BLAISE PASCAL : HEART AND LOGIC — ALEXIS KARPOUZOS

Pascal was well acquainted with what could and could not be known through the mathematical method, the experimental method and reason itself. Through his philosophical investigations, he found that there were strict limits to what we as humans could know. For him, neither the scientific method nor reason more generally could teach individuals the meaning of life or the right way to live.

Pascal also wrote about how humans tried to avoid thinking about their mortality, the extent of their ignorance and their liability to error. Yet he also believed that there was nothing more important for people to consider than their true human nature. In this reasoning, without understanding who we are, it would be difficult to understand how we ought to live.

In Pascal's view, acquiring self-knowledge was a necessary stage on the way to recognizing one's need for living with faith and purpose in something beyond oneself.

Pascal was a mathematician of the first order. At the age of sixteen, he wrote a significant treatise on the subject of projective geometry, known as Pascal's Theorem, which states that, if a hexagon is inscribed in a circle, then the three intersection points of opposite sides lie on a single line, called the Pascal line. As a young man, he built a functional calculating machine, able to perform additions and subtractions, to help his father with his tax calculations.

Pascal's Triangle

The table of binomial coefficients known as Pascal's Triangle

He is best known, however, for Pascal's Triangle, a convenient tabular presentation of binomial co-efficients, where each number is the sum of the two numbers directly above it. A binomial is a simple type of algebraic expression which has just two terms operated on only by addition, subtraction, multiplication and positive whole-number exponents, such as (x + y)2. The co-efficients produced when a binomial is expanded form a symmetrical triangle (see image at right).

Pascal was far from the first to study this triangle. The Persian mathematician Al-Karaji had produced something very similar as early as the 10th Century, and the Triangle is called Yang Hui's Triangle in China after the 13th Century Chinese mathematician, and Tartaglia's Triangle in Italy after the eponymous 16th Century Italian. But Pascal did contribute an elegant proof by defining the numbers by recursion, and he also discovered many useful and interesting patterns among the rows, columns and diagonals of the array of numbers. For instance, looking at the diagonals alone, after the outside "skin" of 1's, the next diagonal (1, 2, 3, 4, 5,…) is the natural numbers in order. The next diagonal within that (1, 3, 6, 10, 15,…) is the triangular numbers in order. The next (1, 4, 10, 20, 35,…) is the pyramidal triangular numbers, etc, etc. It is also possible to find prime numbers, Fibonacci numbers, Catalan numbers, and many other series, and even to find fractal patterns within it.

Pascal also made the conceptual leap to use the Triangle to help solve problems in probability theory. In fact, it was through his collaboration and correspondence with his French contemporary Pierre de Fermat and the Dutchman Christiaan Huygens on the subject that the mathematical theory of probability was born. Before Pascal, there was no actual theory of probability — notwithstanding Gerolamo Cardano's early exposition in the 16th Century — merely an understanding (of sorts) of how to compute "chances" in dice and card games by counting equally probable outcomes. Some apparently quite elementary problems in probability had eluded some of the best mathematicians, or given rise to incorrect solutions.

It fell to Pascal (with Fermat's help) to bring together the separate threads of prior knowledge (including Cardano's early work) and to introduce entirely new mathematical techniques for the solution of problems that had hitherto resisted solution. Two such intransigent problems which Pascal and Fermat applied themselves to were the Gambler's Ruin (determining the chances of winning for each of two men playing a particular dice game with very specific rules) and the Problem of Points (determining how a game's winnings should be divided between two equally skilled players if the game was ended prematurely). His work on the Problem of Points in particular, although unpublished at the time, was highly influential in the unfolding new field.

The Problem of Points

Pascal probability - Fermat and Pascal's solution to the Problem of Points

The Problem of Points at its simplest can be illustrated by a simple game of "winner take all" involving the tossing of a coin. The first of the two players (say, Fermat and Pascal) to achieve ten points or wins is to receive a pot of 100 francs. But, if the game is interrupted at the point where Fermat, say, is winning 8 points to 7, how is the 100 franc pot to divided? Fermat claimed that, as he needed only two more points to win the game, and Pascal needed three, the game would have been over after four more tosses of the coin (because, if Pascal did not get the necessary 3 points for your victory over the four tosses, then Fermat must have gained the necessary 2 points for his victor y, and vice versa. Fermat then exhaustively listed the possible outcomes of the four tosses, and concluded that he would win in 11 out of the 16 possible outcomes, so he suggested that the 100 francs be split 11⁄16 (0.6875) to him and 5⁄16 (0.3125) to Pascal.

Pascal then looked for a way of generalizing the problem that would avoid the tedious listing of possibilities, and realized that he could use rows from his triangle of coefficients to generate the numbers, no matter how many tosses of the coin remained. As Fermat needed 2 more points to win the game and Pascal needed 3, he went to the fifth (2 + 3) row of the triangle, i.e. 1, 4, 6, 4, 1. The first 3 terms added together (1 + 4 + 6 = 11) represented the outcomes where Fermat would win, and the last two terms (4 + 1 = 5) the outcomes where Pascal would win, out of the total number of outcomes represented by the sum of the whole row (1 + 4 + 6 +4 +1 = 16).

Pascal and Fermat had grasped through their correspondence a very important concept that, though perhaps intuitive to us today, was all but revolutionary in 1654. This was the idea of equally probable outcomes, that the probability of something occurring could be computed by enumerating the number of equally likely ways it could occur, and dividing this by the total number of possible outcomes of the given situation. This allowed the use of fractions and ratios in the calculation of the likelhood of events, and the operation of multiplication and addition on these fractional probabilities. For example, the probability of throwing a 6 on a die twice is 1⁄6 x 1⁄6 = 1⁄36 ("and" works like multiplication); the probability of throwing either a 3 or a 6 is 1⁄6 + 1⁄6 = 1⁄3 ("or" works like addition).

Pascal's religion

In fact, Pascal argued that believing in the existence of God is essential to human happiness. For all of his many ideas and accomplishments, he's probably most famous today for Pascal's Wager, a philosophical argument that humans should bet on the existence of God. "If you win, you win everything; if you lose, you lose nothing," he wrote. In other words, he argued, although one cannot know for certain whether or not God exists, we are better off believing in God's existence than not.

Pascal's Wager, Wireless Philosophy.

Pascal saw Jesus as the indispensable mediator between God and humankind. He believed that the Catholic Church was the only religion to teach the truth about human nature and therefore offered the singular route to happiness.

Pascal's preference for Catholicism over any other religion raises a difficult question, however. For why should anyone wager on one religion rather than another? Some scholars, such as Richard Popkin, have gone so far as to call Pascal's attempts to discredit paganism, Judaism and Islam "pedantic."

Whatever one's religious beliefs, Pascal teaches that all individuals have to make a choice between faith in some reality beyond themselves or a life without belief. But a life without belief is also a choice, and in Pascal's view, a bad bet.

Human beings have to wager and to commit themselves to a worldview on which each one would be willing to bet their life. It follows that, for Pascal, human beings could not avoid hope and fear: hope that their bets will turn out well, fear that they won't.

Indeed, people make countless daily wagers — going to the grocery store, driving a car, riding the train, among others — but don't usually think of them as risky. According to Pascal, however, human lives as a whole can also be viewed as wagers.

Our big decisions are risks: For example, in choosing a certain course of education and career or in marrying a certain person, people are betting on a fulfilling life. In Pascal's view, people choose how to live and what to believe without really knowing whether or not their beliefs and decisions are good ones. We simply don't and can't know enough to live without wagering.

The Human Condition

To properly understand Pascal's apologetics, it's important to recognize his motive. Pascal wasn't interested in defending Christianity as a system of belief; his interest was evangelistic. He wanted to persuade people to believe in Jesus. When apologetics has evangelism as its primary goal, it has to take into account the condition of the people being addressed. For Pascal the human condition was the starting point and point of contact for apologetics.

In his analysis of man, Pascal focuses on two very contradictory sides of fallen human nature. Man is both noble and wretched. Noble, because he is created in God's image; wretched, because he is fallen and alienated from God. In one of his more passionate notes, Pascal says this:What kind of freak is man! What a novelty he is, how absurd he is, how chaotic and what a mass of contradictions, and yet what a prodigy! He is judge of all things, yet a feeble worm. He is repository of truth, and yet sinks into such doubt and error. He is the glory and the scum of the universe!{7}

Furthermore, Pascal says, we know that we are wretched. But it is this very knowledge that shows our greatness.

Pascal says it's important to have a right understanding of ourselves. He says "it is equally dangerous for man to know God without knowing his own wretchedness, and to know his own wretchedness without knowing the Redeemer who can free him from it." Thus, our message must be that "there is a God whom men can know, and that there is a corruption in their nature which renders them unworthy of Him."{8} This prepares the unbeliever to hear about the Redeemer who reconciles the sinner with the Creator.

Pascal says that people know deep down that there is a problem, but we resist slowing down long enough to think about it. He says:

Rick Wade examines the contemporary relevance of the apologetics of Blaise Pascal, a 17th century mathematician, scientist, inventor, and Christian apologist. Man finds nothing so intolerable as to be in a state of complete rest, without passions, without occupation, without diversion, without effort. Then he faces his nullity, loneliness, inadequacy, dependence, helplessness, emptiness. And at once there wells up from the depths of his soul boredom, gloom, depression, chagrin, resentment, despair.{9}

Pascal says there are two ways people avoid thinking about such matters: diversion and indifference. Regarding diversion, he says we fill up our time with relatively useless activities simply to avoid facing the truth of our wretchedness. "The natural misfortune of our mortality and weakness is so miserable," he says, "that nothing can console us when we really think about it. . . . The only good thing for man, therefore, is to be diverted so that he will stop thinking about his circumstances." Business, gambling, and entertainment are examples of things which keep us busy in this way.{10}

The other response to our condition is indifference. The most important question we can ask is What happens after death? Life is but a few short years, and death is forever. Our state after death should be of paramount importance, shouldn't it? But the attitude people take is this:

Just as I do Rick Wade examines the contemporary relevance of the apologetics of Blaise Pascal, a 17th century mathematician, scientist, inventor, and Christian apologist. not know where I came from, so I do not know where I am going. All I know is that when I leave this world I shall fall forever into oblivion, or into the hands of an angry God, without knowing which of the two will be my lot for eternity. Such is my state of mind, full of weakness and uncertainty. The only conclusion I can draw from all this is that I must pass my days without a thought of trying to find out what is going to happen to me.{11}

Pascal is appalled that people think this way, and he wants to shake people out of their stupor and make them think about eternity. Thus, the condition of man is his starting point for moving people toward a genuine knowledge of God.

Knowledge of the Heart

Pascal lived in the age of the rise of rationalism. Revelation had fallen on hard times; man's reason was now the final source for truth. In the realm of religious belief many people exalted reason and adopted a deistic view of God. Some, however, became skeptics. They doubted the competence of both revelation and reason.

Although Pascal couldn't side with the skeptics, neither would he go the way of the rationalists. Instead of arguing that revelation was a better source of truth than reason, he focused on the limitations of reason itself. (I should stop here to note that by reason Pascal meant the reasoning process. He did not deny the true powers of reason; he was, after all, a scientist and mathematician.) Although the advances in science increased man's knowledge, it also made people aware of how little they knew. Thus, through our reason we realize that reason itself has limits. "Reason's last step," Pascal said, "is the recognition that there are an infinite number of things which are beyond it."{12} Our knowledge is somewhere between certainty and complete ignorance, Pascal believed.{13} The bottom line is that we need to know when to affirm something as true, when to doubt, and when to submit to authority.{14}

Besides the problem of our limited knowledge, Pascal also noted how our reason is easily distracted by our senses and hindered by our passions.{15} "The two so-called principles of truth*reason and the senses*are not only not genuine but are engaged in mutual deception. Through false appearances the senses deceive reason. And just as they trick the soul, they are in turn tricked by it. It takes its revenge. The senses are influenced by the passions which produce false impressions."{16} Things sometimes appear to our senses other than they really are, such as the way a stick appears bent when put in water. Our emotions or passions also influence how we think about things. And our imagination, which Pascal says is our dominant faculty{17}, often has precedence over our reason. A bridge suspended high over a ravine might be wide enough and sturdy enough, but our imagination sees us surely falling off.

So, our finiteness, our senses, our passions, and our imagination can adversely influence our powers of reason. But Pascal believed that people really do know some things to be true even if they cannot account for it rationally. Such knowledge comes through another channel, namely, the heart.

This brings us to what is perhaps the best known quotation of Pascal: "The heart has its reasons which reason does not know."{18} In other words, there are times that we know something is true but we did not come to that knowledge through logical reasoning, neither can we give a logical argument to support that belief.

For Pascal, the heart is "the `intuitive' mind" rather than "the `geometrical' (calculating, reasoning) mind."{19} For example, we know when we aren't dreaming. But we can't prove it rationally. However, this only proves that our reason has weaknesses; it does not prove that our knowledge is completely uncertain. Furthermore, our knowledge of such first principles as space, time, motion, and number is certain even though known by the heart and not arrived at by reason. In fact, reason bases its arguments on such knowledge.{20} Knowledge of the heart and knowledge of reason might be arrived at in different ways, but they are both valid. And neither can demand that knowledge coming through the other should submit to its own dictates.

The Knowledge of God

If reason is limited in its understanding of the natural order, knowledge of God can be especially troublesome. "If natural things are beyond [reason]," Pascal said, "what are we to say about supernatural things?"{21}

There are several factors which hinder our knowledge of God. As noted before, we are limited by our finitude. How can the finite understand the infinite?{22} Another problem is that we cannot see clearly because we are in the darkness of sin. Our will is turned away from God, and our reasoning abilities are also adversely affected.

There is another significant limitation on our knowledge of God. Referring to Isaiah 8:17 and 45:15{23}, Pascal says that as a result of our sin God deliberately hides Himself ("hides" in the sense that He doesn't speak}. One reason He does this is to test our will. Pascal says, "God wishes to move the will rather than the mind. Perfect clarity would help the mind and harm the will." God wants to "humble [our] pride."{24}

But God doesn't remain completely hidden; He is both hidden and revealed. "If there were no obscurity," Pascal says, "man would not feel his corruption: if there were no light man could not hope for a cure."{25}

God not only hides Himself to test our will; He also does it so that we can only come to Him through Christ, not by working through some logical proofs. "God is a hidden God," says Pascal, " and . . . since nature was corrupted [God] has left men to their blindness, from which they can escape only through Jesus Christ, without whom all communication with God is broken off. Neither knoweth any man the Father save the Son, and he to whosoever the Son will reveal him."{26} Pascal's apologetic is decidedly Christocentric. True knowledge of God isn't mere intellectual assent to the reality of a divine being. It must include a knowledge of Christ through whom God revealed Himself. He says:

All who have claimed to know God and to prove his existence without Jesus Christ have done so ineffectively. . . . Apart from him, and without Scripture, without original sin, without the necessary Mediator who was promised and who came, it is impossible to prove absolutely that God exists, or to teach sound doctrine and sound morality. But through and in Jesus Christ we can prove God's existence, and teach both doctrine and morality.{27}

If we do not know Christ, we cannot understand God as the judge and the redeemer of sinners. It is a limited knowledge that doesn't do any good. As Pascal says, "That is why I am not trying to prove naturally the existence of God, or indeed the Trinity, or the immortality of the soul or anything of that kind. This is not just because I do not feel competent to find natural arguments that will convince obdurate atheists, but because such knowledge, without Christ, is useless and empty." A person with this knowledge has not "made much progress toward his salvation."{28} What Pascal wants to avoid is proclaiming a deistic God who stands remote and expects from us only that we live good, moral lives. Deism needs no redeemer.

But even in Christ, God has not revealed Himself so overwhelmingly that people cannot refuse to believe. In the last days God will be revealed in a way that everyone will have to acknowledge Him. In Christ, however, God was still hidden enough that people who didn't want what was good would not have it forced upon them. Thus, "there is enough light for those who desire only to see, and enough darkness for those of a contrary disposition."{29}

There is still one more issue which is central to Pascal's thinking about the knowledge of God. He says that no one can come to know God apart from faith. This is a theme of central importance for Pascal; it clearly sets him apart from other apologists of his day. Faith is the knowledge of the heart that only God gives. "It is the heart which perceives God and not the reason," says Pascal. "That is what faith is: God perceived by the heart, not by the reason."{30} "By faith we know he exists," he says.{31} "Faith is different from proof. One is human and the other a gift of God. . . . This is the faith that God himself puts into our hearts. . . ."{32} Pascal continues, "We shall never believe with an effective belief and faith unless God inclines our hearts. Then we shall believe as soon as he inclines them."{33}

To emphasize the centrality of heart knowledge in Pascal's thinking, I deliberately left off the end of one of the sentences above. Describing the faith God gives, Pascal said, "This is the faith that God himself puts into our hearts, often using proof as the instrument."{34}

This is rather confusing. Pascal says non-believers are in darkness, so proofs will only find obscurity.{35} He notes that "no writer within the canon [of Scripture] has ever used nature to prove the existence of God. They all try to help people believe in him."{36} He also expresses astonishment at Christians who begin their defense by making a case for the existence of God.

Their enterprise would cause me no surprise if they were addressing the arguments to the faithful, for those with living faith in their hearts can certainly see at once that everything which exists is entirely the work of the God they worship. But for those in whom this light has gone out and in who we are trying to rekindle it, people deprived of faith and grace, . . . to tell them, I say, that they have only to look at the least thing around them and they will see in it God plainly revealed; to give them no other proof of this great and weighty matter than the course of the moon and the planets; to claim to have completed the proof with such an argument; this is giving them cause to think that the proofs of our religion are indeed feeble. . . . This is not how Scripture speaks, with its better knowledge of the things of God.{37}

But now Pascal says that God often uses proofs as the instrument of faith. He also says in one place, "The way of God, who disposes all things with gentleness, is to instil [sic] religion into our minds with reasoned arguments and into our hearts with grace. . . ."{38}

The explanation for this tension can perhaps be seen in the types of proofs Pascal uses. Pascal won't argue from nature. Rather he'll point to evidences such as the marks of divinity within man, and those which affirm Christ's claims, such as prophecies and miracles, the most important being prophecies.{39} He also speaks of Christian doctrine "which gives a reason for everything," the establishment of Christianity despite its being so contrary to nature, and the testimony of the apostles who could have been neither deceivers nor deceived.{40} So Pascal does believe there are positive evidences for belief. Although he does not intend to give reasons for everything, neither does he expect people to agree without having a reason.{41}

Nonetheless, even evidences such as these do not produce saving faith. He says, "The prophecies of Scripture, even the miracles and proofs of our faith, are not the kind of evidence that are absolutely convincing. . . . There is . . . enough evidence to condemn and yet not enough to convince. . . ." People who believe do so by grace; those who reject the faith do so because of their lusts. Reason isn't the key.{42}

Pascal says that, while our faith has the strongest of evidences in favor of it, "it is not for these reasons that people adhere to it. . . . What makes them believe," he says, " is the cross." At which point he quotes 1 Corinthians 1:17: "Lest the cross of Christ be emptied of its power."{43}

The Wager

The question that demands to be answered, of course, is this: If our reason is inadequate to find God, even through valid evidences, how does one find God? Says Pascal:

Let us then examine the point and say: "Either God exists, or he does not." But which of the alternatives shall we choose? Reason cannot decide anything. Infinite chaos separates us. At the far end of this infinite distance a coin is being spun which will come down heads or tails. How will you bet? Reason cannot determine how you will choose, nor can reason defend your position of choice.{44}

At this point Pascal challenges us to accept his wager. Simply put, the wager says we should bet on Christianity because the rewards are infinite if it's true, while the losses will be insignificant if it's false.{45} If it's true and you have rejected it, you've lost everything. However, if it's false but you have believed it, at least you've led a good life and you haven't lost anything. Of course, the best outcome is if one believes Christianity to be true and it turns out that it is!

But the unbeliever might say it's better not to choose at all. Not so, says Pascal. You're going to live one way or the other, believing in God or not believing in God; you can't remain in suspended animation. You must choose.

In response the unbeliever might say that everything in him works against belief. "I am being forced to gamble and I am not free," he says, "for they will not let me go. I have been made in such a way that I cannot help disbelieving. So what do you expect me to do?"{46} After all, Pascal has said that faith comes from God, not from us.

Pascal says our inability to believe is a problem of the emotions or passions. Don't try to convince yourself by examining more proofs and evidences, he says, "but by controlling your emotions." You want to believe but don't know how. So follow the examples of those who "were once in bondage but who now are prepared to risk their whole life. . . . Follow the way by which they began. They simply behaved as though they believed" by participating in various Christian rituals. And what can be the harm? "You will be faithful, honest, humble, grateful, full of good works, a true and genuine friend. . . . I assure you that you will gain in this life, and that with every step you take along this way, you will realize you have bet on something sure and infinite which has cost you nothing."{47}

Remember that Pascal sees faith as a gift from God, and he believes that God will show Himself to whomever sincerely seeks Him.{48} By taking him up on the wager and putting yourself in a place where you are open to God, God will give you faith. He will give you sufficient light to know what is really true.

Scholars have argued over the validity of Pascal's wager for centuries. In this writer's opinion, it has significant weaknesses. What about all the other religions, one of which could (in the opinion of the unbeliever) be true?

However, the idea is an intriguing one. Pascal's assertion that one must choose seems reasonable. Even if such a wager cannot have the kind of mathematical force Pascal seemed to think, it could work to startle the unbeliever into thinking more seriously about the issue. The important thing here is to challenge people to choose, and to choose the right course.

THE KURT GODEL'S PHILOSOPHY — ALEXIS KARPOUZOS

Kurt Gödel (1906–1978) was an eminent Austrian logician, mathematician, and philosopher, renowned for his groundbreaking work in mathematical logic and the foundations of mathematics. His most celebrated contributions include Gödel's incompleteness theorems, which have profound implications not only for mathematics but also for philosophy and our understanding of the limits of human knowledge.

Gödel's Incompleteness Theorems

Gödel's incompleteness theorems are perhaps his most famous work. They can be summarized as follows: Kurt Gödel's First Incompleteness Theorem stands as one of the most significant milestones in the history of mathematics and logic. Presented in 1931, this theorem has profound implications for our understanding of formal systems, the limits of mathematical knowledge, and the nature of truth. This essay delves into the intricacies of Gödel's First Incompleteness Theorem, its mathematical underpinnings, and its philosophical implications.

The Statement of the Theorem

Gödel's First Incompleteness Theorem can be succinctly stated as follows: In any consistent formal system that is sufficiently expressive to encode basic arithmetic, there exist true mathematical statements that cannot be proven within that system. This statement fundamentally challenges the notion that all mathematical truths can be derived from a finite set of axioms and rules of inference.

Mathematical Foundations

The theorem arises from Gödel's ingenious method of "arithmetization," where he encoded statements, proofs, and even the notion of provability itself within the framework of arithmetic. Gödel assigned unique natural numbers to each symbol, formula, and sequence of formulas in the formal system, a process known as Gödel numbering. This allowed him to transform metamathematical concepts into arithmetical ones. Gödel constructed a specific mathematical statement, often referred to as the "Gödel sentence" (G), which asserts its own unprovability within the system. In essence, G is a statement that says, "This statement is not provable." If G were provable, the system would be inconsistent because it would lead to a contradiction. Therefore, if the system is consistent, G must be true but unprovable.

Implications for Formal Systems

Gödel's First Incompleteness Theorem has far-reaching implications for formal systems and the foundations of mathematics: Limits of Formal Systems: The theorem shows that any formal system capable of expressing basic arithmetic cannot be both complete and consistent. Completeness means that every true statement within the system can be proven, while consistency means that no contradictions can be derived. Gödel's theorem demonstrates that achieving both simultaneously is impossible. Impact on Hilbert's Program: At the time, the prevailing belief among mathematicians, led by David Hilbert, was that all mathematical truths could, in principle, be derived from a complete and consistent set of axioms. Gödel's theorem dealt a severe blow to this program, showing that there will always be true statements that elude formal proof.

Provability and Truth: The theorem highlights a crucial distinction between provability and truth. In a consistent system, there exist true statements that are unprovable. This challenges the notion that mathematical truth is synonymous with formal provability, suggesting that truth transcends formal systems.

Philosophical Implications

Gödel's First Incompleteness Theorem has profound philosophical implications, particularly concerning the nature of mathematical knowledge, the limits of human understanding, and the relationship between mathematics and reality.

Mathematical Platonism: Gödel himself was a proponent of mathematical Platonism, the view that mathematical entities exist independently of human thought. The theorem supports this perspective by suggesting that mathematical truths exist in an objective realm, accessible to human intuition but not fully capturable by formal systems.

Human Cognition and Intuition: The theorem implies that human cognition and mathematical intuition play an essential role in understanding mathematical truths. Since formal systems are inherently limited, our intuitive grasp of mathematics allows us to recognize truths that cannot be formally proven.

The Nature of Truth: Gödel's work invites deep philosophical inquiry into the nature of truth itself. It suggests that truth is not merely a matter of formal derivation but involves a more profound, perhaps even metaphysical, aspect of reality. This has implications for various fields, including logic, epistemology, and metaphysics.

The Second Incompleteness Theorem of Kurt Gödel

Kurt Gödel, the towering figure in mathematical logic, not only revolutionized our understanding of formal systems with his First Incompleteness Theorem but also extended his groundbreaking work with a second theorem. Gödel's Second Incompleteness Theorem further elaborates on the inherent limitations of formal mathematical systems, reinforcing the profound insights of his earlier work. This essay delves into the essence of the Second Incompleteness Theorem, its mathematical foundation, and its philosophical implications.

Statement of the Theorem

Gödel's Second Incompleteness Theorem can be succinctly stated as follows:

No consistent system of axioms whose theorems can be listed by an effective procedure (i.e., a computer program) is capable of proving its own consistency.

In simpler terms, the theorem asserts that a formal system capable of arithmetic cannot demonstrate its own consistency from within.

Mathematical Foundations

The Second Incompleteness Theorem builds directly on the first. In his initial incompleteness result, Gödel showed that within any sufficiently powerful formal system, there exist true but unprovable statements. The Second Incompleteness Theorem goes a step further, applying this insight to the system's own consistency.

Gödel's proof involves constructing a specific arithmetic statement that effectively says, "This system is consistent." He demonstrates that if the system could prove this statement, it would lead to a contradiction, assuming the system is indeed consistent. Therefore, if the system is consistent, it cannot prove its own consistency.

Implications for Formal Systems

The Second Incompleteness Theorem has significant implications for the foundations of mathematics and the philosophy of formal systems:

1. Limits of Formal Proofs: The theorem underscores the inherent limitations of formal systems in establishing their own reliability. It highlights that any system powerful enough to encompass arithmetic cannot fully validate itself, placing a fundamental limit on the scope of formal proofs.
2. Impact on Foundational Programs: Hilbert's program aimed to establish a complete and consistent foundation for all of mathematics. Gödel's Second Incompleteness Theorem dealt a decisive blow to this endeavor by showing that no such foundational system can prove its own consistency, thus undermining the goal of absolute certainty in mathematics.
3. Consistency and Incompleteness: The theorem also ties the concepts of consistency and incompleteness together. It illustrates that the quest for a self-proving system inevitably leads to incompleteness, reinforcing the insights from the First Incompleteness Theorem.

Philosophical Implications

Gödel's Second Incompleteness Theorem has profound philosophical ramifications, especially regarding our understanding of knowledge, truth, and the limits of formal reasoning:

1. Philosophy of Mathematics: The theorem challenges the notion that mathematics can be completely formalized and that every mathematical truth can be derived from a finite set of axioms. It suggests that mathematical knowledge involves elements that transcend formal derivation, emphasizing the role of intuition and insight.
2. Foundational Certainty: Gödel's theorem implies that foundational certainty in mathematics is unattainable. It forces philosophers and mathematicians to acknowledge the intrinsic limitations of formal systems and to seek a more nuanced understanding of mathematical truth that goes beyond mere formal proof.
3. Epistemological Limits: The Second Incompleteness Theorem highlights the limits of human knowledge and the boundaries of formal systems. It suggests that there are truths about formal systems (such as their consistency) that cannot be fully captured within the systems themselves, pointing to an inherent epistemological boundary.
4. Reflection on Formalism: Gödel's work invites reflection on the formalist perspective, which seeks to ground mathematics purely in formal systems and symbolic manipulation. The theorem shows that such a grounding is inherently incomplete, suggesting the need for a broader, more holistic view of mathematical practice.

Philosophical Implications

Gödel's work has significant philosophical ramifications, particularly concerning the nature of mathematical truth, the limits of human knowledge, and the interplay between mathematics and philosophy.

Mathematical Platonism: Gödel was a proponent of mathematical Platonism, the view that mathematical objects exist independently of the human mind. His incompleteness theorems support this perspective, suggesting that mathematical truths exist in an objective realm that cannot be fully captured by any formal system. Gödel believed that human intuition and insight could access these truths directly, a stance that contrasts sharply with the formalist and constructivist views dominant in his time.

Limits of Formal Systems: Gödel's theorems highlight the inherent limitations of formal systems, implying that human knowledge cannot be entirely reduced to mechanistic procedures or algorithms. This has profound implications for the philosophy of mind and artificial intelligence, as it suggests that human cognition may involve elements that surpass purely computational processes.

Truth and Provability: Gödel's distinction between truth and provability challenges the notion that all truths can be demonstrated through logical proof. This raises important questions about the nature of knowledge and understanding, emphasizing the role of intuition, insight, and creativity in the discovery of mathematical and philosophical truths.

Philosophy of Mathematics: Gödel's work has influenced various schools of thought within the philosophy of mathematics, including intuitionism, formalism, and constructivism. His ideas have sparked ongoing debates about the foundations of mathematics, the nature of mathematical objects, and the limits of formal reasoning.

Gödel's Philosophical Legacy

Kurt Gödel's contributions to philosophy extend beyond his incompleteness theorems. He engaged deeply with the work of other philosophers, including Immanuel Kant and Edmund Husserl, and explored topics such as the nature of time, the structure of the universe, and the relationship between mathematics and reality.

Gödel's philosophical writings, though less well-known than his mathematical work, offer rich insights into his views on the nature of existence, the limits of human knowledge, and the interplay between the finite and the infinite. His work continues to inspire and challenge philosophers, mathematicians, and scientists, inviting them to explore the profound and often enigmatic questions at the heart of human understanding.

Kurt Gödel's Broader Contributions to Philosophy

Kurt Gödel, while primarily known for his monumental incompleteness theorems, made significant contributions that extended beyond the realm of mathematical logic. His philosophical pursuits deeply engaged with the works of eminent philosophers like Immanuel Kant and Edmund Husserl. Gödel's explorations into the nature of time, the structure of the universe, and the relationship between mathematics and reality reveal a profound and multifaceted intellectual legacy.

Engagement with Immanuel Kant

Gödel held a deep interest in the philosophy of Immanuel Kant. He admired Kant's critical philosophy, particularly the distinction between the noumenal and phenomenal worlds. Kant posited that human experience is shaped by the mind's inherent structures, leading to the conclusion that certain aspects of reality (the noumenal world) are fundamentally unknowable.

Gödel's incompleteness theorems echoed this Kantian theme, illustrating the limits of formal systems in capturing the totality of mathematical truth. Gödel believed that mathematical truths exis t independently of human thought, akin to Kant's noumenal realm. This philosophical alignment provided a robust foundation for Gödel's Platonism, which asserted the existence of mathematical objects as real, albeit abstract, entities.

Influence of Edmund Husserl

Gödel was also profoundly influenced by Edmund Husserl, the founder of phenomenology. Husserl's phenomenology emphasizes the direct investigation and description of phenomena as consciously experienced, without preconceived theories about their causal explanation. Gödel saw Husserl's work as a pathway to bridge the gap between the abstract world of mathematics and concrete human experience. Husserl's ideas about the structures of consciousness and the intentionality of thought resonated with Gödel's views on mathematical intuition. Gödel believed that human minds could access mathematical truths through intuition, a concept that draws on Husserlian phenomenological methods.

The Nature of Time and the Universe

Gödel's philosophical inquiries extended to the nature of time and the structure of the universe. His collaboration with Albert Einstein at the Institute for Advanced Study led to the development of the "Gödel metric" in 1949. This solution to Einstein's field equations of general relativity described a rotating universe where time travel to the past was theoretically possible. Gödel's model challenged conventional notions of time and causality, suggesting that the universe might have a more intricate structure than previously thought. Gödel's exploration of time was not just a mathematical curiosity but a profound philosophical statement about the nature of reality. He questioned whether time was an objective feature of the universe or a construct of human consciousness. His work hinted at a timeless realm of mathematical truths, aligning with his Platonist view.

Mathematics and Reality

Gödel's philosophical outlook extended to the broader relationship between mathematics and reality. He believed that mathematics provided a more profound insight into the nature of reality than empirical science. For Gödel, mathematical truths were timeless and unchangeable, existing independently of human cognition.

This perspective led Gödel to critique the materialist and mechanistic views that dominated 20th-century science and philosophy. He argued that a purely physicalist interpretation of the universe failed to account for the existence of abstract mathematical objects and the human capacity to understand them. Gödel's philosophy suggested a more integrated view of reality, where both physical and abstract realms coexist and inform each other.

Gödel's Exploration of Time

Kurt Gödel, one of the most profound logicians of the 20th century, ventured beyond the confines of mathematical logic to explore the nature of time. His inquiries into the concept of time were not merely theoretical musings but were grounded in rigorous mathematical formulations. Gödel's exploration of time challenged conventional views and opened new avenues of thought in both physics and philosophy.

Gödel and Einstein

Gödel's interest in the nature of time was significantly influenced by his friendship with Albert Einstein. Both were faculty members at the Institute for Advanced Study in Princeton, where they engaged in deep discussions about the nature of reality, time, and space. Gödel's exploration of time culminated in his solution to Einstein's field equations of general relativity, known as the Gödel metric.

The Gödel Metric

In 1949, Gödel presented a model of a rotating universe, which became known as the Gödel metric. This solution to the equations of general relativity depicted a universe where time travel to the past was theoretically possible. Gödel's rotating universe contained closed timelike curves (CTCs), paths in spacetime that loop back on themselves, allowing for the possibility of traveling back in time. The Gödel metric posed a significant philosophical challenge to the conventional understanding of time. If time travel were possible, it would imply that time is not linear and absolute, as commonly perceived, but rather malleable and subject to the geometry of spacetime. This raised profound questions about causality, the nature of temporal succession, and the very structure of reality.

Philosophical Implications

Gödel's exploration of time extended beyond the mathematical implications to broader philosophical inquiries:

Nature of Time: Gödel questioned whether time was an objective feature of the universe or a construct of human consciousness. His work suggested that our understanding of time as a linear progression from past to present to future might be an illusion, shaped by the limitations of human perception.

Causality and Free Will: The existence of closed timelike curves in Gödel's model raised questions about causality and free will. If one could travel back in time, it would imply that future events could influence the past, potentially leading to paradoxes and challenging the notion of a deterministic universe.

Temporal Ontology: Gödel's work contributed to debates in temporal ontology, particularly the debate between presentism (the view that only the present exists) and eternalism (the view that past, present, and future all equally exist). Gödel's rotating universe model seemed to support eternalism, suggesting a block universe where all points in time are equally real.

Philosophy of Science: Gödel's exploration of time had implications for the philosophy of science, particularly in the context of understanding the limits of scientific theories. His work underscored the importance of considering philosophical questions when developing scientific theories, as they shape our fundamental understanding of concepts like time and space.

Legacy

Gödel's exploration of time remains a significant and controversial contribution to both physics and philosophy. His work challenged established notions and encouraged deeper inquiries into the nature of reality. Gödel's rotating universe model continues to be a topic of interest in theoretical physics and cosmology, inspiring new research into the nature of time and the possibility of time travel. In philosophy, Gödel's inquiries into time have prompted ongoing debates about the nature of temporal reality, the relationship between mathematics and physical phenomena, and the limits of human understanding. His work exemplifies the intersection of mathematical rigor and philosophical inquiry, demonstrating the profound insights that can emerge from such an interdisciplinary approach.

The Temporal Ontology of Kurt Gödel

Kurt Gödel's profound contributions to mathematics and logic extend into the realm of temporal ontology — the philosophical study of the nature of time and its properties. Gödel's insights challenge conventional perceptions of time and suggest a more intricate, layered understanding of temporal reality. This essay explores Gödel's contributions to temporal ontology, particularly through his engagement with relativity and his philosophical reflections.

Gödel's Rotating Universe

One of Gödel's most notable contributions to temporal ontology comes from his work in cosmology, specifically his solution to Einstein's field equations of general relativity, known as the Gödel metric. Introduced in 1949, the Gödel metric describes a rotating universe with closed timelike curves (CTCs). These curves imply that, in such a universe, time travel to the past is theoretically possible, presenting a significant challenge to conventional views of linear, unidirectional time.

Implica tions for Temporal Ontology

Gödel's rotating universe model has profound implications for our understanding of time:

Eternalism vs. Presentism: Gödel's model supports the philosophical stance known as eternalism, which posits that past, present, and future events are equally real. In contrast to presentism, which holds that only the present moment exists, eternalism suggests a "block universe" where time is another dimension like space. Gödel's rotating universe, with its CTCs, reinforces this view by demonstrating that all points in time could, in principle, be interconnected in a consistent manner.

Non-linearity of Time: The possibility of closed timelike curves challenges the idea of time as a linear sequence of events. In Gödel's universe, time is not merely a straight path from past to future but can loop back on itself, allowing for complex interactions between different temporal moments. This non-linearity has implications for our understanding of causality and the nature of temporal succession.

Objective vs. Subjective Time: Gödel's work invites reflection on the distinction between objective time (the time that exists independently of human perception) and subjective time (the time as experienced by individuals). His model suggests that our subjective experience of a linear flow of time may not correspond to the objective structure of the universe. This raises questions about the relationship between human consciousness and the underlying temporal reality.

Gödel and Philosophical Reflections on Time

Gödel's engagement with temporal ontology was not limited to his cosmological work. He also reflected deeply on philosophical questions about the nature of time and reality, drawing on the ideas of other philosophers and integrating them into his own thinking.

Kantian Influences: Gödel was influenced by Immanuel Kant's distinction between the noumenal world (things as they are in themselves) and the phenomenal world (things as they appear to human observers). Gödel's views on time echoed this distinction, suggesting that our perception of time might be a phenomenon shaped by the limitations of human cognition, while the true nature of time (the noumenal aspect) might be far more complex and non-linear.

Husserlian Phenomenology: Gödel's interest in Edmund Husserl's phenomenology also informed his views on time. Husserl's emphasis on the structures of consciousness and the intentionality of thought resonated with Gödel's belief in the importance of intuition in accessing mathematical truths. Gödel's reflections on time incorporated a phenomenological perspective, considering how temporal experience is structured by human consciousness.

Mathematical Platonism: Gödel's Platonist views extended to his understanding of time. Just as he believed in the independent existence of mathematical objects, Gödel saw time as an objective entity with a structure that transcends human perception. His work on the Gödel metric can be seen as an attempt to uncover this objective structure, revealing the deeper realities that underlie our experience of time.

Legacy and Continuing Debates

Gödel's contributions to temporal ontology continue to inspire and challenge contemporary philosophers and physicists. His work has spurred ongoing debates about the nature of time, the possibility of time travel, and the relationship between physical theories and philosophical concepts. Gödel's model of a rotating universe remains a topic of interest in both theoretical physics and the philosophy of time, encouraging further exploration of the fundamental nature of temporal reality.

In summary, Gödel's exploration of temporal ontology offers a rich and nuanced perspective on the nature of time. By challenging conventional views and proposing alternative models, Gödel has expanded our understanding of temporal reality and opened new pathways for inquiry into one of the most profound aspects of existence.
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The philosophy of Jorge Luis Borges - Alexis karpouzos

Luis Borges, the Argentine writer, is renowned for his complex and thought-provoking works that often delve into philosophical themes. While Borges himself was not a philosopher in the traditional sense, his writings frequently explore philosophical concepts, particularly those related to metaphysics, reality, and the nature of time and identity.

Borges' philosophy cannot be pinned down to a single set of beliefs or principles. Instead, it manifests as a playful interplay between fiction and philosophy within his diverse body of work. He delighted in blurring the lines between genres, treating literature as non-fiction and vice versa, and often included invented authors and works within his essays.

His fascination with philosophy, especially metaphysics, sets him apart from his contemporaries. Borges appreciated and formulated rigorous philosophical arguments, but also had the unique ability to present abstract ideas imaginatively through metaphors and symbols. For instance, his stories often feature labyrinths, mirrors, and infinite libraries, which serve as symbols for more profound philosophical inquiries into reality, perception, and the infinite.

In his works, Borges frequently references and engages with the ideas of notable philosophers such as Berkeley, Hume, and Schopenhauer. He uses their concepts as a starting point to further explore and sometimes refute or extend their ideas. For example, in "The New Refutation of Time, " Borges discusses Schopenhauer's denial of the reality of our representations and takes it a step further by questioning the reality of time itself.

Overall, Borges' contribution to philosophical literature is significant, and his works continue to inspire and challenge readers and thinkers alike. His approach to philosophy is less about asserting a consistent system of thought and more about exploring the possibilities and paradoxes that arise when one engages deeply with philosophical questions.

The philosophy embedded in Jorge Luis Borges' "El Aleph" is multifaceted, reflecting his deep engagement with metaphysical questions and the nature of reality. The story, which is part of the collection also titled "El Aleph, " revolves around a point in space called the Aleph, which contains all other points in the universe. This point allows the observer to see everything in the universe from every angle simultaneously, without distortion, overlapping, or confusion.

The Aleph symbolizes the concept of infinity and the limitations of human perception and language. Borges uses this narrative device to explore the idea that the universe is ineffable and that experiences shape perception and rationality. The story suggests that language, being sequential, cannot adequately describe the Aleph, which is synchronous and represents an "unimaginable universe" that is infinite.

Borges also touches on the theme of memory and its fallibility. After witnessing the Aleph, the narrator realizes that human memory cannot retain the infinite, and forgetfulness is an inherent part of the human condition. This ties back to the philosophical exploration of time and its inexorable passage, which naturally leads to memories fading away.

In a broader sense, "El Aleph" can be seen as a commentary on the human quest for knowledge and the desire to comprehend the incomprehensible. It challenges readers to consider the limitations of their own understanding and the potential vastness of the universe beyond what can be perceived or described.

Borges' work often blurs the boundaries between the literal and the metaphorical, encouraging readers to reflect on the philosophical implications of his stories. "El Aleph" is a prime example of this, offering a rich tapestry of ideas about infinity, reality, and the power and limits of human cognition.

Jorge Luis Borges' essay "A New Refutation of Time" is a profound exploration of the nature of time and its existence. In this work, Borges challenges the conventional understanding of time as a sequence of events that occur in a linear fashion. He argues that the negations of idealism, which suggest that reality is fundamentally mental or spiritual rather than material, can be extended to time itself.

Borges posits that time may not be a real, objective entity but rather a subjective construction of the human mind. He draws upon various philosophical and literary sources to support his argument, suggesting that time, as we perceive it, is an illusion. The essay delves into metaphysical questions about the continuity of time and personal identity, examining how our perception of time shapes our experience of existence.

The philosophy presented in "A New Refutation of Time" is complex and layered, inviting readers to reconsider their understanding of time and its impact on their lives. Borges' reflections on time have influenced many thinkers and continue to be a topic of discussion in philosophical circles.

In Jorge Luis Borges' essay "A New Refutation of Time, " some of the key arguments include:
The Illusion of Successive Moments: Borges suggests that our experience of time as a continuity of successive moments is a cognitive illusion, not an inherent feature of the universe.
Time and Personal Identity: He explores the idea that time is the foundation of our experience of personal identity, drawing from philosophical and literary sources to support his views.
Time as a Mental Construction: Borges argues that time may not be a real, objective entity but rather a subjective construction of the human mind.
Berkeley's Idealism and Leibniz's Principle of Indiscernibles: He uses these philosophical principles to support his argument that time, as we perceive it, is an illusion.
Parmenides' Proposition: Borges refers to Parmenides' idea that "what is" never was nor will be because it simply exists, which challenges the traditional concept of time.
Time and Movement: He discusses the relationship between time and movement, questioning the conventional belief that time is a measure of change.
Eternity: Borges offers a definition of eternity in the form of a rhetorical question, further complicating the concept of time.
These arguments are part of Borges' broader philosophical inquiry into the nature of reality and existence, as he seeks to demonstrate that time, as we understand and experience it, may be nothing more than an elaborate mental construct.

ONE AND THE MULTIPLE - ALEXIS KARPOUZOS

The relationship between the One and the Multiple in mystic philosophy is a profound and central theme that explores the nature of existence, the cosmos, and the divine. This theme is present in various mystical traditions, including those of the East and West, and it addresses the paradoxical coexistence of the unity and multiplicity of all things.

In mystic philosophy, the One often represents the ultimate reality, the source from which all things emanate and to which all things return. It is the absolute, the infinite, and the unchanging. The One is beyond all attributes and is often associated with the divine or the absolute truth.

The Multiple, on the other hand, represents the manifest world, the diversity of forms, and the realm of change and plurality. It is the world we experience through our senses, the domain of time and space, where differentiation and individuality are apparent.

The relationship between the One and the Multiple is not one of opposition but of emanation and unity. The Multiple is seen as a reflection, expression, or manifestation of the One. In this sense, the diversity of the world doesn't contradict the unity of the One but rather demonstrates it in a myriad of forms.

Mystics seek to understand and experience this relationship through various practices and insights. They aim to transcend the illusion of separation and duality to experience the non-dual reality where the One and the Multiple are recognized as inseparable.

This concept can be illustrated by the metaphor of the ocean and its waves. The ocean represents the One—vast, deep, and all-encompassing—while the waves represent the Multiple—distinct, diverse, and ever-changing. Each wave is unique, yet it is not separate from the ocean. The wave's existence is dependent on and made of the same substance as the ocean. In the same way, each individual entity in the Multiple is an expression of the One.

In summary, the relationship between the One and the Multiple in mystic philosophy is a dynamic interplay that challenges the conventional understanding of separation. It invites a deeper exploration of reality, where the apparent multiplicity of the world is a direct expression of the singular, underlying essence of all that is.

The relationship between the One and the Multiple in the context of mathematical philosophy is a profound topic that touches upon the very foundations of existence and knowledge. It's a theme that has been explored by philosophers and mathematicians alike, often leading to the contemplation of unity and diversity within the structures of reality.

In mathematics, the concept of the One can be seen as the basis of unity from which all numbers derive. It's the identity element in multiplication, the starting point in counting, and the foundation of dimension in geometry. The One is often associated with the concept of monism in philosophy, which posits that there is a single, underlying substance or principle that constitutes reality.

On the other hand, the Multiple represents the infinite variety and diversity of forms and numbers that arise from the One. It's the embodiment of plurality and the complex interplay of different entities that mathematics seeks to understand and describe. This reflects the philosophical stance of pluralism, which acknowledges the existence of multiple realities or truths.

The interplay between the One and the Multiple can be seen in the mathematical concept of sets. A set can be thought of as a unity, a whole composed of distinct elements. Yet, each element within the set also retains its individuality, contributing to the diversity of the set's composition. This duality is mirrored in the philosophical exploration of the universal and the particular, where the universal represents the One, and the particulars represent the Multiple.

In the realm of mathematical philosophy, this relationship often leads to questions about the nature of mathematical objects: Are they discovered as part of an objective reality (the One), or are they constructed by the human mind from a multitude of experiences (the Multiple)? This debate resonates with the philosophical inquiry into the nature of truth and reality.

Reflecting on the One and the Multiple can also lead to a deeper understanding of the self and the cosmos. Just as the number one is integral to the existence of all other numbers, the individual self can be seen as a unique expression of the universal whole. Similarly, the cosmos can be viewed as a grand unity composed of a multiplicity of forms and phenomena.

Gödel's incompleteness theorems have a fascinating connection to the philosophical concepts of the One and the Multiple. These theorems, which are pivotal in mathematical logic and philosophy of mathematics, articulate the inherent limitations of formal axiomatic systems, particularly those sufficient to express the arithmetic of natural numbers.

Gödel's incompleteness theorems have a fascinating connection to the philosophical concepts of the One and the Multiple. These theorems, which are pivotal in mathematical logic and philosophy of mathematics, articulate the inherent limitations of formal axiomatic systems, particularly those sufficient to express the arithmetic of natural numbers⁴.

The first incompleteness theorem reveals that within any such consistent system, there are propositions that are true but cannot be proven within the system itself. This reflects the idea of the Multiple in that there is an abundance of mathematical truths, a multiplicity that exceeds the unifying framework of any one system. It suggests that the realm of mathematical truth is more extensive than any single formal system can fully capture.

The second incompleteness theorem extends this by showing that a system cannot prove its own consistency. This relates to the concept of the One, as it implies that a system's complete self-understanding, its unity and coherence, is unattainable from within. It must look beyond itself, to an external vantage point, to ascertain its consistency.

In the context of the One and the Multiple, Gödel's theorems imply that the One (a consistent formal system) is inherently incomplete and cannot encompass the Multiple (the totality of mathematical truths). This resonates with philosophical discussions about the relationship between unity and plurality. Just as a single philosophical system cannot capture the entirety of truth, a single formal mathematical system cannot encapsulate all mathematical truths.

Moreover, Gödel's work suggests that the pursuit of a single, unified theory of everything in mathematics—a One that encompasses all Multiples—is an inherently Sisyphean task. There will always be more truths (Multiples) than can be derived from any given set of axioms (the One).

In essence, Gödel's incompleteness theorems provide a formal underpinning to the philosophical notion that the One cannot exist without the Multiple, and vice versa. They are interdependent, with the One giving rise to the Multiple, and the Multiple necessitating the One for its expression and comprehension. This interplay is a dance of limitations and possibilities, where the boundaries of logic, mathematics, and philosophy blur into one another.

The relationship between the One and the Multiple in the context of the mathematics of zero is a deeply philosophical inquiry that bridges the abstract world of numbers with the existential questions of being and non-being.

Zero, in mathematics, is a symbol of absence, a representation of nothingness, yet it holds a pivotal position as a number. It is the void from which all things emerge and to which they return. In the philosophy of mathematics, zero is the paradoxical junction where the One and the Multiple converge and diverge.

From the perspective of the One, zero can be seen as the origin—the singular point that precedes the existence of numbers. It is the empty set, the foundation upon which the edifice of mathematics is constructed. As the identity element in addition, zero maintains the integrity of numbers, for adding zero to any number leaves it unchanged, reflecting the immutable nature of the One.

Conversely, when we consider the Multiple, zero represents the infinite potentiality of creation. It is the canvas upon which the integers, both positive and negative, express their multitude. Zero is the balance point, the fulcrum around which the symphony of numbers dances. It embodies the plurality of possibilities, the beginning of the number line that stretches infinitely in both directions.

Philosophically, zero challenges our understanding of existence. It is both something and nothing—a number that quantifies the absence of quantity. This duality echoes the philosophical struggle to comprehend how the One gives rise to the Multiple. How does the unity of being manifest the diversity of the cosmos? Zero offers a mathematical metaphor for this mystery, as it encapsulates the transition from non-existence to existence, from the undifferentiated One to the differentiated Multiple.

In the realm of set theory, zero corresponds to the empty set—a set with no elements. This set is unique in that it is the only set that contains nothing, yet it is the foundation upon which all other sets are built. The empty set is the mathematical embodiment of the One, and all other sets, containing multiple elements, arise from it.

Reflecting on zero in the context of Gödel's incompleteness theorems, we find a resonance with the idea that the One (a consistent formal system) cannot capture the entirety of the Multiple (the totality of mathematical truths). Zero, as the foundation of numbers, similarly suggests that from the nothingness of the One, the infinite complexity of the Multiple emerges—yet it can never be fully encompassed or expressed by any single system.

In conclusion, the mathematics of zero offers a profound reflection on the relationship between the One and the Multiple. It serves as a bridge between the abstract and the concrete, the known and the unknowable, challenging us to ponder the origins of existence and the nature of reality itself.

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THE PHILOSOPHY OF GILLES DELEUZE - ALEXIS KARPOUZOS

Gilles Deleuze, a prominent figure in postmodern French philosophy, made significant contributions to various fields, including metaphysics, aesthetics, and literary theory. Gilles Deleuze's philosophy invites us to embrace creativity, multiplicity, and the perpetual process of becoming. His ideas continue to inspire scholars, artists, and thinkers across disciplines, bridging Western philosophy with Eastern mysticism and inviting us to question established norms

Let's delve into some key aspects of his philosophy:

Concept Creation: Deleuze conceived of philosophy as the creation of concepts. His writings take the form of precise deductions of these concepts. Unlike traditional philosophical systems, which often seek to establish universal truths, Deleuze's approach is dynamic and inventive. He engages with thinkers such as the Stoics, Leibniz, Hume, Kant, Nietzsche, Spinoza, and Bergson, extracting insights and weaving them into his own unique framework.

Gilles Deleuze, a prolific philosopher, believed that philosophy is not merely about analyzing existing ideas but rather about creating new concepts. Let's explore this fascinating aspect of Deleuze's thought:

Philosophical Encounters:

Deleuze didn't approach art, literature, or cinema as mere subjects of study. Instead, he engaged in philosophical encounters with them. These encounters inspired him to develop fresh concepts, emphasizing the creative aspect of philosophical thinking.



Constructivist Stance:

Deleuze considered himself a constructivist. For him, philosophers are concept creators. Each encounter with philosophy should lead to the birth of novel ideas, expanding our understanding of the world.

Leibniz and Unusual Concepts:

Leibniz, whom Deleuze admired, exemplifies this creative approach. Leibniz's rationalist philosophy involved inventing unusual concepts. These concepts weren't pre-existing; they emerged through Leibniz's intellectual activity.

God as a Creator of Concepts:

Leibniz believed that God created the world through calculation. He used examples like tiling (arranging figures to fill space while minimizing emptiness) to illustrate his concept of creation.

In summary, Deleuze's philosophy encourages us to be active creators of concepts, just as painters create lines and colors.

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Difference and Repetition: Deleuze's magnum opus, "Difference and Repetition" (1968), explores the interplay between difference and repetition. He argues that difference is fundamental to reality, and repetition is not mere duplication but a creative force. Deleuze challenges conventional notions of identity and sameness, emphasizing the productive potential of difference.

Gilles Deleuze's "Difference and Repetition" is a seminal work that challenges traditional Western metaphysics and offers a fresh perspective on concepts like identity, repetition, and creativity. Let's explore some key ideas from this remarkable book:



Pure Difference: Deleuze argues that difference is fundamental to reality. Unlike classical philosophy that seeks universal truths, he emphasizes divergence and decentering. Each moment contains unique differences, and these differences shape our understanding of the world.



Complex Repetition: Repetition, for Deleuze, isn't mere duplication. Instead, it involves displacement and disguising. While repetition contributes to generality and thought, it's the differences within each repetition that account for change and novelty. Concepts and things derive meaning from these differences.



Shift Away from Hegel and Marx: "Difference and Repetition" played a crucial role in shifting French thought away from Hegel and Marx toward Nietzsche and Freud. Deleuze's exploration of difference challenged established philosophical norms and paved the way for new perspectives.



The Image of Thought: Deleuze critiques the traditional "image of thought," which often relies on fixed identities and binary oppositions. Instead, he encourages us to embrace multiplicities, complexities, and the perpetual process of becoming.



Asymmetrical Synthesis of the Sensible: Deleuze introduces the concept of asymmetrical synthesis, emphasizing the interplay between perception and affect. Reality is a dynamic, ever-changing process, not a static being. Copies are never identical; they're something new.



"Difference and Repetition" invites us to rethink how we perceive reality, emphasizing creativity, multiplicity, and the constant flux of existence. Deleuze's work continues to inspire thinkers across disciplines, bridging philosophy with art, literature, and mysticism.

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Multiplicities and Becoming: Deleuze rejects fixed identities and embraces multiplicities. He sees reality as a complex web of interconnected processes, where entities are constantly becoming. His collaboration with psychoanalyst Félix Guattari resulted in the influential works "Anti-Oedipus" (1972) and "A Thousand Plateaus" (1980). These texts explore desire, capitalism, and the rhizomatic structure of thought.

Gilles Deleuze's concept of multiplicity is a fundamental departure from traditional metaphysical notions. Let's explore it in more detail:



Multiplicity Defined:

Deleuze draws upon ideas from mathematician Riemann and philosopher Bergson. Multiplicity, for him, isn't a mere combination of many elements; it's an organization intrinsic to the many itself.

Unlike the One-Many dialectic, where unity and opposition dominate, multiplicity thrives on differences within and between multiplicities.

Substance vs. Multiplicity:

Substance theory (from Aristotle to Spinoza) often reduces the world to a unity (the One) or a variety (the Many). Deleuze challenges this.

He replaces substance with multiplicity, asserting that even the One is a multiplicity. Instead of rigid oppositions, we find a rich variety of differences—difference becomes the key.

Desiring-Production and Multiplicity:

In "Anti-Oedipus," Deleuze links multiplicity to desiring-production. Desiring-production is pure multiplicity—an irreducible affirmation beyond unity.

Multiplicity accounts for the dynamic, ever-changing nature of reality, resisting fixed essences.

In summary, Deleuze's multiplicity invites us to embrace difference, complexity, and the perpetual becoming of existence. It's a departure from traditional metaphysics, opening new vistas for thought and creativity.

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Rhizome and Non-Hierarchical Thinking: Deleuze introduces the concept of the "rhizome," an alternative to hierarchical structures. Rhizomes grow horizontally, connecting diverse elements without a central point. This idea challenges traditional tree-like models of knowledge and encourages a multiplicity of connections and pathways.

The concept of rhizome in Gilles Deleuze and Félix Guattari's philosophy:



What Is a Rhizome?

A rhizome is a descriptive or epistemological model that contrasts with hierarchical structures. Unlike a tree-like system with a central root and branches, a rhizome has no fixed order or hierarchy. In a rhizome, any element can connect to any other, creating a network of multiplicities. It defies linear thinking and embraces complexity.



Non-Hierarchical Connections:

Rhizomes mark a horizontal conception where diverse elements link without respect for specific species. For instance, Deleuze and Guattari connected desire and machines, giving rise to the intriguing concept of "desiring machines". Rather than following a predetermined path, a rhizome allows for nomadic growth and propagation. It resists chronology and organization, favoring a dynamic, interconnected system.



Rhizome vs. Tree:

While trees represent hierarchical models, rhizomes work with planar and trans-species connections. They emphasize multiplicity and interbeing. Just as water spreads across available spaces, a rhizome's surface can be interrupted and moved, leaving no trace but seeking equilibrium.

In summary, the rhizome challenges traditional thinking, encouraging us to explore networks, multiplicities, and the perpetual middle ground between things.

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Affect and Percept: Deleuze emphasizes affect (intensity) and percept (sensory experience) over representation. He explores how affective forces shape our encounters with the world. His collaboration with Claire Parnet resulted in the captivating book "Dialogues" (1977), where he discusses these concepts in depth.

Gilles Deleuze's concepts of affect and percept within his materialist philosophy:

Percept:

Percepts go beyond mere perceptions. They are independent of the state of those who experience them. Unlike perceptions, percepts exist in themselves, with their validity transcending individual lived experiences. They are self-sufficient entities.

Affect:

Affects exceed ordinary feelings or affections. They possess a force that surpasses the strength of those who undergo them. Deleuze's notion of affect emphasizes intensity, vitality, and the transformative power of forces that shape our existence1.

In summary, percepts and affects are essential components of Deleuze's philosophy, emphasizing the immanence of forces and the dynamic interplay between sensation, intensity, and lived reality

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Virtuality and Actualization: Deleuze distinguishes between the virtual and the actual. The virtual contains potentialities, while the actual represents realized states. His work on cinema, particularly the concepts of the "movement-image" and the "time-image," exemplifies this exploration of virtual and actual dimensions.

Let's delve into the fascinating concept of virtuality and actualization in Gilles Deleuze's philosophy.



Deleuze's exploration of virtuality is deeply rooted in the work of French philosopher Henri Bergson. Rather than framing it solely as a realm of mere possibilities waiting to be actualized, Deleuze considers the virtual as a dynamic and productive field. Here are some key points:



Virtual vs. Actual:

Deleuze distinguishes between the virtual and the actual. These are not opposing realms but interconnected aspects of reality. The virtual refers to an ideal yet real dimension. It is not merely potential; it possesses full qualities of the real. The actual, on the other hand, unfolds from the virtual through processes of actualization or differentiation.



Bergson's Influence:

Deleuze credits Henri Bergson for developing the notion of the virtual to its highest degree. Bergson's concept of "duration" aligns with the virtual. Duration is inseparable from the movement of its actualization.

Example: Reflection in a Mirror: Consider a reflection in a mirror. It exists fully, regardless of whether we perceive it. The mirror image is already there, waiting for no further actualization. Yet our perception of it remains real.

Political Implications:

Brian Massumi highlights the political implications of virtuality.

The virtual is inaccessible to the senses but can be felt through its effects. Massumi uses topological figures to illustrate virtuality, emphasizing its imaginative presence. In summary, Deleuze's virtuality is not a passive waiting room for actualization; it's a vibrant force shaping our experience.



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The Topology of Deleuze: A Virtual Continuum

Deleuze's ontological categories include a virtual continuum—a dynamic interplay between pure extension and thought. This continuum, akin to Spinoza's substance, embodies two powers: the power of being and the power of thinking1. Throughout his writings, Deleuze employs various terms to describe this continuum: "intensive spatium," "ideal or metaphysical surface," "plane of consistency," and "plane of immanence." These diverse labels emphasize different aspects of the same underlying concept.


Pre-Extensive Milieu: Deleuze characterizes this continuum as a pre-extensive, non-qualified "milieu" or "space-stratum." It envelops complexes of differential relations, pure intensities, and singularities. Unlike empirical fields, it doesn't correlate with consciousness and its objects, nor does it dissolve into undifferentiated chaos.

Topological Model: Deleuze consistently employs a "topological model" to describe the properties of this transcendental field. He draws inspiration from Michel Serres and Merleau-Ponty, emphasizing topological categories like position, junction, and connection. In this framework, places matter more than what fills them, defining a non-extensive, pure spatium1.

Surface and Co-Existence: Deleuze's transcendental field constitutes a surface—a topological surface. It connects internal and external spaces without regard to distance. This property echoes Simondon's argument that all organization presupposes an absolute outside and inside, leading to relative intermediary exteriorities and interiorities.

In summary, Deleuze's topology transcends traditional notions of space. It's a dynamic, pre-individual field where intensities and relations coexist, shaping our experience beyond empirical confines.

WE ARE DEAD AND ALIVE— ALEXIS KARPOUZOS

When day comes we ask ourselves where can we find light in this never ending shade? The loss we carry, a ocean we must wade. We braved the belly of the beast. I have seen you in millions of places. I met you in a million forms. We met among the ruins, the ashes and the bones, we lost them all, but we found each other, I saw your lion heart, and it pulled me. I saw the creation and the destruction in your eyes. I see you here in the mud, on the rock, in the rays of the rising sun. We are Dead and alive, we saw a thousand Christs go by As they went up to Calvary but The dove it found no resting place. You were where our solar system was formed, you whispered something to me for eternal love and then you fell from my hands and everything became fire. All the myths always showed you.

We are man and woman, plant and stone, amorphous and form, swallow and eagle, snake and gazelle, fantastic creatures of the depths. They crucified us, beat us, tied us to poles and burned us, wrapped us in gold and silver jewelry, then exalted someone in the world and then we were ridiculed. We stood together in front of the executive detachment, our bodies pressed against each other for the last time, flesh by flesh, as we became utensils for the spirit.
But don't forget You are my brother, my sister, my child. I took care of you from infancy and you took care of me. We were lovers and friends, we recognized each other with countless disguises, here on one side and there on the other. And in the end, there were no sides at all, only this magnificent loop, this One Circle — majestic, magnificent, royal, timeless, utterly mysterious and towering above all things. Print me in your heart, love is as strong as death ". Does not matter. You are inside me and I am inside you and we will compose again a humanity committed to all cultures, colors, characters, and conditions of man.

THERE IS A LAND — ALEXIS KARPOUZOS

There is a land by faith I've seen
Where skies no clouded regions know;
Where they know not the sorrows of time
and no shadows fall to blight the view
That land no want has ever known,
Nor pain nor sickness nor distress;
there, Death, the last enemy, is slain;
There those who meet shall part no more,
And those long parted meet again.
There's a land far away..
Beyond these wild winds and gloomy skies,
Beyond Death's cloudy portal,
There is a land where beauty never dies
And love becomes immortal;
A land whose light is never dimmed by shadow,
Whose fields are ever vernal,
Where nothing beautiful can ever fade,
But blooms for aye eternal.

THE MESSAGE — ALEXIS KARPOUZOS
When the dimension of time is added to the dimensions of space and the 4-dimensional universal space-time continuum is shaped, two spirits which are located in different spots of space-time will be able to be one, or structural elements of One Universal Spirit, which would compose an inseparable and ineffable being

Humanity is unaware of its great strength, and it is unaware of its great perils. It is these two things that have brought a New Message into the world for the protection and the advancement of humanity. World seeks to protect human civilization and to give it new life, a new purpose and a new direction, to warn you of the great adversities you face now that are greater than anything your ancestors ever had to deal with. Therefore, so much depends on human response — human responsibility, the ability to respond. So much depends upon your awareness and decisions, and your ability to recognize you are living in a time of evolution, in a time of transition into a more difficult and more hazardous world. So much depends upon human intelligence, the intelligence of individuals who can respond to a New Evolution and can share its wisdom and its guidance with others. So much depends upon the power and the presence of Knowledge within each individual, a power and a presence that is so unknown and that it is not heeded by most. You have to love humanity and have great faith in humanity to believe that humanity will make the right decisions and follow the path that will provide a new way forward. You have to love humanity and have faith in the spirit of humanity and in the promise and the talents of humanity despite its tragic and prolonged mistakes.

Do not think another race in the universe will come to save you, for those that claim to do so are only here to take advantage of your weakness and your naiveté. Do not think that if human civilization fails, something better or greater can be established as a consequence. Do not underestimate the power of the time in which you live and the great adversities that you now face that still remain unknown to so many.

Do not lose faith in the power and presence of Knowledge within you and within others to recognize and to respond to this and to see the great opportunity to forge a new union in the human family — a union built by necessity, a union built in the fire of necessity, a union built by the recognition that together you can succeed, where in the past you have failed. The warning is upon you, but the blessing is upon you as well. For life loves humanity and does not want to see you fail or lose your freedom as you emerge into a Greater Community of life in the universe.

You must have this love and this faith and this commitment to humanity as well. If you do, you will begin to experience the power and the grace of Knowledge within yourself. You will see that you too have come into the world at this time specifically to make a unique contribution to certain people in certain circumstances. And though you may not yet realize who these people are or what these circumstances are, you will feel the power and presence of Knowledge moving you, freeing you, reshaping your life, recasting your commitments, moving you in a new direction.

the future of humanity will be a time of transitionMay this blessing be yours to experience, for it is a blessing truly. May these times arouse a newer, deeper commitment and a deeper courage. May you see that your future is before you to be decided at this time of great transition. May you recognize that you as an individual must make these decisions and not simply rely upon others to make them for you. May you recognize that the power and the grace of Knowledge live within you, beneath the surface of your mind. Within your heart, you know things the mind cannot understand, and that your true identity exists beyond the realm and the reach of the intellect, in the power and the presence of Knowledge. May you hear these words with your heart with an open mind to see the great love that they demonstrate and the great respect and trust that they offer to you, who do not yet have your own self-respect and trust. May New Evolution illuminate your life and give you strength and courage to navigate the difficult times ahead, and to speak as one voice in this world, and to forge the foundation for a greater future for the human family.

The planetary thought and the nixilism — Alexis karpouzos

The Greek philosopher and revolutionary Kostas Axelos tried in his vast philosophical production to distance himself from orthodox and Stalinist Marxism, seeing Marx as the thinker of technology and providing a metaphysical interpretation of the Trier revolutionary on the basis of the scant indications offered by Heidegger in some of his works. Over time, having put aside his youthful revolutionary fury, he arrived at a post-metaphysical thought, which set itself the objective of thinking about the becoming of being, renouncing, moreover, any attempt to provide men with points of reference of an ethical nature.

Kostas Axelos' reflection, from the beginning, stood out — as well as for its radicality — for the ambitious objective it set itself: the overcoming of Marxism and Heideggerian post-metaphysics in view of a thought to come, capable of healing the bleeding wound of nihilism and joyfully facing the challenges of planetary technology. Already in the years in which he was editor-in-chief of the magazine Arguments, founded in 1956 by Edgar Morin, the Greek philosopher and revolutionary examined the existing relationship between Marx and Heidegger, who connect, from his point of view, on the theoretical ground of concept of estrangement, understood as an essential negative of the history of metaphysics: as economic and social alienation in Marx, as constitutive uprootedness of the subject in Heidegger. But if the first explains the technical essence of modern man, the second recognizes in technology the status of a historically determined form of truth. It is therefore necessary to deduce the emancipatory perspective from Marx, from Heidegger the temporal relativity of this perspective, and the opening onto a broader project of liberation. It is on this basis that it is possible for us to grasp, in the problematic conjunction of Marxism and Heideggerism, the traces of a future thought: «Through Marx and Heidegger, and through them we can at the same time go beyond them. This reflection also introduces an anticipatory thought" (1). The critical project of Marxism allows us to take Heidegger as the one who, questioning the history of philosophy, indicated the urgency of a new way of thinking, of a liberation of thought from the representative status of classical metaphysics, and from the will to technical power of the subject. And, vice versa, the Heideggerian approach to the problem of metaphysics allows us to see in Marx the overcoming of philosophy in technology.

The starting point of the Axelosian interpretation of Marx is, therefore, represented by the Letter on "Humanism", in which Heidegger underlines the need to reach a productive dialogue with Marxism, a dialogue which neither Sartre nor Husserl have reached, since who have not recognized «the essentiality of the historical dimension in being» (2). Marx, for Heidegger, in experiencing alienation and considering the entity in its totality as work material, managed to penetrate an essential dimension of history, superior to any type of historiography. The true essence of materialism, consequently, does not lie in the affirmation that everything is matter, but rather in conceiving reality as that which man continually transforms, thus imprinting his own mark on all beings, reduced to mere background: «The essence of materialism is hidden», observes Heidegger, «in the essence of technology, about which much is written, but little is thought. In its essence, technique is a destiny, within the history of being, of the truth of being that rests in oblivion. In fact, it dates back to the techne of the Greeks not only in name, but comes in an essential historical sense from the techne understood as a way of alethèuein, that is, of making being manifest" (3). But, while techne, for the Greeks, is co-essential to nature, in the sense that natural arising and poietic producing, cosmic happening and active operating, are determined by the same thing which remains, ultimately, enigmatic, in the modern era first, and even more so in the planetary one, it opposes nature, trying to dominate it. In other words: modern thought, according to Heidegger, which, on this point, greatly influences Axelos, carries forward the work of dissolving the unity of the totality of physis, already called into question by Christianity, placing the ego of the subject as res cogitans and opposing it to the objective world of the res extensa, understood as the set of things that are in front of the man who takes possession of them and shapes them. Fundamental, then, is Descartes' thought, for which the subject, the res cogitans, must, through representation, dominate the res extensa, in order to use it rationally. Man becomes the "measure" of the entity, in the sense that he gives the entity the measure, determining what can be considered as an entity. From this it is clear that the notion of objectivity, very important in modern philosophy, always refers to that of the subject: objective reality is that which appears as such to the subject, which is why what constitutes it is the certainty that the representing subject has of it.

From that moment on, being has the fundamental and exclusive property of presence, the essence of truth is given by the certainty of representing, the entity is increasingly subjugated by man who methodically exploits it. Descartes therefore takes the first decisive step in that process which will slowly lead to the philosophical becoming of the world as the mundane becoming of philosophy: physics begins to transform into technique, and man, the human subject, who aims at control totality of the entity through the ratio, is itself posited as an object. According to this reconstruction of Western philosophy, Kantian philosophy, which fits into the path traced by Descartes, places the transcendental ego by trying to found it: «this thinking and acting ego», as Axelos observes, taking up Heideggerian arguments, «constitutes things as objects of experience, that is, as objects. The transcendental of objectivity includes transcendental subjectivity and is both founded by it. Transcendental subject and transcendental object are referred to each other and are rooted in the same" (4). The criticism of Kantian pure reason, far from being understood as a criticism of Cartesian reason, is seen as its strengthening, as a further enthronement of the subject, increasingly aimed at the conquest of beings in its entirety, at total domination and scientific about reality.

The fulfillment of metaphysics begins with the Hegelian metaphysics of absolute knowledge understood as the will of the spirit. Hegel, in fact, reviving the entire Western philosophical tradition in his thought, understands philosophy as the "awareness" of universal becoming that leads to the Absolute Spirit; Spirit which, alienated in nature, returns to itself and recognizes itself as what it actually is. The Spirit, which has the prerogative of neutralizing any force of disintegration, is, therefore, the power that stands up to time by occupying the place of the future and reuniting it with that of the beginning; it, therefore, is History, unlike nature, which has no history, «because in it universality is only an internal without actual development. There are indeed living individuals, but in them life can only express itself as an abstract universal, as the negation of any particular specificity. In other words, the meaning of organic life is death, the annihilation of everything that aspires to give itself a separate existence" (5). The life of the spirit, on the contrary, is that life that does not fear death, but, on the contrary, tolerates it and maintains itself in it: it knows how to face the negative and assimilate it. Philosophy, as a phenomenological description of the vicissitudes of the Spirit, must acquire knowledge of principles and general points of view, thus presenting itself as a science and no longer as a love of knowledge. It must be real and absolute knowledge of the Absolute Spirit, because only the spirit for Hegel is real, that is, Being. Thought presents itself, therefore, as the engine of becoming, which, in turn, is the unity of being and non-being, a process of revelation of the absolute, the will of the Spirit. Marx precisely questions this, replacing the spirit and ideas with the productive forces and their real movement, maintaining that the true reality is not that posed by thought, but is constituted by the social being understood as the result of the historical process, determined from practice. He is, therefore, seen as the one who, with the overthrow of the Hegelian metaphysics of absolute knowledge, underlined the importance of technical praxis, through which man sets out to conquer the entire planet and philosophy begins to become worldly. . But, at the same time, through the concept of alienation, Marx thematized the disorientation of modern man, who is no longer able to make sense of himself and his actions. In short, Marxian thought, like all Western metaphysics, does not have access to the truth of being, which continues to be veiled, denying itself to thought. We must not forget, in relation to this, that the history of metaphysics presents itself, for Heidegger, as the history of the oblivion of being, which occurs by withdrawing and remains, for this reason, in hiding. At no moment in history, which is the history of being, has the truth of being been thought of. Indeed, history, as the history of being, begins precisely with the oblivion of being, with a thought that thinks only the truth of beings, leaving the truth of being unthought: «thought is constantly set in motion by a single fact: that in Western history, from the beginning, it is thought to be silent with respect to its being, but without the truth of its being being thought of king, so that this is not only rejected to thought as a possible apprehension, but it is so in such a way that Western thought itself, in the form of metaphysics, hides the fact of this rejection, even if it is not aware of it" (6). The oblivion of being, therefore, does not derive from a lack of thought, or from our negligence, since it has its roots in the essence of being which tends to withdraw into itself: metaphysics, consequently, is denied the truth of being, at least until it reaches the era of its fulfillment. And that era, for Heidegger, is the era of deployed technique, in which only the truth of beings emerges and being is totally forgotten, covered by the productive/destructive fury of man, who is preparing to become the undisputed master of the world. Everything bends in the face of the inexhaustible power of technique, which presents itself in the form of Gestell, of imposition: « Gestell, imposition, indicates the meeting of that request which requires, that is, provokes, man to to reveal the real, in the way of use, as a background". (7) But, Heidegger observes, while stating that technique is imposition, it is not possible to fully understand its essence, it is not yet possible to identify the direction in which it proceeds, to pose the problem of what is intimately pushes her. Technology has invaded every aspect of man's life, from politics to art, to religion, it has transformed his interiority, but individuals are still far from correctly asking themselves the question about his true essence. It is therefore necessary to question our history, which, for Heidegger, is the history of being, of his oblivion. That is, entering into dialogue with those essential thinkers who can help us understand the desert that is growing around us; search in the work of a philosopher for what has always remained unthought, hidden. But who, in his works, has thematized the alienation of the human being increasingly distant from the land of being? Who discovered in advance the power of planetary technique by penetrating its essence, who destroyed philosophy by bringing it before the tribunal of material and transformative praxis? That thinker, for Heidegger, is Marx, who, although he did not complete metaphysics, although he did not "go beyond" it, laid the foundations for its destruction, for the worldliness of philosophy.

Starting from these assumptions, Axelos constructs his discourse in the monograph Marx, penseur de la technique: de l'aliénation de l'homme à la conquête du monde. The Greek philosopher presents, in the footsteps of Heidegger, a particular interpretation of Marxian thought, seen as a continuation of the metaphysics of subjectivity inaugurated by Descartes at the dawn of modernity and carried forward by Kant and Hegel. A metaphysics of subjectivity, the Marxian one, which has its nerve center in the question of the economic, political and ideological alienation of man and which leads us directly to the problem of nihilism and world technicization. However, the aim of the Axelosian interpretation of Marx and its constant comparison with the Heideggerian Seinsfrage is not at all to lay the foundations for the construction of a new ethical-political edifice. With the death of God and the humanization of nature, with the affirmation of a one-dimensional thought incapable of opposing the existing state of things, with the decline of the messianic utopias that offered the hope of a future redemption, it has failed, according to Axelos , the possibility of pushing towards new horizons, so much so that the very notion of horizon has become problematic, and mediocrity and ambiguity have taken over: we act without knowing why, we build cities that tend to become necropolises and we end up populating deserts. Mystical impulses crystallize in churches, revolutionary movements in bureaucratic states, research of thought in sclerotized universities, the existential adventures of individuals in autarchic and hypocritical families. Man's actions are meaningless, the time in which he finds himself living is so poor that he is unable to even recognize his own indigence, that it presents itself as a radical lack of future: the sense of total being in its becoming sinks into nothing and there is no longer any foundation, purpose, meaning or idea. Dominated by this game we are forced to realize that truth and true life have left us, without ever having existed. Our being becomes dark and we become problems to ourselves. The main question of politics and ethics, What to do?, although insistently posed again, has, for Axelos, only one answer: abandoning oneself to the becoming of being, to what the Greek thinker defines as the game of the world.

Man must learn to live without thinking that he can make humanity better, playing a game made up of acceptance and renunciation, of reclamation and reconciliation. ion, of observation and contestation, a game that allows him to play in detachment and indifference: Classical morality resided in intentions. Future ethics will reside in problematization and the executions that perform therein. It will be neither the ethics of maxims nor that of sentences. These are of little use to us in times of peace, a little more in times of war. The problematic ethics of the future will be aphoristic, because aphorisms delimit the fields of life and the fields of death. And this ethic will certainly suffer its repercussions in the banal and in chatter. The positive was given to us. The negative has been generated. How to (re)find what has not yet existed and will never fully exist: the constructive, globalizing and questioning, integrating and problematizing power? First of all, we must abandon the conflict of points of view to those who dedicate themselves to it with the tepid fury in which particular and partial points of view, political, aesthetic and ethical, crackling of different opinions caused by the crossing of wooden sabers are confronted[ …]The throne of knowledge, conscience and self-certainty on which the human being has painfully installed himself is not yet completely worm-eaten. Innovative behaviors will therefore still appear as transgressors with respect to the rules of the adaptation game. The contradictory aspects of everything have already become problematic. But everything that comes too soon for this fraction of space-time fails to make itself recognized. The clarity of notions, concepts, categories and definitions, so desired, playing within large, relatively transparent systems, is contained in a sort of opacity that no one dares call superior, because it is not. Every production scheme is certainly worn out by what it does not account for. Hence the task of deconstructing it. (8)

The ethics of planetary man problematizes his very being, introducing us to a beyond of ethics. It invites man to play the great Game, to abandon himself to it, without expecting anything in return, without asking or looking for points of reference, rules, more or less firm, to follow: «one cannot do anything other than play on and with the game of the two senses of the word game: playing like a door plays on its own axis and playing like a game. The more or less explicit systematics of the rules of the human game and its transgressions, i.e. the ethical problematic, would harmonize man's participation in the game of the world in the form of an always unstable equilibrium." (9) Immerse yourself in the infinite ocean of becoming, taking the place of the gods: this is the only possibility, at the dawn of an era in which nothing is as it seems, in which everything can, from one moment to the next , turn into its opposite. Man thus finds himself abandoned to his finiteness as a cosmic player, who cannot win or lose, who cannot set himself goals or hope for a radical change in his existence. He must have the wisdom of Ecclesiastes which proclaims the infinite emptiness of everything; he will have to quench his thirst at the source of Heraclitean wisdom, which he understands the One-All as becoming marked by the rhythm of time, as becoming of time as a royal and infantile game. In other words, he will have to experience nihilism to the fullest, putting an end to the era of subjectivity, abandoning the search for meaning, experiencing the efforts of vanity, opening up to the repeatable, to the old and the new, reaching a discordant agreement with the game of the world, renouncing every revolutionary attempt, since every revolution is always restorative: in fact the planetary man, as Axelos conceives him, has lost the Archimedean point, constituted by that "inhabitation of the future" of which Bloch speaks, which alone can constitute the support for the deployment of authentic ethical action.