By claiming that emotional thinking is superior, I do not mean that “logical” thinking ought to be rejected, and replaced by passions running wild. I am just saying what I am saying, and no more. Not, just the opposite, “*logical*” thinking ought to be embraced. However, there are many “logical” types of thought in existence (as Pascal already pointed out). Including the emotional type. They are entangled.

Emotional and logical thinking can be physiologically distinguished in the brain (the latter is mostly about axons; the former about the rest).

Any “logical” thinking is literally, a chain made of points. (And there are no points in nature, said a Quantum Angel who passed by; let’s ignore her, for now!)

Elliptic Geometry In Action: Greeks, 240 BCE, Understood The Difference Between Latitude & Geodesic (Great Circle). (Traditionally, one quotes Eratosthenes. However, it’s Pytheas of Marseilles who first did this elliptic geometry computation… A century earlier. Pytheas also discovered the Polar Circle, sea ice, and maybe Iceland, among other things boreal…) Whether to develop, or not, this sort of mathematics and physics was, fundamentally, an emotional decision. Involving in particular the emotional worth of the axioms involved.

Some say that hard logic, and mathematics is how to implement “correct thinking”. Those who say this, do not know modern logic, as practiced in logic departments of the most prestigious universities.

In truth, overall, logicians spent their careers proposing putative, potential foundations for logic. Ergo, there is no overall agreement, from the specialists of the field themselves, about what constitute acceptable foundations for “logic”.

It is the same situation in mathematics.

Actually dozens of prestigious mathematicians (mostly French) launched themselves, in the 1950s into a project to make mathematics rigorous. They called their effort “Bourbaki”.

Meanwhile some even more prestigious mathematicians, or at least the best of them all, Grothendieck, splendidly ignored their efforts, and, instead, founded mathematics on Category Theory.

Many mathematicians were aghast, because they had no idea whatsoever what Category Theory could be about. They derided it as “Abstract Nonsense”.

Instead it was rather “Abstract Sense”.

But let’s take a better known example: Euclid.

There are two types of fallacies in Euclid.

The simplest one is the logical fallacy of deducing, from emotion, what the axioms did not imply. Euclid felt that two circles which looked like they should intersect, did intersect. Emotionally seductive, but not a consequence of his axioms.

Euclid’s worst fallacy was to exclude most of geometry, namely what’s not in a plane. It’s all the more striking as “Non-Euclidean” geometry had been considered just prior. So Euclid closed minds, and that’s as incorrect as incorrect can be.

To come back to logic as studied by logicians: the logicS considered therein, are much general than those used in mathematics. Yet, as no conclusion was reached, this implies that mathematics itself is illogical. That, of course, is a conclusion mathematicians detest. And the proof of their pudding is found in physics, computer science, engineering.

So what to do, to determine correct arguments? Well, direct towards any argument an abrasive, offensive malevolence, trying to poke holes, just as a mountain lion canines try to pass between vertebras to dislocate a spine.

That’s one approach. The other, more constructive, but less safe, is to hope for the best, and launched logical chains in the multiverses of unchained axiomatics.

Given the proper axioms, (most of) an argument can generally be saved. The best arguments often deserve better axiomatics (so it was with Leibnitz’s infinitesimals).

So, de facto, people have longed been using not just “inverse probability”, but “inverse logic”. In “inverse logic”, axioms are derived from what one FEELS ought to be a correct argument.

Emotions driving axiomatics is more metalogical, than axiomatics driving emotions.

***

To the preceding philosophy professor Massimo Pigliucci replied (in part) that:

*“Patrice, *

“…Hence, to think critically, one needs enough facts. Namely all relevant facts.”

*Enough facts is not the same as all the relevant facts, as incorrectly implied by the use of “namely.” *

“It is arrogant to think that other people are prone to “logical fallacies”.”

*It is an observation, and facts are not arrogant. *

“A Quantum Wave evaluates the entirety of possible outcomes, then computes how probable they are.”

*Are you presenting quantum waves as agents? They don’t evaluate and compute, they just behave according to the laws of physics.*

“just as with the Quantum, this means to think teleologically, no holds barred”

*The quantum doesn’t think, as far as I know. *

“Emotional Thinking Is Superior Thinking”* *

*I have no idea what you mean by that. Superior in what sense? And where’s the bright line between reason and emotion?*

“Any “logical” thinking is literally, a chain made of points”

*No, definitely not “literally.” *

*It may not follow from the axioms, but I am having a hard time being emotionally seductive by intersecting circles. *

“Euclid’s worst fallacy was to exclude most of geometry, namely what’s not in a plane.”

*That’s an historically bizarre claim to make. Like saying that Newton’s worst fallacy was to exclude considerations of general relativity. C’mon. *

“as no conclusion was reached, this implies that mathematics itself is illogical*” *

*Uhm, no. *

“to hope for the best, and launch logical chains in the multiverses of unchained axiomatics”* *

*Very poetic, I have no idea what that means, though.”*

***

Massimo Pigliucci is professor of philosophy at CUNY in New York, and has doctorates both in biology and philosophy. However, truth does not care about having one, or two thousands doctorates. It would take too long to address all of Massimo’s errors (basically all of his retorts above). Let me just consider two points where he clings to Common Wisdom like a barnacle to a rock. The question of Non-Euclidean geometry, and of the Quantum. He published most of the answer below on his site:

Dear Massimo:

Impertinence and amusement help thought. Thank you for providing both. Unmotivated thought is not worth having.

**The Greeks discovered Non-Euclidean geometry. It’s hidden in plain sight.** It is a wonder that, to this day, so many intellectuals repeat Gauss’ self-serving absurdities on the subject (Gauss disingenuously claimed that he had discovered it all before Janos Bolyai, but did not publish it because he feared the “cries of the Beotians”… aka the peasants; Gauss does not tell you that a professor of jurisprudence had sketched to him how Non-Euclidean geometry worked… in 1818! We have the correspondence.).

The truth is simpler: Gauss did not think of the possibility of Non-Euclidean geometry (although he strongly suspected Euclidean geometry was not logical). Such a fame greedster could not apparently resist the allure of claiming the greatest prize…

It is pretty abysmal that most mathematicians are not thinking enough, and honest enough, to be publicly aware of Gauss’ shenanigans (Gauss is one of the few Muhammads of mathematics). But that fits the fact that they want mathematics to be an ethereal church, the immense priests of which they are. To admit Gauss got some of his ideas from a vulgar lawyers, is, assuredly, too painful.

That would be too admit the “Prince of Mathematics” was corrupt, thus, all mathematicians too (and, indeed, most of them are! Always that power thing; to recognize ideas have come out of the hierarchy in mathematics is injurious to the hierarchy… And by extension to Massimo.)

So why do I claim the Greeks invented Non-Euclidean geometry? Because they did; it’s a fact. It is like having the tallest mountain in the world in one’s garden, and not having noticed it: priests, and princes, are good at this, thus, most mathematicians.

The Greek astronomer Ptolemy wrote in his Geography (circa 150 CE):

*“It has been demonstrated by mathematics that the surface of the land and water is in its entirety a sphere…and that any plane which passes through the centre makes at its surface, that is, at the surface of the Earth and of the sky, great circles.”*

Not just this, but, nearly 400 years earlier, Eratosthenes had determined the size of Earth (missing by just 15%).

http://en.wikipedia.org/wiki/Eratosthenes

How? The Greeks used spherical geometry.

Great circles are the “straight lines” of spherical geometry. This is a consequence of the properties of a sphere, in which the shortest distances on the surface are great circle routes. Such curves are said to be “intrinsically” straight.

Better: Eusebius of Caesarea proposed 149 million kilometers for the distance of the Sun! (Exactly the modern value.)

Gauss, should he be around, would whine that the Greeks did not know what they were doing. But the Greeks were no fools. They knew what they were doing.

Socrates killed enemies in battle. Contemporary mathematicians were not afraid of the Beotians, contrarily to Gauss.

Aristotle (384-322 BC) was keen to demonstrate that logic could be many things. Aristotle was concerned upon the dependency of logic on the axioms one used. Thus Aristotle’s Non-Euclidean work is contained in his works on Ethics.

A thoroughly modern approach.

The philosopher Imre Toth observed the blatant presence of Non-Euclidean geometry in the “*Corpus Aristotelicum*” in 1967.

Aristotle exposed the existence of geometries different from plane geometry. The approach is found in no less than SIX different parts of Aristotle’s works. **Aristotle outright says that, in a general geometry, the sum of the angles of a triangle can be equal to, or more than, or less than, two right angles.**

One cannot be any clearer about the existence on Non-Euclidean geometry.

Actually Aristotle introduced an axiom, *Aristotle’s Axiom*, a theorem in Euclidean and Hyperbolic geometry (it is false in Elliptic geometry, thus false on a sphere).

Related to *Aristotle’s Axiom is Archimedes’ Axiom* (which belongs to modern Model Theory).

One actually finds non trivial, beautiful NON-Euclidean theorems in Aristotle (one of my preferred frienemies).

Non-Euclidean geometry was most natural: look at a sphere, look at a saddle, look at a pillow. In *Ethika ad Eudemum*, Aristotle rolls out the spectacular example of a quadrangle with the maximum eight right angles sum for its interior angles.

Do Quantum Wave think? Good question, I have been asking it to myself for all too many decades.

Agent: from Latin “*agentem*”, what sets in motion. Quantum waves are the laws of physics: given a space, they evaluate, compute. This is the whole idea of the Quantum Computer. So far, they have been uncooperative. Insulting them, won’t help.

Patrice Ayme’