I Mood Therefore I Think

SYSTEMS OF MOOD ARE CRUCIALLY ENTANGLED WITH IDEAS:

***

***

MOODS COME BEFORE IDEAS:

  The philosopher Foucault became professor at the most prestigious Collège de France in 1970 as a “Historian of Systems of Thought“. That was an admission, by the power that be, that there are such things as Systems of Thought, and that they are most important. I don’t know if Foucault did that much of a good job (I find his analysis of the Franks extensive but rather superficial, and worst, rather conventional; but, at least Foucault had the merit to think that the founders of the West were worth studying).

  The idea the Collège de France had,  of studying Systems of Thought, is crucial. (By the way the CdF was founded in 1530 CE, all its lectures are free, and the professors the foremost world experts.) All comes from there. Even the hardest sciences.

  Just as one studies arithmetic, or organic chemistry, one could, or should study any system of thought, from fly fishing to Islam. They have lots in common.

  Foucault’s “genealogy of knowledge“, was similar to Nietzsche‘s “genealogy of morals“. A colleague of Foucault was Maurice Merleau-Ponty. His phrase: “No preconceived notions, but the idea of free thought” is burned in golden letters above the main hall of the building of Collège de France. But is free thinking an idea, or a mood?

  Ideas are central to logic, but what do they do? They connect notions, that’s all what logic is, and that’s the job of axons in the brain, basically. Yet, the axonal network is only part and parcel of the brain.

  In a related effort at understanding, David Hume held that reason alone cannot move us to action. Action come from passion. Reason alone is merely the “slave of the passions,” i.e., reason pursues abstract and causal relations solely in order to achieve passions’ goals and that reason provides no impulse of its own. (Treatise Of Human Nature.)

  My opinion is more extreme. Just as in Quantum Physics, particle and wave are entangled concepts, logic and passion are also entangled in Brain Physics, at any single moment, or during each other’s blossoming.

 Not only are moods involved in thinking, but moods have to be attributed to entities involved in logic, for conceiving better what is going on. If nothing else, I observed this with top mathematicians and physicists, who I had the good fortune to observe in their natural environment for quite a while.

  These creators view themselves as the most rational people in the world, but they are pretty much dominated by passions, not just as a motivations, but also as a way, the way, of thinking. When addressing terms in equations, Fields medal level mathematicians will talk about, “these guys”. Top mathematicians need to make mathematics into an anthropological milieu, with mathematical terms running around in their heads like little beings, with moods of their own… I would even venture to say that it is this animation of mathematics that makes the top mathematicians: they are at the zoo, herding terms from equations.

  Modern brain imagery and studies show that neurons and neuroglia are entangled deeply together. Clearly neurons embody logical connections, and glias partake in entangled emotional support. Both make (their won, but entangled) networks.

  The mood behind Damasio’s  Somatic Markers Hypothesis, and similar work, supports all this. Damasio pointed out that Descartes made an error by concentrating just on logic, and forgetting emotions in the scaffolding of logic. But I go much further, be it only because I point out that, on (meta)logical grounds alone, emotion, and only emotion, can provide logics with the universes they need to exist.

  Thus we need to dig deeper. To study thought, we need to study the passions, which often come as culturally imprinted Systems Of Mood.

***

AMERICAN ROBOTS DREAM OF FINANCIAL SHEEP; USA WEALTH ADMIRATION MOOD:

  Systems of mood are all over civilization. For centuries, Christians and Muslims screamed:”God Is Great!” Often while slicing each other up. They were both expressing, and reinforcing, a mood. A large part of this mood was apparently that slicing each other up, was the best of all possible worlds. (A more careful consideration shows that the most enthusiastic God Is Great screamers were part of military aristocracies which profited handsomely from the political systems that God Is Great served so well… Thus God/Allah was part of a mutually reinforcing triangle of oppression)

  When Obama became president, he arrived with the mood that financiers were most admirable: his “friend” Jamie Dimon, he much “admired for his gigantic portfolio, which he [Obama] could certainly not manage“.  It’s not just that Obama wants apparently a lucrative job of consultant at JP Morgan. It’s worse than that: he is sincere.

  Dimon was born and raised a financial plutocrat, third generation (at least). Dimon made his most important financial investment in a plot with the central bank of the USA, which was so famous, among banksters, that it got its own name, the “Jamie deal” (buying Bear-Stearns for peanuts, thanks to his always so generous friends, Ms and Mr. American Taxpayers!)

  Obama is still deep in his mood of admiring Lord Dimon.

  On May 15, 2012, episode of ABC’s The View, Obama responded to JPMorgan Chase’s recent $5 billion (or is it 9 billions?) trading losses by defending Dimon against allegations of irresponsibility, saying, “first of all, JP Morgan is one of the best managed banks there is. Jamie Dimon, the head of it, is one of the smartest bankers we’ve got”.

  Notice the imparted mood: Dimon is not just the “first of all“, but “we” all own Dimon as a sort of national treasure… Dimon got the treasury, the Fed, and apparently the president, by the balls (if any), but Dimon “we’ve got”! He is ours! Lucky us: we owned Dimon all along, we just did not notice. Dimon is our man, he works for us. Soon we will dreaming we sleep in the 17 rooms mansion he had in Chicago …It reminds me of the song of the Temptations: “Just my imagination![running away from me]“…

  Well, “best managed” is not the “first of all” of Dimon. On the face of it, very few banks, worldwide, have been as badly managed as JP Morgan. How many banks, worldwide, may have got maybe 100 billion of subsidies from taxpayers? Very few. Out of 8,000 USA banks, or so, nearly none needed taxpayer help. Same in Europe with more than another 10,000 banks. And certainly at most a handful of banks, worldwide got help on the scale of JP Morgan (OK, Dimon, a screamer, screamed that he did not need the help; watch what they do, not what they scream about).

  Obama should, please try to get out of his bankster admiration mood. Dimon is using taxpayer money. That’s the “first of all“, about Dimon, for those who approach the situation with the right mood, the objective mood. 

  Let’s Paul Krugman say it. Dimon is “the point man in Wall Street’s fight to delay, water down and/or repeal financial reform. He has been particularly vocal in his opposition to the so-called Volcker Rule, which would prevent banks with government-guaranteed deposits from engaging in “proprietary trading”, basically speculating with depositors’ money. Just trust us, the JPMorgan chief has in effect been saying; everything’s under control. Apparently not.”

  The key point, notes Krugman, “is not that the bet[s] went bad; it is that institutions playing a key role in the financial system have no business making such bets, least of all when those institutions are backed by taxpayer guarantees”.

  And, a fortiori, when those plutocrats’ heavens use taxpayer money directly, which is exactly what expanding the “monetary base” or “quantitative easing” amounts to. (Krugman did not mention these, because he is partial to them… He has to. But he knows…)

  Someone like Obama is desperately into the mood of believing Warren Buffet is his father, or something like that. Dreams of his father.

  Yes, fathers are important, in the plutocratic universe: Dimon got a gold plated career from the start; his father, a stockbroker, executive VP at American Express helped… Although the fact that Obama’s father was at Harvard, also helped him, no doubt, Harvard having instituted the prerogative of inheritance as part of its global reach of plotting pseudo intellectuals.

  I documented in “Sage of Obama” Obama’s mood of embarrassing adulation of riches. That deep desire to confuse financial wealth and wisdom, shared by all too many Americans (millions of whom partake in calling Buffet, a miserable financial conspirator, who, in a just world, would be the object of a warrant of arrest from Interpol, the “Sage of Omaha“).

  In Mexico, by the same token, we have Carlos Slim, plutocrat, son of plutocrat, and made much richer, as all real plutocrats, by being serviced by the state. Slim bought Telmex, Telecommunication Mexico, from the state, for not much, allowing him now to control now 90% of telecoms there, while charging some of the highest rates in the world. A conspiracy theorist may believe that happened because many politicians and bureaucrats got paid under the table. That is why conspiracy theorists are the enemies of philanthropists.

  Indeed, there again, the only reason Slim is not in jail is that the mood has been carefully sown that he is a “philanthropist“, and that such titans can only be admired (and they could never have conspired to buy Telmex because, just because, we told you, everybody knows, that conspiracy theorists are crazy.)

  Obama tasted of wealth enough when he was a child, to want much more. Something about having four in-house servants… That put him in the mood of respecting wealth. A mood that became much more extensive in the USA after Ronald Reagan was elected king.

  Being a prisoner of such a mood of adulation of the richest, one could not expect Obama to prosecute banksters with the vigor presidents Reagan and Bush Senior had shown with the Saving & Loans conspiracy.

  Contrarily to its ill repute of being cool and remote, science is completely entangled with systems of mood. Examples are found in fundamental physics (Big Bang, Foundations Quantum). reciprocally, science can be brought to bear on Systems of Mood. OGMs and the attitude relative to nuclear energy are two obvious examples.

***

THE NEOLITHIC OUGHT TO BE FELT AS THE REIGN OF GENETICALLY MODIFIED ORGANISMS:

  A tale of two moods. Some are going around, hysterically decrying GMOs, feeling very progressive (the headquarters of the anti-GMO agitation being France, although that may change now that the Socialists are in power). I personally think that any GMO that could potentially, and plausibly, gravely threaten the environment should be outlawed. That’s a good mood to have, indeed.

  And yet, another, even better mood to have, is to realize that, without GMOs we would still be in a pre-Neolithic state. And that Earth could carry, optimistically, only a few million people (and they would be eating each other a lot).

  Indeed nearly all we eat, plants, nuts, fruits, animals, are Genetically Modified Organisms. So we should feel gratified to enjoy GMOs. (The most correct and deepest mood in that arena of thought.)

  Considering that civilization would never have appeared without GMOs, a meta-mood ought to be called upon: to be against GMOs is uncivilized.

  So in connection with GMOs, three moods are justified:

1) Potentially dangerous GMOs ought to be outlawed. (Caution!)

2) No GMOs, no civilization. (Gratitude!)

3) Throwing all and any GMOs out with rage is inhuman, the royal road to total destruction. (Defiance Against Chimps On A Rampage!)

***

FUNDAMENTAL MOOD BEHIND SCIENCE: OFF WITH THEIR HEADS!

  Science has been distinctively unpopular under tyrants. Examples abound: Imperial Rome, which was crafty enough to cover its anti-intellectual mien with extravagant generosity to philosophers obsequious to the plutocratic system. The Catholic Church in the Middle Ages, Stalin, Hitler, were also great enemies of science…

  Science and technologies are often the butt of fierce moods. Some people have written to me of their hatred for the LHC at CERN (which just discovered the Higgs field). Some even identified CERN (a French acronym) with Hitler’s weapon programs, in the vain hope to ruffle me in the wall street Journal comments.

  I will explain in a future essay that the mood against nuclear energy is actually a mood that contradicts the reality that our planet is life giving because Earth is the largest fission nuclear reactor in the universe we know of.

  Once this fact gets to be well known and understood by the world’s masses, no doubt the mood about nuclear energy will change, from revulsion to adoration. Nuclear energy! Our savior! Our creator! Our shield! Our continent churner! Our CO2 storage device!

  Why so much hatred against new knowledge? Because new ideas threaten the established order, which is, first of all, a mental order. The mood that what we know leaves much to be desired, is intrinsically threatening to all and any established authority. If we know more than the authority why is it not us the authority? If we do not ask this question, the authorities certainly will, thus suspect and dislike us.

***

STRANGE MOODS EVERYWHERE, ONE, OR MORE, PER TRIBE:

  The Big Bang is another mood. Never has so much rested on so little. It just, feels good. Just like Genesis. Same mood, Fiat Lux.

  As far as I am concerned, established observations are compatible with a 100 billion years old universe. (Not 13.7 billions! They get to 13.7 billion by macerating the data with a special Big Bang sauce) But of course, the mood among the Very Serious Scientists is that, if you say such a thing, you are ignorant. The VSS are not known for condescending to be fully honest with the public.

  Never mind that Big Bang theory necessitates the Inflationary Universe, zillions of new universes everywhere, all the time. On the face of it, that’s the most insane idea, ever. Well, if you think so, you are just not in the right mood, and we know of no conference nor seminar you will ever be invited to. VSS are not in the mood to talk to you.

  Once I gave a seminar (at Stanford) on Black Holes (in a joint math-physics seminar), and I explained that the theory crucially depended up hypothetical Quantum effects, that I made explicit, and which were usually ignored. Thus the logic had unexamined bifurcations, and the standard Black Hole theory could not be viewed as conclusive. A (future) Fields Medal accused me of “meditation“. He was in the mood of embracing only what it was fashionable to embrace (sure it helped him to get the Fields Medal).

  The Big Bang has a great advantage: precisely because it rests on a great mystery (universes out of nothing, everywhere!) that deep revelation is impenetrable to the masses, and thus unites, and empowers the priesthood.

Along similar lines, the Nicean version of Christianism insisted that 1 = 3 (the mystery of the “Trinity”, justly derided by Arians, Copts, and, later, Muslims).

  The more absurd the belief, the more mysterious, the more distinguishing, unifying and empowering to the oligarchy that holds it. Such is the mystification mood.

  And I do say such a thing, because I lived in many cultures, and I have seen many, where dozens of millions of people are very much into the mood of deliberately believing into something stupid. They are in the mood of imposing upon themselves a crazy mood.

  Why?

  Simply because distinctively outrageous moods define, enforce and encourage an even more rewarding mood, the tribal mood. Tribes made humanity possible. They made the many into a super organism. The tribal instinct is tied to deep psychobiology to make it not just irresistible, but something to crave for.

  This why there are these insane moods supporting the local sport team (whatever sport, whatever team, whatever locale it is).

  The tribal mood is why the British view themselves as living in democracy, while refusing to live in republic, or with a written constitution, and call “Glorious Revolution“, the ignominious invasion that gave rise to the present rather plutocratic regime. Britain: not a thing public (res publica), but public rule (demokratia)? There again we find the mood of the absurdity that binds.

  On a less quaint note, an Israeli commission of eminent jurists suggested to validate all West Bank settlements, even the wildest, and less authorized. In other words, the ancient Israeli jurists are trying their best to make Israel hated worldwide. Why? Because hatred is a mood that reinforces the tribe. Moods within moods. 

***

MORE NUCLEAR MOODS:

  General Electric and Hitachi have applied for a licence to make a Uranium laser enrichment plant, a new technique that would allow to make nuclear bombs cheaper and more discreetly. There was great anxiety about releasing the details. An expert pointed out in the journal Nature, though that the main secret was already out: namely that Uranium laser enrichment worked. The details are less important than the mood: it can be done.

  Similarly, in World War Two, the top Nazi physicists were not in the mood of believing that one could make nuclear bombs, so they did not push for such a program. Whereas the French war Ministry was sure, as early as January 1938, in great part because of (Nobel laureate) Irene Joliot-Curie’s fierce temperament, that a nuclear bomb could be made.

  Similarly, Japanese scientists conveyed to their fascist government the mood that nuclear bombs were possible, and the Japanese military started no less than three different nuclear bomb programs, in an effort to nuke before being nuked.

  And of course, in the USA, Einstein wrote to president FDR, in the summer 1940, conveying his certainty that a bomb could be made (now that the French nuclear scientists had escaped to England). After the war, Churchill, suspecting French nuclear scientists were commies, eager to tell all to Stalin, wanted to jail them all (another funny mood; instead the PM was defeated in elections). In truth, French intellectuals, led, once again by Irene Joliot-Curie, confirmed to their dismay that, after all, Stalin was just another fascist, and were not in the mood of collaborating in any further bomb program, now that the Nazis had been defeated. The French military cooperated with Israeli scientists instead, to develop bombs. Israelis, for some reason, were in the mood…

***

THOROUGH THOUGHTFULNESS STARTS WITH HONEST MOODS:  

  Some will say: “Wait a minute! are you not regressing? Did not Socrates say that the correct way of thinking was by piling up little reasonings such as: ‘Socrates Is A Man, All Men Are Mortal, Therefore Socrates Is Mortal’?

  All I can say is that I have seen lions hunting, and their reasonings, on the fly, were much more clever than that. (The antelopes were pretty smart too.) This sort of reasonings a la Socrates were amusements. A 2 year old can understand them (I enjoy a two year old). The obsessions with these infantile reasoning covered up the truth. Athens’ truth. The truth of the plutocratic friends Socrates lived from, as Rousseau would later live off rich women.

  The truth was that Socrates was a man, because he was not a slave. That was the real mood of Athens, and, to be obsessive about: [(a>b>c)>(a>c)] was just a way to change the conversation, from the mature, to the infantile.

  Fundamentally a contradiction of moods stabbed through the heart all of Athens’ logical systems, just as it would with the Roman republic later, with the same result: collapse. Athens’ principal mood, the mood of the rule of a free people resting upon the mood enforcing the massive enslavement of others, for no good reason, but happenstance, was itself a happenstance waiting for no good.

  Everybody is dominated by moods, but nobody with contradictory moods goes very far. And the same holds for societies. No logic in the world will change that. Why? Because logic always needs a universe in which to unfold. And that is provided by moods (the Incompleteness Theorems in metamathematics say nothing else).  

  Those who want to think better will work on their moods first. It’s harder than to work on ideas. Philosophers will view any, all packaged, already prepared mood, with even more suspicion than an unexamined idea. The unexamined mood is not worth having. Yes, I always lived that way. Early on in life, I acquired the mood of respecting, somewhat, but not trusting, at all, the naïve way the natives felt about their perception of their universe.

***

Patrice Ayme

LARGEST NUMBER

If It’s Physically Impossible, It’s Impossible: THE INTEGERS USED SO FAR ARE INCONSISTENT WITH THE UNIVERSE.

***

Abstract: The senior, extremely experienced, and justly famous Princeton mathematics professor, Edward Nelson, tried to prove that arithmetic was inconsistent. But he assumed something while deriving his attempted proof, which was not true.

I have more basic, and much more drastic claims:

There is a largest number. All and any logic is bounded, and local. Full real logic involves qubits, not bits. Only thus is infinity recovered, through non local methods. A situation with realistic logic exists, which closely parallel that encountered in geometry, before the invention of local differential geometry. Local logic can be integrated, using a connection.

In other words, if you can’t build them, don’t pretend inobservable castles in the air exist, and compute with them, to boot! Basic number theory and logic have to become much more subtle.

***

THERE IS A LARGEST NUMBER. AND LOGIC IS LOCAL.

A well known theorem in primary school is that there is an infinity of numbers. Indeed, suppose there is not, and N is the largest number. Then the number (N+ 1) is even larger, Quod Erat Demonstrandum. 

Simple. That’s what all mathematicians say. But is the reasoning truly valid? Indeed, what is N?

In Cantor’s theory of cardinals, N is the set of sets which have, well, N elements. This is not exactly as circular as it sounds. As John Von Neumann pointed out, one can build up a set with no element (by decree: we just say there is such a thing; it’s an axiom, the axiom of the empty set.)  Symbolize it by 0.

Then we can consider the set whose only element is the empty set: symbolize it by {0}. So when you look inside, inside the brackets, all you see is 0, the empty set. Call that set “one”, or 1.Then look at the set having as elements only 0 and 1. One can symbolize it by {0, 1}, that is: {0, {0}}. Call it 2. And so forth.

N+1 would be the set having as elements 0 and N: {0, N}. This way we get all the numbers and the successor operation, +1. So far, this is standard fare, known to all research mathematicians.

However, suppose G* is the apparent number of particles, virtual or not, in the known universe (using the Planck Length which terminates renormalization, and bounds on energy density coming from bounds on gravitational curvature, one can estimate G*; G* is not infinite because the knowable universe is bounded, be it only because, far away enough, space recedes beyond light speed). Contemporary logic and mathematics have ignored this situation, just like Euclid ignored the fact that he did not have a non local definition of a straight line (although he needed it).

Now in the preceding construction of G*, written only as a symbol made of 0s, and the brackets {s and }s, one gets, on the right hand side of G*, well, a large number of symbols }s, namely G* of them: G* }s! That means one would have as many brackets }s than there are particles, virtual or not, in the universe. But what are the }s made of? Particles, virtual or not.

So just thinking of G* is impossible: G* would require all the particles in the universe to symbolize it.

Some will say: hey, wait a minute, you are confusing mathematics and engineering. In mathematics one generally prove that a would be mathematical object, BAD, does not exist by arriving at a contradiction. Given a set of axioms, AXIOM, supposing the existence of that object, a supplementary axiom, gets to a proposition A such that: A –> Non A.

In other words, honorable mathematical proofs consists in demonstrating that the theory made of AXIOM + BAD is “inconsistent“.

Another thing mathematicians do a lot of, as Terry Tao just did to his professor Nelson, who was his logic professor at Princeton, is to show that a proposed reasoning does not work, because something which was supposed to effect that reasoning, and was viewed as obvious, is not obvious, or is even wrong.

Tao seemed to have found that a sub theory had got to have had a greater Komolgoroff complexity than Nelson had supposed; by an enumeration argument. Nelson’ perfect answer: “You [Terry] are quite right, and my original response was wrong. Thank you for spotting my error. I withdraw my claim [That Peano Arithmetic is inconsistent].”

My main reasoning here to establish the existence of the largest number G*, is the ultimate enumeration argument. One cannot construct (G* +1) because one has run out of matter.

Some will say: ah, but to prove mathematics, one uses only the inner experience, whereas you used a mixed approach. Well mathematicians do the same. Euclid famously supposed a number of hidden hypotheses besides his axioms. For example that two circles intersected. The only way to justify that is through Analytic Geometry (17C) resting on the concept of continuum (19C)… In other words, on the construction of the real numbers, itself resting on the conventional (and as we saw, erroneous) construction of the integers.

To hammer the point some more. Princeton’s Wiles proved Fermat’s Last Theorem by using some powerful hypotheses about infinity. It is supposed to be a heroic task beyond human achievement to convert the proof into first order logic… And, in any case, it is not clear what axiomatics Wiles really used (did he use an “inaccessible cardinal”, in a vital way, or not?) However, as long as the axiomatics is not clear, one cannot assert one has a proof, but just the sketch of one.  

Notice that the main strategy in philosophy is to precisely show that a time honored reasoning does not work, because something viewed as obvious is not actually obvious, or that is actually completely wrong.

But here we have done something more radical. We have a symbol which cannot possibly exist. No axiomatics can build it. How could something one cannot even symbolize exist in mathematics?

The limitations on logical systems are also severe and go beyond simply being limited to coding with a finite number of symbols or numbers. The length of the implication chains and the length of the descriptions of the propositions, themselves or the numbers describing them are all bounded. (So all diagonalization arguments a la Cantor, including all Gödel theorems fail, etc.)

Thus any logical language is limited, there is a limit to any (local) logical universe.

We will call that the Logical Horizon, or Golo Horizon (Golo being the male dominant baboon in West African language; there is only that much that a Golo can understand, due to the nature of his neurological universe).

The situation with the Logical Horizon is analogous to the horizon in a differentiable manifold given by the exponential map. Except here it applies to logic itself. Conclusion: arithmetic, and logic are both local.

(This will have consequences to all domains of thought which use mathematics either technically, or as a source of models or inspiration; that includes philosophy.)

So what happens to the various notions of infinity found in logic? Well, they will have to be reconsidered carefully.

Another notion which can wiped out, is that information is more important than matter: Wheeler famously said at some point that he wanted to reduce physics to information. Or, as he put it, “it from bit“.

This is a bad joke if there ever was one. Wheeler knew plenty of Quantum Physics (he was Feynman’s teacher, and co-conspirator at Princeton, after all). Plenty enough to know his joke was deeply misleading. I am myself often reduced to dubious jokes of kindergarten level such as Bushama, Obabla. “It from bit” is much worse. Whereas the Bush-Obama era is a solid evidence of reducing taxes on the superrich, giving public money to banksters, warring in Afghanistan, throwing away the constitution, and civilization as obsolete, while describing the whole thing as the opposite of what it is, there is no evidence whatsover for “It From Bit“.

All the evidence there is, consists in people thinking that “digital” is superior to “analogue“. True, monkeys have digits, and they are superior, but that’s roughly where the analogy, and the fun, stops. 

“It from Bit” is exactly the erroneous conclusion to draw out of Quantum Physics. “Bit” is an artificial idea. The real world does not have “bits”, anymore than it has “digits”. As we just saw, numbers are very limited.

Any “bit“, the smallest piece of information, is a convened packet of energy. In its smallest form it is the presence, or absence, of a photon, neutrino or electron. So any information stream is actually an energy stream. There is a finite number of bits. Fundamentally, because they are about particles, namely, in my vision of the Quantum, very special manifestation of the continuous Quantum reality.

Reality is all about Quantum Physics, which deals in “qubits“, not bits. Qubits entangle with each other, are non local, and provide with an infinity beyond integers. These three complexities that qubits have, simple bits are deprived of. And of course three complexities to be essential ingredients in non local logic.

Information is made of energy and energy is bounded, locally and to infinity, and so are mathematics and logic.

Dedekind famously entitled his work on numbers:”Was sind und was sollen die Zahlen“. “What are and what ought to be the numbers”. He made the famous commentary:”God created the positive integers, and the rest is the work of man.”

However, we just saw that the constraints of the real world are so strong that the numbers cannot be whatever. Maybe, as god does not exist, it could not even create the numbers. Or is it that man created the integers, but, since he was not god, could not finish the task?

Or maybe we just found a proof of the inexistence of god? Behind this joke is a serious point: the idea of god contained that of infinity. However, we just saw that infinity cannot be obtained on the cheap, by piling up numbers in one spot.

***

HOW LOGIC WILL BECOME LOCAL: THE GEOMETRICAL ANALOGY.

The situation as it is in logic, and as I expect it to evolve, is similar to what happened with Euclid. Euclid stricly made geometry on an infinite flat plane, something which obviously did not exist in his world. Or in any world at all. Similarly we just saw that conventional logic and arithmetic do not exist in any world at all. However, qubits are non local, entangled. That allows us to do the same with logic (demonstration some other time).

Let’s go back to the genesis of full geometry. Let’s suppose Euclid honestly tried to draw straight lines on a sphere. Suppose the Earth was an ideally smooth sphere, and one had a bit of straight line on the ground, Bit(1), and a point X off it. Euclid’s postulates said two strange things.

First that the bit of straight line, B(1) could be extended in a full straight line, L(1). That seemed obvious on the plane, but it was NOT obvious on a sphere (so Euclid spoke of easier things).

To do this properly, Greek mathematicians would have needed to first find the essence of the idea of a line. That was to minimize length. Now ancient Greeks had to find out what lines minimized length locally, on a sphere. As it turns out those lines are what are called great circles.

To figure those out several notions, several subtleties, to extend the notion of straight line to a sphere, a new style of logic had to be introduced,  establishing what is now known as differential geometry. This immense field of mighty subtleties started in the first half of the Nineteenth Century, with the work of Gauss, Bolay and Lobachevsky, but fully blossomed only a century later, with the implementation of Riemann’s program for gravitation by many mathematicians (and to which Einstein contributed enthusiastically).

The notion of tangent vector was indispensable: this is the direction V in which Euclid would have pointed, when at point x on that sphere called the Earth. The great circle tangent to V is the intersection of the sphere with the plane in (normal three dimensional) space containing  V and the vector from the center of the Earth to x.

This can all be demonstrated in various way, the most modern being that the connection on the sphere is the trace of the (“Levi-Civitta”) connection in normal three dimensional space when it is equipped with the normal basic distance known to the Egyptians (the square root of the sum of the squares of the differences of coordinates).

So poor Euclid, trying to extend his bit of line B(1) into a full line L(1), on the sphere, would have been forced to invent geodesics (but that taxed Euclid’s imagination, so he decided to ignore the obvious fact that the Earth was not flat, just like the obnoxious servants of militarized plutocracy nowadays.)

After discovering that great circles locally minimized distance, our imaginary Euclid, if he had tried to implement his fifth postulate (“Through a point y there is one and only one line, L(2), which never meets L(1)”), would have encountered miserable failure. However, the very nature of the geodesics-as-great-circles would have made clear why: great circles always intersect.

The ancient Greeks could have found out much of the preceding. Actually Euclid’s immediate predecessors had introduced the first elements of Non Euclidean geometry, with subtle considerations of various angles in possible triangles. Euclid’s obsessive development of plane geometry was made at the exclusion of the mathematics of his predecessors. It was a rigorous step forward into backwardness.

Why did Euclid do his flat Euclidean geometry, exclusively? Well, I believe, because of the conquest of the Hellenistic world by fascist plutocratic generals of Alexander the Great, who established dictatorships that would last centuries (and similar successor regimes which lasted millennia). A mood set on intellectuals which made it clear that revolutionary thinking was out. And it stayed pretty much out until the European Middle Ages, when the rise of local effective democracy reconstituted progressively the combative originality of the Greek City-States, prior to the Hellenestic degeneracy (while socialized fascism, friendly to demography, but not to revolutionary thinking, installed itself over Vietnam, China, Korea and Japan). 

Euclidean geometry was more fascist than the Non Euclidean sort. After all fascism wants rigid, flat, or, better uninformed, uncritical, unidimensional minds, just obsessed by corporate monetary profits. That is why Tom Friedman publishes best seller after best seller, and editorial after editorial in the New York, while that august publication seemed to wisely decide blocking my comments since the “Occupy Wall Street” movement has blossomed. More than 50 comments blocked already, and counting… It was the same in 2003 with the Iraq war…

Euclid’s geometry was a physical impossibility on the ground, and that should have given a hint to Euclid’s contemporaries (instead of having to wait 21 centuries, for the obvious). But they had other worries.

We have a similar situation with numbers now.  Logic is bounded, finite, and so are numbers, locally. To reach global implications, we have to connect local logics in a global whole.

We have an advantage on the Greeks, to figure more advanced mathematics (and civilization!): we have the Internet, disseminator of truth! And so far just out of reach of the fascist government, in most places. However, have no illusions: so it was with Athens until the well named Antipater took control, after striking a deal with the plutocrats.

***

Patrice Ayme

***

Science Better Accelerate With The Universe

DARK ACCELERATION Of The Universe, QUASICRYSTALS: New Science. NOT ENOUGH TO AVOID COLLAPSE. Yet.

Abstract: The Accelerating Universe is one of the greatest discovery in physics for more than 70 years. It requires new physics to explain it, some sort of anti-gravitation at a very large scale. First established with supernovas, it seems to have been confirmed independently by a galactic survey. “Dark Acceleration” would be a better way to call it than “Dark Energy” (for two reasons: it expands the dark, and we don’t know for sure that it is caused by “energy” in the conventional sense).

This astounding discovery of that the universe is taking off, is typical of how revolutionary science is created. Checking supernovae was just supposed to be routine, a sort of boring science anti-scientists condemn. It was supposed to confirm what the Big Bangists all knew so well: the universe was expanding less fast than in the past, as the Big Bang theory had proven it.

Helas, it was not so! We are faced with something not just unpredicted, but thoroughly unpredictable by conventional physics.

[Some embryonic theories predicted the accelerating expansion, as those I call Dimensional Leakage (they have no official name I know of) and TOW.] 

Revolutionary science is all about finding out the unexpected. Revolutionary science keeps on being found, so we  have faith, we the faithful, the scientists, that much more revoltuionary science has to be found. The Accelerating Universe further bolsters our faith that the present physics with its Standard Model, a noble, useful, fruitful, but naïve and feeble attempt, missed one or more dimensions in the needed logic of what is going on.

How does one find the unexpected? Well, by using the most corrosive logic, propped by the most drastic imagination, and the most careful observations.

The supernova surveys belong to the later, extreme experimental care, and so does the OPERA experiment on neutrinos, and other efforts at CERN. Lesson? All of science has to be funded, if we want to find the unexpectable. Don’t just listen to the fashionable ones. Actually fashionable physics has rigorously not scored for several decades now. It’s not “Not even wrong!“, as physicist pointed out (Their millions of “predictions” are all over the place, so, whatever happened, they said so! Besides, none can be checked!)

Interestingly, the Nobel committee emphasized the point by offering the chemistry Nobel to an Israeli, “for the discovery of quasi-crystals“.

Intelligence is the only thing which can save Israel, and its neighbors, so it is a good indication that the 62 year old nation of seven million got ten Nobel prize winners. This is all the more striking that quasiperiodic patterns are a discovery made by the Islam led civilization, which long controlled Israel and Palestine.

Surviving in this universe we created for ourselves we require us to master the unexpected. So finance the most profitable activity of the government: fundamental research.

When the regime in the USA (the monneyed Congress) cancelled the Super Conducting Super Collider, it saved less than 10% of the bonuses it paid with public money to Wall Street in the first year after the 2008 collapse of deregulated finance.

And what did the Wall Street pirates spent their money on? $40,000 Champagne bottles? Whereas the ever better telescopes use ever better technology which no other human activity requires at this point. $40,000 Champagne has no future, but the new technology used in astronomy does. It could even save our lives (two small asteroids bracketed the Earth this summer, and one of them so close it was severely deflected; the silly ones will say that it does not matter, but if the 1908 bolide had exploded over Wall Street rather than over a desolated part of the Siberian forest, there would have been no more New York).

Let alone all the medical research, say on cancer, which was not financed, because it is $40,000 Champagne which got funded instead. The Medecine Nobel Prize honored progress in immunology. One of the recipients died of pancreatic cancer before the announcement. He had been using his discovery, dentritic immune cells, to activate his own defenses against his cancer. A few days later, the artistic technology integrator of Silicon Valley, Steve Jobs, died of the same cancer, after 7 years of a long battle. Some claimed that Jobs got a liver transplant in 2009 that the average job seeker would never have got. So money is never far from it all. There is clearly need for more biomedical research, for those who prefer their lives and those of their loved ones. But who are they, those who care about life? Who are they relative to those, the immense majority, who prefer, by far, seeing 20 something traders  buy a $40,000 Champagne bottle and then cause with it another $50,000 in damage to a restaurant? (That happened in Paris.)

***

***

EINSTEIN WAS ALL CONFUSED BY THE EXPANSION OF THE UNIVERSE:

I was a bit surprised that the physics Nobel was given for the faster-than-expected expansion of the universe. Not that the discovery was not important. Just the opposite extraordinary claims require extraordinary proofs. Clearly the committee decided to encourage research in the area, by making it more prestigious. Inasmuch as our leading politicians seem cowardly, unable to decide anything important, the Nobel committee has adopted the opposite strategy, bold, fully appropriate.

The Accelerating Universe is the most stunning discovery in experimental physics since, perhaps, the accidental confirmation of De Broglie’s matter waves by two American experimentalists. In the case of matter waves, theory (from De Broglie, in conceptual depths never touched before or since) preceded the discovery. Same thing with antimatter (Dirac predicted it).

But the Accelerating Universe was predicted by no prominent theory. It was an experimental find completely outside of standard theory, just as it was the case with neutrinos.

Neutrinos were also very important, because they meant that there was a new interaction at work, the weak force. Before the neutrinos and their weak force, Einstein knew of only two forces: gravitation and electromagnetism. So he said lots of things about space, and time, trying to tie both of the latter, with the preceding two. As if they could have eyes only for each other. Cute. Romantic. (An integrated 5 dimensional theory, Kaluza-Klein integrated both).

However, with the discovery of the weak force, it looked as if adding one, or more, dimensions was lurking in the distance. Einstein was unable to pursue his dream, and it would be taken again, long after his death, by adding dimensions frantically, using the mathematical theory of fiber bundles.

The basic idea of the Einstein gravitational field equation is:

[Curvature of spacetime = Energy (in) spacetime].

(The idea was spawned by Riemann, a generation before Einstein’s birth.) Both curvature and energy are locally defined, so this is an equation at every point, and it has to be integrated to give a geometry of the universe over a sizable bit.)

Just as it is, even in this grossest of approximations, there are a lot of problems with this would-be equation (the right hand side is not well defined, as Einstein himself remarked, and depends upon the left hand side, as I would perfidiously add).

A problem Einstein saw was that, as his equation was, the curvature would be unstable in time: it would either collapse the universe, or then the universe would have to expand. But Einstein came out at night he saw that the stars were not moving: Einstein believed that the universe was static. So Einstein added a little constant on the left hand side of his equation, the Cosmological Constant, k, to prevent the universe to move.

This was a singular example of lack of imagination, because proper motions were first demonstrated by Edmund Halley in 1718 for the three bright stars: Sirius, Aldebaran, and Arcturus. Halley compared his measurements of their positions to those made by Hipparchus of Rhodes (300 BCE). In 2000 years the motions built up to the point that they became apparent to naked eye observers.

One should have suspected galaxies also moved. Not just that, but Hubble, operating with the Hooker telescope at Mt Wilson, next to Los Angeles, then the largest in the world (until 1948, with its 2.5 meters, just 10 centimeters larger than the Hubble Space telescope), soon discovered that distant galaxies were going away from us wherever he looked, and the further they were, the faster they went. [Speed Galaxy = (Constant) (distance galaxy)].

Einstein coquettishly proclaimed his recourse to k, the cosmological constant, the “greatest blunder of his life“. Because otherwise, he pointed out, he would have been able to predict what Hubble found. (I rather think that the greatest blunder of his life was to abandon his family, but that’s just me.)

I am giving all these sordid details to show that scientific inventions are not always what they are cracked up to be, even from the best and brightest. It was a sort-of-a-prediction, because, in all intellectual honesty, Einstein did not know whether the cosmological constant was zero or not, one way or another.

Actually, if he had been really crafty, the way he liked to be crafty, he could have said:”It is possible, considering my cosmological constant can vary all over the place, that the universe will be found to accelerate more or less, like Hubble saw, or even shrink, on an even larger distance, here or there”. So Einstein made the biggest blunder in his life, twice, with his own cosmological constant, by his own weatherwane standards, depending how the cosmic wind was blowing through his brain.

Because an Accelerating Universe was found: type 1a Supernovas at large cosmological distance flee faster and faster from us.

***

POSSIBLE CAUSES OF DARK ACCELERATION OF THE COSMIC EXPANSION:

Of course installing an Einstein’s Cosmological Constant in Einstein gravitational equation is purely descriptive.  Nobody has any certainty about its source. The Cosmological Constant depicts DARK ACCELERATION. I call expansion beyond the Hubble prediction, Dark Acceleration, because we don’t know what it is due to.

People use interchangeably “Dark Energy” and what I call Dark Expansion, but the concepts are different. Dark Expansion is a fact, but it is not necessarily due to “Dark Energy”.

So “Dark Energy” is an abuse of language. Actually there are at least two imaginable mechanisms where Dark Acceleration is not caused by Dark Energy.

The DARK ENERGY idea proposes that somehow energy is injected in space which allows it to expand faster. Imaginable explanations for Dark Energy could be:

VACUUM ENERGY, which has been brandished as the source of the Dark Expansion. In Quantum Field Theory, the vacuum is full of energy. Nobody knows how much. All we know is that there is some energy in the vacuum (we have some indirect theoretical-experimental proofs, and direct measurements such as the Casimir effect). Evocations of “Vacuum Energy” are generally not accompanied of suggestions for a mechanism to expand the universe with it.

What do we observe though? A faster expansion. It could be due to a weakening of gravity at large distances (interestingly, inside galaxies and galactic clusters, gathering of the mysterious Dark Matter makes gravity stronger).

A mechanism to weaken gravity has been proposed by suggesting that space has more than three dimensions, and that gravity, somehow, would be leaking in one or more of these dimensions. One could call it DIMENSIONAL LEAKAGE. Dimensional leakage has also been proposed to explain the possible supraluminal speed of the very high energy neutrinos coming out of CERN (high energy neutrinos would jump into an extra dimension which shortens their trajectory through the “bulk“).

A final explanation for accelerated expansion could be TOW (Totally Objective wave), the author’s pet theory. TOW rests on the idea that its (hypothetic) Quantum Interaction proceeds at absolute speed TAU (more than ten billion times the speed of light!) Even though, the Quantum Interaction is overwhelmed by large cosmological distances: when a graviton, coming from way too far, singularizes itself, it loses part of its energy. Thus, according to TOW, gravity should weaken at large distances (just as light does).

The fact that there is no ready explanation for the Dark Acceleration shows that the hubristic crowd sing-songing on the rooftops about the “end of physics” a while ago, did not have much imagination. (Feynman was told about TOW, and was very appreciative, by the way.) 

***

VINDICATED, A RIDICULED SCIENTIST WINS NOBEL:

When Israeli scientist Dan Shechtman claimed to have stumbled upon a new type of crystalline structure that seemed to violate the then known of the laws of nature, his “peers” and some giants of chemistry (Pauling) mocked him, insulted him and exiled him from his research group (“Danny, go away!“). “I was thrown out of my research group. They said I brought shame on them with what I was saying, I never took it personally. I knew I was right and they were wrong.”

Indeed, he just received the 2011 Nobel Prize in chemistry.

The lesson? “A good scientist is a humble and listening scientist and not one that is sure 100 percent in what he read in the textbooks” Shechtman said. I would add that the greater the thoughts, the fewer the peers.

The shy, 70-year-old Shechtman said he never doubted his findings and considered himself merely the latest in a long line of scientists who advanced their fields by challenging the conventional wisdom and were shunned by the establishment because of it. And the greater the idea, the greater the shunning.

In 1982, “Metallic Phase with Long-Range Orientational Order and No Translational Symmetry” by Dan Shechtman et al. demonstrated “Order with No Translational Symmetry”, the key here. Translational symmetry is what Pauling wanted to see, because he learned it in his kindergarten, way back when. That, or no symmetry at all, namely a glass, as Shechtman had expected to find. But Shechtman had serendipitously discovered what are now called “quasicrystals” – atoms arranged in patterns that seemed forbidden by nature…

Although they were clearly authorized in Islamic art since the Middle Ages… which should have been enough of an hint: if even the Islamists allowed them, assuredly their existence could not be denied. True, at that point Islam was very open minded, and early in that “Golden Age”, most of the thinkers were actually not Muslims, but Jews, Zoroastrians, with probably a vast complement of atheist Neoplatonists. Theocratic fascism, as among the Franks, would grow later (and simultaneously with the Franks, as fascists, on both sides, realized that the Bible was an inspirational celebration of holocausts and other injustices that kept on being rewarded in high places, and thus provided business opportunities).

Quasiperiodicity was recorded from an Al-Mn alloy which has been rapidly cooled after melting (which probably means the quasi periodicity is higher energy than full periodicity).

The art in Isfahan (a fantastic city I highly recommend, by the way, not just artistically speaking, but for the presence of immensely old wisdom breathing through the stupendous beauty displayed in mosques and other buildings) showed that quasicrystals were logically permissible. They preceded the work of the British mathematician Penrose by nine centuries, and, definitively constitute the original discovery.

It seems pretty obvious to me that the mere possibility of these, as exhibited in the mosques, did most of the conceptual work. And how could not chemical bonds glue all the atoms all together, once we had showed the quasiperiodic pattern was possible? It’s not quasicrystals which were surprising, but how people could think they could not exist. 

The discovery “fundamentally altered how chemists conceive of solid matter,” the Royal Swedish Academy of Sciences said. It redefined the notion of crystal in textbooks. Previously, a crystal had “a regularly ordered, repeating three-dimensional pattern,” according to the International Union of Crystallography. The new definition, adopted in 1992, states that a crystal is any solid with a “discrete diffraction diagram” — that is, something that produces patterns, whatever a pattern is, just as Shechtman saw.

Quasiperiodic tiling, Masjid-e-Jameh, Isfahan

Since Shechtman’s discovery, more quasicrystals have been found. A Swedish company found them in most durable steel, used in products such as razor blades and thin needles made specifically for eye surgery, the academy said. Quasicrystals are studied for use in new materials converting heat to electricity.

They have also been discovered in nature, in a Siberian river. As it seems obvious that they will prevent the propagation of fractures, one may expect to see plenty of quasicrystals in a future near us, except of course, if the banksters devour our civilization first, like they tried last time and the time before that, with their dangerous marionettes.

Quasi periodic structures are all over nature. They depict the subquantum world. Indeed the waves produced by dropping four or more stones into a pond always form a quasicrystal (or more exactly a quasiperiodic pattern), because there is a mathematical theorem saying this. Schechtman was aware of the  theorem, and when he saw the 10 fold quasiperiodicity, he knew.

Matter waves continually interfere, creating quasi periodic existence waves all over.

***

SCIENCE: EVERY BODY’S BUSINESS. OR THE WORLD WILL GET OLD:

We are living in a scientific civilization, whether we like it or not, whether we are superstitious or not. The French ministry of ecology (headed by an experienced politician, a young and charging polytechnicienne who is piling up elected offices) just forbid by decree (executive order) fracking all over France (the National Assembly had already voted in that direction). The reason? Existing techniques have not demonstrated that they were ecologically sound, in the fullness of time. In other words, it was a purely scientific decision.

In the USA, fracking is practiced massively: whether the technique will lead to corruption of the aquifers is of no import, because, as Suskind’s book “Confidence Men” demonstrates, corruption is of the essence, and, now that it owns the White House, it may as well own the dark underground.

The essentiality of serious science is funded by states, as there is no monetary profit in it (learn, Tea Party!). But the science we have is not enough: we will soon be using several times this planet sustainable productivity, so we need to become much more efficient. This can be achieved only by considerable scientific progress, in all domains, from plate tectonics, to material science, to the most esoteric biology, to quantum algorithms, to whatever.

I say “to whatever” because there is no telling where the scientifically and, or, socially important breakthrough will occur.

For example nobody would have guessed that surveying supernovas would smash the hubristic certainty of those prestigious scientists who recited their new religion, the Theory Of Everything, the claim that they, the glorious ones, had figured out the entire universe (just like the proverbial god, before he got to know man). Not by coincidence, the peak of their hubris was around 1998, just when the accelerating universe started to make its presence known.

We are in scientific civilization, thus those who are anti-science are actually anti civilization. This is true even in the fossil fuel industry.

Even that fossil pursuit, trying to keep the fire going, as Homo Erectus already did, a million years ago, is scientific. Ever more primitively scientific, but still scientific, and involving many scientific issues, some potentially disastrous. Hydrocarbon burning is a massive scientific bet that the recombining with oxygen in the air of much of the hydrocarbons buried over 400 million years, will have no serious adverse consequence(s).

The evolving scientific evidence is that the consequences are many, and potentially extremely cataclysmically adverse. But as too many people in the leading countries are trained for superstitious, or sport analogy reasoning rather than scientific reasoning, especially in the USA, nothing much is done: other countries cannot afford to become uncompetitive with the USA and its 10,000 Chinese factories.  

Those who are not pro-science, being anticivilizational, are actually pro-world war, and pro-holocaust.

Should science not jump ahead, very soon, the coming holocaust is easy to compute: soon, on present trends, we will be using more than two sustainable earths, with eight billion people gnawing the shrinking resources (we are 7 billion now). Thus, if we do not augment our science considerably, we will have to cut the world population by half. 

Let’s insist on that point: we are exhausting the existing resources. For example we are well post easy-to-extract peak oil. The only reason we are not past vulgar peak oil, is that we are using increasing energetically expensive (energy is the only currency that counts by itself) and technologically expensive means.

The Romans did the same in their mines, with ever more slaves pushing ever more their primitive digging technology, to its bitter end, devastating the ecology for millennia. And they persisted, until they could not anymore. Exhausted, the mines closed for nearly 2,000 years. The Romans had no plans for that event. Nor did they have plan B.

Rome had been most technologically inventive as a republic. That’s how it vanquished everybody.

Having captured a Carthaginian ship, they, those Roman peasants, reverse engineered it, and made invincible fleets of ships. However as Rome progressively degenerated in the fascist dictatorship known as the Roman empire, innovation was the first victim, as proven by the fact that the Germans and Hellenized Persians became increasingly hard to beat on the open battlefield… because they had superior weapons (in particular, composite bows the arrows of which penetrated Roman armor).

Plutocratic, fascist imperial Rome did not want to understand that it was running out of science, considering the problems it faced. Plutocracy want the people to owe them, and the last thing it wants is to owe the people, and especially ideas! The official line in Rome was just to whine that the “world was getting old“. Rome was running out of resources, among other problems, its ore mines exhausted. When the Muslim army invaded, Rome needed to melt the metallic roofs of Rome to make weapons.

The master problem was of course that Rome was running out of moral, and thus intellectual, supremacy. When one treats one’s engineers, and engineering, badly, one runs out of engineering badly. (Something one can observe nowadays in the USA.) 

A similar shock between the demands of society and insufficient science and technology happened, roughly at the same time, to the Mayas (who confronted a dreadful drought). The Mayan ecology, construction technology, hydrology, agriculture and forestry science, although all sophisticated, and established for centuries, if not millennia, all came all too short, considering the crisis. Plutocracy got all enraged, and fought against itself, the way sharks do when they run out of food, and Mayan civilization imploded.

Science is about what really is, and why. With (more or less great) certainty. Thus science creates models and theories of great explanatory power, which can be emulated in other domains (sometimes simply as metaphors). Science transforms confusion and, or, phenomenological wealth into an harmonious explanatory whole. In other words, it can be inspirational, a leader to democracy, sociology, economy, even literature, poetry, etc.

It would be a dark future without new, really revolutionary science. Quasicrystals and the accelerating universe tell us, with certainty, that much revolutionary science is still to be discovered. Science has to pursue its dark expansion in the unknown, emulating the universe which harbors it. No choice.

***

Patrice Ayme

Final PROOF OF INCOMPLETENESS.

 

IT’S ALL IN THE MIND, AND THE MIND IS LOCALLY COUNTABLE, BUT THE UNIVERSE IS NOT.

***

In a nutshell: The discovery of a general theory of incompleteness, in the last century, is one of the greatest advance in civilization, 25 centuries in coming. It ought to have a gigantic impact on general human understanding, and action, greater than any other scientific theory, but it has failed to do so, so far, because it has stayed all too esoteric.

I give a new, neurological approach to incompleteness, designed in part to remedy this. Verily, mathematics is not out there, but thoroughly inside the mind (this contradicts Plato). Just as symbolic systems are limited, so is the mind, in the same exact way, it turns out (although the mind found a way out of this limitation). And the limit is countability. And that’s where incompleteness comes from.

***

Introduction and abstract:

Incompleteness is a common characteristic of all axiomatic systems (Turing demonstrated this by a variant of Cantor diagonalization). Therein a formidable weapon against human hubris. Too bad so few philosophers, and, a fortiori, politicians, have heard of it. (If the politician realized how incomplete his mind is, he may think enough before going out, and killing innocent people with robots, to the point that he may find other ways, like talking to people with the discourse that kills the conflict, rather than the babies.)

In any case, incompleteness has been hard to understand, because it is a major advance in understanding, and, as all such advances, it leaves the savage mind behind. New, and fresher generations will be exposed to incompleteness early, and find it as natural as zero. The concept zero. How? By finding, as usual, a natural approach to the existence of the new concept, and how to build a reproduction of it in one’s mind, hard wired from the start.

Indeed, Zero is, because zero does. Zero does what nothing else does, namely introduces nothingness into the computational realm (take that, Sartre! Initially the Babylonians, 3,000 years ago, just left an empty space for zero. Later they put various marks.)

This is how, and why, people find the concept of zero so natural, and many other basic mathematical concepts, so natural, although, well, they took the best minds millennia to develop. People find them natural, because the new concepts do natural, and helpful things. It took millennia to find the correct approach to zero. In the end, zero is an axiom. But, a very useful one. We are going to do the same thing for incompleteness in this essay, with a simple observation which has vast consequences, and will be taken as completely obvious in the future (just as zero is now obvious, except zero is more of a convention, and we are going to make an observation).

To make a new concept self obvious, one always has first to find the correct, that means the simplest and most elegant approach.

Godel’s work on incompleteness was complicated in its details (because he imposed onto himself a minimalist setting, working only with integers).

Much less complicated is Cantor’s much more crucial breakthrough (reproduced below), and others’ work on incompleteness (Turing, Chaitin). Complication went down, as intelligence went up.  Therein this essay a new demonstration of incompleteness in mathematics, using the (author’s) neuromathematical approach (taken for granted here as a background, an admittedly unfair but necessary short cut).

***

NEUROLOGY IS MORE SOPHISTICATED THAN EXISTING LOGIC:

The neuromathematical approach claims that any mathematical theory is a neuronal geometry (you can call this an axiom, if you wish, but, one day it will be proven in the lab in minute detail, so it’s a conjecture, Tyranosopher’s conjecture). This goes much beyond the usual theory of neural networks, which has no geometry, and the simplest of topologies (mathematical semantics is used here, common readers can ignore it, or look up Wikipedia).

Neuronal geometry is given by a METRIC, itself given by the time it takes to process neuronal logic (see annex; dendrites, neuroglia, synapses, firing rates, and the finite speed of action potentials are involved in this delay in communication, hence in the distance function; neurological signal speed plays the role of the speed of light in physics, which is also the distance function).

The crucial point is that any neuronal geometry rests on a countable network. Hence it is clearly incomplete. This is the essence of my argument. One cannot make simpler than that.

Of course some will smirk that it cannot be that simple. How to generate the richness of mathematics (let alone poetry!) from this madly simple picture? Well the point is that it is not that simple, it’s an immensely complicated configuration space of very high dimension (from neurotransmitters), endowed with geometrodynamics (right there, it’s much more complex than General Relativity, a very simple geometrodynamics, with just 4 dimensions and a fixed topology).

The geometrodynamics allows each neuronal geometry to morph into a neurology next door, topologically inequivalent to it (with a different genus; so neurology is also endowed with a topologicodynamics, differently from the much poorer General Relativity, which is stuck with just one topology). This is how the space of all neuronal geometries can mimic (what Cantor called) the power of the continuum (see below).

***

THE IMPORTANCE OF INCOMPLETENESS:

The realization that Incompleteness Is A Non Compressible Feature Of Understanding has been a major philosophical and scientific advance (arguably the greatest, and not just of the twentieth century). It has been a new notion, so enormous that it surfaced slowly over 25 centuries (!). Only now have we reached a final understanding of what is going on.

In its modern version, due initially to Gödel, incompleteness showed up as the Gödel’s incompleteness theorem, which states that the theory of numbers includes undecidable propositions. So propositions exist that can neither be proven, nor disproven (this is similar to the parallel postulate being neither proven nor disproven from the other axioms of Euclid; it’s just saying there will always be the equivalent of the parallel postulate, propositions that can neither be proven, nor disproven, from any previous set of axioms, in any thought system.)

***

HOW THE GREEKS STUMBLED REPEATEDLY OVER INCOMPLETENESS: LIAR, IRRATIONALS, PARALLELS, ZERO.

Incompleteness is the opposite of the all knowing god, it shows that one such being could never be (that is why the Greeks had gods all over, probably, they were smart, they guessed the truth). The first inklings that something was amiss in the theory of human knowledge came from (first) the paradox of the liar. The paradox of the liar surfaces in self referential statements that make logic literally short circuits. An example of the liar paradox is the statement: "this sentence is false!" Indeed, if it is true, it is false, so it is false, but that is true, etc.

It sounds stupid to worry about such things, but it is not, when one tries to establish perfect logic as some Greeks, and later, philosophers in Paris around 1100 CE, tried to do (the philosophers in Paris, being also theologians, were trying to find the thinking of god, which had to be perfect, hence their great rigor).

The paradox of the liar resurfaced brutally in the heart of mathematics in 1900 after Friedrich Frege wrote down what he thought was perfect axiomatics for arithmetic. Bertrand Russell found the paradox of the liar just below Frege’s surface, causing a serious crisis (which is not fully resolved yet: when set theory is taught, what is taught is so called "naïve set theory", which ignores most serious problems, see annex, where it is revealed that the foundations of mathematics are rather fluid…).

Another way in which incompleteness appeared, a little while later, was with the apparition of irrational numbers (as they came to be known). The Greek mathematicians, building up on the work of their Babylonian and Egyptian predecessors, thought they had a full axiomatic of arithmetic, with just their pathetic little integers. They had connected that with their axiomatization of geometry, through the concept of length.

The Egyptians lined up their pyramids perfectly with the true north (within three-sixtieths of a degree), thus demonstrating they knew how to measure stuff. Geometry was used mostly to determine property extent, obviously important in the periodically flooded rich arable land of the Nile valley. It had come to be that numbers were used to measure length, and both concepts had been identified, through the concept of ratio of integers (giving fractions of a measuring unit).

Pythagoras, a Greek in Southern Italy, proved the theorem that the square of the hypotenuse was equal to the sum of the squares of the sides of the triangle. Soon after, he and his students found that the diagonal of a square of side one, whatever it was, was not the ratio of two integers. It could be calculated with an arbitrary precision, but the process was never ending.

The Greeks had thought that all numbers were "rational numbers", which supposedly made sense because they were… well, may be, not numbers that one could count on one’s fingers, the integers, but, at least, ratios of integers.

Thus the notion of rational number was incomplete, in the following sense. The Greeks had hoped that all and any length was a number, AND they had also hoped that any number was the ratio of two integers. That was a lot of hope they wanted to believe in. Suddenly the world was not something the Greeks mastered anymore.

If one made the esthetic decision that a number was always a ratio of integers, as the Greeks did initially, then, they found to their dismay that not all lengths were numbers. But then, if one made the decision that all and any length had to correspond to a "number", the notion of number had to be extended, beyond "ratio", to include all the hypotenuses of all and any triangle (thus the ir-ratio numbers). That was philosophically maddening, completely, well, irrational. Indeed then what was a number? Were there still other definitions extending further the notion of number? Where did these extensions stop? OK, so suppose, as the Greeks ended up doing, that any length was a number, and that so was any ratio of lengths.

According to this new definition, the ratio of circumference to diameter of a circle, named pi, is a number. Could it be represented as a length obtained by Greek instruments, line and compass? If not, how to compute it?

Meanwhile the Greeks stumbled on the concept of zero. Instead of completing their mathematics in that direction, they passed the concept to the Indians (who, with a more numerically aggressive religion, friendly to big numbers, were not afraid to develop it, while using a more advanced notation perhaps following the Chinese).

Meanwhile the postulate was made by Euclid that one could NOT deduce from the rest of Euclid’s axioms whether one, and only one parallel to a given line passed by a given point exterior to it.

This so called "parallel postulate" nagged the Greeks, and everybody else, for 2,100 years (it was not as “self obvious” than the other axioms of Euclid, so people had a feeling that it ought to be a theorem, hence demonstrable form the “self obvious” axioms).

It should not have nagged all mathematicians for 2100 years, but it did. People tried, in vain, to prove the postulate (from the other axioms). Finally, starting in 1829 with Lobachevski, geometries were found that satisfied all of Euclid axioms, except the parallel postulate. (Exotic geometries were so scandalous that Gauss claimed he did not publish his research because he feared the "cries of the Beotians", the Beotians were peasants north of Athens known for being civilizational retards.)

Exotic geometries should have been obvious, as long as one had made the esthetic, not to say hedonistic, decision to make Euclidean geometry on a pillow, or a saddle, or a sphere…

Why to restrict oneself to a plane? Was the world flat? No, and the Greeks knew it was not, they even had measured the size of the Earth with great precision (so big was Earth’s size that Columbus was not believed when he said he could sail to China, because it was known that China was beyond the range of existing sail boats… But America was not, and the Vikings had traded ivory, and even timber, from there for 5 centuries…)

So it was irrational to restrict oneself to flat geometry (and Euclid’s predecessors knew this, but as the world veered into Macedonian fascist domination, full blown thinking became the enemy of the sovereign, and thus was forced to adopt a low profile: in a world where Gold Man Sacks, little men learn to be stupid servants).

***

 

NEW GEOMETRIC MODELS COMPLETED THE PICTURES:

In the meantime, "numbers" which, when multiplied by themselves gave a negative product, had been found to be useful to solve equations. Those "imaginary" numbers led to real solutions, and had many other esthetic advantages. For example any polynomial equation of degree n had exactly n roots (d’Alembert’s theorem). Very pretty, very handy. Finally a magnificent, trivial and beautiful interpretation of "imaginary" numbers was found (1806). That was more or less coincidental with the discovery by Faraday that a moving magnet created an electric current (1821).

A stupid journalist asked Faraday what was the use of that effect, to which the great man replied: "What is the use of a new born baby?" (1821.) Faraday law of induction is of course at the basis of all of the world’s industry now. Tellingly a madly rotating turbine in an electric power plant, or windmill, describes a geometry that exactly depicts imaginary numbers (Argand’s diagram, 1806). So an extension of the concept of number that would have driven the Greeks completely mad, at first sight, had a natural geometric description… Simply geometry was not just about the technology of line and compass. Now we have the technology of turbines, and, or Quantum Mechanics, and those are all about “imaginary” numbers, which are not imaginary anymore than turbines or Quantum Mechanics.

(Both turbines and Quantum have to do with electromagnetic waves, that’s their connection: the famous 2-slit experiment, in optics or electromagnetism, is also the basis of Quantum Mechanics,)

So the parallel postulate was solved by being more open to what one meant by geometry… In general understanding further is similar to what happened with the parallel postulate: suppose more stuff, to get richer abstractions, abstractions that can do more. The world is rich, a richer mind, correctly made, can model it better.

Then it turned out that pi, although calculable (its square being an infinite series of predictable rationals), was transcendental (that is, it was not the solution of a polynomial equation).

***

THEN CANTOR INVENTED THE CANTOR DIAGONALIZATION PROCESS:

Mathematicians massaged the Cantor diagonalization process for the  century that followed its establishment in Cantor’s mind, extracting juicy theorems and spectacular results from it (with generally trivial proofs, see annex). It is very simple.

Cantor supposed that all the real numbers could be counted like sheep, from top to bottom, and so he lined them across, developing each real number in its full decimal expansion horizontally. OK, the bottom was down to infinity. That gigantic array he obtained is also called a matrix.

Cantor ended visually with a gigantic matrix of integers, let’s call it the CANTOR MATRIX. Cantor labeled that gigantic matrix as R(n, m): R(n,m) being the mth integer in the decimal expansion of the nth real in the Cantor matrix.

Then Cantor built a real C by considering the diagonal R(n,n) of his giant Cantor matrix. He defined C by giving an algorithm for its decimal expansion, namely a way to compute C: the nth digit of C would be C(n) where C(n) would be R(n,n) plus a (perhaps variable, or not) non zero integer. To define things precisely, say: C(n) = R(n,n) + 1. In other words, the nth decimal of the made up number C can never be the nth decimal of… any number: at this point, the conclusion is obvious: C cannot exist. But let’s pound it, the way mathematicians like to do.

Indeed, since Cantor had supposed that the reals could be lined up like sheep, the number C ought to be in the list, as the kth (say) number. In other words: C = R(k). Hence we should have the nth digit of C, C(n), equal to the nth digit of R(k). But that is R(k,n). But C(n) was constructed to be R(n,n) + 1. In particular, C(k) is then both R(k,k) and R(k,k) +1. So, either 0 = 1, or the initial hypothesis at the root of the whole contraption, that it was possible to build the Cantor Matrix, containing all the integers, was FALSE.

A more intuitive way to look at the proof is this: suppose each number is viewed as a mountain range, each point in the decimal expansion being viewed as an altitude, anything between 0 and 9. A Cantor mountain range is made up, by modifying one of the heights of each mountain range at some point, and gluing all such modifications along to obtain a mountain range guaranteed to be different from all those lined up initially. This means that the Cantor modification is geometric in nature (height being a distance). As we will see neurology can do more, because it can not only make geometric changes, but topological ones (changing its genus with all sorts of surgery) .

What has exactly happened here, in this Cantor diagonalization trick? Well, I claim, something neurological happened.

***

FORMAL INCOMPLETENESS:

Now go forward another generation or two to Godel and Turing. Godel demonstrated that, as long as one had basic integers, with multiplication and addition, a sentence could be made that would say:"I am not demonstrable".

Turing generalized this, and Chaitin, generalizing in turn ideas of Leibnitz and Borel, found a probabilistic approach. Borel had observed that chance could not be defined, because, if it were, it won’t be chance anymore. This may sound too philosophical, but, remember, mathematics is about philosophy. Or, as I point out, neurophilosophy.

***

INCOMPLETENESS IS HOW WE REACH FOR THE STARS:

For 2,000 years, mathematicians were mystified by parallels, but all they had to do is look at any curved surface to realize that they were mystified erroneously: the problem was not what parallels did, or did not do, but how they should be defined. Same for the concept of numbers, same for the concept of chance.

The paradox of the liar was a big subject of (non trivial) reflection in the depth of the Middle Ages, between Paris and Oxford (circa 1100-1400 CE). It is still alive and well; Bertrand Russel used a variant of it to show that the axiomatics of mathematics were self contradictory (circa 1900). He considered the set B all of which elements are sets which are not elements of themselves. Now if B is an element of B, it is not an element of B. And if it is not an element of B, it is an element of B.

Godel used a variant of the liar argument.

OK, so what is the verdict? Can we progress by introducing much more powerful semantics and abstraction? What do I mean? Imaginary numbers were hard, until it was realized that they corresponded to rotations in the plane (Argand diagrams, rotating electric fields). Then that a number multiplied by itself could be minus became trivial. Similarly, curved geometry, irrational numbers, became obvious, once looked at the right way. This is the case for the zero, or negative numbers, everybody take those for granted.

Abstraction consists into forgetting the details, and concentrating on an essence, which becomes the new definition. Incompleteness is made to work in reverse.

Neurologically, abstraction corresponds to establishing a shorter, less energetically and less temporally costly neural network going more directly to the meat of the matter (don’t forget you are dealing with the mind of a killer ape, meat is where it’s at).

It is my opinion that pieces of mathematics correspond to subsets of neuronal architecture (I should say neuroglial architecture, because glial cells, which make up 90% of the human brain, are involved). Any of subset of neuronal architecture is countable (actually, although large, it is finite, say involving a skeleton of at most a of a trillion trillions pieces of networks (yes, ten to the power 24), counting everything, even dendrites). So basically a mathematical reasoning is a neural network (a subset of all paths possible with a trillion trillions pieces). But the neural network can be changed in a non trivial way, TOPOLOGICALLY speaking (topology is the science of neighborhoods, forgetting about distance measured by number: distance gets measured only by the notion of neighborhood -literally, not by a number).

Although any given neural network is countable, it can readily morph into something completely different, geometrically, or topologically. [Neural] countability is next to the infinity of the continuum. According to me, this is the essence of Cantor diagonalization: any countable array gives rise to elements not in it.

And it is the essence of incompleteness: any mathematical theory is, by essence a COUNTABLE neural network, and thus misses most of math. Realized mathematics will always be of measure zero in the set of all possible mathematics.

Notice that the neural networks can vary geometrically (which is in a way what Cantor did), but can also do much more, because they can morph into some which are not topologically equivalent (and do this all the time, since their connectivity varies through new neuronal, axonal, dendritic and glial geometry).

Now, of course, if even any mathematical reasoning is that incomplete, a fortiori all and any reasoning. Thus the preceding result has impact on all knowledge and cognition.

***

LIAR AND NEUROLOGY:

And what of the paradox of the liar in all this? (Another version of it is: "The following sentence is true. The preceding sentence is false.")

Well, Russell solved it with his theory of "type", a hierarchy avoiding self reference. I think the solution is just to realize that neurology has a hierarchical organization that can be called "meta" (loosely corresponding to Russel’s hierarchy, or one founded by Von Neumann, which starts with the empty set, then the infinity, or inductive et axiom: if y is in it, so is y U {y}).

"Meta" enables abstraction. It basically consists, given a neuronal set S into a set of higher neurons, H, which, observing the quasi-simultaneity of some sorts of firings, draw the consequences, in the form of new axonal chains between S and H that short circuits the long axonal chains confined to S (this corresponds to the logician Alonso Church’s definition of abstraction).

Say neuron A, after a long chain of intermediaries, makes neuron Z fire always; then a neuron B appears that connects directly A and Z, shortening the axiomatic/program structure: such is the abstracting process, reproduced in mathematics by forgetting (some of) the details (of course, it is the same abstracting process which is used all over).

The liar paradox disappears in neurology, because neuromathematics eliminates self referential loops. These cannot happen (neuro)logically (neurons don’t short circuit, be it only because they cannot fire immediately again, let alone the fact that neurology sees no interest in close by, loopy circuitry).

***

INCOMPLETELY YOURS:

In any case, such is my resolution of incompleteness. All and any theory is countable, but the universe is not. And neither is the mind (thus the mathematics). The related liar’s problem is done with by the geometrodynamics.

Some will say: what sort of proof is that? But what is the proof of zero? Or the proof of irrationals? Or the proof of hyperbolic geometry? Of course, there is none, they are just choices, and then observations we make in life.

I realize this is incomplete, and (not yet) demonstrable in its entirety, but, as I was saying, all and any theory is incomplete…

***

Patrice Ayme

***

1) Annex on why neurology has intrinsic geometry: Generally, neurology is viewed as set of neural networks. Neural networks are almost trivial things: a directed graph with edge weights, and perhaps a "transfer function" at each vertex. The interesting content is in algorithms that progressively improve a solution to an inverse problem — calculating edge weights that result in desired couplings between input and output edges. The picture here goes completely beyond that, since NEUROLOGY BECOMES VARIABLE GEOMETRY, AND EVEN VARIABLE TOPOLOGY.

indeed, neuronal logic incorporates a temporal hierarchy, given by the time it takes to process the logic. Neurology, among other things, is logic + time delayed causality (notice the analogy with special relativity, and, or field theory, be it electrodynamics or gravitational, where a crucial point is delayed, hence local, causality).

Neuronal geometry is thus given by the time it takes for logical processes (neuronal firing and the propagation of signals down axonal-dentritic-glial chains is far from instantaneous, because not only are nerve impulses slow, but the signal is reprocessed along the way, with typically a glial cell’s foot interfering into each synapse, which is itself, all by itself, a geometrical computer).

*

2) Annex on how the debate progressed in Paris circa 1100 CE: The notions above address, and are an attempt at solving, once and for all, directly the debate between realism (Champeaux, Archdeacon of Paris, teacher of Abelard) versus nominalism (Roscelin, preceding teacher of Abelard), versus conceptualism (Abelard). Those thinkers, circa 1100 CE, all knew each other, and were busy going well beyond Aristotle’s metaphysical uselessness in the debate on "universals" (ideas). Champeaux thought "universals" were real, out there (a position started with Plato, I guess, where it made strictly no sense). Roscelin thought "universals" were all in the mind. Abelard was in between. The position above, neuromathematics, is that universals are real, but all in the mind. This is how the universe teaches us to become human. (BTW, this shows that the European thought system had gone well beyond the Greeks by 1100 CE, and thus was not dependent of getting reacquainted with the Greeks, contrarily to what is generally depicted. The argument can even be made that forgetting the forgettable details of Greek thought is exactly what the doctor ordered.)

*

Cantor diagonalization has many spectacular applications. For example, suppose we considered a number m, and then suppose we enumerated its properties: P(1) could be whether it is even, P(2) could be whether it is prime, P(3) could be whether it is normal, etc… Each property could be expressed by an expansion as a sequence of 0s and 1s, as is done in computer science. Then one could consider diagonals, and tweak them as Cantor did, getting properties not found in the original list. Conclusion: the properties of any given number are not countable. (That puts an ironical light on the physicists searching for a "theory of everything" and the believers who believe in just one got: they should take the diagonal of god, see what happens…)

*

The foundations of mathematics have proven to be a jungle: Many foundational systems have been elaborated (ZFC, MK, T-G, NBG, etc.), to try to have enough logical power to support the elaborated reasonings of some mathematicians (such as Grothendieck), or theories such as category and model theory, while avoiding paradoxes. The final word is not in, but the implicit morality modern mathematicians have extracted from it is that your foundation depends upon your construction, just as in the building industry. All foundations are local, only the mind is global.

CATASTROPHIC PROBABILITY THEORY.

 

CATASTROPHIC RISK = [SCIENTIFIC PROBABILITY] + [PHILOSOPHICAL PROBABILITY].

***

Abstract: I propose to search for truth in a different way, which is more appropriate to potential catastrophes. Political application of this new method will be life saving, on a planetary scale. The new method, being intrinsically teleological, is closer in spirit to Lamarck than to Descartes.

***

Overview: WHY THE RECENT ENORMOUS CATASTROPHES OF WESTERN CIVILIZATION, IN SPITE OF ALL THE SCIENCE?

I promote a new way of establishing certainty which should have tremendous impact on how decisions are taken, especially in politics. There is plenty of evidence that the Obama administration, far from following such a method, is taking decisions the old fashion way, with catastrophic consequences presently unfolding.

Humanity is at the crossroads between a radiant future, and a holocaust of six billion, after ruining the planet within a generation or two. The choice is now. It depends upon finding truth, and plenty of it. Probabilities are a big help to ascertain truth. But the usual method for computing probabilities is fatally flawed, and leads people astray by creating false certainties galore.

The probability computation of a rare event from a complicated theory should be added to a new term, the probability that the theory itself is wrong, multiplied by how probable it is to get the same rare event, should it be so. Running to this new term is how CERN physicists answered their critics about destroying the world with their accelerator. They did not trust their own entangled mathematical theories (GUT, QCD, and General Relativity).

Because a complex theory has a high probability to be false, the new factor should dominate when probabilities are low, the sort of probabilities characteristic of catastrophes. The new factor is conventionally ignored. We suggest to use inverted, catastrophic logic to evaluate it.

The conventional term is built with rigorous mathematics, it depends upon complicated data. It is organized like clockwork. But, like clockwork, it is good only if all its ingredients, empirical and logical, are 100% exact. A tall order. The logic breaks down if a piece is dead wood, instead of refined crystal.

Not only that, but the conventional term can be called the scientific term. Science is common sense articulated around elements of reality that independent probability analysis has established as true, certain, irrefutable. Thus this conventional first term is built only with elements of perceived reality and conventional logic that are well known, and can only lead to a conclusion that is well known.

But catastrophes are not well known before their time.

The new term is built with the philosophical method. One can call it the philosophical term. The philosophical method is common sense, articulated around potential elements of reality that destroy the preexisting paradigms. Those elements of potential reality may have been perceived, or guessed, but they have not been tested enough  to be considered certain. They could have been perceived just once. But these elements, should they turn out to be real, are game changers (an example below are dark comets; famous examples of  game changers in physics are the double slit experiment (Young), the electric current generated by moving magnet (Faraday), and the photoelectric effect (Hertz)). Of course game changers are why there are catastrophes, so the philosophical term is more appropriate to look out for potential disasters.

The philosophical term uses as ingredients what could go wrong, boosted with maximal imagination, and weighing more gravely the riskier outcomes and ominous warnings.

So one may abstract this new approach to risk evaluation with the following equation: risk of catastrophe = [scientific probability] + [philosophical probability].

***

***

CONVENTIONAL PROBABILITY COMPUTATIONS LEAD ASTRAY; THE EXAMPLE OF ASTEROIDS VERSUS DARK COMETS:

Conventional probabilities twist logic perversely. The tendency is to compute what can easily be computed, while ignoring the rest.

An example is given by studies of potential collisions of celestial bodies with our planet. To compute such a probability, scientists naturally look at SITUATIONS THEY CAN COMPUTE WITH. That means asteroids. Asteroids can be seen. Never mind if there is something worse lurking out there. OLD PROBABILITY THEORY MEANS OUT OF COMPUTATION, OUT OF MIND. Asteroids are space rocks on well established trajectories. The reason that they are well established, is that their former colleagues collided with the planets already. The survivors are well behaved, they follow intricate avoidance trajectories in resonance with the planets. Hence collisions with asteroids are rare. Accordingly, low probabilities are found.

Another reasonable approach to estimate collision probability seemed to be to look to the ground, and search for impacts, and estimate how many collisions there have been. The problem with that approach is that it also selects for the same space rocks, because only big rocks can reach the ground. The end result is that two probabilistic approaches come up with the same probability (roughly), giving a false sense of security. Both approaches are biased towards asteroids. One bias was computational (asteroid trajectories can be computed), and the other bias is observational (asteroids can be seen and their impacts can be observed).

The correct catastrophic question is not to ask how often a space rock would hit, but instead how often the earth is likely to be hit by something. So catastrophic logic asks: what could that something be? The philosophical method gives an immediate answer. PHILOSOPHY GUESSES GENERAL RULES FROM RARE EXAMPLES. (Whereas science makes laws from the systematic return of the same.)

In 1983, comet IRAS-Araki-Alcock passed within 300 times the diameter of the Earth. That was the closest known approach in 200 years by such a big object. It was detected only 2 weeks out, because it was so dark. Comet IAA had only 1 percent of its surface active. It was going at a relative speed of 44 kilometers per second, and its impact with Earth would have caused an explosion of 200 million Megatons of TNT. Yes, more than ten billion times Hiroshima. Most of civilization would have been taken out. Preceding statistics had underestimated such an occurrence by at least one hundred times.

Comet Borrelly, visited by NASA’s Deep Space 1 probe in 2001, was found to have extremely dark patches over much of its surface. The enormous explosion a century ago at Tugunska in Siberia, enough to kill 30 million people if it happened nowadays over a mega city, was probably a piece of comet (besides the fact that it exploded at 8,000 meters, leaving no debris, showing it was not very stony, it happened just the day when the Earth was crossing a well known meteoritic stream made of comet remnants).

There is some evidence of a double cometary impact off Indonesia and Norway during the Sixth century, more exactly in 536 CE, with nefarious worldwide consequences (frosts, darkness). A huge trailing fireball over North America 12,900 years ago, scorching half of the continent, may have blazed across the skies too (archeology finds a soot layer, and micro diamonds, continent wide, with massive changes all over).

So this is a case where finding the probability of whatever can be computed has been far from helping. Planners have years to consider any scary asteroid. So they have not been too worried about asteroids. But asteroids are not the problem.

A planetary defense system with warnings of only a few hours, should instead be set up to be ready to handle comets. A thermonuclear armed missile could certainly do away with a Tugunska style impactor, even with an hour’s warning (a 100 meters across cometary fragment would not be dangerous if blown to pieces). In any case, a serious theory of dark comets is necessary. Some scientists (Napier and Asher, 2009) are pointing out that a bit of (philosophical) thinking shows that comets become dark as they age. So many may be lurking out there. This is the correct approach: instead of computing asteroids to no end, thinking outside of the box.

***

THE GREATEST MISTAKES OF HUMANKIND HAVE BEEN PHILOSOPHICAL, NOT SCIENTIFIC:
The best example of a huge philosophical mistake is given by the Greco-Roman civilization. That civilization had decided to develop slavery instead of higher technology, condemning their civilization to no exit. (The Franks corrected that mistake, by outlawing slavery at the outset.)

Another philosophical mistake arose from the young British PM  hostility to the French revolution (which, after all, was just following in the footsteps of the earlier English revolutions, in which Louis XIV of France refused to partake, although the English king begged him to.) That unjustified anger brought 25 years of wars all over Europe, and many millions dead. (British Prime Minister Lloyd George recognized the Pitts for what they were, really the pits, 130 years after the facts.)

The USA has made colossal philosophical mistakes in the last 40 years (many of them reminiscent of the Greco-Romans, because they overemphasized the exploitation of man by man as the major engine of the economy, such as when Nixon decided to make health of Americans a profit center.)

USA president Obama has an armada of scientific advisers (or at least 4). All together they have pushed for a colossal (120 billion dollars says New Scientist) stimulus in science and technology. That is an excellent decision, very favorable to civilization. As long as it has legs (and is not just a jerk to the system).

But Obama has no philosophical adviser. This reflects the deeply erroneous belief that philosophy is so easy, any lawyer can do it.

In truth, it’s the other way around: precisely because it deals with certainties, it is science that is much easier than philosophy. Science looks like magic to the commons only because of massive deficiencies in the educational system.

But scientific progress gets really hard when it needs new philosophy to progress. ANY MAJOR SCIENTIFIC PROGRESS WAS FIRST OF ALL A PHILOSOPHICAL JUMP. There are no exception, “major philosophical jump” is nearly the definition of “major scientific revolution”. And this extends to mathematics: any ultra major mathematical advance was first of all a philosophical jump. (Those who screams too loud at this point may calm down if submitted to the intricacies of the foundations of Quantum Mechanics; in QM, not even light moves at the speed of light: see Feynman’s “QED”, page 89.)

Philosophy is much harder than science, because it is common sense applied to rare or heretofore undescribed events or patterns. It is the hard edge of science, far out. Newton knew this. So did Einstein. Lamarck was the discoverer of the theory of biological evolution, the discoverer of the immense age of the earth (from studying the evolution of clams), and the discoverer of the order of invertebrates, among other things. Although Lamarck was one of the world’s first biology research professor, and although he was very famous during the French revolution, as conservatism returned, he became the butt of jokes, died in misery, and his spirit ate crow for two centuries after that. Why? Because, as a scientist of civilizational class, he had stepped on the philosophical toes of conventional wisdom. And so did his science. Several celebrated British students of Lamarck (Wallace, Darwin) could not restore their teacher’s post mortem reputation. Lamarck kept on being the object of ridicule, until, on Darwin’s 200th birthday, epigenetics came back in force.

Studying philosophy and history or sociology is in full contradiction with the official philosophy of the USA, the fact Americans are supposed to “trust in” and “under” God. So philosophy and generally examining life is done much less, and the incapacity of conducting the most elementary logic reigns. When asked why he does not nationalize the banks, Obama and his entire government revert to the logic of the cave people, or as bin Laden would say: “it’s not our culture, it’s not traditional, we have more than five banks…” Silly juxtaposition, I know, but telling that it can be done. No pride of rationality, no glory in the human spirit: in God we trust, under God we are…

And sure enough, Obama has not reached philosophical clarity in Iraq and Afghanistan. Instead he pursues the exact same mistakes of his predecessor, with renewed military enthusiasm. Why? Because he has not evaluated the enormously high probabilities for the disastrous outcomes the USA is cruising towards, in both cases. How am I so sure? Well, because the catastrophic scenarios in both cases are all too likely and way too stable in their catastrophic natures.

If he wanted to go a good job of reflection, Obama would have to reason catastrophically, and backwards out of the imaginable pitfalls. But he may be unable do this anymore than he has been able to do it with his tiny little bank problem that he is ruining the world’s economy with.

To think that Iraq will not someday take its vengeance and stabs the USA is in the back for what it did, is the sort of naivety those who have never lived history are affected by.

So the USA is sinking in depression, while its army tries to ravage the Middle East. It’s like Roman history never happened. Shame. Remember Julian, Barack, and stop before knowing an ominous fate. Do not ask what God will do for you, imagine what catastrophe will do to your country. Please reason backwards from the worst possible outcome.
***

NO PROBABILITY IS AN ISLAND:
What we just saw is that planetary defense, “Spaceguard”, is oriented towards unlikely asteroids because asteroid theory is easier. Whereas the real danger is Dark Comets. This was pretty obvious ever since the Tugunska explosion. To be alarmed by this one has to use the philosophical method (because it was a one time event, just as comet IAA was a one time event).

Policy decisions and legislation depend on probabilities. Any probability of an event “E” is computed using a reasoning. the probability P depends upon the reasoning, R. P is justified only if the reasoning R is correct. So the real probability of E is not P, but the product of P with the probability p[R] that R is correct. So it’s [(P) (p[R])]. Now there is also a probability that the reasoning R is false and that the event E happens nevertheless. Call that Q. In the end the total probability of E is: P (p[R]) + Q (1- p[R]) = (P – Q) p[R] + Q.

Notice that the more complex the theory R, the higher is the probability p[R] that the reasoning R will be false, and so the probability coming out of such a reasoning R, if it very small, is irrelevant. Some critics have used this approach to claim that the CERN accelerator in Geneva is dangerous. The reasoning R, in the case of CERN, is QCD, plus dubious science such as the physics of black holes. There is no way that this unholy assemblage is all true. Conscious of this, CERN physicists changed tactics, and discarded the basic theory behind their accelerator to answer their critics.

If P is very small (like the probability of blowing up the world by turning on the CERN LHC accelerator), and Q is not, P becomes irrelevant. The probability of the event E is then roughly Q (1 – p[R]). Now a complicated theory like QCD (the theory behind CERN) gives us only a very low probability that R is right. So p[R} is roughly zero too. Everything gets dominated by a probability that has nothing to do with what mathematically oriented statisticians look at usually.

Instead of playing the game of going into the probably erroneous details of modern physics, CERN physicists argued that, even if the theory were false, the probability of an adverse outcome was insignificant, because the universe has collisions at these sorts of energies all the time (so the universe would make mini black holes and strange matter, all the time, if they could be made at these energies; since it does not, these energies are safe).

***

IF ONE DOES NOT HAVE ALL THE GASES, ONE CANNOT COMPUTE GLOBAL HEATING:

OK, let's focus minds with a particular case; the probability of a catastrophic outcome for greenhouse warming, turning it into global heating. It has been computed many times. Each time the results are worse, and there is a good reason for that.

What happens every time is that each greenhouse probability computation depends upon greenhouse gases. The most well known is CO2 (Carbon dioxide), the next one, a greater threat, is CH4 (Methane). Another powerful greenhouse gas is H2O (Water), as anybody who has spent a night in the desert will testify. The hotter it gets, the more H2O goes up, and the more CH4 comes out of permafrost, so the greenhouse effect augments non linearly (the more it augments, the faster it augments).

There are other greenhouse gases, found in minute quantities, but they are much more powerful as warming agents. Methane, over 100 years has 25 times the Warming Potential of CO2, and Nitrous Oxide, 298 times. Over ten years, the warming potential of methane is much higher. Incorporating those one gets 15% of the warming. But recently some gases were found that had the Warming Potential of 20,000 times CO2 (yes, twenty thousands times). This changes seriously the probability of massive heating. That is the type of situation where a probability computation fails because of missing ingredients. The reasoning R is not false intrinsically, but with not all the data in, it becomes so.

So what do I suggest? Well, start with what we know that could go wrong. That is exactly the most complete initial conditions to evolve logic from. One does not expect anything less from an airplane pilot. If she started with hoping for the best, and only considered what she can compute, instead of preparing for the worst, and the unexpected, she would not live very long. Nor would we.

***

THE TRAP OF COMPUTATION IN GREENHOUSE HEATING:

As I said above, Spaceguard has been looking at asteroids because asteroids can be computed. The same occurs with "Global Warming". The IPCC (International Panel on Climate Change) did not incorporate  the possible melting of Greenland and Antarctica in its predictions. This is astounding, because this is the most dangerous proximal effect of global heating. Greenland is all white and sticks low in the blue Atlantic ocean, it does not require much brain power to decide it will melt. If it does, it's 6 meters of water...

Why did the IPCC neglect this melt? Well, you guessed it: too hard to compute. So the IPCC ignores it outright, like Spaceguard ignored dark comets outright.

The correct catastrophic probability approach is to observe that total melting has happened before, and if it happen again, it would raise oceans by 72 meters. Thus it's possible, and a great danger, so it should be viewed as an end one wants to avoid. Plus there is also the corroborating danger is that the oceanic level is rising 3 mm a year, faster than the IPCC projections. 

*** 

WHEN IMAGINATION BEATS INSPECTION:

There can be logical flaws in the most precise reasoning. Mathematics is full of perfect reasonings that were found wanting later on. More exactly, by changing the logic R, much more was found later. It was not that R was really "false". No, R was often found to be incomplete, to rest on unjustified, non explicit steps. (This is why it has been found hard to get computers to check mathematical proofs: they are not that logical, after all!) Overall, mathematical reasonings often worked not just from logic, but also from convention, and tradition.

Example. It was long thought there was no number that, multiplied by itself, would produce a negative number. It was a sort of reasoning by inspection: the probability of seeing such a number was zero, well, because no one had seen one before. BUT INSPECTION OF A BOX DOES NOT MEAN THERE IS NOTHING OUTSIDE OF THE BOX.

Indeed, an Italian surgeon-mathematician, Cardano, came around and solved some equations by assuming during the computation that there were numbers with negative squares. It turns out that there was just a psychological block: if one assumed such numbers with negative squares existed, the whole world was easier to interpret, and, ever since one has assumed that they exist, and these numbers allowed to visualize better electricity, electromagnetism and Quantum mechanics. Mathematicians now view these "complex numbers" as most natural, and physics cannot be done without. 

The same happened with curved geometry. Although it is everywhere, and it preceded Euclidean geometry, Euclid did to it what it did to the zero (another Greek invention that had to emigrate to India to be treated well). Euclid had the reasoning bias: he admitted as a worthy object of study what he could present a lot of logic about, instead of making the philosophical jump to realize that he was after small logical fry, instead of really big game.

***

WHEN ERROR CAN BE FRUITFUL, BECAUSE SOMETHING IS RIGHT INSIDE:

False logics can lead to true results. An example is infinitesimal calculus (in its original Leibnitz version). As Berkeley pointed out, the logic made little sense. But certainly the results were true all over, and soon used in engineering. It would take three centuries, and a big advance in logics (model theory) to justify Leibnitz' infinitesimals directly; Cauchy gave a different rigorous version of calculus, using limits, 150 years after Leibnitz' initial invention.

An example of a falsehood bringing truth, from Physics: the establishment by Maxwell of the equations for electromagnetism used the hypothesis of the luminiferous ether. That was shown later not to exist. It came to be implicitly understood that the waves created the space, to some extent. Dirac systematized the madness, by predicting spin and antimatter just this way: he imposed the simplest wave equation on electrons, Maxwell-like, but Dirac did not care if he did not have a (non local) space for his equation.

***

TO BE SAFE, USE CATASTROPHIC LOGIC:

So what to deduce for policy making? Well, when trying to compute the probability of some catastrophe, one should start from the worse possible conclusion, and try to find out if there was a way to get, backwards, from there, to the present conditions. That is how the CERN physicists argued: there was no way to get to a catastrophe at CERN, because the universe is trying all the time, and fails to get there, namely to a catastrophe.

In climate research, the opposite is found; the planet Earth was often in its HOT MODE. So it certainly can get there. In that mode, dinosaurs and crocodilians lived in the polar regions. The tipping point out of the hot mode is the concentration of CO2 below which Antarctica freezes over (~ 425ppm CO2 equivalent). This is below a level of greenhouse gases we have already passed, on the way up. So we could have utter planetary destruction in 30 years, we are in mortal danger, and one should bring CO2 emissions to zero right away, because CO2 emissions is how we got there. Instead Obama is proposing to "capture carbon", a wild goose chase where the mentally challenged nebulously runs around emitting gas, hoping for the best.

In the case of Afghanistan, or the Middle East and South Asia in general, the worst outcomes are absolutely terrifying: they involve nuclear war, and even nuclear world war. That's bad. But what is worse is that it is easy to build scenarios to get from there to where we are, and conversely. Thus policy searching for better outcomes should be focused on poisoning these scenarios before they can unfold. Much fewer nukes worlwide is an obvious solution.

In economy, the advisers of Obama view as "improbable" that the economic conditions will get to 20% further down in housing prices, and more than 10% unemployment. They define those as "depression conditions". Of course at the present rate of collapse, they will be there in three months, but they say no, because they say it would be improbable. It is true that it did not happen in the last 70 years, so making a theory from that duration, it can't happen. (So, having decided it will not happen, they do nothing to prevent it, and one can safely predict it is going to be a disaster!)

Far from this circular logic, the catastrophic approach is to extend present trends, and realize that a deep depression is likely. Then, having realized the enormity of the disaster at hand, one has to go backwards, and see what could derail that scenario.

Well, the only thing that could derail catastrophe is basically what the advisers of Obama refuse to do: dump the rich, refuse to keep on sending them taxpayer money, and use the money of the People to give cheap housing, cheap health care, and jobs, jobs, jobs.

***

A NEW THEORY OF CERTAINTY, AS PROPOSED ABOVE, TO SAVE OUR LIVES:

The old fashion way of computing probabilities ignores the fact that any probability computation is in a logical universe, and that the probability that this logical universe represents all relevant ingredients for the situation at hand has itself to be evaluated.

One just needs a glance at the crashing world financial system to see what fatally flawed means. In modern times manmade disasters that had been viewed as very unlikely have been happening with increasing frequency. Examples are a few holocausts, two world wars and two great depressions (without counting the present one).

On a slightly larger time scale, there were important break downs of the Western moral code. The wars that happened around the French revolution were totally unnecessary, and killed millions, in France, and more, around Europe. After the initial hatred of the Old regime against the Rights of Man had caused an invasion of France, the blood kept flowing for 25 years because one "unlikely" event followed another. By contrast vast periods in history have been very calm (some are found in the Middle Ages!)

In North America, the American Revolution brought more slavery for Africans and holocaust for the Native Americans. That would have been viewed by the proponent of the Enlightenment, who wanted American independence, as another "unlikely" event. How likely was the American Civil War, the bloodiest civil war known? And how likely was it that Great Britain would imitate USA methods in South Africa as it forced the (white) Boers to surrender their republic?

Important advances were made in logic in the twentieth century. A basic trick was used throughout, GOING META. Going meta makes a theory of the theory, it manipulates it from the outside. The same basic trick can be carried over in the realm of computing probabilities. The disregard of the moral code by the Nazis or the USA (official reintroduction of torture) were unimaginable. They looked totally improbable before they occurred.

The usual approach to what is probable is dangerously misleading, because it WILL ALWAYS underestimate catastrophically a potential catastrophe (strategic, nuclear, financial, climatic, etc.). This new meta law of probability is pernicious: it says that, the worst the potential catastrophe, the rarer it will be, and the more the usual way of computing probability of said catastrophe is logically flawed.

So what to do I suggest to do when trying to find how probable a catastrophe is? Well, I propose to backtrack from the catastrophe itself: do not ask your theory how unlikely catastrophe is, ask catastrophe what it can do for your theory. As I showed, knowing dark comets exist, and knowing that so does the hot mode of Earth's climate, radically augments the probability of an unhappy outcome.

***

Conclusion: The way truth theory is generally used is wrong. It ignores that any logical system carries its own truth inside, but that this truth has no value outside. Conventional probability computations of catastrophes are flawed, or, more exactly, meta-logically flawed. Yes, there is such a concept as “metalogical” and it led to the greatest advances in logic in 2,500 years. In metalogics, one applies logic to a logic from the outside. Any logic has countless metalogics surrounding it. Incompleteness theorems in logics say that there are a number of ways to build them up. To be logical is necessary, to be metalogical is prudent. Probability without metalogics is folly.

The philosophical probability weighs these metalogics according to the threats they represent. So it proceeds from the end (telos: end, goal, result). That's why it is teleological. "Teleological" has not been philosophically very popular in recent centuries, because it smacks of God, intelligent design, or animals thinking about how they should evolve (the later being the oft made parody of Lamarckism).

But a moment's thought reveals that all intelligence is teleological: the animal wants to save its life, so it rushes into the hole. It does not rushes into the hole first, and then wants to save its life. The flight in the hole is teleologically motivated: any Cretaceous critter fleeing for its life has not experienced yet catastrophic doom between the jaws of T Rex, five meters away. But it reasons BACKWARDS from a theory of catastrophe it has.

So, in the end, I am just suggesting to be a bit more systematic about something  the ancestors of rats understood 200 million years ago.

RISK OF CATASTROPHE = [SCIENTIFIC PROBABILITY] + [PHILOSOPHICAL PROBABILITY]. But the worst it can get, the more the term that dominates is the philosophical one.

***

Patrice Ayme.

http://patriceayme.com

http://patriceayme.wordpress.com

http://tyranosopher.blogspot.com

***

 

Addendum: More on Spaceguard.

Of course, ideally, any impact warning system should have a capacity for tracking and changing orbits of potentially harmful near-Earth asteroids and comets years ahead of impact. Unfortunately we don’t know how to do this, and years may not be available: cometary trajectories can be both hidden and chaotic (as they fall off the Oort cloud or graze giant planets).

The kinetic energy of the comet IAA was 44 squared by unit of mass, that is 2,000 (two thousand) per unit of mass times that of a speeding bullet. Hard to deflect: we could do little  with a comet within 2 weeks of impact, considering that its mass is of the order of a thousand billion tons.

But what happens if sometimes around 5:47 am on some month of June in the close future, as we cross the same comet debris stream again, astronomers discover that a Tunguska sized impactor is within two hours of hitting New York City? The Tunguska object was between 30 and 50 meters across, and exploded with about 15 Megatons of TNT equivalent (1,000 times Hiroshima), flattening 2,000 square kilometers of forest (an area greater than Washington DC).

Interestingly, we could do something about it, with existing technology.  Any explosion of a thermonuclear bomb in the vicinity of such an object, more than ten seconds before impact, should be mitigating (even with the EMP, as long as people close their eyes!).

***

Final word from T.S. Eliot:

“What we call the beginning is often the end
And to make an end is to make a beginning.
The end is where we start from.”

***