One of the most important results in mathematics and logic in the past century came from Kurt Gödel, an Austrian logician and mathematician. His two incompleteness theorems force a sort of white-knuckled humility upon mathematicians everywhere. However, for the most part, outside of mathematicians, logicians, and philosophers, Gödel’s results remain largely unknown and seem largely irrelevant but the theorems actually have far-reaching impacts for almost any discipline. The two theorems are as follows:
Theorem 1: No consistent algorithm/axiomatization of formal arithmetic is complete.
Theorem 2: For all systems of arithmetic (A), if A is consistent, then there is no proof in A of the consistency of A.
These might seem highly technical and irrelevant, but let’s look at what the theorems actually mean and what some of their consequences are.
To understand either theorem requires some knowledge of terminology:
Arithmetic is one of the mathematical sciences (actually, the mathematical science of the integers)
An algorithm is a system of steps or procedure for solving a problem
An axiomatization is a set of axioms (or self-evident truths) which forms the starting-point for a mathematician desiring to arrive at new conclusions in mathematics
A set of axioms is consistent if and only if none of the axioms are mutually exclusive. That might seem somewhat abstract, but it makes sense. For instance, let’s say one of my axioms was “a certain number exists” and then another of my axioms was “that same number does not exist.” This is an absurdity and this set of axioms would be considered inconsistent.
A set of axioms is complete if and only if that set of axioms can be used to arrive at every truth in its particular science. For instance, a set of arithmetic axioms (or an axiomatization for arithmetic) is complete if it can be used to show every arithmetical truth.
It is desirable that an axiomatization be consistent, because if it is inconsistent, then it will arrive at contradictory conclusions. In other words, an inconsistent axiomatization leads to false conclusions, which is obviously undesirable. It is desirable that an axiomatization be complete so that we can arrive at every true conclusion regarding our particular science (for instance, arithmetic).
Again, this may seem abstract, but I am reaching a point, so please bear with me. Noting all of this information, then, informally we may state Gödel’s incompleteness theorems as follows:
No set of axioms can be proven to be consistent using that set of axioms. Furthermore, even if you could prove that it was consistent, it couldn’t be complete. That is to say, there is no way to show that a set of axioms will never lead to falsehood. However, say you knew you had a consistent set of axioms. That set of axioms cannot possibly arrive at all mathematical truth.
Okay, now to the fun part. Why is all this crazy stuff about mathematical systems of axioms important? The implications of these theorems are huge. Since those theorems are true, it appears to be true that, at least through mathematical methods, not every mathematical truth can be shown to be true. In other words, there are true things that we cannot prove. And if that’s true in mathematics, it’s certainly true in other disciplines as well. This is a strong argument for the existence of objective truth, because it shows that there are truths that are true even though it is impossible for us to prove them – that are true independent of our own minds.
It is also an important conclusion for refuting people who deny the existence of immaterial reality. Why? People who deny immaterial reality deny it on the basis that it can’t be proven using the empirical sciences (with the exception of certain sophists who choose to deny the existence of any reality at all - material or immaterial). But we know there are truths that can’t be proven via deductive reasoning. It is a mathematical fact (which, ironically enough, has been proven). Obviously this, of itself, is not sufficient for showing the existence of immaterial reality, but it is enough to silence those who argue against its existence which might open them up to our own arguments for the existence of immaterial reality.
To conclude, then, Gödel proved his two incompleteness theorems which (together) show that a complete knowledge of mathematical truths is outside the capability of any mathematical system (that is to say there are mathematical propositions that are true that are also unprovable). This has repercussions for our debates with people who believe in the supremacy of mathematics or physics rather than theology and her handmaiden philosophy. As such, even though the theorems themselves might be a little too technical to bring into a discussion, its consequences are (if not essential) highly useful for the Catholic participating in the New Evangelization.
Mariae et Jesu Semper Servus Sum,
Joe
Source: Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures, 2nd Edition, written by James Robert Brown
Showing posts with label materialism. Show all posts
Showing posts with label materialism. Show all posts
Thursday, October 07, 2010
Friday, November 30, 2007
Authority, Obedience, and Selfishness
The title of this post reminds me of the “which one doesn’t fit?” question often found on IQ tests and the like. Clearly, the answer to the above would be ‘selfishness’, but why doesn’t it fit? If we recognize that selfishness is insoluble with authority and obedience, why do the terms ‘authority’ and ‘obedience’ cause such discomfort in people? I would propose that a population conditioned by an environment and culture absolutely obsessed with individualism claiming that this individualism is what makes us ‘great’ has not a chance of escaping the pitfalls that are bound to occur – pitfalls characterized by the eventual rejection of obedience as a virtue and selfishness as a vice. When this occurs, recognition of authority as a good becomes a virtue held by a small minority, those who are diagnosed as sheltered, parochial, and even simplistic by the multitude of self-made pop psychologists who have it all figured out. They just can’t seem to figure out why their own marriages are broken, their own children are misfits, and the list could go on.
Evidence of this as the prevailing mindset of many today (although none would ever admit to it), is most clearly seen in the elementary and high school classroom, a veritable observatory of the many and varied errors the modern family has imposed upon itself with the children serving as guinea pigs for the new human cultural experiment performed by mad scientists that go by the title of ‘mother’ and ‘father’. In this experiment, a dual hypothesis is being tested: children will be better suited to reach the goal with less guidance, and they will be able to assimilate into the real world in a productive way without being conditioned to recognize the need for authority and more importantly the need to accept another’s authority.
There is an insidious movement of parents away from the role of teacher, guide, and disciplinarian and towards the role of friend. This establishes a relationship in which authority has no place, and the results of this new experiment are quite telling. Since mothers and fathers have decided to be simply peers to their children, there has been a corresponding rise in apathy towards those things that create a stable society – education, self-discipline, faith, and marriage. This should raise a question in our minds: Why would lack of authority on the part of parents within the home lead to such a result?
To answer such a question, we must first understand the nature of authority and, therefore, its purpose. Etymologically, ‘authority’ comes from the Latin auctor and auctoritas, meaning ‘model, teacher, and progenitor’ and ‘security, full power, and decree’, respectively. The purpose of authority is to provide a standard by which we learn and model our lives. It is meant to securely and with power guide those in submission to it into a life of goodness. Clearly authority can be abused in the most horrid of ways; nevertheless, this fact does not take away from its necessity for cultural stability. More specifically, authority in the home plays an even greater role. The role of the father, and consequently his authority, is meant to provide an image of the paternity of God to his children, thus forming the child’s notion of God and the submission due to Him by virtue of His authority. St. Paul assures us of this notion in Ephesians 3:14,15 where he writes, “For this reason I bow my knees before the Father, from whom all fatherhood in heaven and on earth is named.”
The natural consequence of such a lack of authoritative parenting, most clearly seen when they choose to be their children’s peers, is a corresponding lack of acknowledgement of God as Father. This rejection may not occur in an explicit fashion but can certainly be discerned in the wayward decisions of those who adopt such a way of life. How could a child conditioned to reject the notion of valid authority somehow innately accept the authority of God? Faith, therefore, is no longer part of the equation. Neither is the concept of self-sacrifice, for if mom and dad have taken on the role of ‘friend’, then at what point will little Suzy or little Johnny ever experience the need to submit their will and desire to the greater good? This sets up children for nothing but failure, but not only children as individuals, but also the society of which these children will ultimately take control. With no appreciation for self-sacrifice and an overdose of selfishness, lasting and fruitful marriages become near-impossibilities, and as we are experiencing now, the breakdown of marriage translates into breakdown of the culture.
Another symptom of such rampant selfishness is the unveiling of its high-maintenance handmaiden that goes by the name of Materialism, which should be no surprise to the thinking individual. The materialist philosophy is a natural outgrowth of such an environment in that the one who rejects sacrifice for the greater good accepts only consumption for the benefit of self.
At the risk of sounding sheltered or parochial, I would gladly forgo the pleasure of being my child’s friend if it meant saving them from adopting the status of a selfish, materialistic, and faithless divorcee.
Evidence of this as the prevailing mindset of many today (although none would ever admit to it), is most clearly seen in the elementary and high school classroom, a veritable observatory of the many and varied errors the modern family has imposed upon itself with the children serving as guinea pigs for the new human cultural experiment performed by mad scientists that go by the title of ‘mother’ and ‘father’. In this experiment, a dual hypothesis is being tested: children will be better suited to reach the goal with less guidance, and they will be able to assimilate into the real world in a productive way without being conditioned to recognize the need for authority and more importantly the need to accept another’s authority.
There is an insidious movement of parents away from the role of teacher, guide, and disciplinarian and towards the role of friend. This establishes a relationship in which authority has no place, and the results of this new experiment are quite telling. Since mothers and fathers have decided to be simply peers to their children, there has been a corresponding rise in apathy towards those things that create a stable society – education, self-discipline, faith, and marriage. This should raise a question in our minds: Why would lack of authority on the part of parents within the home lead to such a result?
To answer such a question, we must first understand the nature of authority and, therefore, its purpose. Etymologically, ‘authority’ comes from the Latin auctor and auctoritas, meaning ‘model, teacher, and progenitor’ and ‘security, full power, and decree’, respectively. The purpose of authority is to provide a standard by which we learn and model our lives. It is meant to securely and with power guide those in submission to it into a life of goodness. Clearly authority can be abused in the most horrid of ways; nevertheless, this fact does not take away from its necessity for cultural stability. More specifically, authority in the home plays an even greater role. The role of the father, and consequently his authority, is meant to provide an image of the paternity of God to his children, thus forming the child’s notion of God and the submission due to Him by virtue of His authority. St. Paul assures us of this notion in Ephesians 3:14,15 where he writes, “For this reason I bow my knees before the Father, from whom all fatherhood in heaven and on earth is named.”
The natural consequence of such a lack of authoritative parenting, most clearly seen when they choose to be their children’s peers, is a corresponding lack of acknowledgement of God as Father. This rejection may not occur in an explicit fashion but can certainly be discerned in the wayward decisions of those who adopt such a way of life. How could a child conditioned to reject the notion of valid authority somehow innately accept the authority of God? Faith, therefore, is no longer part of the equation. Neither is the concept of self-sacrifice, for if mom and dad have taken on the role of ‘friend’, then at what point will little Suzy or little Johnny ever experience the need to submit their will and desire to the greater good? This sets up children for nothing but failure, but not only children as individuals, but also the society of which these children will ultimately take control. With no appreciation for self-sacrifice and an overdose of selfishness, lasting and fruitful marriages become near-impossibilities, and as we are experiencing now, the breakdown of marriage translates into breakdown of the culture.
Another symptom of such rampant selfishness is the unveiling of its high-maintenance handmaiden that goes by the name of Materialism, which should be no surprise to the thinking individual. The materialist philosophy is a natural outgrowth of such an environment in that the one who rejects sacrifice for the greater good accepts only consumption for the benefit of self.
At the risk of sounding sheltered or parochial, I would gladly forgo the pleasure of being my child’s friend if it meant saving them from adopting the status of a selfish, materialistic, and faithless divorcee.
Friday, November 23, 2007
The Truth about Security and the Security of Truth
Materialism, being the raison d’etre of our present society, has proven itself to be the driving force for most decisions of most people. It is embraced as the means by which to obtain security, albeit an extremely false security as it is sought in that which is devoid of Truth.
Security is freedom from fear and anxiety as well as freedom from the danger of loss. This is clearly a good in the truest sense. There is not a sane person alive that would opt for insecurity over security. While understanding of how to obtain it may differ profoundly from one person to the next, the fact remains that security is good by its very nature.
To hold that security is good by its very nature should lead one to an even greater understanding of both God and security. Being that God is all-good, it is safe to conclude that the nature of security, which is good, receives its goodness from a certain source, that is, Security Itself – God. God also being Truth would necessitate that security can only be found in Truth. Through a simple method of substitution, we come to the realization that to be united to God is to be united to Security; in God, there is no fear or anxiety, nor is there any danger of loss. Conversely, to be separated from God (Truth) is to be united to fear, anxiety, and loss.
On a practical level, one can quickly ascertain the effects of materialism on the prevailing understanding of security and the means by which to obtain it. What is the reason that we buy and consume at an almost obsessive level? Why must our houses be turned into warehouses for our ever-expanding collection of stuff (for which we pay a great price) instead of a home for an ever-expanding family? The answer is that there is a certain perception of good in the possession of many things: increase in social status, establishment of a certain reputation, and an increase in options by which we amuse ourselves. Ultimately, these things are seen as synonymous with security – security for one’s reputation and social status, security in one’s ability to provide for oneself or family, even security to possess more through the inordinate amount of work done in order to make money that will be used to be even more stuff.
The fundamental problem in the above philosophy is that the very thing that it seeks is the very thing that it destroys. It seeks security as the ultimate goal, yet in amassing a great collection of goods, that person has only increased the potential for loss. For the materialist, the greatest good is the possession of many things; therefore, the greatest fear of the materialist must necessarily be the loss of those things. At this point, what should be clear is that materialism is marked by increased danger of loss and the accompanying fear and anxiety. This marks the complete loss of security. In fact, one of the most telling signs of the insufficiency of materialism at gaining security is the constant need to gain more….and more….and more. Why more if security has been achieved? Can we not rest on our laurels once we’ve arrived at the goal? Clearly, the Goal has not been achieved. It has been avoided completely, and it is this Goal that haunts those who insist on substituting Him with every form of idol known to man, from statues in the ancient world to sex and money today. Francis Thompson may be the poster-child for this essay. In The Hound of Heaven, he wrote:
“I hid from Him, and under running laughter.
Up vistaed hopes I sped;
And shot, precipitated,
Adown Titanic glooms of chasmed fears,
From those strong Feet that followed, followed after…
They beat – and a Voice beat
More instant than the Feet –
‘All things betray thee, who betrayest Me.’”
To seek security in anything but the very source is to deny ourselves of that which we seek. Indeed, the more we seek it elsewhere, the more bitterly it flees and betrays us. We have assurance of the converse in the words of the Word Itself: “Seek first the Kingdom of God and all these things shall be added unto to you.” In the non-canonical but creatively poetic words of Thompson, He reveals Himself to us as the goal that we desperately seek yet at the same time exclude in the words:
“Ah, fondest, blindest, weakest,
I am He Whom thou seekest!
Thou dravest love from thee, who dravest Me.”
Security is freedom from fear and anxiety as well as freedom from the danger of loss. This is clearly a good in the truest sense. There is not a sane person alive that would opt for insecurity over security. While understanding of how to obtain it may differ profoundly from one person to the next, the fact remains that security is good by its very nature.
To hold that security is good by its very nature should lead one to an even greater understanding of both God and security. Being that God is all-good, it is safe to conclude that the nature of security, which is good, receives its goodness from a certain source, that is, Security Itself – God. God also being Truth would necessitate that security can only be found in Truth. Through a simple method of substitution, we come to the realization that to be united to God is to be united to Security; in God, there is no fear or anxiety, nor is there any danger of loss. Conversely, to be separated from God (Truth) is to be united to fear, anxiety, and loss.
On a practical level, one can quickly ascertain the effects of materialism on the prevailing understanding of security and the means by which to obtain it. What is the reason that we buy and consume at an almost obsessive level? Why must our houses be turned into warehouses for our ever-expanding collection of stuff (for which we pay a great price) instead of a home for an ever-expanding family? The answer is that there is a certain perception of good in the possession of many things: increase in social status, establishment of a certain reputation, and an increase in options by which we amuse ourselves. Ultimately, these things are seen as synonymous with security – security for one’s reputation and social status, security in one’s ability to provide for oneself or family, even security to possess more through the inordinate amount of work done in order to make money that will be used to be even more stuff.
The fundamental problem in the above philosophy is that the very thing that it seeks is the very thing that it destroys. It seeks security as the ultimate goal, yet in amassing a great collection of goods, that person has only increased the potential for loss. For the materialist, the greatest good is the possession of many things; therefore, the greatest fear of the materialist must necessarily be the loss of those things. At this point, what should be clear is that materialism is marked by increased danger of loss and the accompanying fear and anxiety. This marks the complete loss of security. In fact, one of the most telling signs of the insufficiency of materialism at gaining security is the constant need to gain more….and more….and more. Why more if security has been achieved? Can we not rest on our laurels once we’ve arrived at the goal? Clearly, the Goal has not been achieved. It has been avoided completely, and it is this Goal that haunts those who insist on substituting Him with every form of idol known to man, from statues in the ancient world to sex and money today. Francis Thompson may be the poster-child for this essay. In The Hound of Heaven, he wrote:
“I hid from Him, and under running laughter.
Up vistaed hopes I sped;
And shot, precipitated,
Adown Titanic glooms of chasmed fears,
From those strong Feet that followed, followed after…
They beat – and a Voice beat
More instant than the Feet –
‘All things betray thee, who betrayest Me.’”
To seek security in anything but the very source is to deny ourselves of that which we seek. Indeed, the more we seek it elsewhere, the more bitterly it flees and betrays us. We have assurance of the converse in the words of the Word Itself: “Seek first the Kingdom of God and all these things shall be added unto to you.” In the non-canonical but creatively poetic words of Thompson, He reveals Himself to us as the goal that we desperately seek yet at the same time exclude in the words:
“Ah, fondest, blindest, weakest,
I am He Whom thou seekest!
Thou dravest love from thee, who dravest Me.”
Monday, October 22, 2007
"Christian" Materialism and Ambition Prevalent in Education
About six months back I attended a cultural performance at the Jetson Center for Youth, a Juvenile correctional facility outside of Baton Rouge,Louisiana. From my previous visits and observations this facility does a good job at both disciplining and motivating the youth, and I could notice a positive difference in the individual I was visiting. Furthermore, this cultural program gave the teenagers an opportunity to perform for friends and family showcasing their art, music, speech, and acting. Yet in the midst of this experience I noticed something very alarming.
At the beginning of the cultural program a local and young preacher opened the ceremony in prayer followed by a sermon. The sermon entailed his own struggles to transcend a troubled youth and rely on the grace of God. The gripping ‘truth’ that he emphasized was that he once struggled with drugs, behavior, and academically, but once he was saved all of these problems ceased and allowed him to achieve success. He proposed that because he was saved he begin receiving all A’s in college. He showed the audience his expensive watch, jewelry, and clothes. He proposed that once you are saved you no longer have problems and all your worldly concerns will magically vanish. His message embodied a gospel of success mentality that would be reinforced several times throughout this cultural program.
The drama program greatly disappointed me. Within the spirit of fundamental evangelical Protestantism the theatrical production was nothing more than “are you saved?” skits. Literally, in multiple skits, one boy or group of boys was approached by another group of boys and questioned about their assurance of eternal salvation in which the unsaved group climatically converted in the end asking Jesus to be their personal Lord and Savior.
The public speaking segment was the most revealing. The sequence involved several young men giving speeches about their aspirations and dreams. I heard ambitions about being famous athletes, boxers, music entertainers, successful businessmen, etc. The most consistent aspiration was to become financially wealthy and able to afford many material luxuries. What I did not hear were desires to be a good man, a contributing member of society, a good husband, and/or father. All the dreams appeared selfish and materially oriented.
These teenagers have developed a very material notion of success supported by a certain understanding of Christianity. The dichotomy becomes painfully obvious. The Christian life has been reduced to the acquisition of earthy pleasure and divorced from the acceptance of suffering that does not aid in this material quest. In other words, although personal suffering occurs, it will always translate into some material award. Even then I could not help thinking that I would rather this state funded facility emphasis Greek virtue over this flawed Christian perspective. At least then they would have a context to properly reevaluate certain values and live a more authentic notion of the Christian life.
The discipline and motivation provided did seem to inspire these teenagers to action but to what end. They learn the means to become selfish and achieve worldly wealth and success? What happens when the circumstances are stacked against them? Do they have the tools to cope and seek happiness with whatever circumstances of life they are given? The problem is the ideal of happiness being communicated is very material. The American dream has become a material dream. They are automatically placed in competition with their peers and community in which they live. To be successful they must elevate themselves above others. But not everyone can get to the top within this paradigm. Even those who get to the top will be greatly disappointed when they find that material goods only lead to the illusion of happiness. I can’t help believing that this type of education is setting young people up for personal failure in their spiritual lives and depression. They are told that certain things will make them happy but what happens when the illusion fades and they are still stuck with their unhappy selves.
The personal good of an individual is tied within the good of the community. Where is the context of virtue ethics that will associate personal achievement with communal living? Happiness is not in material possessions but in relationships- to God and neighbor. I believe it is too often thought, “I must become rich in order to be a philanthropist and help my neighbor”. On the contrary the opposite is true. “I must help and love my neighbor in order to truly become rich.”
At the beginning of the cultural program a local and young preacher opened the ceremony in prayer followed by a sermon. The sermon entailed his own struggles to transcend a troubled youth and rely on the grace of God. The gripping ‘truth’ that he emphasized was that he once struggled with drugs, behavior, and academically, but once he was saved all of these problems ceased and allowed him to achieve success. He proposed that because he was saved he begin receiving all A’s in college. He showed the audience his expensive watch, jewelry, and clothes. He proposed that once you are saved you no longer have problems and all your worldly concerns will magically vanish. His message embodied a gospel of success mentality that would be reinforced several times throughout this cultural program.
The drama program greatly disappointed me. Within the spirit of fundamental evangelical Protestantism the theatrical production was nothing more than “are you saved?” skits. Literally, in multiple skits, one boy or group of boys was approached by another group of boys and questioned about their assurance of eternal salvation in which the unsaved group climatically converted in the end asking Jesus to be their personal Lord and Savior.
The public speaking segment was the most revealing. The sequence involved several young men giving speeches about their aspirations and dreams. I heard ambitions about being famous athletes, boxers, music entertainers, successful businessmen, etc. The most consistent aspiration was to become financially wealthy and able to afford many material luxuries. What I did not hear were desires to be a good man, a contributing member of society, a good husband, and/or father. All the dreams appeared selfish and materially oriented.
These teenagers have developed a very material notion of success supported by a certain understanding of Christianity. The dichotomy becomes painfully obvious. The Christian life has been reduced to the acquisition of earthy pleasure and divorced from the acceptance of suffering that does not aid in this material quest. In other words, although personal suffering occurs, it will always translate into some material award. Even then I could not help thinking that I would rather this state funded facility emphasis Greek virtue over this flawed Christian perspective. At least then they would have a context to properly reevaluate certain values and live a more authentic notion of the Christian life.
The discipline and motivation provided did seem to inspire these teenagers to action but to what end. They learn the means to become selfish and achieve worldly wealth and success? What happens when the circumstances are stacked against them? Do they have the tools to cope and seek happiness with whatever circumstances of life they are given? The problem is the ideal of happiness being communicated is very material. The American dream has become a material dream. They are automatically placed in competition with their peers and community in which they live. To be successful they must elevate themselves above others. But not everyone can get to the top within this paradigm. Even those who get to the top will be greatly disappointed when they find that material goods only lead to the illusion of happiness. I can’t help believing that this type of education is setting young people up for personal failure in their spiritual lives and depression. They are told that certain things will make them happy but what happens when the illusion fades and they are still stuck with their unhappy selves.
The personal good of an individual is tied within the good of the community. Where is the context of virtue ethics that will associate personal achievement with communal living? Happiness is not in material possessions but in relationships- to God and neighbor. I believe it is too often thought, “I must become rich in order to be a philanthropist and help my neighbor”. On the contrary the opposite is true. “I must help and love my neighbor in order to truly become rich.”
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