Archive for the ‘Axiomatics’ Category

Logic Is Not Everything: It Can Be Anything

February 24, 2016

A common mistake among many of the simple ones, is that, as long as we keep calm and use logic, we can solve any conflict. It was understandable that one could do such a mistake due to naivety and inexperience, before the Twentieth Century. However, we have now, black on white and well known, demonstrations to the contrary, in formal systems studied by professional logicians. Besides, as The French Republic is demonstrating in Libya again, in collaboration with the USA, war has a logic which squashable critters don’t have.

Yes, I am also thinking of the famous Incompleteness Theorems, but, obviously, not only. There is way worse.

This Means All Important Choice Have to Do With Love, Esthetics, Will, Power, Craziness, The Proverbial Human Factors. Logic Can't Go Where The Heart Rules. Or Then Go Into METAlogic.

This Means All Important Choice Have to Do With Love, Esthetics, Will, Power, Craziness, The Proverbial Human Factors. Logic Can’t Go Where The Heart Rules. Or Then Go Into METAlogic.

Before I get in incompleteness and further evils, let me recap some of the traditional approach. I thank in passing Massimo P, for calling my attention to this.

The value of logic, February 23, 2016, Massimo

logicThis is going to be short and rather self-explanatory, with no additional commentary on my part necessary at all. Here is the full transcription of Epictetus’ Discourses, II.25, a gem to keep in mind for future use:

“When one of his audience said, ‘Convince me that logic is useful,’ he said, Would you have me demonstrate it?

‘Yes.’

Well, then, must I not use a demonstrative argument? And, when the other agreed, he said, How then shall you know if I impose upon you?

And when the man had no answer, he said, You see how you yourself admit that logic is necessary, if without it you are not even able to learn this much–whether it is necessary or not.”

Actually Epictetus uses “logic”, it seems to me, rather in its original sense, a discourse. Yet Massimo, like the moderns, will tend to use logic as it was meant in, say, 1900, just before Bertrand Russell objected to Frege’s forgetfulness of the “Liar Paradox” in his formal system justifying arithmetic. A further exploitation of the Liar Paradox brought the incompleteness theorems.

Logic is indeed how human beings communicate. Logic enables debate, and debate is the equivalent of sex, among ideas. It generates entirely new species. However post Gottlob Frege and Bertrand Russell, logic has progressed much.

Modern studies in logic show formal logic can be pretty much anything. Formal systems contradicting the most cherished axioms have been found to be consistent. Some have cute names, such as “paradoxical logic”, “fuzzy logic”, “linear logic”. Thus, Logical Pluralism has been discovered. “Classical Logic” (which neither complete nor coherent) is just a particular case. In some logics, a proposition can be neither true, nor false.

Along the line of ones of the greatest logician and mathematicians of the 20th Century, Generalized Tarski Thesis (GTT):
An argument is valid if and only if in every case in which the premises are true, so is the conclusion.
So yes, madness can be logical. “Logic”, per se, is not much of a constraint. It’s only a set of coherent rules to draw a conclusion. The only constraint is to keep on talking.
The Ancient Greeks would have been very surprised.

So if logic is not the end-all, be all, what is?

Knowledge. Knowledge of the details. In other words, knowledge of evil. That is why, when I demonstrate, using knowledge, that Marcus Aurelius, supposedly the big time Stoic philosopher, makes the apology of Intellectual Fascism, I hear the cries of the Beotians, whose pathetic logic have crushed underfoot. What happened? I went outside of their logic. So they insult me. (Should I spurn them, and make them feel that I have nothing to say, or insult them back, by showing them, and others, what idiots they are? Sitting on one’s hand in front of rabid fascism is neither wise, nor safe!)

In other news, French special forces are operating on the ground in Libya, helping, among other things, very precise US strikes. This is a case of using the same logic as the enemy. The Islamists terrorize and kill: a logic which is pretty drastic. It can also be adopted. Let see how it goes, when the country with the greatest, longest military tradition, adopts it too. (France has a long history of drastic war against invading Islamists, since the Battle of Toulouse, in 721 CE)

Speaking of the enemy, the French Internal Revenue Service is forcing Google Inc., the famous monopoly. to pay back taxes. Google has sent the clown it uses as CEO to Paris. Google was transferring profits it made in France through Ireland, and then Bermuda. The bill? 1.6 BILLION Euros. Great Britain has a similar situation and economy, but is asking for only a tenth of that. Such is plutocracy: greater in the UK than in France.

Ah, the French Defense Ministry is not denying media reports that the French army is in combat, on the ground, in Libya. Instead, the French government has announced the start of an inquiry for finding out who compromised National Defense. In other words, it wants it to be known. Or there is much more coming, which it wants to hide. Or then American paranoia is contagious.

In other news, Belgium, historically a part of Gaul, has closed its border with France, as if Belgium were Great Britain, and France, full of Huns (instead of only Afghans). Amusingly, and a testimony of how much old gripes have subsidized, the Franco-German border stays open.

Logic is not all. Facts are much more important, including facts on the ground.

Patrice Ayme’

The MATHEMATICAL MIND HYPOTHESIS

April 25, 2015

Abstract: A new view is seen (“theo-ry”) for the relationship of mind and universe, and mathematics is central. The Mathematical Mind Hypothesis (MMH). The MMH contradicts, explains, and thus overrules Platonism (the ruling explanation for math, among mathematicians). The MMH is the true essence of what makes the Mathematical Universe Hypothesis alluring.

***

What’s the nature of mathematics? I wrote two essays already, but was told I was just showing off as a mathematician, and the subject was boring. So let me try another angle today.

The nature of mathematics is a particular case of the nature of thinking.

For a number of reasons, deep in today’s physics, as I have (partly) explained in “Einstein’s Error”, many physicists are obsessed with the “Multiverse”, an extreme version of which is the “Mathematical Universe Hypothesis” (MUH), exposed for example by Tegmark, a tenured cosmologist at MIT. Instead of telling people what happened in the first second of the universe, as if I considered myself to be god, I prefer to consider dog:

Dogs LEARN To Choose “y” According To Least Time

Dogs LEARN To Choose “y” According To Least Time

[Dogs can also learn to solve that Calculus of Variation problem in much more difficult circumstances, if the water is choppy, the ground too soft, etc. To have such a mathematical brain allowed the species to catch dinner, and survive.]

The “Multiverse” has its enemies, I am among them. Smolin, a physicist who writes general access books, has tried to say something (as described in Massimo’s Scientia Salon’ “Smolin and the Nature of Mathematics”).

“Smolin,” Massimo, a tenured philosophy professor also a biology PhD, told me “as a counter [to Platonism], offers his model of development of mathematics, which does begin to provide an account for why mathematical theorems are objective (the word he prefers to “true,” in my mind appropriately so).”

My reply:

Smolin is apparently unaware of a whole theory of “truth” in mathematical logic, and of the existence of the work of famous logicians such as Tarski. When Smolin was in the physics department of Berkeley, so was the very famous Tarski, in the mathematics department. Obviously, the young and unknown Smolin never met the elder logician and mathematician, as he is apparently still in no way aware of any of his work.

Thus, what does Smolin say? Nothing recent. Smolin says mathematics is axiomatic, and develops like games. That was at the heart of the efforts of Frege’s mathematical logic, more than 115 years ago. (Bertrand Russell shot Frege’s theory down, by applying the 24 centuries old Cretan Paradox to it; interestingly, Buridan had found a rather modern solution to the problem, in the 14C!) To help sort things out, it was discovered that one could depict Axiomatic Systems with sequences of numbers. Could not Axiomatics then be made rigorously described, strictly predictive?

Gödel showed that this approach could not work in any system containing arithmetic. Other logicians had proven even more general results in the same vein earlier than that (Löwenheim, Skolem and contemporaries). Smolin is now trying to reintroduce it, as if Löwenheim, Skolem, Gödel, and the most spectacular advances in logic of the first half of the Twentieth Century, never happened.

Does Mr. Smolin know this? Not necessarily: he is a physicist rather than a mathematician (like Tarski, or yours truly).

Smolin: “Both the records and the mathematical objects are human constructions which are brought into existence by exercises of human will.”

Smolin: Math brought into existence by HUMAN WILL. Mathematics as will and representation? (To parody Schopenhauer.)

So how come the minds of animals follow mathematical laws? Dogs, in particular, behave according to very complicated applications of calculus.

http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/do-dogs-know-calculus-of-variation

How come ellipses exist? Have ellipses been brought into existence by Smolin’s “human will”? When a planet follows (more or less) an ellipse, is that a “construction which has been brought into existence by exercises of human will”?

Some will perhaps say that the planet “constructs” nothing. That I misunderstood the planet.

Massimo’s quoted me, and asserted that there was no value whatsoever to the existence of mathematical objects:

I had said: “How come enormously complex and subtle mathematical objects, which are very far from arbitrary, exist out there?”

Massimo replied: “They don’t.”

And that’s it. It reminded me the way God talked in the Qur’an. It is, what it is, says Allah, and his apparent emulator, Massimo. Massimo did not explain why he feels that the spiral of a nautilus does not exist (or maybe, he does not feel that way, because it clearly looks like a spiral). According to Smolin, the spiral is just a “construct of human will”.

If the spiral is a construct of human will, why not the mountains, and the ocean?

I am actually an old enemy of mathematical Platonism. However, I don’t throw the baby with the bath.

I agree that the “Mathematical Universe Hypothesis”, and Platonism in general are erroneous. However that does not mean they are deprived of any value whatsoever.

Ideas never stand alone. They are always part of theories. And idea such as Platonism is actually a vast theory.

MUH is: ‘Our external physical reality is a mathematical structure.’

I do not believe in the MUH. Because of my general sub-quantic theory, which predicts Dark Matter. In my theory, vast quantum interactions leave debris: Dark Matter. That process is essentially chaotic, and indescribable, except statistically (as the Quantum is). propose a completely different route: our mind are constructed by (still hidden) laws which rule the universe. Call that the MATHEMATICAL MIND HYPOTHESIS (MMD).

Here is the MMD: Our internal neurological reality constructs real physical structures we call “mathematics”.

This explains why a dog’s brain can construct the neurological structures it needs to find the solutions of complex problems in the calculus of variations.

Dogs did not learn calculus culturally, by reading books. Indeed. Still they learned, by interacting with the universe. (It’s unconscious learning, but still learning. Most learning we have arose unconsciously.)

From these interactions, dogs’ brains learn to construct structures which solve very complicated calculus of variations problems. As explained by the Mathematical Mind Hypothesis, (hidden) physics shows up in neurological constructions we call mathematics. And those structures, constructed with this yet-unrevealed, not even imagined, physics, are not just mathematical, but they are what we call mathematics, itself. That’s why dogs know mathematics: their brain contain mathematics.

Patrice Ayme’

Technical Note: Some may smirk, and object that my little theory ignores the variation in neurological structure from one creature to the next. Should not those variations mean that one beast’s math is not another beast’s math?

Not so.

Why? We need to go back to Cantor’s fundamental intuition about cardinals, and generalize (from Set Theory to General Topology). Cantor said that two sets had the same cardinal if they were in bijection. (Then he considered order, and introduced “ordinals”, by making the bijection respect order.)

I propose to say two neurological structure are mathematically the same if they produce the same math. (Some will say that’s obvious, but it’s not anymore obvious than, say, “Skolemization“.)

[Last point: I use “neurology” to designate much more than the set of all neurons, dendrites, synapses, axons and attached oligodendrocytes. I designate thus the entire part of the brain which contributes to mind and intelligence (so includes all glial cells, etc.). That ensemble is immensely complex, in dimensions and topologies.]