Painting… Praying… Reading…

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During the European Middle Ages, literacy and written texts were largely the province of religious institutions. Richly illustrated manuscripts were created in monasteries for use by members of religious institutions and by the nobility. Some of these illuminated manuscripts were embellished with luxurious paints and pigments, including gold leaf and ultramarine, a rare and expensive blue pigment made from lapis lazuli stone.

In a study published in Science Advances, an international team of researchers led by the Max Planck Institute for the Science of Human History and the University of York shed light on the role of women in the creation of such manuscripts with a surprising discovery — the identification of lapis lazuli pigment embedded in the calcified dental plaque of a middle-aged woman buried at a small women’s monastery in Germany around 1100 AD. Their analysis suggests that the woman was likely a painter of richly illuminated religious texts. (1)

Reading. Writing. Praying.

We see the evidence.

To prove that something happened.

We analyze the dental plaque.

To know what this woman did.

And yet, all her efforts are cancelled.

By our lust for proof.

By our eagerness for knowledge.

For the books she helped write, called on for a different kind of knowledge. Knowledge not based on books or proof. Knowledge not based on what you see or hear. But wisdom based on the unseen and the unprovable. For it is that which is the only thing worth seeking in this irrational life governed by the unseen and the unprovable.

That woman did write or supported the writing of holy books. And she did so without the need to prove that to anyone. Her belief was strong enough not to ask for such earthly manifests of recognition. For she recognized the true essence of herself in the humility of a God who came to Earth as a Man and who was recognized by only a few fishermen.

So, the next time you open such a book, remember.

It is not a book written to be read.

But a book which is already read and that is why it was written…

Gödel’s incompleteness theorem: The non-Cretan way out…

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Gödel’s incompleteness theorem is well known for proving that the dream of most mathematicians to formulate foundations for a complete and self-consistent theory of mathematics is a futile exercise.

Gödel proved that any set of axioms you could posit as a possible foundation for math will inevitably be incomplete; there will always be true facts about numbers that cannot be proved by those axioms. He also showed that no candidate set of axioms can ever prove its own consistency (1).

In essence, the incompleteness syllogism by Gödel starts from talking abour logical propositions (or mathematical propositions if you like) and ends up with a proposition that talks about the validity of… itself. This proposition which we might as well call reads something like “I cannot be proved”.

This leads to a dead-end.

If it can be proved, then it means that it cannot.

And vice versa.

So it is essentially a logically true proposition (since indeed it cannot be proved) but which cannot actually be proved within the axiomatic system at hand.

Hence, the incompleteness.

Essentially this is something the ancient Greeks have thought of thousands of years ago; something which they formulated in the famous Epimenides paradox. Epimenides was a man from Crete who said the following simple thing: “All Cretana are liars”.

Well, this ends up in the same dead-end as the proposition mentioned above. If Epimenides is truthful, then he is a liar since he is Cretan and all Cretes are liars. If he is a liar, then he is telling the truth! And, thus, he is a liar!

A self-reference paradox which essentially destroys the hope of mathematicians around the world for a consistent and full way to formulate mathematics. It is weird, but also important to mention here, that self-reference is the basis of our existence. Consciousness, our ability to speak about our self and our own existence and being, is the foundation of our essence as human beings. Without that, we would be nothing than complex machines.

But how can this dead-end be surpassed or perhaps by-passed?

Well, it cannot actually.


You ctu right through it.

I was in a discussion the other day about the above topics and when the Epimenides paradox was mentioned, one immediate reaction that I got was the simple “So the solution is that he is not from Crete” (!)

What?! I answered. But I told you he was a Cretan.

Sure. He was.


What is he wasn’t?

Then there wouldn’t be any paradox!

In the same sense…

What if the logical proposition…

“I am false”

is not a… proposition?

Then all problems would be solved!

But if it is not a logical or mathematical proposition then what is it? Well, as I said above, self-reference is not mathematics per se. It is more of a metaphysical reference to existence and being. A proposition talking about… itself is no more a proposition but an attempt to speak with the abyss. It is more God talking to humans than humans trying to talk with God. Such a thing could be many things, but ‘simply’ a logical (mathematical) proposition not.

But this is gibberish, one might counter-argue.

Sure, it can be.

(Gibberish like the Russel way out of his paradox?)

If you really think a Cretan would ever call himself a liar.

Sure, it can be.

If you accept that a proposion can ever referto itself.

But it cannot.

In a cosmos where only humans can talk for themselves.


In a cosmos where mathematics cannot prove themselves.


In a world where endless-loop paradoxes exist.


In a life which is full with nothing but them.

Paradoxes were the end of the hopes of mathematicians. They alone can be the ones which will instil hope in the once again.

Look around Cretan.

Tell me.

If you cannot prove that there is a sea…

Will you ever lie that you are swimming?

(Scientific) Pressure.

Johannes Kepler, famed German astronomer and mathematician, first suggested in 1619 that pressure from sunlight could be responsible for a comet’s tail always pointing away from the Sun, says study co-author and UBC Okanagan engineering professor Kenneth Chau. It wasn’t until 1873 that James Clerk Maxwell predicted that this radiation pressure was due to the momentum residing within the electromagnetic fields of light itself.

“Until now, we hadn’t determined how this momentum is converted into force or movement,” says Chau. “Because the amount of momentum carried by light is very small, we haven’t had equipment sensitive enough to solve this.”

Now, technology has caught up and Chau, with his international research team from Slovenia and Brazil, are shedding light on this mystery. (1)

We always admire how science “proves” or “measures” things today. But what is really astounding is how some people came up with a theory or an explanation which is now “proved” (too heavy word, but that is another discussion) correct, without having – back then – the tools to ‘prove’ it or measure anything.

Science moves on in leaps.

Leaps made by giants.

Then the rest of the people just try to catch up.

And slowly try to “prove” things.

When they do, the giants will be another step forward.

Don’t spend too much time proving that there is a cliff.

Just have faith.

You don’t have to pass over the cliff.

You are already on the other side.

Make that leap…

There was a man who once said there is nothing to prove…

Do you really need to prove it in order to believe it?

Intuitionism (Constructivism) vs. Logicism vs. Platonism.

Does infinity exist?

Is the whole larger than the parts?

Are all the numbers either negative, positive or zero?

Phenomenally simple questions. With no definite answer!

Is everything “out there” for us to discover? (Platonism)

Is everything we can “write on paper” true? (Logicism)

Or only the things we can construct do exist? (Intuitionism/ Constructivism)

For every truth, there has been a debate. For every given axiom, there has been a completely different and opposite one. For every solution, there has been a controversy lost in the depths of time.

Search for the obvious.

Ask the “easy” questions.

Be careful of what we “know”.

It is usually the cloak of what we do not.

Science and Faith, two good friends…

If no human can check a proof of a theorem, does it really count as mathematics? That’s the intriguing question raised by the latest computer-assisted proof. It is as large as the entire content of Wikipedia, making it unlikely that will ever be checked by a human being.

“It might be that somehow we have hit statements which are essentially non-human mathematics,” says Alexei Lisitsa of the University of Liverpool, UK, who came up with the proof together with colleague Boris Konev.

The proof is a significant step towards solving a long-standing puzzle known as the Erdős discrepancy problem. It was proposed in the 1930s by the Hungarian mathematician Paul Erdős, who offered $500 for its solution. (1)

And now, we just have to BELIEVE the computer. To have FAITH in his results.
But isn’t that what we always do when we believe someone has proved a theorem – especially in a sector where we are not experts in?

Believing is a much more essential part of science than you might think….
I can prove it to you. Trust me!

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