Plato was right

Today's guest post is by Marek Wolf

When in 1997 I visited Isaac Newton Institute in Cambridge I bought a book "Men of Mathematics" written by E.T. Bell, edition from 1937 [1]. In the five story bookshop there were a few shelves with classics: "Mathematical Principles of Natural Philosophy" by Isaac Newton, "On the Origin of Species" by Charles Darwin, "Principles of Quantum Mechanics" by Paul A.M. Dirac from 1958 and the above-mentioned book by E.T. Bell.

Chapter nine of Bell's book  is devoted to Leonhard Euler (1707--1783). On page 147, citing de Morgan, Bell tells an anecdote, about Euler who was a deeply religious Calvinist. At the same time there lived Denis Diderot, who, as a progressive philosopher,  was atheist. The two men  met one day and Diderot asked Euler if he has a proof that God exists. Euler wrote some formula and said: "Hence God exists. Sir, please reply". Diderot embarrassed quickly walked away.

We do not know what the formula was, but here, I will present two formulas in the same spirit, which I often expose  students to during the first calculus lecture. The first is following: 

 Add two first odd numbers: 1+3 = 4, we added two numbers and the result is 4=22. Next we add five: 1+3+5=9, the summations of three first odd numbers gives 9=32. Next we add 7 and obtain 16=42, again the square of a number of added terms. It continues to be true up to to infinity. In formula we can write it as

The simplest proof is by induction.

The second example is following:

 We add cubes of consecutive natural numbers. Thus we have

1 + 23 = 1 + 8 = 9= 32 = (1+2)2 ,

1 + 23 + 33 = 36 = 62 = (1 + 2 + 3)2 ,

1 + 23 + 33 +  43 = 100 = 102 =  (1+2+3+4)2 : the sum of cubes of the consecutive integer numbers is a square of the sum of these numbers.

And again so on to infinity.

Isn't it miraculous and mysterious? It seems to hide some deep secret. As an equation we write

This identity is sometimes called Nicomachus’s theorem, after Nicomachus of Gerasa (c. 60 – c. 120 CE). About four centuries ago, Johannes Faulhaber (1580--1635) developed formulas for the power sums. He published many papers on this topic, initially he found particular sums up to exponent 7 in 1614. Subsequently Faulhaber gave a formula expressing the sum of the k-th powers of the first n positive integers. In no other case is a sum of powers of consecutive natural numbers equal to square of another sum. In [2] Sheila M. Edmonds proved that no higher power of n sums would be expressed as the square of the sum of powers of 1, 2, ...,, n. More precisely she proved that if

then p=3 and q=1.

These formulas (1) and (2) are the absolute and eternal truth. It means that they were true even before mankind appeared on the Earth. They were true also before the Big Bang (if you believe in it). It means that there is a world containing such mathematical truths independent of our real Universe. This idea was first put forward by the Greek philosopher Plato (428/427 or 424/423 – 348/347 BC). This system of beliefs in philosophy is called "Platonism". 

 What is the reality of mathematical theorems? Has the theorem existed anywhere until the moment it is formulated by a mathematician, or is it only the appearance of it in the head of a human that creates the reality it concerns?

Many mathematicians consider themselves Platonists.

Today Euler could argue using the axiom of choice. It is a kind of mathematical joke: choice axiom is equivalent to the existence of God. The choice axiom is a statement from set theory and was formulated in 1904 by Ernst Zermelo. The axiom of choice says that given any collection of non--empty sets (even if the collection is uncountable), it is possible to take one element from each set and construct a new set from these elements. Assuming the axiom of choice,   Zermelo proved that each set can be well-ordered. It means that in each set there exists the first element with respect to some ordering relation. In particular a set of all causes can be well ordered.  Hence there exists the first cause. 

 According to Saint Thomas Aquinas the first cause of everything was God. In the opposite direction it is simpler: God is an omnipotent being and thus can choose from each set by one element and the axiom of choice follows. In 1963 Paul Cohen (1934 -- 2007) proved that the axiom of choice is independent of other axioms of the set theory. It means it is possible to have consistent mathematics with or without the choice axiom. Therefore it "means" it is possible to have a world with God and a world without God.

German mathematician Don Zagier in his habilitation lecture in 1977 said "... upon looking at prime numbers one has the feeling of being in the presence of one of the inexplicable secrets of creation." [3]


[1] E.T. Bell, "Men of Mathematics. The lives and achievements of great mathematician from Zeno to Poincare ", A touchstone book, 1937, New York, London, Sydney

[2] Sheila M. Edmonds, Sums of Powers of the Natural Numbers, Mathematical Gazette, Vol. 41, No. 337 (Oct., 1957), pp. 187-188

[3] D.Zagier, The first 50 million prime number,  Math. Intellig. no. 0 (1977), p.8, left column

Marek Wolf

Thinking With a Forked Brain

This is a continuation of the previous post "Psychological interlude - Authority in Science", which is a continuation of "Quantum Magic - Incoherent Decoherence", which is a continuation of ... etc.

It seems to me that the thinking of those who write that “Today there seem to be no phenomena which contradict quantum theory – perhaps with the sole exception that there are definite (“classical”) phenomena at all!” is highly compartmentalized. Such a person “thinks with a forked brain, ”exactly as is the case with right-wing authoritarians. The Dictionary of Psychology tells us also that:

"So the picture of authoritarian followers after all these years of research is far from flattering – unless you are a potential dictator."

Indeed, the picture of many of the quantum theorists emerging from their “decoherence” papers is not very flattering. We will see it better after we delve into the subject somewhat deeper.

Further Incoherence

So, let us continue with the preface of the book about Decoherence. First we find a statement that quantum theory, “from a conservative point of view … may well be called inconsistent.” Next comes the admission that standard textbook arguments “are insufficient for several reasons”, and that “conventional treatments” (of the classical limit of quantum theory) “are flawed for a simple reason: they do not represent any realistic situation.”

So, we now know how really bad quantum theory is. It is in terrible shape, everything that is definite and real contradicts it. So, what to do?

The answer, from reading the contributions to this book and to other books and papers, lies in “decoherence”.

But what is this decoherence? Is it something that is objective? Or is it something that occurs only in the minds of some theoretical physicists? In order to answer this question we need to look into these minds.

What is this decoherence that is so fashionable nowadays? According to the French Wikipedia:

The quantum decoherence is a physical phenomenon capable of explaining the transition between them quantum physical rules and classic physical rules as we know them, on a macroscopic level. More particularly, this theory provides an answer, recognized as being the most complete to date, to the paradox of Schrödinger cat and at quantum measurement problem.

The theory of decoherence was introduced by Heinz Dieter Zeh in 1970

So, we learn that “it is a physical phenomenon”. Is it a definite phenomenon or not? If it is definite, then, according to Erich Joos, it would contradict quantum theory. That would be bad. So, I am deducing that it is not a very definite phenomenon. But, it is said that it is a “physical phenomenon”. So, let us check what Mr. Zeh has to say. According to Zeh’s article in the Decoherence book, this decoherence occurs when “the complete information about the passage of the particle is carried away (into the “environment”) in some physical form.”

Information and Environment: What is hidden by Decoherence

So, we have two new terms, one is information, the other one is “environment” – put in quotation marks by Zeh. Why in quotation marks? I checked the number of occurrences of “environment” in the book, and the result is 372. But I could not find even one precise definition. What I have found, however, is that

In quantum cosmology (where no environment exists), decoherence is only meaningful with respect to local observers (subsystems).

What is real? Is Nobel Prize real?

That there may be a problem with “environment of the Universe”, I can imagine, but then he goes to say that decoherence is meaningful only with respect to a local observer. But what exactly is “local observer”? And does that mean that before there were any observers in the universe, this mysterious decoherence did not happen? Is decoherence really something physical or it happens only in the mind of a physicist?

Can Quantum Phenomena Solve the Mystery of Consciousness?

 It is, in fact, a physicist who separates, in his mind, the Universe, in two parts; one part he calls “the system”, another part he calls “environment”. Then he starts calculating and comes up with some numbers. Another physicist will make the division in a different way, and will come up with a different set of numbers. So, it seems that all this decoherence business is another way of saying that quantum theory really is not able to get rid of subjectivity, and all this fashionable decoherence program is just a way to hide this fact under a pretentious name. Is that the case?

Well, I will show that it is even worse than that. But we need to talk about Schrödinger’s cat first.

To be continued

P.S.1. What is REAL? Here is an example:

Mark Knopfler & Emmylou Harris -
I Dug Up A Diamond
Real Live Roadrunning
Official Live Video

My wife likes this... so this is for her. Very REAL lyrics are here.

P.S.2. The first seven pages of ``Notes Aether Theory''.

P.S.3. Worth reading, it is REAL: More on my case, the law on misinformation, and why other doctors don't challenge their medical boards

P.S.4. 02-06-23 19:10 It’s the movie they really don’t want you to see: #WhatIsAWoman?

P.S.5. 03-06-23 8:38 Yesterday a Reader wrote to me asking for a help with one of the ``inventions'' of a Polish mystic-mathematician Hoene-Wronski. I knew the name, but nothing more. So I checked the internet resources. Wikipedia tells us, in particular, that:

In science, Hoene-Wroński set himself an extraordinary task: the complete reform of philosophy as well as that of mathematics, astronomy and technology. He elaborated not only a system of philosophy, but also applications to politics, history, economics, law, psychology, music and pedagogy. It was his aspiration to reform human knowledge in an "absolute, that is, ultimate" manner.

In 1803, Wroński joined the Marseille Observatory, and began developing an enormously complex theory of the structure and origin of the universe. During this period, he took up a correspondence with nearly all of the major scientists and mathematicians of his day, and was well respected at the observatory. In 1803 Wroński "experienced a mystical illumination, which he regarded as the discovery of the Absolute."^[3]

Well, I have had my own illumination in Marseille as well. And similarly to Wronski i saw that at the foundations of the universe is an `open loop process', a `multi-threaded algorithm', not an "equation". EEQT was one of the off-springs of this illumination.

Additionally I was writing in my journal after the illumination the following

So work, work, work. Work gives results. Sooner or later. Work gives satisfaction. It gives happiness. I think, for example, of those moments when I did something. When I have accomplished something. Then it is good, then there is happiness. On the other hand, I think about the remark of M. Blau that I am somehow unhappy, dissatisfied. Why? Because I don't produce anything, I don't create anything. So every thing made gives happiness. And happiness is probably the sum of things produced. Almost certainly. Happiness comes from giving.  From giving the best of yourself. Birds probably know this - they sing. Flowers know - they smell and give colors. The sun knows - it gives light. And I should know and remember it myself. To give others the best. And one more thing: that flowers do not look if someone smells. Flowers do not look if someone is present, if someone is watching. Whether someone is around. Flowers only look at whether they smell good. They give. And it's accumulating somewhere. They give to the Creator. And it is somewhere. It's not going to die. It is already there. Irreversibly. Some part of non-being is transformed into being? Are we sure it is so? But shouldn't one make an effort to make one's work available to others? Yes and no. The fact is that sharing no longer adds to the sum of happiness. Which sum depends solely on the number of things completed and released. So creation is important. Hence the happiness of young Werther's creative anguish.

Of course, today we are taught that birds sing and flowers smell for mundane reasons - ``the goal of attracting attention.'' Simple evolution," they say. One can look at it that way too, but one loses infinitely more with such a limited understanding of reality.

Two weeks later, now in Florence, still influenced by the ``illumination" I was writing this:

Here we return to the domain. Inside, nothing can be created. All creation takes place on the boundary. It is the boundary that is most interesting. It is through the boundary that opening takes place, it is on the boundary that causality takes place, it is on the shore that time flows. There is no time in the middle. The middle is Euclidean, in the middle nothing does age, but also nothing is born.

P.S.6. Perhaps I should not worry so much about non-orientability. Here is an interesting paper:

Nontrivial topological states on a Möbius band, and here is the video of making the double cover of the Mobius band

The double cover is orientable. And there is a general theorem stating  that every unorientable manifold has a double cover that is orientable. So the double cover is the answer! I knew it!!! More here: Understanding the orientable double cover.

Information Universe

 I will be developing this post during the coming days. It will be based on this interview:

Alexey Krugly about Quantum Gravity and the Information Universe

See also this lecture (in Russian)

* Transcript of the talk, Russian, doc format. Automatically generated subtitles, sometimes really funny

** Russian subtitles in .srt format (with timestamp)

*** English subtitles (automatic generation and translation) in .srt format

I am now reading the transcript and highlighting the essential sentences. Will replace the doc when finished.

Kruglov mentions in Particular David Finkelstein and his "Space-Time Code" ideas. These ideas has been developed in a monograph "Quantum Relativity", Springer 1996. At some point I was very excited about this approach, where Finkelstein started with "At the beginning there was a Word". The idea is, more or less, that the Universe is resembling something similar to the Matrix in the movie, a computer game running by some algorithm. Which leaves, however, unanswered th question: where is the Programmer who selects the algorithm to be run. Kruglov, in his talk, stresses the fact that only quite a small number of algorithms leads to "interesting" and self-organized and self-propagating structures. Kruglov, Wolfram ot Conwey (the creator of Game of Life) are doing essentially the same, though Kruglov is using random events while Cellular Automata and Game of Life are using deterministic algorithms. All of them are far away from the notions of awareness and consciousness. There is more to life then just self-organization and self-replication.

And with the next post we will be back to a gentle math. Esoteric concepts must, at some point, give us inspiration for creating rigorous abstract structures that can be used as mathematical models. It is much like a painter who converts personal and subjective ideas and visions into framed paintings that can be shared with other people. The painter needs paints, brushes and canvas. A physicist needs integer numbers and algebras.