(Second Part of “Causality Explained”)
Axiomatic Systems Are Fragile:
Frege was one of the founders of mathematical logic and analytic philosophy. Frege wrote the Grundgesetze der Arithmetik [Basic Laws of Arithmetic], in three volumes. He published the first volume in 1894 (paying for it himself). Just before the second volume was going to press, in 1903, a young Bertrand Russell informed Frege of a dangerous contradiction, Russell’s paradox (a variant of the Cretan Liar Paradox). Frege was thrown in total confusion: a remedy he tried to apply reduced the number of objects his system could be applied to, to just ONE. Oops.
Frege was no dummy: he invented quantifiers (Second Order Logic, crucial to all of mathematics). It is just that logic can be pitiless.
Neurons are (part of) the solution to the problem of thinking, a problem so deep, we cannot conceive of it. A second independent evolution of neuronicity would certainly prove that.
Truer Axiomatics Is Simpler, More Powerful:
Russell and Whitehead, colossal mathematicians and philosophers, decided to demonstrate 1 + 1 = 2. Without making “Cretan Liar” self-contradictions.
They wrote a book to do so. In the second volume, around page 200, they succeeded.
I prefer simpler axioms to get to 1 + 1 =2.
(Just define the right hand side with the left.)
It would be interesting that philsophers define what “causing” means, and what “causality” is, for us. Say with explicit examples.
I want to know what cause causes. It’s a bit like pondering what is is.
Some creatures paid as philosophers by employers know 17th century physics, something about billiards balls taught in first year undergraduate physics. (I know it well, I have taught it more than once.) Then they think they know science. All they know is Middle-Ages physics.
These first year undergraduates then to explain the entire world with the nail and hammer they know so well.
They never made it to Statistical Mechanics, Thermodynamics, etc. And the associated “Causality” of these realms of knowledge.
Axiomatics Of Causality With The Quantum:
How does “causality” work in the Quantum Mechanics we have?
You consider an experiment, analyze its eigenstates, set-up the corresponding Hilbert space, and then compute.
“Billiard Balls” is what seems to happen when the associated De Broglie wave has such high frequency that the eigenstates seem continuous.
So Classical Mechanical “causality” is an asymptote.
Know How To Dream… To Bring Up New Axiomatics
Human beings communicate digitally (words and their letters or ideograms), and through programs (aka languages, including logic and mathematics).
All of this used conventions, “rules”, truths I call axioms, to simplify… the language (this is not traditional, as many of these axioms have had names for 25 centuries).
So for example, I view the “modus ponens” (if P implies Q and P happens, then Q) as an axiom (instead of just a “logical form” or “rule of inference”).
The reason to call basic “logic forms” “axioms” is that they are more fragile than they look. One can do with, or without them. All sorts of non-classical logics do without the “excluded third law” (for example fuzzy set theory).
With such a semantic, one realizes that all great advances in understanding have to do with setting up more appropriate axioms.
Buridan’s Revolution, Or An Axiomatics Revolution:
In the Fourteenth Century, the intellectual movement launched by Buridan, included Oresme and the Oxford Calculators. They discovered inertia, momentum (“impetus”), graphs, the law of falling bodies, the heliocentric system (undistinguishable from the geocentric system, said Buridan, but we may as well stick to the latter, as it is in Scripture, said Buridan, wryly).
Buridan’s revolution is little known. But was no accident: Buridan refused to become a theologian, he stuck to the faculty of arts (so Buridan did not have to waste time in sterile debates with god cretins… differently from nearly all intellectuals of the time). Much of Buridan is still in untranslated Medieval Latin, that may explain it, after centuries of Catholic war against him.
These breakthroughs were major, and consisted in a number of new axioms (now often attributed to Galileo, Descartes, Newton). The axioms had a tremendous psychological effect. At the time, Buridan, adviser to no less than four Kings, head of the University of Paris, was untouchable.
The philosopher cum mathematician, physicist and politician, died in 1360. In 1473, the pope and king Louis XI conspired to try to stop the blossoming Renaissance.
More than a century after his death, Buridan’s works, his new axioms, were made unlawful to read. (However Buridan was mandatory reading in Cracow, and Copernic re-published the work, while he was on his death bed).
The mind, the brain, is quite fuzzy (in the sense of fuzzy set theory; the dreaming part). Axioms, and axons enable to code it digitally. So mathematization, and programmation are intrinsic human mental activities.
We Are All Theoretical Scientists Of The Mathematical Type:
Human beings continually draw consequences from the axioms they have, through the intermediary of giant systems of thought, and systems of mood (mentality for short).
When reality comes to drastically contradict expected consequences, mentality is modified, typically in the easiest way, with what I call an ANTI-IDEA.
For example when a number of physics Nobel laureates (Lenard, Stark) were anxious to rise in the Nazi Party, they had to reconcile the supposed inferiority of the Jews with the fact that Einstein was a Jew. They could not admit either that Poincare’ invented Relativity, as he was also of the most hated nation (and of the most anti-German fascism family in France!).
So they simply claimed that it was all “Jewish Science” (this way they did not have to wax lyrically about why they had collaborated with Einstein before anti-Judaism).
When brute force anti-ideas don’t work after all (as became clear to Germans in 1945), then a full re-organization of the axiomatics is in order.
An example, as I said, is fuzzy set theory. It violates the Excluded Third Law.
But sometimes the reconsideration may be temporary. (Whether A and Non-A holds in the LOGIC of Quantum Mechanics, the Einstein-Schrodinger Cat, is a matter of heated debate.)
The removal of old logical axioms can be definitive. For example the Distributive Law of Propositional Calculus fails in Quantum Logic. That has to do with the Uncertainty Principle, a wave effect that would be etched in stone, were it not even more fundamental.
Verdict? Neurons, Axons, And Axioms Make One System:
We have been playing with axioms for millions of years: they reflect the hierarchical, axon dominated, neuron originated most basic structure of the nervous system.
Well, the neuronal-axonal skeleton of minds is probably the lowest energy solution to the problem of thinking in the appropriate space. It has just been proposed neurons evolved twice:
We do not just think axiomatically, but we certainly communicate axiomatically, even with ourselves. And the axiomatics are dynamical. Thus causes learn to fit effects.
The fact this work is subjective, in part, does not mean it does not have to do with nature. Just the opposite: causality is nature answering the call of nature, with a flourish.
Human mentality is a continual dialogue between nature inside (Claude Bernard) and nature outside.
Changing axioms is hard work: it involves brain re-wiring. Not just connecting different neurons, but also probably modifying them inside.
Mathematicians have plenty of occasions to ponder what a proof (thus an explanation) is. The situation is worse than ever, with immense proofs only the author gets (Fermat’s Last Theorem was just an appetizer), or then computer-assisted proofs (nobody can check what happened, and it’s going to get worse with full Quantum Computers).
Not all and any reasoning is made to be understood by everybody. (Mathematicians have to use alien math they don’t really understand, quite often.)
Yes, thinking is hard. And not always nice. But somebody has to do it. Just remember this essence, when trying to communicate with the stars: hard, and not always nice.