Uncle Petros and Goldbach's Conjecture: A Novel of Mathematical Obsession
by
by
Trust the Young
Number theory has nothing to do with the real world, unless you happen to be a number theorist. Then it is the real world; everything else is illusory. Number theory has no application to anything except... well, numbers. It eschews other branches of mathematics as pedestrian. Physics, engineering, and even geometry, although they use numbers, are simply diversions for the less talented, that is to say inferior, intellect. Mere calculation is trivial even if it is arduous and complex. Number theory’s only concern is with the relationship of numbers to each other. It seeks the hidden, often mysterious, connections that exist, and have always existed, among these most abstract of all ideas. Since numbers are eternal, infinite, and everywhere, they are easily taken for divine (For more on the significance and literary import of number theory, see: https://www.goodreads.com/review/show...).
And who knows? Numbers may well be divine. Among other reasons because the fundamental logic of numbers is as elusive as the theology of God. Apparently, just like any other believers, mathematicians make an act of faith every time they form an hypothesis, attempt a proof, or demonstrate a theorem. The reason that faith is necessary is that when the foundations of mathematics are probed far enough, they are shown to be, not built on sand, but entirely absent. All of mathematics, including number theory, floats above an intellectual void known as Godel’s Incompleteness Theorem. Kurt Godel showed in the 1930’s that no logical system, of which basic arithmetic is one, can contain the axioms necessary to ensure its own reliability. The consequences of Godel’s discovery are even more profound for mathematics than the discovery of Quantum Uncertainty is for physics.
So Incompleteness raises some important issues which Doxiadis uses as the substrate of his story. If mathematics is suspect, what hope is there for any other science? Or for human thought in general? The Incompleteness Theorem seems to suggest that everything is relative, that intellectual discipline is a fraud, that science is some kind of con game which we pretend to take seriously. And yet, mathematicians, scientists, engineers, and the check-out assistants in the supermarket continue to work away with numbers as if nothing were amiss. Is their apparent faith different from that of Christians, Jews, Muslims, and Mormons who also create doctrines, liturgies, and ethics over an abyss empty of ultimate reason and without sufficient reasons they can articulate?
Uncle Petros introduces these issues implicitly in his concern with the so-called Goldbach Conjecture, an hypothesis first formulated by an 18th century mathematician: “Every number greater than 2 can be written as a sum of two primes.” When he started his career, Uncle Petros was unaware of the Conjecture’s relation to the Incompleteness Theorem since the latter had yet to be formulated. But he had a certain faith, not just in his own mathematical ability but, more importantly, in the power of mathematics to ultimately prove or disprove what was Goldbach’s intuitive guess. One could accurately call this ‘blind’ faith but not as a reproach and not in the manner of religious faith, which when it is blind is often dangerous.
The first difference between religious and mathematical faith, of course, is that mathematical hypotheses such as that of Goldbach are based on empirical observation; mathematics is very much an empirical discipline. Every specific number greater than two which has been analysed can be shown to be the sum of two primes. But such empirical proof is inadequate for mathematics no matter how often it is observed. There could be some number among the infinity of numbers for which this general rule doesn’t apply. Hence the importance of the Conjecture if it can be proven abstractly as an invariable condition for any number at all and not just specific numbers.
This is where the mathematician’s second difference with religious believers comes into play. Mathematical logic works backwards from an hypothesis to discover the logic by which the hypothesis can be derived from communally accepted axioms. This logic can be both positive - if X, then Y etc. - or it can be negative - if not X, then Y or not Y etc. This latter form is that of a logical reductio ad absurdam, a contradiction which both affirms and denies a conclusion. Almost everything about religious faith is subject to a reductio ad absurdam. For example the proposition to reject the religious claim that ‘God created everything’ is compatible with both the axioms ‘the world is inherently good’ and ‘there is tremendous evil in the world.’ Creation is both good and not good with or without a divine origin. Mathematicians would view such a theological conclusion therefore as meaningless.*
The final difference in matters of faith concerns the fundamental axioms and their status in mathematics and religion. The difference here is somewhat surprising. Theology relies on fixed axioms from which it the derives its conclusions. These axioms are given the status of revealed truth and are the dogmatic focus of religious faith. Mathematicians since Godel on the other hand know that the fundamental axioms of their science are somewhat arbitrary. Euclid’s axioms of three dimensional space, for example, can be replaced by the n-dimensional space of Riemannian geometry with no loss of logical rigor. Euclid is not proven wrong but merely shown to be a special case of the more general conditions of Riemann.
This is an illustration as well of a crucial difference in method. Mathematics seeks to continuously extend the generality of its conclusions by discovering more and more inclusive axioms from which to work. Religious faith seeks to fix the axioms in order to preserve, limit and restrict doctrinal conclusions. Put another way: mathematics looks into the abyss of the Incompleteness Theorem and considers it an horizon to be striven toward painfully, incrementally, and with ho chance of complete success. This takes intellectual and, dare one say it, moral courage. The theologian looks into a similar abyss and considers that the horizon has arrived at his location. There is nothing to explore, nothing to find beyond the axioms which have been set by tradition. This sort of faith isn’t one of courage but stubbornness. It ramifies ‘conclusive’ conclusions because it takes its axioms for granted.
So the faith of the mathematician is not the faith of the religious believer. The faith required to attack the apparent intractability if the Goldbach Conjecture has nothing to do with the faith that proclaims One God or a religious ethic of forgiveness, or retribution, or community. The mathematician must have perseverance; the theologian merely persistence. The mathematician is concerned about expanding the world we know, the theologian with limiting what we know of the world. One requires genuine creative faith; the other futile insistence. Doxiadis quotes the 19th century mathematician, David Hilbert: “We must know, we shall know! In mathematics there is no ignorabimus!" Compare that with the commitment of religious faith by the first Christian theologian, Paul of Tarsus: “But if there be no resurrection of the dead, then is Christ not risen: and if Christ be not risen, then is our preaching vain, and your faith is also vain.”**
Both mathematicians and theologians tend to be obsessives. Both types are probably born not made. But mathematicians reach their intellectual peak in their relative youth. The best theologians are usually old men. I suspect the reason for this has to do with the paradoxical character of experiential wisdom. Young mathematicians have an intuitive grasp of reality unsullied by too much knowledge of what others call the world; while old theologians have been pickled in cultural history and traditions of thought that distort much of reality.
Uncle Petros himself tends to slide into quasi-theological reverie about mathematics in his dotage after suffering the slings and arrows of family as a failed mathematical prodigy, and especially after the publication of Godel’s Incompleteness Theorem. In a sense he has lost his faith in mathematics and replaced it with an entirely different kind of faithful resignation to cultural inevitability. This is why Petros loves chess: as in theology, its axioms are fixed, its rules invariably traditional.
It takes his nephew to help Uncle Petros understand that Godel freed mathematics from theological pretensions. There is no firm foundation for mathematics. Even David Hilbert thought there was but he was wrong; and so was Uncle Petros if he thought that firm intellectual foundations were necessary for his continuing commitment to mathematics. As Doxiadis has Alan Turing say, “Truth is not not always provable.” But this doesn’t invalidate mathematics or its method. Unlike theology, the commitment to mathematics is to an entirely unknown, and unnamed, future state of knowing. The content of that state is not, and cannot be, specified much less proven. Doxiadis compares it to being in love.
This is an important message from a young mathematician to an old one about a very different sort of tradition and a very different sort of faith, one that does away with “attainable goals.” And it is a message that makes Doxiadis’s little book a lot bigger than its size.
———————————————————————————————————-
*Neil Gaiman, in his American Gods captures the character of the reductio in theology rather well: “I believe in a personal god who cares about me and worries and oversees everything I do. I believe in an impersonal god who set the universe in motion and went off to hang with her girlfriends and doesn't even know that I'm alive. I believe in an empty and godless universe of causal chaos, background noise, and sheer blind luck.”
** For those who have forgotten their school Latin ‘ignorabimus’ is future tense: “we will be ignorant.” St. Paul, significantly, is concerned solely about the present but only in terms of the past in his exhortation.
Number theory has nothing to do with the real world, unless you happen to be a number theorist. Then it is the real world; everything else is illusory. Number theory has no application to anything except... well, numbers. It eschews other branches of mathematics as pedestrian. Physics, engineering, and even geometry, although they use numbers, are simply diversions for the less talented, that is to say inferior, intellect. Mere calculation is trivial even if it is arduous and complex. Number theory’s only concern is with the relationship of numbers to each other. It seeks the hidden, often mysterious, connections that exist, and have always existed, among these most abstract of all ideas. Since numbers are eternal, infinite, and everywhere, they are easily taken for divine (For more on the significance and literary import of number theory, see: https://www.goodreads.com/review/show...).
And who knows? Numbers may well be divine. Among other reasons because the fundamental logic of numbers is as elusive as the theology of God. Apparently, just like any other believers, mathematicians make an act of faith every time they form an hypothesis, attempt a proof, or demonstrate a theorem. The reason that faith is necessary is that when the foundations of mathematics are probed far enough, they are shown to be, not built on sand, but entirely absent. All of mathematics, including number theory, floats above an intellectual void known as Godel’s Incompleteness Theorem. Kurt Godel showed in the 1930’s that no logical system, of which basic arithmetic is one, can contain the axioms necessary to ensure its own reliability. The consequences of Godel’s discovery are even more profound for mathematics than the discovery of Quantum Uncertainty is for physics.
So Incompleteness raises some important issues which Doxiadis uses as the substrate of his story. If mathematics is suspect, what hope is there for any other science? Or for human thought in general? The Incompleteness Theorem seems to suggest that everything is relative, that intellectual discipline is a fraud, that science is some kind of con game which we pretend to take seriously. And yet, mathematicians, scientists, engineers, and the check-out assistants in the supermarket continue to work away with numbers as if nothing were amiss. Is their apparent faith different from that of Christians, Jews, Muslims, and Mormons who also create doctrines, liturgies, and ethics over an abyss empty of ultimate reason and without sufficient reasons they can articulate?
Uncle Petros introduces these issues implicitly in his concern with the so-called Goldbach Conjecture, an hypothesis first formulated by an 18th century mathematician: “Every number greater than 2 can be written as a sum of two primes.” When he started his career, Uncle Petros was unaware of the Conjecture’s relation to the Incompleteness Theorem since the latter had yet to be formulated. But he had a certain faith, not just in his own mathematical ability but, more importantly, in the power of mathematics to ultimately prove or disprove what was Goldbach’s intuitive guess. One could accurately call this ‘blind’ faith but not as a reproach and not in the manner of religious faith, which when it is blind is often dangerous.
The first difference between religious and mathematical faith, of course, is that mathematical hypotheses such as that of Goldbach are based on empirical observation; mathematics is very much an empirical discipline. Every specific number greater than two which has been analysed can be shown to be the sum of two primes. But such empirical proof is inadequate for mathematics no matter how often it is observed. There could be some number among the infinity of numbers for which this general rule doesn’t apply. Hence the importance of the Conjecture if it can be proven abstractly as an invariable condition for any number at all and not just specific numbers.
This is where the mathematician’s second difference with religious believers comes into play. Mathematical logic works backwards from an hypothesis to discover the logic by which the hypothesis can be derived from communally accepted axioms. This logic can be both positive - if X, then Y etc. - or it can be negative - if not X, then Y or not Y etc. This latter form is that of a logical reductio ad absurdam, a contradiction which both affirms and denies a conclusion. Almost everything about religious faith is subject to a reductio ad absurdam. For example the proposition to reject the religious claim that ‘God created everything’ is compatible with both the axioms ‘the world is inherently good’ and ‘there is tremendous evil in the world.’ Creation is both good and not good with or without a divine origin. Mathematicians would view such a theological conclusion therefore as meaningless.*
The final difference in matters of faith concerns the fundamental axioms and their status in mathematics and religion. The difference here is somewhat surprising. Theology relies on fixed axioms from which it the derives its conclusions. These axioms are given the status of revealed truth and are the dogmatic focus of religious faith. Mathematicians since Godel on the other hand know that the fundamental axioms of their science are somewhat arbitrary. Euclid’s axioms of three dimensional space, for example, can be replaced by the n-dimensional space of Riemannian geometry with no loss of logical rigor. Euclid is not proven wrong but merely shown to be a special case of the more general conditions of Riemann.
This is an illustration as well of a crucial difference in method. Mathematics seeks to continuously extend the generality of its conclusions by discovering more and more inclusive axioms from which to work. Religious faith seeks to fix the axioms in order to preserve, limit and restrict doctrinal conclusions. Put another way: mathematics looks into the abyss of the Incompleteness Theorem and considers it an horizon to be striven toward painfully, incrementally, and with ho chance of complete success. This takes intellectual and, dare one say it, moral courage. The theologian looks into a similar abyss and considers that the horizon has arrived at his location. There is nothing to explore, nothing to find beyond the axioms which have been set by tradition. This sort of faith isn’t one of courage but stubbornness. It ramifies ‘conclusive’ conclusions because it takes its axioms for granted.
So the faith of the mathematician is not the faith of the religious believer. The faith required to attack the apparent intractability if the Goldbach Conjecture has nothing to do with the faith that proclaims One God or a religious ethic of forgiveness, or retribution, or community. The mathematician must have perseverance; the theologian merely persistence. The mathematician is concerned about expanding the world we know, the theologian with limiting what we know of the world. One requires genuine creative faith; the other futile insistence. Doxiadis quotes the 19th century mathematician, David Hilbert: “We must know, we shall know! In mathematics there is no ignorabimus!" Compare that with the commitment of religious faith by the first Christian theologian, Paul of Tarsus: “But if there be no resurrection of the dead, then is Christ not risen: and if Christ be not risen, then is our preaching vain, and your faith is also vain.”**
Both mathematicians and theologians tend to be obsessives. Both types are probably born not made. But mathematicians reach their intellectual peak in their relative youth. The best theologians are usually old men. I suspect the reason for this has to do with the paradoxical character of experiential wisdom. Young mathematicians have an intuitive grasp of reality unsullied by too much knowledge of what others call the world; while old theologians have been pickled in cultural history and traditions of thought that distort much of reality.
Uncle Petros himself tends to slide into quasi-theological reverie about mathematics in his dotage after suffering the slings and arrows of family as a failed mathematical prodigy, and especially after the publication of Godel’s Incompleteness Theorem. In a sense he has lost his faith in mathematics and replaced it with an entirely different kind of faithful resignation to cultural inevitability. This is why Petros loves chess: as in theology, its axioms are fixed, its rules invariably traditional.
It takes his nephew to help Uncle Petros understand that Godel freed mathematics from theological pretensions. There is no firm foundation for mathematics. Even David Hilbert thought there was but he was wrong; and so was Uncle Petros if he thought that firm intellectual foundations were necessary for his continuing commitment to mathematics. As Doxiadis has Alan Turing say, “Truth is not not always provable.” But this doesn’t invalidate mathematics or its method. Unlike theology, the commitment to mathematics is to an entirely unknown, and unnamed, future state of knowing. The content of that state is not, and cannot be, specified much less proven. Doxiadis compares it to being in love.
This is an important message from a young mathematician to an old one about a very different sort of tradition and a very different sort of faith, one that does away with “attainable goals.” And it is a message that makes Doxiadis’s little book a lot bigger than its size.
———————————————————————————————————-
*Neil Gaiman, in his American Gods captures the character of the reductio in theology rather well: “I believe in a personal god who cares about me and worries and oversees everything I do. I believe in an impersonal god who set the universe in motion and went off to hang with her girlfriends and doesn't even know that I'm alive. I believe in an empty and godless universe of causal chaos, background noise, and sheer blind luck.”
** For those who have forgotten their school Latin ‘ignorabimus’ is future tense: “we will be ignorant.” St. Paul, significantly, is concerned solely about the present but only in terms of the past in his exhortation.
posted by The Mind of BlackOxford @ June 11, 2018
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