Proof – Harmonia Philosophica

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.

Unless…

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.

But…

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.

Gibberish.

In a cosmos where mathematics cannot prove themselves.

Gibberish.

In a world where endless-loop paradoxes exist.

Paradoxes.

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?

Is there a black elephant in the room?

Many people are discussing these days the racism topic.

A black man – George Floyd – was some days ago killed due to – as it seems – police brutality. This sparked a series of protests some of which were violent.

People started to talk against the protests and that violence. Others responded that the violence had started actually from the police. But no matter the details, the question of whether racism is a problem in the US was prevalent in all discussions.

For me, this is an elephant sitting in the room.

All we have to do is see it.

But as in the elephant example, what I see is not what you see…

For some, there is no solid ‘proof’ of racism in the US. And to be fair to the other countries as well, I will extend this to all other countries: For those people who deny the existence of racism altogether, there is no solid evidence for racism against blacks in the West in general (cannot really speak for other countries). There is no ‘proof’ that this group of people is treated unfairly.

The arguments the proponents of this ‘There-is-no-racism’ view have, vary from wrong to ridiculous – with a strong tendency to the latter.

RELATED ARTICLE: The source of ethics

So for example in the case of George Floyd, I have seen many people asking “How do I know that the police officer did what he did because of racism? Perhaps he is doing the same to white people as well”. That could be a good counter-argument. If we had actually data to support it. If not (which is currently the case), then this counter-argument is just a generalized counter-argument based on ‘doubt’ which we anyway have for everything in life – even for things we see in front of our eyes.

Hey, it is raining.

(But how can I be sure that it rains?)

This counter argument used by those denying the existing racism against blacks (and gays and women etc) is the cornerstone of hypocrisy. They see something (in that case we even have a video) but they choose not to see it because of something else that… might be true. They do not know whether the police officer could do something like that to whites as well. They just assume it. And then based on that assumption, they build their theory.

Convenient is it not?

Sure is.

But simple things are never simple.

The denial of so many in the face of something so evident is based on something very fundamental in human thought: Our inability to prove (or accept the proof) of anything, unless we experience it. This is something constantly mentioned by Harmonia Philosophica, but with regards to science philosophy mainly. Here it is – wrongly – used to justify injustice. Yes, it is true that there can be no positive proof not only of racism in this case but of any other logical proposition of any kind.

The truth is a philosophically elusive notion and cannot be attained, even for the most trivial and fundamental of scientific beliefs. For every scientific theorem or theory is based on unapproved axioms. Change these axioms and you will reach to a different theory (if that sounds weird, then read more Harmonia Philosophica). So if it is not possible to even prove that 1+1=2 (really, even this is based on axioms), how could one prove that the death of a black man under police brutality was because of racism?

And yet, we know it.

People denying the obvious are in this case doing nothing more than using a loophole in our inherent thinking mechanism to deny what in other circumstanced they would accept at ease. It would be right to say “I don’t know” or “I am not sure”, if you indeed said that every time you are not sure about something. But as said above, we are never 100% sure about anything, nor can we be in any way.

RELATED ARTICLE: Why you can’t be an agnostic

So what is this?

How can we be sure?

The answer is already answered.

By your everyday actions.

By your everyday choices.

By life itself.

Life which transcends the theory and makes us open our eyes beyond the limits of philosophy. Because no matter how you “do not know” whether there is racism, you do choose to be afraid of a black running in the street. No matter how much you are not “sure” of, automatic face detection algorithms you design keep on detecting more black people as ‘criminals’. No matter how much you are not ‘sure’ that there is racism, you still need to revert to nonexistent hypotheses to prove that what you see was not true. Regardless of your inability to pinpoint racism against black people, you do feel weird when standing next to a black person. No matter whether you are – philosophically – certain that black people do have the same rights as other people, it is still true that we have a disturbingly extremely low representation of black people in high-level positions.

Because you see, the greatest problem with science per se, is its inability to testify for the obvious. Even when it is raining, there will always be possible to claim that everything is an illusion and even build a theory based on that illustrious assumption. (Why not? Scientists today even talk about multiple universes which we will never anyway see – and they even get paid for that research) Even when Achilles is running to overcome the turtle, philosophy will still be able to ‘prove’ that Achilles will never overcome it. And even when black people are dying outside, science will still struggle with statistical models to ‘prove’ whether there is racism or not…

So beware of people expressing ‘doubt’.

They are usually the ones with the greatest beliefs.

And when they close their books.

And when they walk outside…

They will be astounded to see…

That Achilles did overcome the turtle.

That the policeman did step on Floyd’s neck…

And some of them, some of them who still have a soul, will come to realize what they knew but they have forgotten. That whatever they know, is because they really don’t know anything at all…

Hey!

Who put that black elephant in the room?!

Important Notes

Similar to the argument “How can we know it?” is the opposite argument “But I can prove to you that there is no racism”. In such arguments people tend to use examples of how for example the police was once also brutal to a white person. This is the opposite of what was described above: As it is impossible to prove anything beyond the shadow of a doubt, it is also very easy to claim that you have ‘proved’ something with the relevant assumptions. Again in that case, the very same fundamental limitation of science and our way of thinking is exploited: That whenever we think to prove something, we need to start from somewhere. And that somewhere is always a not-proved axiom/ assumption. With proper models and assumptions, one can even ‘prove’ that our universe does not exist.
No case is like the other. I have sure not covered all of the here. There are also cases which are similar to the case of black people suffering racism (e.g. women or gay people suffering discrimination) who are not also mentioned here. This is because the purpose of this post is not to generalize or prove anything. The are sites which do that in a much better way while providing all the necessary data for all these types and cases of racism and discrimination. The purpose of this article, since Harmonia Philosophica is a philosophy portal, is to show that philosophy and theory are good but up to a point. And from that point onward, life itself is much more important.
I have deliberately taken a stand not to refer to the violence in the protests during the previous days. Again, I believe there are many sites which cover these facts in an excellent and very professional manner. The goal of this article is to provide some philosophical insight. Nothing more. And, I hope, nothing less.

(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.