Archive for the ‘Mathematics’ Category

CATEGORIZING the MIND

October 27, 2014

What is the mind made of? We have progressed enormously as far as the brain objects are concerned: neurons, axons, dendrites, glial cells, neurohormones, various organs and substructures in the brain, etc.

But is there a broad mathematical framework to envision how this is all organized? There is! Category Theory! It turns out it’s a good first order approximation of mind organization. At least, so I claim.

Category Theory is about diagrams. Category Theory has been increasingly replacing advantageously Set Theory. It’s not only because Category Theory does not have to ponder the nature of objects, elements, sets.

Category Theory was long derided as “abstract nonsense” and “diagram chasing”. But it gives very deep, powerful theorems.

http://en.wikipedia.org/wiki/Category_(mathematics)

I claim the powerful theorems of Category Theory should translate directly into… neurology.

Amusingly, although I accused Aristotle to have demolished democracy and fostered plutocracy through his beloved pets, the mass murdering criminal plutocratic psychopaths, Alexander and Antipater, I recognize humbly that it’s the same Aristotle who invented categories (thus making him a great thinker, and justifying an Aristotle cult among those who need to have cults to feel good about themselves)…

Aristotle’s meta-idea about categories was just to talk about the most fundamental notions:

http://en.wikipedia.org/wiki/Categories_(Aristotle)

The present essay was suggested, and is an extension of what the honorable Bill Skaggs seems to have wanted to say, in Scientia Salon, in his “Identity A Neurobiological Perspective”. (As far as I can comprehend.)

However, forget Theseus’ ship and Hollywood’s Star Trek “Transporter”. As I said in “Quantum Identity Is Strong”, Quantum Identity is not erasable, and makes those time honored examples impossibly disconnected with reality. The notion of identity has thus to be found elsewhere (as we intuitively know that there is such a notion).

According to modern Quantum Field Theory, we are made, at the most fundamental level, of fluctuating fields. They come and go, out of nowhere. So, that way, we are continually been deconstructed and rebuilt. The question naturally arises: what is preserved of me, as a set of Quantum Fields? Well, the most fundamental mathematical structure is preserved.

The same seems to hold, to a great extent, in neurobiology, as neuro circuitry, to some extent, seems to come, go, and come back.

Thus we are all like old wooden Greek ships, perpetually falling apart, and rebuilt.

To some extent, this is what happens to species, through reproduction: cells split, and reproduce themselves, thanks to DNA.

A species has identity. Yet that identity is made of DISCONTINUOUS elements: the individuals who incarnate the species, who are born, and then die. And others appear, just the same, sort of. How is that possible?

A species’ identity is its structure. Just as a neurology, or an elementary particle identity is its structure. Not just a geometric structure, not just a topological structure, but its structure, as the most fundamental notion, as a category.

So what is preserved? Shape. And how to morph said shapes… Naturally (there is a notion of natural transformation, in Category Theory).

Historically, analyzing shape was systematized by the Greeks: Euclidean geometry, cones, etc. Then, at the end of the Nineteenth Century, it was found that geometry studied shapes mostly by studying distance, and yet, even if distance was denied consideration, there was a more fundamental notion of shape, topology. That was the structure of shapes as defined by neighborhoods.

Two generations later, Category Theory arrived. Category Theory is about morphisms, and the structures which can be built with them. Please listen to the semantics: structures, morphing… This is all about shapes reduced to their most basic, simplest symbolic expression. It’s no wonder that it would come in handy to visualize neurological structures.

A morphism is a pair of “objects” (CT leaves unspecified what the “objects” are). To model that neurologically, we can just identify ‘objects’ to neurons (or other neurological structures), and morphisms to axons (although dendrites, and more, could be included, in a second stage, when the categoretical modelling become more precise).

The better model is category theory. When are two diagrams equivalent? When are they IDENTICAL? Cantor defined as of the same cardinal two sets in a bijection (a bijection is a 1 to 1, onto map).

Category Theory defines as identical the same diagram (a drawing reduced to its simplest essence). Say: A>B>C>D>A is the same as E>F>G>H>E.

Thus, when are two diagrams identical in category theory? When they are modelled by the same neuronal network. (Or, more exactly, axonal network: make each arrow “>” above, into an axon.) And reciprocally!

Discussing the mind will involve discussing the most fundamental structures constituting it. What better place to start, than the most basic of maths? Especially if it looks readily convertible in neural networks.

Category Theory is the most fundamental theoretical structure we know of. It is the essence of identity, and identification. In conclusion, two objects are identical, neurologically, and in fundamental physics, if they are so, in category theory.

Time to learn something categorically new!

Patrice Ayme’

***

Note: No True Isolated Rocks: In other news, and to address a point of Bill Skaggs, whether a rock can be truly isolated is an open problem, experimentally speaking.

According to the theory of gravitation of Einstein and company, a rock cannot be isolated. Why? Because the rock is immersed in spacetime. Spacetime is animated by gravitational waves: this is what the Einstein Field Equation implies. Now, according to an unproven, but hoped-for principle of fundamental physics, to each force field is associated a particle. In the case of gravity, that hoped-for particle is called the graviton. “Particle” means a “particular” effect. Thus, an isolated rock, according to established theory, and hoped-for theory, ought to be adorned occasionally with a new particle, a new graviton, thus ought not to be isolated.

In my own theory, Objective Quantum Physics, on top of the preceding standard effect, resolving Quantum Entanglements, ought to create even more particles in “isolated” rocks.

Universe: Not Just Mathematical

August 14, 2014

Some claim the “Universe is mathematical”. Their logic is flawed. I show why.

Max Tegmark, a MIT physics professor, wrote “Our Mathematical Universe”. I present here an abstract I concocted of an interview he just gave to La Recherche. Followed by my own incisive comments. However absurd Tegmark may sound, I changed nothing to the substance of what he said:

La Recherche (France; Special Issue on Reality, July-August 2014): Max, you said “Reality is only mathematical”. What do you mean?

Tegmark: The idea that the universe is a mathematical object is very old. It goes all the way back to Euclid and other Greek scientists. Everywhere around us, atoms, particles are all defined by numbers. Spacetime has only mathematical properties.

La Recherche: Everything is math, according to you?

Formulation Before Revelation of Mathematization

Formulation Before Revelation of Mathematization

Tegmark: Think about your best friend. Her great smile, her sense of humor. All this can be described by equations. Mathematics explain why tomatoes are red and bananas yellow. Brout, Englert, Higgs predicted a boson giving mass to all other particles. Its discovery in 2012 at CERN in Geneva led to the 2013 Nobel Prize in Physics!

Tyranosopher [unamused]: Notice, Max Tegmark, that the “Nobel” thoroughly excites you. You brandish it, as if it were a deep reality about the universe. But, in truth, the Nobel is strictly nothing for the universe. It’s just a banana offered by a few self-interested apes to other self-fascinated apes. The Nobel has more to do with the nature of apish society, rather than that of the universe. In other words, we ask you about the nature of the universe, and you answer with the Authority Principle among Hominidae. You may as well quote the Qur’an.

Tegmark [unphazed]: There are an enormous number of things that equations do not explain. Consciousness, for example. But I think we will make it. We are just limited by our imagination and our creativity.

La Recherche: According to you, there is no reason that part of the world escape mathematics?

Max Tegmark: None whatsoever. All properties are mathematical! We potentially can understand everything!

La Recherche: As a Platonic mathematician, you consider mathematical concepts are independent of all and any conscious act?

MT: I am an extreme Platonist, as I think that not only mathematical structures are real, but they are all what reality is.

Relativity and Quantum Physics confirmed that reality is always very different from what one believes. Very strange and very different from our intuition. Schrodinger’s equation, the fundamental equation of Quantum Mechanics, shows that a particle can be in several places at the same time. Thus one does not try to describe the motion of this particle, but the probability of its presence in such and such a place.

But, a century later, physicists are still in deep disagreement about what it all means. I think this interpretation keeps dividing people, because they refuse to admit what goes against their intuition.

Tyranosopher: Notice, Max Tegmark, that you presented as a fact (“a particle can be in several places at the same time”) something you admit later is only an “interpretation”. That’s dishonest: an “interpretation” is not a “fact”.

Tegmark [livid]: The strength of mathematics comes from the fact that they have no inhibition. Strangeness does not stop them.

Tyranosopher: Indeed, that’s why, as a trained mathematician, I am very insolent.

La Recherche: Max Tegmark, is it your mathematical approach that makes you defend another controversial idea, that of multiple universes?

Max Tegmark: I really believe that human beings never think big enough. We underestimate our capability to understand the world through mathematics, but also our capacity to apprehend its dimensions. To understand that we live on a planet with a diameter of a bit more than 12,000 kilometers was a first, enormous, step. That this planet is infinitesimal in this galaxy, itself one out of billions, was another enormous step. The idea of multiverses is more of the same. We discover again, and more, that what we understand is only a speck of something much larger. That much larger thing is the Multiverses, of types I, II, III, and IV.

Tyranosopher: La Recherche’s Interview then proceeds further, but let me unleash a fundamental critique here.

I am a deadly enemy of the Multiverse, as I believe that it rests on an ERROR of interpretation of Quantum Physics (the one Tegmark presented as a fact above, before admitting that it was, well, only an interpretation). The fact that it is another desperate scaffolding erected to save the Big bang theory does not make it better.

Now for the notion that the universe being full of math. This is understood to mean that the universe is full of equations. Equations were invented in the Sixteenth Century. Many, if not most, equate mathematics with the art of equating.

What’s an equation? It’s something that says that two things independently defined, one on the left side of the equal sign, the other on the right side, are equal. Great. What could be simpler: what is different is actually the same!

Notice this, though: before you can equate, you must define what you are equating. On both sides.

An equation equates concepts independently defined. Ultimately, definitions are not mathematical (see on the Nature of Mathematics, to follow soon). At best, definition is metamathematical. Our metamathematical universe? End of Mr. Tegmark’s naivety.

When we get down to it, it’s more our philosophical universe, before it’s our mathematical universe: no definitions, no equations.

How can a physicist make such a gross logical mistake? Are they not supposed to be smart? (OK, it’s smart to sell lots of books).

What allows to make that logical mistake? Education, or lack thereof. Many a mathematician will make the same mistake too. The problem is that neither conventional mathematicians, nor, a fortiori, physicists, are trained logicians. They just play some in the media.

Who needs a multiverse? It seems the universe of science is already too large for many physicists to understand.

Patrice Ayme’

Show Strength To Negotiate With Iraq, Putin

August 13, 2014

Putin is doing what the Kaiser did, a century ago, and for roughly the same reasons: trying one’s luck with war is better than suffering destitution. Like the Kaiser, a century ago, Putin hopes to win, because the democracies are weak, in weapons and resolve. What could go wrong?

Ah, yes, this is less haughty a subject than the first female mathematician, to be given a Fields Medal. Moreover, she is Iranian. She studies practical things, like counting “simple” geodesics on hyperbolic surfaces, depending upon their length. (I am not joking when I say this sort of research is practical: another of the 2014 Field medalist research has already been applied, to… surveillance; Fields Medals were attributed this year to understandable mathematics… Instead of the sort of crazy math I view with a jaundiced eye, as it depends upon infinity all too much.)

Mathematician Mirzakhani In Isfahan

Mathematician Mirzakhani In Isfahan

… In Isfahan with her parents. Isfahan is one of the world’s most beautiful cities. The artful architecture above is typical. Visiting such places, one can only be awed by the splendor of the human spirit, and feel compelled to contribute.

Now back to the dismal subject of the Twenty-first Century, out of control Czar. This is a serious problem.

Yet, here we talk about what makes all these fun and game possible, namely the pursuit of civilization. It depends upon crazy people and insane ideas, been kept in check.

The Kaiser was afraid of the Socialists inside the Parliament (Reichstag) that Bismarck had set-up. The Parliament’s power was fictitious. The Socialists wanted to make it real. Such was the inside pressure.

The outside pressure was an admission, by the heads of the military, the Kaiser himself, and his chancellor, top deputies, advisers: democracies were superior to the authoritarian, exploitative regime they profited from. Those German oligarchs recognized that the economic, political, and financial alliance between France, a democracy, and democratizing Russia, was increasing in economic, and thus military power, in a way that the German plutocracy could not match.

It would have been too much against their mentalities to do the right thing, and try to do what the Romanov Czar was doing in Russia: democratizing. Instead they decided to “work on the press”, and prepare the Germans for the world war they had decided to gamble everything on. That was December 11, 1912. (For a 2004 perspective of mine on that, see “To Make War, All You Need Is Love.”)

On June 1, 1914, Colonel House, the adviser of President Wilson of the USA, saw the Kaiser, and proposed him an alliance, against the “racially inferior” French. In exchange, the Kaiser would limit his battle fleet built-up (which upset the unable-to-keep-up British).

Of these little facts, these devils that truly propel history, conventional historians never speak: that is how they earn their keep. Well esteemed professors, their fate is little better than that of mice, scurrying for crumbs below the masters’ tables.

Putin is losing in Eastern Ukraine. The Ukrainian military has regained much territory, and cut off Donetsk, a city of more than a million, from the Russian military. Defeat is not what Putin wants, he wants an unending war, but one which he wins. Putin is proposing to his captive Russian public opinion, to send a “humanitarian mission” inside Ukraine. In other words, he is preparing a naked invasion.

What can the West do?

Go back to basics. Putin decided to attack Ukraine, after he saw that Western plutocrats, his natural allies, had enough control of the West to prevent a justified strike against that major satanic creature, Assad, son of Assad. That was shown by the defection of the British first. The Assad family has major plutocratic connections in London. Then, while French pilots were already strapped in their seats, Obama called off the attack.

If Assad, a dictatorial monster who started a huge civil war against pacific civilians who just wanted him to go, could get away with a poison gas attack inside Damascus that killed more than 1,500 civilians, obviously, Putin could get away with anything.

Putin wanted the Black Sea oil. That it belonged to Ukraine was a detail, now that Obama and the UK had demonstrated that Western civilization was in recess, and plutocracy reigned. Putin could do what he does best: grabbing what he needs. Like the Kaiser, he could see that gas and oil is all what held his empire together (most of the ex-Soviet “republics” have shown signs of exaggerated affection towards the European Union).

If the Kaiser and his generals had been persuaded that they would lose the war, they would have not started it: after the French nearly annihilated the German army at the Battle of the Marne, Von Molkte, who had done more to start that war than anyone else, was so deeply depressed, that he could not do anything anymore (he was secretly replaced). His mood had completely changed from the one six weeks earlier, when he mobilized the entire German army, catching the world by surprise.

So, if we want peace, we have to persuade Putin he will lose, should he pursue his policy of invasion and annexation. The best way to do this is to intervene in the situation of the Yazidi, an ancient, non-Muslim group hard pressed by the ISIS in Iraq.

This may seem a surprising position for someone who was vociferously against the invasion of Iraq in 2003.

However this is now. The ISIS, outwardly Islamist, and full of Jihadists, is getting lots of its backbone from the old army of Saddam Hussein. A demonstration of military power to help the Yazidi could, and ought, to be turned into a negotiation with some of the officers of that old army, and those who regret aspects of the Iraqi state that worked better under Saddam.

In other words: strike, but then negotiate.

There was never any serious negotiation with the secular Iraqi state, in the nearly quarter of century the USA has made war to it (in the hope of some USA plutocrats to grab its oil). Even the Neocons will have to admit that this time has come.

Of course, this is all very dirty, it’s how the sausage of civilization is made. But, if good people do not make it, to the best of their abilities, evil will be in charge.

And then, in the worst possible case, all intellectual pursuits will collapse, as happened after the Roman state streaked out of control, burned and crashed. Next time would be worse.

Patrice Ayme’

Axiom of Choice: Crazy Math

March 30, 2014

A way to improve thinking is to imagine more, and be more rigorous. What a better place to exert these skills than in mathematics and logic? Things are clearer there.

The crucial Axiom Of Choice (AC) in mathematics has crazy consequences. After describing what it is, and evoking some of its insufferable consequences, I will expose why it ought to be rejected, and why the lack of a similar rejection, at the time, in a somewhat similar situation, may have help in the decay of Greco-Roman antiquity.

This is part of my general, Non-Aristotelian campaign against infinity in mathematics and beyond. The nature of mathematics, long pondered, is touched upon. A 25 centuries old “proof” is mauled, and not just because it’s fun. There is deep philosophy behind. Call it the philosophy of sustainability, or of finite energy.

Intolerably Crazy Math From Axiom of Choice

Intolerably Crazy Math From Axiom of Choice

The Axiom of Choice makes you believe you can multiply not just wine, fish and bread, but space itself: AC corresponds, one can say, to a wasteful mentality.

The Axiom of Choice says that, given a collection C of subsets inside a set S, one can consider that a set exists, made of elements, each one of them is an element in exactly one of the subsets. That sounds innocuous enough, and obvious. And obvious it is, if one thinks of finite sets. However, if C is infinite, it gets boringly complicated.

Moreover, AC has a consequence: given a unit sphere, one can cut it in disjoint pieces, and reassemble those pieces to build two unit spheres. Banach and Tarski, both Polish mathematicians working in what’s now Western Ukraine, the object of Putin’s envy and greed, demonstrated this Banach-Tarski paradox. It’s viewed as an object of wonder in General Topology.

I prefer to view it as an object of horror. (The pieces are not Lebesgue measurable, that means not physical objects. Such non measurable objects had been found earlier by Vitali and Hausdorff)

Punch line? The Axiom Of Choice (AC) is central to all of modern mathematics. Position of conventional mathematicians? The fact that AC is so useful, all over mathematics, proves that AC can be fruitfully considered to be true.

My retort? Maybe what you view as fruitful mathematics is just resting on a false axiom, or, at least one against nature, and thus, is just plain false, or against nature. One may be better off, studying mathematics that is not against nature..

As I showed earlier, calculus survives the outlawing of infinity in mathematics. That pretty much means that useful mathematics survives.

You see a problem with mathematics, even the simplest arithmetic, is that, once one has admitted the infinity postulate, thanks to the Cantor Diagonal process, one can always find undecidable propositions (this is part of the Incompleteness Theorems of mathematical logic: Gödel, etc.).

That means a field such as Euclidean geometry is infinite, in the sense that it has an infinite number of non-provable theorems. Each can be decided both ways: false, or true. Each gives rise to two mathematics.

Yet, even modern mathematicians will admit that studying Euclidean geometry for an infinite amount of time is of little interest. Proof? They don’t do it.

Yet, what’s the difference with what they are doing?

Mathematics is neurology, and neurology can be anything, but infinite. Think about what it means. Yes, mathematics is even cephalopod neurology, with the octopus’ nine brains. Fractals, for example, are part of math, but far from the tradition of equating angles or algebraic expressions.

It’s a big universe out there. The number one consequence to draw from the history of science, is that scientists make tribes. Quite often those tribes go astray… for more than 1,000 years (see notes). Worse: my making science, and, or mathematics, uninteresting, they may lead to a weakening of public intelligence.

I would suggest that effect, making science, and mathematics priestly and narrow minded, contributed to the powerful anti-intellectual tsunami that struck the Roman empire.

Greek mathematicians had excluded all mathematics as unworthy of consideration, but for a strict subset of “Euclid’s Elements” (some of the present Euclid Elements were added later). The implementation of those discoveries were made by others (Indians, and to some extent, Iranians and Arabs).

It turned out that these more practical mathematics, excluded by Euclid, because they were viewed as non rigorous and primitive, led to deeper and more powerful insights.

The irony was that Euclid’s Elements, in the guise of rigor, were using an axiom that was not needed, in general, the parallel axiom. That axiom, by supposing too much, killed the imagination.

I suggest nothing less happening nowadays, with the Axiom of Choice: it’s one axiom too far.

Patrice Aymé

Technical notes:

Up to a recent time, if one was not a Supersymmetric (SUSY) physicist, it was impossible to find a job, except as a taxi cab driver. There was a practical axiom ruling physics: the world had got to be supersymmetric.

Now the whole SUSY business seems to be imploding as the CERN’s LHC came up empty, and it dawned on participants that there was no reason for an experimental confrontation in the imaginable future… I have studied SUSY, and I have a competitive theory, where there are two hints of experimental proofs imaginable (namely Dark Energy and Dark Matter).

I said the AC was one axiom too far, but actually I think infinity itself is an axiom too far. I exposed earlier what’s wrong with the 25 centuries old proof of infinity (it assumes one can use a symbol one cannot actually evoke, because there is no energy to do so!).

The geocentric astronomy ruled from Aristarchus of Samos (who proposed the heliocentric system, 3C BCE) until Buridan (who used inertia, that he had discovered to make the heliocentric system more reasonable; ~1320 CE; Copernic learned Buridan in Cracow, Poland). It could be viewed as an axiom.

Hidden axioms are found even in arithmetic, for example the Archimedean Axiom was used by all mathematicians implicitly, before Model Theory logicians detected it around 1950 (it says, given two integers, A and B, a third one can be found, D, such that: AD > B; if not fulfilled one gets non-standard integers).

PROOF IS PHYSICAL

November 27, 2013

”Information is physical”. Always. Of course. What else?

Yet, the mystery is far from dispelled, as we don’t know what “physical” is. We don’t know, what physics is, for sure. Some roll out the Quantum, and say:”here is physics: it from bit”. However, we are not certain of what the Quantum is (= we don’t know whether quantum theory is “complete” or not; ultimately it’s a Physical Problem, experimentally determined; Von Neumann thought he had a “formal” proof, but he was wrong).

Are there Physical Problems that are not Mathematical Problems? Or Physics Proofs that have not Mathematical Proofs? Well, at this point, there are. Take general fluid flow. Be it water inside a fluid, or a meteor going hypersonic, these Physical Problems exist, and have solutions, that the physical objects themselves are Physical Proofs. It is not clear that they have Mathematical Solutions, let alone Mathematical Proofs.

Theorems From Physics? claims that:

“mathematical theorems are not supposed to be contingent. This is a fancy philosophical term for propositions that are “true in some possible worlds and false in others.” In particular, the truth of a mathematical proposition is not supposed to depend on any empirical fact about our particular world.”

With all due respect, that’s theology. Conventional theology, so called “Platonism”, but still theology. For me Plato, and his modern parrots are seriously obsolete, and “an embarrassment, for these people are friends”, as Aristotle put it.

I can show that the proof that square root of two is irrational contains assumptions made on an empirical basis (along the lines of mn = nm, actually; similarly, the choice between Presburger arithmetic and Robinson, or Peano, or Ayme arithmetics, can be viewed as empirically driven.)

However, what is an achieved mathematical proof? Just a neural arrangement. Similar neural arrangements in the minds of noble primates called mathematicians. Thus, a mathematical proof is a physical object constructed similarly in the minds of many. So a mathematical Proof is a Physical Proof, just as the fluid in a tube is a Proof of a Physical Problem, the flow problem. And similar tubes have similar “proofs”, once similar fluids similarly flow.

So any Mathematical Proof is a Physical Proof.

***

Patrice Ayme

***

Notes:

1) Could Quantum Theory be Wrong?

(Meaning not as perfect as it is taken to be.) Actually the main objection I have against the Quantum-as-it-is is exactly the same as the objection Isaac Newton had against his own theory of gravitation: instantaneous interaction at a distance with nothing between made no sense, said Newton.

(Einstein remedied this partly by proposing that gravitation was a field propagating at the speed of light.)

2) The preceding was a comment of mine on the “Gödel Lost Letter and P=NP” site in Theorems From Physics?

And most notably the following passages: “The philosopher in us recoils dogmatically at the notion of such a “physical proof”…  Imagine that someone shows the following: If P is not NP, then some physical principle is violated. Most likely this would be in the form of a Gedankenexperiment, but nevertheless it would be quite interesting. Yet I am at a loss to say what it would mean. Indeed the question is: “Is this a proof or not?”

Actually this is exactly the general method I used to prove there is a largest number. Basically, I said, if there is infinity, there is a violation of the conservation of energy principle. Oh, by the way, if you want to know, in my system, the proof of P = NP is trivial (as everything is polynomial; four words proof, so I should the Clay Prize, hahaha)…

NON ARISTOTELIAN

November 5, 2013

It’s a NON ARISTOTELIAN WORLD:

Tyranosopher: Finite Logic should be called Non Aristotelian Logic. As I will show.

Simplicius Maximus, a contradictor: I have two objections to your finite math madness. First it makes no sense, and, secondly, even if it did, it would be pointless. 

Tyranosopher: I love contradictions. I squash them, then drink their juicy parts. OK, bring it on. Let’s start with the contradiction you found. A French contributor, Paul de Foucault, already made the objection that m/0 = infinity. 

Sounds good. However, it violates Peano Arithmetic (PA). PA is the arithmetic common to all metamathematics. But for mine, of course. (I violate much, with glee, including the pairing axiom!)

In PA, a.0 = 0 is one of the two axioms defining multiplication. So we see that if x = m/0, we would have x.0 = m. In other words, m = 0.

That’s not surprising: a number called “infinity” is not defined in PA

Simplicius Maximus: OK, fine. Here is my objection. It’s well known that the square root of two is irrational. Even Aristotle knew this, but you apparently don’t. And then you give the world lessons about everything. You are a charlatan. 

T: What do you mean by irrational?

SM: Ah, you see? It means square root of two cannot be equal to m/n, where m and n are integers. Let’s abbreviate square root two by sqrt(2). Irrational means the expansion of sqrt(2) never ends. 

T: Why? 

SM: Here is the proof. Suppose sqrt (2) were rational. That means: m/n = sqrt (2). Let’s suppose the terms m and n are as small as possible. That’s crucial to get the contradiction. 

T: Fair enough.

SM: Now, square both sides.  

T: That means, more exactly, that you contrive to multiply the left hand side of the equation by m/n and the right hand side by sqrt(2).

SM: Happy that you can follow that trivial trick. That gives us the equation: mm/nn = 2.  

T: As sqrt (2) sqrt (2) = 2. Indeed. By the way, you made an unwarranted assumption, so I view your reasoning as already faulty, at this point

SM: Faulty? Are you going mad? 

T: I will dissect your naïve error later. But please finish, Mr. Aristotle. 

SM: Call me Aristotelian if you wish. Multiplying both sides of the equation by nn, we get: mm =  2 nn. That implies that m is even. Because if m were odd, m = 2u + 1, then mm = 4uu + 4u + 1 , the sum of an even number (4uu + 4u) plus 1… And that, the sum of an even number with one, is odd. Hence m = 2a.

But then 2a2a = 2 nn, or: 2 aa = nn. Thus n is even (same reasoning as before: the square of an odd number cannot be even). So we see that both m and n are even, a contradiction, as we assumed m and n were the smallest integers with a ratio equal to sqrt (2). 

T: This proof is indeed alluded to in Aristotle, and was interpolated much later into Euclid’s elements. The official Greek mathematicians did not like algebra. 

SM: I see that, although you don’t know math, you know historiography.

Tyranosopher: I do know math, I’m just more rigorous than you, august parrot.

Simplicius Maximus: Me, a parrot? Me, and 25 centuries of elite mathematicians who are household names, dozens of Field Medalists are also of the avian persuasion? How can you be so vain and smug? 

Tyranosopher: Because I’m smarter.

SM: Really? Smarter than Aristotle? 

T: That’s an easy one. People like Aristotle spent a lot of time, all too much time, with politics, not enough with thinking. OK, let’s go back to your very first naive mathematical manipulation. You took the square of both sides. 

SM: Of course I did. 

Tyranosopher: You can’t do that.

SM: Of course I can.

Tyranosopher: No. In FINITE math, a = b does not imply that aa = bb

SM: Why?

T: Because aa could be meaningless. It could be too big to have meaning. It’s a added to itself a times. If, as we compute aa, we hit the greatest number, #, we must stay silent, as Wittgenstein would have said. 

In FINITE math, the infinite set of integers N does not exist. Only what can be finitely constructed exist. Because there is no way to construct the set N, as it would be infinite (if it existed; that’s a huge difference between what I propose, and what David Hilbert proposed). In my system, integers and rational numbers are constructed,  according to the principles I exposed in META, layer by layer, like an onion

SM: Wait. There are other proofs of the irrationality of square root of two.

T: Yes, but it’s always the same story: at some point, multiplication is involved, so my objection resurfaces.   

SM: OK, all right. Let me go philosophical. What’s the point of all this madness? Trying to look smarter because you are so vain, at the cost of looking mad? Do you realize that you are throwing out of the window much of modern mathematics?

T: Calm down. Entire parts of math are left untouched, such as topology, category theory, etc. My goal is to refocus all of math according to physics, and deny any worth to the areas that rest on nothing.

All too many mathematicians have engaged in a science as alluring as the counting of angels on a pinhead in the Middle-Ages. 

SM: Dedekind said: “God created the integers, and the rest was man’s creation.” 

T: Precisely, God does not exist, so nor does the infinite set of the integers, N. This will allow mathematicians to refocus on what they can do, and remember that there is a smallest scale, and it would, assuredly change the methods of proof, in many parts.

SM: Such as? 

T: Take the Navier Stokes fluid equation: one has to realize that, ultimately, the math have got to get grainy. This would help physics too, including all computations having to do with infinities. 

SM: You are asking for a mad jump into lala land.

T: We are already in lala land. Finding the correct definitions is even more important than finding the correct theorems (as the latter can’t exist without the former). The reigning axiomatic theory, ZFC (Zermelo Fraenkel Choice) requires an infinite number of axioms. What’s more reasonable? An infinite number of axioms, or my finite onion?

The answer is obvious. It’s a NON ARISTOTELIAN WORLD.

In my not so humble opinion, the consequences are far reaching.

***

Patrice Ayme

FINITE CALCULUS

October 31, 2013

If we want to get real smart, we will have to let no reason unturned. Foundations of calculus have been debated for 23 centuries (from Archimedes to the 1960s’ Non Standard Analysis). I cut the Gordian knot in a way never seen before. Nietzsche claimed he “made philosophy with a hammer”, I prefer the sword. Watch me apply it to calculus.

I read in the recent (2013) MIT book “The Outer Limits Of Reason” published by a research mathematician that “all of calculus is based on the modern notions of infinity” (Yanofsky, p 66). That’s a widely held opinion among mathematicians.

Yet, this essay demonstrates that this opinion is silly.

Instead, calculus can be made, just as well, in finite mathematics.

This is not surprising: Fermat invented calculus around 1630 CE, while Cantor made a theory of infinity only 260 years later. That means calculus made sense without infinity. (Newton used this geometric calculus, which is reasonable… with any reasonable function; it’s rendered fully rigorous for all functions by what’s below… roll over Weierstrass… You all, people, were too smart by half!)

If one uses the notion of Greatest Number, all computations of calculus have to become finite (as there is only a finite number of numbers, hey!).

The switch to finitude changes much of mathematics, physics and philosophy. Yet, it has strictly no effect on computation with machines, which, de facto, already operate in a finite universe.

In the first part, generalities on calculus, for those who don’t know much; can be skipped by mathematicians. Second part: original contribution to calculus (using high school math!).

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WHAT’S CALCULUS?

Calculus is a non trivial, but intuitive notion. It started in Antiquity by measuring fancy (but symmetric) volumes. This is what Archimedes was doing.

In the Middle Ages, it became more serious. Shortly after the roasting of Johanne’ d’Arc, southern French engineers invented field guns (this movable artillery, plus the annihilation of the long bow archers, is what turned the fortunes of the South against the London-Paris polity, and extended the so called “100 year war” by another 400 years). Computing trajectories became of the essence. Gunners could see that Buridan had been right, and Aristotle’s physics was wrong.

Calculus allowed to measure the trajectory of a canon ball from its initial speed and orientation (speed varies from speed varying air resistance, so it’s tricky). Another thing calculus could do was to measure the surface below a curve, and relate curve and surface. The point? Sometimes one is known, and not the other. Higher dimensional versions exist (then one relates with volumes).

Thanks to the philosopher and captain Descartes, inventor of algebraic geometry, all this could be put into algebraic expressions.

Example: the shape of a sphere is known (by its definition), calculus allows to compute its volume. Or one can compute where the maximum, or an inflection point of a curve is, etc.

Archimedes made the first computations for simple cases like the sphere, with slices. He sliced up the object he wanted, and approximated its shape by easy-to-compute slices, some bigger, some smaller than the object itself (now they are called Riemann sums, from the 19C mathematician, but they ought to be called after Archimedes, who truly invented them, 22 centuries earlier). As he let the thickness of the slices go to zero, Archimedes got the volume of the shape he wanted.

As the slices got thinner and thinner, there were more and more of them. From that came the idea that calculus NEEDED the infinite to work (and by a sort of infection, all of mathematics and logic was viewed as having to do with infinity). As I will show, that’s not true.

Calculus also allows to introduce differential equations, in which a process is computed from what drives its evolution.

Fermat demonstrated the fundamental theorem of calculus: the integral was the surface below a curve, differentiating that integral gives the curve back; otherwise said, differentiating and integrating are inverse operations of each other (up to constants).

Arrived then Newton and Leibnitz. Newton went on with the informal, intuitive Archimedes-Fermat approach, what one should call the GEOMETRIC CALCULUS. It’s clearly rigorous enough (the twisted examples one devised in the nineteenth century became an entire industry, and graduate students in math have to learn them. Fermat, Leibnitz and Newton, though, would have pretty much shrugged them off, by saying the spirit of calculus was violated by this hair splitting!)

Leibnitz tried to introduce “infinitesimals”. Bishop Berkeley was delighted to point out that these made no sense. It would take “Model Theory”, a discipline from mathematical logic, to make the “infinitesimals” logically consistent. However the top mathematician Alain Connes is spiteful of infinitesimals, stressing that nobody could point one out. However… I have the same objection for… irrational numbers. Point at pi for me, Alain… Well, you can’t. My point entirely, making your point irrelevant.

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FINITUDE

Yes, Alain Connes, infinitesimals cannot be pointed at. Actually, there are no points in the universe: so says Quantum physics. The Quantum says: all dynamics is waves, and waves point only vaguely.

However, Alain, I have the same objection with most numbers used in present day mathematics. (Actually  the set of numbers I believe exist has measure zero relative to the set of so called “real” numbers, which are anything but real… from my point of view!).

As I have explained in GREATEST NUMBER, the finite amount of energy at our disposal within our spacetime horizon reduces the number of symbols we can use to a finite number. Once we have used the last symbol, there is nothing anymore we can say. At some point, the equation N + 1 cannot be written. Let’s symbolize by # the largest number. Then 1/# is the smallest number. (Actually (# – 1)/# is the fraction with the largest components.)

Thus, there are only so many symbols one can actually use in the usual computation of a derivative (as computers know well).  Archimedes could have used only so many slices. (The whole infinity thing started with Zeno and his turtle, and the ever thinner slices of Archimedes; the Quantum changes the whole thing.)

Let’s go concrete: computing the derivative of x -> xx. it’s obtained by taking what the mathematician Cauchy, circa 1820, called the “limit” of the ratio: ((x + h) (x + h) – xx)/h. Geometrically this is the slope of the line through the point (x, xx) and (x + h, (x + h) (x + h)) of the x -> xx curve. That’s (2x + h). Then Cauchy said: “Let h tend to zero, in the limit h is zero, so we find 2x.”  In my case, h can only take a number of values, increasingly smaller, but they stop. So ultimately, the slope is 2x + 1/#. (Not, as Cauchy had it, 2x.)

Of course, the computer making the computation itself occupies some spacetime energy, and thus can never get to 1/# (as it monopolizes some of the matter used for the symbols). In other words, as far as any machine is concerned, 1/# = 0! In other words, 1/# is… infinitesimal.

This generalizes to all of calculus. Thus calculus is left intact by finitude.

***

Patrice Ayme

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Note: Cauchy, a prolific and major mathematician, but also an upright fanatic Catholic, who refused to take an oath to the government, for decades, condemning his career, would have found natural to believe in infinity… the latter being the very definition of god.

META

October 8, 2013

I pursue my (energy motivated) program of turning all mathematics and logic, FINITE. I define the appropriate notion of META. Not just that, but I use the notion to make any logic into a chrono-logy. (A Chronology/Semantic Hierarchy evades the logical paradoxes.)

This is extremely advanced material, well beyond the edge of what’s commonly understood, using implicitly the implicated order from my sub-Quantum theory. Still most of the notions used below are easy to understand!

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The notion of “META” is fundamental for the analysis of any system of thoughts or emotions. What’s going meta? I claim: Any theory has meta-theories associated to itself.

If one looks at the literature of meta, it’s a big mess. Recently it was encumbered by a sensation author obsessed by “strange loops” (Douglas Hofstadter, in books starting in 1979 with Gödel, Escher, Bach…)

Studying meta with “strange loops” is older than Aristotle (see the Cretan paradox below).

However the notion of meta I introduce here is much more general (although it contains the “strange loops” thingy, it also evades it, see below!)

To understand the essence of meta, one has to go back to bare-bone logic.

Given a language L, one can talk within that language L. However, what’s L made of? L = (LOG, TRUTH, U). “LOG” is the logic, U the Universe of objects the logic applies to. The logic consists in a set of assemblies that can be applied again and again to objects of U and make constructions. “TRUE” is a label applied to some Well Formed Formulas (WFF) within LOG. (Not all WFF are TRUE.)

Example: suppose LOG is the usual logic, and U consists only of the set made of 3 elements: eat, banana, good. Then ((eat, banana) –> good), a Well Formed Formula from LOG and U, could be the (one and only) TRUE formula (all WFFs are true in some purely formal sense).     

Metalogic and metamathematics, as usually understood, arose when Cantor showed that the Real Numbers were uncountable. Cantor was the metamathematician per excellence (he invented cardinal and ordinal theories). Cynics would say that’s why Cantor became crazy: he went a few “meta” too far.

Relatively simple modifications of (one of) Cantor’s proof(s), his diagonalization trick, led to the revelation that any logical system that contains the usual arithmetic is incomplete: statements can be made that are neither true nor false (which statements, that’s not clear; although Cantor’s Continuum Hypothesis is one of them…).

From my point of view, the problem with the most honorable, and usual, metalogic is that it uses infinity to go from logic to metalogic. I believe only in finite stuff. (Still the Cretan/Liar paradox, that started the field, 26 centuries ago, looks finite, although it truly is not really…)

However one can define meta easily in a finite (or not!) setting:

TRUE, (by definition the set of all true WFFs) is a subset of WFF, the set of all WFFs. (LOG2, TRUTH2, U2) is meta relative to (LOG1, TRUTH1, U1) if and only if each of three sets of the latter is a subset of the corresponding set of the former, one of them strictly (say TRUTH 2 includes TRUTH1, or U2 includes U1).

So meta carries as a useful concept in the finite realm, and has nothing to do with confusing causal loops.

How is the 26 centuries old Liar paradox solved in this scheme? That’s the paradox presented by the statement:

“This statement is false.”

Well, that deserves its own essay. Let’s just say I was chuckling all the way about how clever I was, until I discovered that my first solution was exactly the one found by Buridan seven centuries ago, and the second one, using my theory of meta above, resulting in a semantic hierarchy, was somewhat similar in spirit to that of Alfred Tarski.

Buridan’s solution is excellent (he notices that “This statement is false” is equivalent to A and non A, so is obviously false). However this is too ad hoc. One needs to handle contradictions where the implication chain is longer (A –> B –> Non A). Thus:

My hierarchy idea is to build the Language L by layers, like an onion, starting with a core (L, TRUE, U). One assumes that the initial TRUE of WFFs is non contradictory. Call that SEMANTIC (0). And then one grows TRUE by using L and U, one implication (or operation) of L at a time. Operating L once on TRUE, one gets TRUE (1). Either TRUE (1) has a self contradiction, or not. If it does, stop: (L, TRUE, U) admits no META. If it does not, call it SEMANTIC (1), and proceed to (L, TRUE(2), U). And so on. The iteration operation gives a notion of time (like a clock in a computer). L(n + 1) is richer than L(n), etc.

Thus META allows to build a hierarchy of logics, and semantics. To say that a theory is “meta” relative to another can be rigorously defined.

Progress in understanding is always achieved by climbing up the Semantic Hierarchy of meta.

***

Patrice Ayme

I Mood Therefore I Think

July 13, 2012

SYSTEMS OF MOOD ARE CRUCIALLY ENTANGLED WITH IDEAS:

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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.

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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.

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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!)

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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.

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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. 

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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…

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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.

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Patrice Ayme

LARGEST NUMBER

October 10, 2011

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

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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 (as a result).

I have more basic, and much more drastic claims:

There is a largest number. Or more exactly, numbers can’t be too large (in sheer size of the numbers of digits needed to express them). 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 unobservable castles in the air exist, and compute with them, to boot! Basic number theory and logic have to become much more subtle.

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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 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 (established in the 17th Century) resting on the concept of continuum (19th Century)… In other words, on the construction of the real numbers, in the second half of the nineteenth century, 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, over the millennia, 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. It’s naturally one of the main ways a philosophical attitude by civilization class scientists impacts science. 

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; also the nickname of somebody dear to me).

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. This means that any physical theory, even a classical one, is indeterminate, just from that.

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.” Dedekind made “cuts”… A Quantum event (there no classical events (except in an approximate sense).

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.

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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.

Real numbers are not real. Really.

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Patrice Ayme

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