in practice rest on computations, in particular electronically driven computations, which do NOT use real numbers, but only finite numbers.

Saying the Uncertainty Principle is not correct flies in contradiction to proven facts. However, its establishment does not depend upon real number theory (if one digs real deep).

PA

September 22, 2015 at 12:22 am |

My problem is how can you describe a banana using math?

September 22, 2015 at 4:26 am

Math is just a language elaborated by mathematicians in the last 10,000 years to describe logical operations. It provides with particular models in MODEL THEORY, itself a subset of metamathematics. With all due respect, your problem is deep and my time shallow. There are zillions of courses out there at all levels, on the subject of bananas versus math, from preschool to research level…

September 17, 2015 at 1:40 am |

Patrice says brain is real, neural networks are real, and there your numbers!

September 23, 2015 at 8:04 am |

“Mathematics as used in practice rest on computations,…” Mathematics is used in electronic engineering. That does not mean math works. Real numbers or finite numbers are both false, because they are not objects of nature. They cannot work. Engineering works because of lots of patches and kludges. You have to see the blog site for many examples to support the idea that math is false. Uncertainty Principle has never been tested; it cannot be tested, because it has false assumptions. Again proof is given in the blog site. Truth is very difficult to find. Ayn Rand said – “Truth is not for all men, but only for those who seek it.”

September 23, 2015 at 8:48 pm |

Even the notoriously clueless Rand knew integers were real…