## Posts Tagged ‘Theorems’

### Causality Explained

March 29, 2015

WHAT CAUSES CAUSE?

What Is Causality? What is an Explanation?

Pondering the nature of the concept of explanation is the first step in thinking. So you may say that there is nothing more important, nothing more human.

I have a solution. It is simplicity itself. I go for the obvious model:

Mathematics, logic, physics, and the rest of science give a strict definition of what causality, and an explanation is.

How?

Through systems of axioms and theorems.

Some of the sub-systems therein have to do with logic (“Predicate Calculus”). They are found all over science and common sense (although they will not be necessarily present in systems of thought such as, say, poetry, or rhetoric).

WHEN A IMPLIES B, IN A LOGOS, ONE OUGHT TO SAY THAT A “CAUSES” B.

A and B are propositions. They do not have to be very precise.

Precision Is Not Necessarily The Smartest. Semantic Web Necessary.

As it turns out, except in Classical Computer Science as it exists today (Classical CS by opposition to Quantum CS, a subject developing in the last 20 years), propositions are never precise (so a degree of poetry is everywhere, even in mathematics!) Propositions, in practice, depend upon a semantic web.

A could be: “Plate Tectonic” and B could be “Continental Drift”. That A causes B is one of axioms of present day geophysics.

Thus I define causality as logical implication.

To use David Hume’s example: flame F brings heat H, always, and so is supposed to cause it: F implies H. Hume deduced causality from observation of the link (if…then).

More detailed modern physics shows that the heat of flame F is agitation that can be transmitted (both a theorem about, and a definition of, heat). Now we have a full, detailed logos about F and what H means, and how F implies H, down to electronic orbitals.

Mathematicians are used to make elaborate demonstrations, and then, to their horror, discover somewhere something that cannot be causally justified. Then they have to reconsider from scratch.

“Causes” in mathematics are also called axioms. In practice, well known theorems are used as axioms to implement further mathematical causality. A mathematician using a theorem from a distant field may not be aware of all the subtleties that allow to prove it: he would use distant theorems he does no know the proof of, as axioms. Some mathematician’s, or logician’s axiom is another’s theorem.

(Hence some hostility between mathematicians and logicians, as much of what the former use the latter proved, but the former have no idea how!)

Causality, by the way, reflects the axonal geometry of the brain.

The full logic of the brain is much more complicated than mathematics, let alone Classical Computer Science, have it. Indeed, brain logic involves much more than axons, such as dendrites, neurotransmitters, glial cells, etc. And of these, only axonal geometry is simple enough to be approximated by classical logic… In first order.

Mathematics is causation. And the ultimate explanation. Mathematics makes causation as limpid we can have it.

This theory met with the approval of Philip Thrift (March 27, 2015): “I agree exactly with the words Patrice Ayme wrote — but with “mathematics”→”programming”, “mathematical”→”programmatical”, etc.”

I pointed out later to Philip that Classical Programming was insufficient to embrace full human (and quantum!) logic. He agreed.

However the preceding somehow made Massimo P , a professional philosopher, uneasy. He quoted me:

“Patrice: “To claim that mathematics is not causal is beyond belief. Mathematics is all about causality.”

Massimo: It most obviously isn’t. What’s causal about Fermat’s Last Theorem? Causality implies physicality, and most of pure math has absolutely nothing whatsoever to do with physicality.

Patrice: “Causes” in mathematics are also called axioms.”

Massimo: “You either don’t understand what causality means or what axioms are. Or both.”

Well, once he had released his emotional steam, Massimo, a self-declared specialist of “physicality” [sic] did not offer one iota of logic in support of his wished-for demolition of my… logic. I must admit my simple thesis is not (yet) in textbooks…

Insults are fundamentally poetic, illogical, or pre-logical. Massimo is saying that been totally confused about causality and explanations is a sacred cow of a whole class of philosophers (to whom he had decided he belongs). Being confused about causality started way back.

“All philosophers, “said Bertrand Russell,” imagine that causation is one of the fundamental axioms of science, yet oddly enough, in advanced sciences, the word ’cause’ never occurs … The law of causality, I believe, is a relic of bygone age, surviving, like the monarchy, only because it is erroneously supposed to do no harm …”

Russell was as wrong as wrong could be (not about the monarchy, but about “causation”). He wrote the preceding in 1913, when Relativity was well implanted, and he, like many others, was no doubt unnerved by it.

Poincare’ noticed, while founding officially “Relativity” in 1904, that apparent succession of events was not absolute (but depended upon relative motions).

Indeed.

But, temporal succession is only an indication of possible causality. In truth causality exists, if, and only if, a logical system establishes it (moreover, said logic has to be “true”; that, assigning a truth value, is, by itself is a separate question that great logicians have studied without clear conclusions).

When an explanation can be fully mathematized, it is finished. Far from being “abstract”, it has become trivial, or so suppose those with minds for whom mathematics is obvious.

Mathematics is just like 2 + 2 = 4, written very large.

Fermat’s Last Theorem is not different in nature, from 2 + 2 = 4… (But for something very subtle: semantic drift, and a forest of theorems used as axioms to go from side of Fermat’s theorem to the other.)

To brandish mathematics as unfathomable “abstract” sorcery, as was done in Scientia Salon, is a strange, but not new, streak.

There in “Abstract Explanations In Science” Massimo and another employed philosopher pondered “whether, and in what sense, mathematical explanations are different from causal / empirical ones.”

My answer is that mathematical, and, more generally logical, explanations are the model of all explanations. We speak (logos) and thus we communicate our thoughts. Even to ourselves.

The difference between mathematics and logic? Mathematics is more poetical. For example, Category Theory is not anchored in logic, nor anywhere else. It is hanging out there, beautiful and useful, a castle in the sky, just like all and any poem.

Such ought to be the set-up on the nature of what causality could be, to figure out what causality is in the physical world. Considering that Quantum Entanglement is all over nature, this is not going to be easy (and it may contain a hidden clock).

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