The Incompleteness Theorem

The Incompleteness Theorem

In 1931 and while still in Vienna, Gödel published his famous incompleteness theorems in "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme" (called in English "On formally undecidable propositions of Principia Mathematica and related systems"). In that article, he proved for any computable axiomatic system that is powerful enough to describe the arithmetic of the natural numbers (e.g. the Peano axioms or ZFC), that:

If the system is consistent, it cannot be complete.

The consistency of the axioms cannot be proven within the system.

These theorems ended a half-century of attempts, beginning with the work of Frege and culminating in Principia Mathematica and Hilbert's formalism, to find a set of axioms sufficient for all mathematics. The incompleteness theorems also imply that not all mathematical questions are computable.

In hindsight, the basic idea at the heart of the incompleteness theorem is rather simple. Gödel essentially constructed a formula that claims that it is unprovable in a given formal system. If it were provable, it would be false, which contradicts the idea that in a consistent system, provable statements are always true. Thus there will always be at least one true but unprovable statement. That is, for any computably enumerable set of axioms for arithmetic (that is, a set that can in principle be printed out by an idealized computer with unlimited resources), there is a formula that obtains in arithmetic, but which is not provable in that system. To make this precise, however, Gödel needed to solve several technical issues, such as encoding statements, proofs, and the very concept of provability into the natural numbers. He did this using a process known as Gödel numbering.

The Power to Integrate....

Wittgenstein, who ... looked back nostalgically on a well-ordered world where everyone had his place, found modernity uncultured because it had lost its power to integrate, and left individuals in a state of confusion. The only ones who can keep their balance and personal creativity are those whom Nietzsche calls the strong men, that is the most moderate, who need neither convictions nor religion, who are able not only to endure, but to accept a fair amount of chance and absurdity, and are capable of thinking in a broadly disillusioned and negative way without feeling either diminished or discouraged (p. 296).by Jacques Le Rider’s (1993)

What is the Use of Studying Philosophy?

What is the use of studying philosophy if all that it does for you is enable you to talk with some plausibility about some abstruse questions of logic, etc., and if it does not improve your thinking about the important questions of everyday life .... You see, I know that it is difficult to think well about “certainty”, “probability”, “perception”, etc. But it is, if possible, still more difficult to think, or try to think, really honestly about your life and other people’s lives.-Wittgenstein

Compassion

Compassion (from Latin: "co-suffering") is a virtue —one in which the emotional capacities of empathy and sympathy (for the suffering of others) are regarded as a part of love itself, and a cornerstone of greater social interconnectedness and humanism —foundational to the highest principles in philosophy, society, and personhood.

There is an aspect of compassion which regards a quantitative dimension, such that individual's compassion is often given a property of "depth," "vigour," or "passion." More vigorous than empathy, the feeling commonly gives rise to an active desire to alleviate another's suffering. It is often, though not inevitably, the key component in what manifests in the social context as altruism. In ethical terms, the various expressions down the ages of the so-called Golden Rule embody by implication the principle of compassion: Do to others what you would have them do to you.

The Dialectical Forms of Government

The dialectical forms of government

Plato spends much of the Republic narrating conversations about the Ideal State. But what about other forms of government? The discussion turns to four forms of government that cannot sustain themselves: timocracy, oligarchy (also called plutocracy), democracy and tyranny (also called despotism).

Timocracy
Socrates defines a timocracy as a government ruled by people who love honor and are selected according to the degree of honor they hold in society. Honor is often equated with wealth and possession so this kind of gilded government leads to the people valuing materialism above all things.

Oligarchy
These temptations create a confusion between economic status and honor which is responsible for the emergence of oligarchy. In Book VIII, Socrates suggests that wealth will not help a pilot to navigate his ship. This injustice divides the rich and the poor, thus creating an environment for criminals and beggars to emerge. The rich are constantly plotting against the poor and vice versa.

Democracy
As this socioeconomic divide grows, so do tensions between social classes. From the conflicts arising out of such tensions, democracy replaces the oligarchy preceding it. The poor overthrow the inexperienced oligarchs and soon grant liberties and freedoms to citizens. A visually appealing demagogue is soon lifted up to protect the interests of the lower class. However, with too much freedom, the people become drunk, and tyranny takes over.

Tyranny
The excessive freedoms granted to the citizens of a democracy ultimately leads to a tyranny, the furthest regressed type of government. These freedoms divide the people into three socioeconomic classes: the dominating class, the capitalists and the commoners. Tensions between the dominating class and the capitalists causes the commoners to seek out protection of their democratic liberties. They invest all their power in their democratic demagogue, who, in turn, becomes corrupted by the power and becomes a tyrant with a small entourage of his supporters for protection and absolute control of his people.

Ironically, the ideal state outlined by Socrates closely resembles a tyranny, but they are on opposite ends of the spectrum. This is because the philosopher king who rules in the ideal state is not self-centered but is dedicated to the good of the state insofar as the philosopher king is the one with knowledge.

Vita Contemplativa

Vita Contemplativa

It may already be September. I drank tasteless coffee
in the cafe garden of the Museum-Insel
and thought about Berlin, its dark waters.
Black buildings that have seen so much.
But peace reigns in Europe, diplomats doze,
a pale sun, the summer dies serenely,
spiders weave its shining shroud, the plane trees'
dry leaves write memoirs of their youth.

So this is the vita contemplativa.
The black walls enclosing white sculptures.
The bust of a Greek beauty. So this is it.
An altar before which no one prays.
So this is the vita contemplativa.
Narkissos--a Roman copy of a Greek boy
on prosthetic limbs of bronze (a veteran of which war?).
Then a kuros with his pouch of testes (a vanished phallus).

We seem to occupy a desert island.
Time moves deliberately, without haste.
Helpless rapture, so this is the vita contemplativa.
Bliss. An instant with no hour, as the poet said,
the poet killed in Lublin by a bomb.
But what if, in this or a different city,
the vita activa surged again, what would Artemis,
fourth century B.C.E., do then? Hermes? Narcissus?

Parchment faces stare at me with envy--
I still make mistakes, they can't.
Comparing day and night, so this is it.
Sleep and waking, mind and world, this is it. Joy.
Tranquility, taut attention, the levitating heart.
Lucent thoughts smoulder in black walls.
So this is it. What it is, we do not know.
We dwell in the abyss. In dark waters. In brightness.

By Adam Zagajewski, Translated From the Polish by Clare Cavanagh

Vandana Shiva

"[How do I do it?] Well, it's always a mystery, because you don't know why you get depleted or recharged. But this much I know. I do not allow myself to be overcome by hopelessness, no matter how tough the situation. I believe that if you just do your little bit without thinking of the bigness of what you stand against, if you turn to the enlargement of your own capacities, just that itself creates new potential. And I've learned from the Bhagavad-Gita and other teachings of our culture to detach myself from the results of what I do, because those are not in my hands. The context is not in your control, but your commitment is yours to make, and you can make the deepest commitment with a total detachment about where it will take you. You want it to lead to a better world, and you shape your actions and take full responsibility for them, but then you have detachment. And that combination of deep passion and deep detachment allows me to take on the next challenge, because I don't cripple myself, I don't tie myself in knots. I function like a free being. I think getting that freedom is a social duty because I think we owe it to each not to burden each other with prescription and demands. I think what we owe each other is a celebration of life and to replace fear and hopelessness with fearlessness and joy."
— Vandana Shiva