https://medium.com/humanist-voices/if-youth-knew-if-age-could-15-all-things-bright-and-wonderful-and-unknown-what-do-we-know-34476c70a775
Scott Douglas Jacobsen: Of all possible mathematical knowledge, what do we really know? You were a distinguished professor in the past. We have written a text on this.
Herb Silverman: Here is what we know about mathematics. Mathematicians start with axioms (assumptions) and see what conclusion may logically be deduced (proved) from these axioms. The nineteenth century mathematician Leopold Kronecker once said, “God created the integers, all else is the work of man.” I interpret this statement to be more about the axiomatic approach than about theology. Mathematicians often begin with axioms that seem “self-evident,” because they are more likely to lead to real-world truths, including scientific discoveries and accurate predictions of physical phenomena. But if at least one axiom is false, then the conclusion may not be scientifically applicable. Unlike with applied mathematicians, theoretical mathematicians are not so concerned with whether their axioms are true. Axioms in some branches are contradictory to axioms in others. In non-Euclidean geometry, we replace Euclid’s parallel axiom with a different axiom. The axioms in Euclidean geometry have led to discoveries on planet Earth; results from the axioms in non-Euclidean geometry were applied many years later by Einstein for his general theory of relativity, when he showed we live in a non-Euclidean four-dimensional universe, consisting of three-dimensional space and one-dimensional time.
There is a lot we don’t know, and never will know. Just about any problem solved in mathematics seems to raise additional questions that we would like to solve. So I expect there are infinitely many questions that we would like answers to, which won’t be found in a finite amount of time. There might even be infinitely many possible theories, not all of which humans can ponder. With or without machines, even now the majority of scientific discoveries are barely comprehensible (or incomprehensible) to most human beings.
Speaking of infinity, which is a theoretical construct created by humans, the number “infinity” does not exist in reality (as a real number). My math students sometimes falsely treated infinity as a real number, and such misuse often got them into trouble.
The concept of infinity is useful to help solve many math problems involving limits in calculus. For instance, we know there are infinitely many positive integers because the integer n+1 is larger than n for any integer n. What happens to the sequence {1/n}, n = 1, 2, 3, …? The sequence gets arbitrarily close to 0, and we say that the limit of the sequence is 0.
Here’s a limit example for an infinite series: 1/2 + 1/4 + 1/8 + 1/16 + … = 1.
Also, we can’t draw a “perfect” circle, we can just imagine one. Imagine a polygon with an ever-increasing number of equal sides. As the number of sides approaches infinity, the polygon will become a circle as the limit of an infinite number of infinitesimally small sides. No matter how accurate a computer’s rendering of a circle might be, it will only be an imperfect approximation.
Mathematics has played a major role in bringing about innovations. Many mathematical theories and models of real-world problems have helped scientists and engineers grapple with seemingly impossible tasks. The eighteenth century mathematician Gauss said, “Mathematics is the queen of sciences.” He said this because mathematics is essential in the study of all scientific fields. Galileo referred to mathematics as the language in which the natural physical world is written. When scientific statements are translated into mathematical statements, including about the structure of the universe, we apply mathematics to solve scientific problems.
Jacobsen: How much do we not know? Even with this, what can we say for certain about particular categories of things, as simply falsehoods?
Silverman: We often get into trouble when we apply mathematical concepts to God. Most religious people believe in an infinite God with infinite power who has lived for an infinite time. Just as finite humans created infinity, so finite humans created God and gave him infinite attributes. God had to be presumed infinite, because a finite god would be limited. However, we can show mathematically that there can’t be a largest infinity. The German mathematician Georg Cantor showed that every subset of an infinite set has a higher cardinality (more elements). In other words, there are infinitely many infinities. So, any infinite god could theoretically be replaced by a more powerful infinite god.
Infinity, like gods, is not sensible (known through the senses). Just as infinity does not exist in reality, it does help solve some math problems. Lots of humans believe in a (nonexistent) god who helps them solve human problems.
Mathematicians, unlike most theologians, recognize that their axioms are just made up. So, a perfectly valid and logical proof may have nothing to do with reality if the axioms are not true. Most ancient religions are also loosely based on axioms. Their most common axiom is “God exists,” which is not as self-evident as it appeared to be in a pre-scientific world. A “God axiom” might give comfort to some, but it lacks predictive value.
Besides their practical uses, numbers have cultural significance throughout the world. For example, in Western society, the number 13 is regarded as unlucky. Some people also believe in numerology, which attributes a divine or mystical significance to numbers. One such example, espoused by many Christian fundamentalists, is fear of the number 666, which they refer to as the Mark of the Beast. Numerology is also associated with the paranormal and astrology. Of course, numerology is a pseudoscience, a superstition that uses numbers to give their subject a veneer of scientific authority.
Jacobsen: Where does this bring humility into the equation?
Silverman: Kurt Gödel, a mathematician/logician, made a rather disturbing groundbreaking discovery in mathematics. Gödel showed that with just about any set of axioms there must be at least one true but unproveable statement. In other words, not all true statements in mathematics have formal proofs. Furthermore, we have no way of knowing in advance whether a statement is really hard to prove (or disprove), or whether it is impossible. For instance, mathematician Andrew Wiles proved Fermat’s Last Theorem 358 years after Fermat proposed it in 1637. The proof was difficult but provable. We don’t know if questions about the beginning of our universe and multiverses are really hard to answer completely or are logically unanswerable. Or maybe the human mind is not bright enough to figure it out.
Gödel's incompleteness theorem suggests to many that a Theory of Everything (an all-encompassing, coherent, theoretical framework of physics that fully explains and links together all physical aspects of the universe) is unattainable. In fact, Gödel's theorem seems to imply that theoretical mathematics is inexhaustible. No matter how many problems we solve, there will always be other problems that can’t be solved within the existing rules.
Jacobsen: Thank you for the opportunity and your time, Dr. Silverman.
Speaking of infinity, which is a theoretical construct created by humans, the number “infinity” does not exist in reality (as a real number). My math students sometimes falsely treated infinity as a real number, and such misuse often got them into trouble.
The concept of infinity is useful to help solve many math problems involving limits in calculus. For instance, we know there are infinitely many positive integers because the integer n+1 is larger than n for any integer n. What happens to the sequence {1/n}, n = 1, 2, 3, …? The sequence gets arbitrarily close to 0, and we say that the limit of the sequence is 0.
Here’s a limit example for an infinite series: 1/2 + 1/4 + 1/8 + 1/16 + … = 1.
Also, we can’t draw a “perfect” circle, we can just imagine one. Imagine a polygon with an ever-increasing number of equal sides. As the number of sides approaches infinity, the polygon will become a circle as the limit of an infinite number of infinitesimally small sides. No matter how accurate a computer’s rendering of a circle might be, it will only be an imperfect approximation.
Mathematics has played a major role in bringing about innovations. Many mathematical theories and models of real-world problems have helped scientists and engineers grapple with seemingly impossible tasks. The eighteenth century mathematician Gauss said, “Mathematics is the queen of sciences.” He said this because mathematics is essential in the study of all scientific fields. Galileo referred to mathematics as the language in which the natural physical world is written. When scientific statements are translated into mathematical statements, including about the structure of the universe, we apply mathematics to solve scientific problems.
Jacobsen: How much do we not know? Even with this, what can we say for certain about particular categories of things, as simply falsehoods?
Silverman: We often get into trouble when we apply mathematical concepts to God. Most religious people believe in an infinite God with infinite power who has lived for an infinite time. Just as finite humans created infinity, so finite humans created God and gave him infinite attributes. God had to be presumed infinite, because a finite god would be limited. However, we can show mathematically that there can’t be a largest infinity. The German mathematician Georg Cantor showed that every subset of an infinite set has a higher cardinality (more elements). In other words, there are infinitely many infinities. So, any infinite god could theoretically be replaced by a more powerful infinite god.
Infinity, like gods, is not sensible (known through the senses). Just as infinity does not exist in reality, it does help solve some math problems. Lots of humans believe in a (nonexistent) god who helps them solve human problems.
Mathematicians, unlike most theologians, recognize that their axioms are just made up. So, a perfectly valid and logical proof may have nothing to do with reality if the axioms are not true. Most ancient religions are also loosely based on axioms. Their most common axiom is “God exists,” which is not as self-evident as it appeared to be in a pre-scientific world. A “God axiom” might give comfort to some, but it lacks predictive value.
Besides their practical uses, numbers have cultural significance throughout the world. For example, in Western society, the number 13 is regarded as unlucky. Some people also believe in numerology, which attributes a divine or mystical significance to numbers. One such example, espoused by many Christian fundamentalists, is fear of the number 666, which they refer to as the Mark of the Beast. Numerology is also associated with the paranormal and astrology. Of course, numerology is a pseudoscience, a superstition that uses numbers to give their subject a veneer of scientific authority.
Jacobsen: Where does this bring humility into the equation?
Silverman: Kurt Gödel, a mathematician/logician, made a rather disturbing groundbreaking discovery in mathematics. Gödel showed that with just about any set of axioms there must be at least one true but unproveable statement. In other words, not all true statements in mathematics have formal proofs. Furthermore, we have no way of knowing in advance whether a statement is really hard to prove (or disprove), or whether it is impossible. For instance, mathematician Andrew Wiles proved Fermat’s Last Theorem 358 years after Fermat proposed it in 1637. The proof was difficult but provable. We don’t know if questions about the beginning of our universe and multiverses are really hard to answer completely or are logically unanswerable. Or maybe the human mind is not bright enough to figure it out.
Gödel's incompleteness theorem suggests to many that a Theory of Everything (an all-encompassing, coherent, theoretical framework of physics that fully explains and links together all physical aspects of the universe) is unattainable. In fact, Gödel's theorem seems to imply that theoretical mathematics is inexhaustible. No matter how many problems we solve, there will always be other problems that can’t be solved within the existing rules.
Jacobsen: Thank you for the opportunity and your time, Dr. Silverman.
