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

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Photo by Spiros Kakos from Pexels

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?

Old philosophers, science and the poison of knowledge…

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A friend recently asked: how can we seriously read philosophers from before the 18th century, now that we know of their lack of knowledge regarding the cosmos and the universe? How can we read them and draw any serious conclusions now that we know that they knew almost nothing that we currently do, based on our supreme technology and modern science?

And my answer was: Actually it is only those philosophers whom we need to read! Because their thought was pure and not yet poisoned by the knowledge we think we have.

My friend was stunned. But what about all this knowledge we have amassed?! All the things stubbiness know for the universe? All the things physics knows about the workings of the cosmos? My friend was not the exception. It is really unfortunate that so many people believe that science today has proved things regarding the truth of our cosmos instead of what it is really doing: formulating theories to model the cosmos based on specific unproven assumptions.

But what do I mean by that?

Let’s take for example the field of astronomy and the infamous cosmological principle. This is a principle which governs astronomy today and which in two words holds the belief that the universe is homogenous and isotropic. This principle is based on observations and on this principle many theories are built by modern astronomers.

So far so good, one might say. Except the fact that nothing of the above is true.

What is true is that there are indeed observations which support the cosmological principle, but there are also observations which refute it. (See https://en.m.wikipedia.org/wiki/Cosmological_principle or https://philpapers.org/rec/KAKFGT)

So why do we hold that principle? one might ask. The answer to that would be more shocking to someone not acquainted with epistemology: Science continuously used unproven theses as a starting point of theories! This is not bad nor good. It is just the way science works. What is wrong is to take these starting point as “true” even though they never meant to have any relation to what philosophy calls “truth” or “reality”.

Scientific models are just… scientific models!

Nothing more.

Think of a glass dropping to the floor for example. This is something we all observe. Let’s now try to formulate a theory to explain this observation. The modern atheist will hold the belief that the explanation of why the glass is dropping to the floor is something “objective” and based on “facts” and data. But he would be wrong. For the observational data is just… observational data. The theory to interpret that data is something else. And for the glass dropping to the floor we have many!

Ancient Greeks thought perhaps that Zeus made the glass drop. Then came Newtown. And we explained the observation with the help of an invisible all existing field called gravitational field. And then Einstein changed everything and now we have not a field but curved spacetime!

See?

Same observation, three different theories!

But are those theories equally valid? And do they all adhere to the data equally successfully?

The answer is yes, if we wish so! Even the theory which wants Zeus to bring the glass down to earth can be formulated in such a way that there is full compliance with the observational data in hand. (E.g. by starting that Zeus makes the glass fall with an acceleration equal to g) In the same way the theory of Newton can be also as accurate as the latest theory of Einstein if we make it so. The problem is that scientists rarely tend to update the details of old theories, so people tend to believe that these theories were abandoned because they were less accurate. A grave misunderstanding which is based on the arrogant ideas that we know more than the people before us. And yet the ancient Greeks could easily predict celestial phenomena centuries in the future even while believing that the gods were moving the planets in the celestial sphere…

To the modern atheist all this is crazy of course.

People who believe in scientism today can hear nothing which could refute their perfect idea of science as a method to reach the “truth”. Not even Godel could change their mind.
Going back to the cosmological principle, today’s believers (in science and scientism) truly believe that this is a fact we hold true on the basis of observations. My friend and his friends could not even consider an alternative. So here we are. Men who do not know if Mars had water, but who do know with certainty that the whole universe is isotropic and homogenous! It would be comical if not so terribly arrogant…

At the end it is not a matter of data or knowledge. It is a matter of the ability to think freely without just following what others say.

Today’s atheists and proponents of scientism would be the greatest followers of the institutional church during the middle ages. Because what makes them blind today is not a lack of knowledge for something specific, but the arrogance of a man who does not want to admit that others might be able to see things clearer than him. These people would follow the Pope in whatever he said, in the same way they now follow the opinion of the majority regarding the truth of science.

The same people would swear that you can only draw one parallel from a straight line.

They would argue fiercely in favor of the fifth axiom of Euclid and would mock anyone trying to attempt to utter a different opinion.

At the end, we will discover if Mars has water…
At the end, we will “know” that no parallel lines can be drawn…
At the end we will draw multiple parallel lines…

Do you see?
There is nothing there.
Except for the things you see…

Satellite galaxies. Dark matter. Simulating the unknown…

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The mystery of the missing satellite galaxies has bedeviled astronomers for more than a decade. While dark matter has proven to be exceptionally good at explaining the formation of large galaxies and clusters, it routinely runs into trouble when attempting to describe tiny structures such as satellite galaxies.

Dozens of tiny galaxies known as satellite galaxies orbit the Milky Way, but theorists predict there should be hundreds. Now a team of astronomers offers a new resolution to this conflict between observation and theory: Maybe dark matter, the mysterious substance thought to bind galaxies together, isn’t quite so dark. The researchers propose that radiation might have stirred up dark matter in the early universe, preventing the formation of satellites. (1)

We do not know what “dark matter” is.
But we simulate it.

We are so dogmatically stupid.
And nothing can simulate that…

Dogmas, science, assumptions and the need for Philosophy!

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Horgan: What’s your opinion of the recent philosophy-bashing by Stephen Hawking, Lawrence Krauss and Neil deGrasse Tyson?

Rovelli: Seriously: I think they are stupid in this. I have admiration for them in other things, but here they have gone really wrong. Look: Einstein, Heisenberg, Newton, Bohr…. and many many others of the greatest scientists of all times, much greater than the names you mention, of course, read philosophy, learned from philosophy, and could have never done the great science they did without the input they got from philosophy, as they claimed repeatedly. You see: the scientists that talk philosophy down are simply superficial: they have a philosophy (usually some ill-digested mixture of Popper and Kuhn) and think that this is the “true” philosophy, and do not realize that this has limitations.

Here is an example: theoretical physics has not done great in the last decades. Why? Well, one of the reasons, I think, is that it got trapped in a wrong philosophy: the idea that you can make progress by guessing new theory and disregarding the qualitative content of previous theories. This is the physics of the “why not?” Why not studying this theory, or the other? Why not another dimension, another field, another universe? Science has never advanced in this manner in the past. Science does not advance by guessing. It advances by new data or by a deep investigation of the content and the apparent contradictions of previous empirically successful theories. Quite remarkably, the best piece of physics done by the three people you mention is Hawking’s black-hole radiation, which is exactly this. But most of current theoretical physics is not of this sort. Why? Largely because of the philosophical superficiality of the current bunch of scientists. (1)

Couldn’t say it more eloquently

Modern science is a science with no compass. Or even worse to be exact: A science which does not accept it has a compass EVEN THOUGH it has one!

A science which has specific philosophical dogmas as foundations (e.g. materialism, atheism, nihilism) but does not even acknowledge it!

Be aware of the hidden assumptions on which you base your thought.

They are there. No matter how hard you try not to accept it.

Watch out for the assumptions…

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A new study from scientists at the University of Miami Rosenstiel School of Marine and Atmospheric Science and colleagues confirms rising levels of water vapor in the upper troposphere — a key amplifier of global warming — will intensify climate change impacts over the next decades. The new study is the first to show that increased water vapor concentrations in the atmosphere are a direct result of human activities.

“The study is the first to confirm that human activities have increased water vapor in the upper troposphere”, said Brian Soden, professor of atmospheric sciences at the UM Rosenstiel School and co-author of the study. (1)

But WHY is water vapor a “KEY AMPLIFIER OF GLOBAL WARMING”?
Is that a fact?
Or a conclusion based on assumptions?

Search on your own.
Question what seems obvious.
Even if someone tells you 1+1=2 ask him WHY?
It is the simple things which hide the greatest traps.

Hint:See Climate models, IPCC, assumptions: Why “climate change” is in our mind. for more on what is “right” and “wrong” in the climate models.