Ask Dr. Silverman 1 — The Philosophy of Mathematics: Its Faces and Facets
Herb Silverman is the Founder of the Secular Coalition of America, the Founder of the Secular Humanists of the Lowcountry, and the Founder of the Atheist/Humanist Alliance student group at the College of Charleston. Here we talk about the philosophy of mathematics, science, and theology in a new series on the philosophy of mathematics.
Scott Douglas Jacobsen: Mathematics works within the constraints of structure or order, relationships between and within structure and order, and the changes in structure or order and the relationships between structure and order.
Philosophy of mathematics deals with the meanings of mathematics, whether its interpretations or its assumptions and derivations. It asks fundamental questions relatable to the structure of the universe, as these question the basic operations behind science in many ways.
What are some basic principles of mathematics? How does this relate to the philosophy of mathematics and, more generally, the philosophy of science? Where does mathematics reach a limit and philosophy of mathematics some extra legs, in some fundamental and important ways?
Dr. Herb Silverman: I’ll begin by describing some differences between mathematics and the sciences, mathematics and philosophy, and how they approach fundamental questions like the structure of the universe. I’ll then bring in theological approaches to the same fundamental questions.
Mathematicians begin with assumptions (axioms) and try to discover what may logically be deduced from the axioms. Theoretical mathematicians are not concerned with whether the axioms are true. Axioms in some branches are contradictory to axioms in others. 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.
The 19th-century mathematician Gauss referred to mathematics as the “queen of the sciences,” perhaps because mathematics is essential in the study of all scientific fields. And physicist Eugene Wigner wrote “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Unlike mathematicians who are free to begin with any consistent set of axioms, scientists always begin with axioms (hypotheses) they believe to be true. Using the scientific method (collecting data and testing empirically), they hope to find sufficient evidence for their hypotheses to be elevated to theories (like gravity, natural selection, big bang, general relativity). When scientific statements are translated into mathematical statements, including about the structure of the universe, we apply mathematics to solve scientific problems. Perhaps that’s why Galileo referred to mathematics as the language in which the natural physical world is written.
Philosophy of mathematics looks into questions about mathematical theories and practices, which may include the nature or reality of numbers, the nature of different mathematical disciplines, limits of formal systems, and why mathematics coming from human minds can have such a relationship with reality. Philosophers, like scientists and unlike many mathematicians, care about whether their axioms are true and what implications they have in the real world. Philosophers, like most mathematicians and unlike scientists, stay mostly in their mind and don’t draw conclusions based on applying the scientific method. College courses in logic are taught in either philosophy or mathematics departments.
Kurt Gödel, a mathematician/logician, made a ground-breaking discovery in mathematics that also has implications to both science and philosophy. And it’s a rather disturbing discovery.
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 it was proposed by Fermat in 1637. The proof was difficult but provable. We don’t know if questions about the beginning of our universe and multiverses is really hard to answer completely or is logically unanswerable. Or maybe the human mind is not bright enough to figure it out. I’d say my cat is incapable of learning integral calculus, just as humans might be incapable of answering some deep questions about the universe. And then there’s artificial intelligence.
To give more complete answers about math, science, and philosophy, I’d have to be an expert in all those fields, which I am not. I’m a research mathematician specializing in Complex Variables, but I’m not an expert. I’m not quite an expert in the subfield of Geometric Function Theory, but I might be considered an expert in a much smaller subfield of GMF in which I proved the first theorems. But very few mathematicians work in that area, which has no known applications to other branches of mathematics or usefulness other than to help me get tenure.
Deciding on who are “experts” in a field is not clear cut, but I think the number of experts on any topic is inversely proportional to the evidence available on that topic. And by that criterion, we are all experts on God because there is absolutely no evidence for her/his existence. Anyone can make up stuff about God or quote stuff from books made up by others. In fact, acknowledging my ignorance qualifies me as a top God expert. To paraphrase Socrates: “He who believes he knows something when he knows nothing is less wise than he who knows he knows nothing.”
This brings me to debates I’ve had with some religious people about fundamental questions. They differ considerably from discussions I’ve had with scientists and philosophers. When these theists were given contradictory or unanswerable questions that didn’t match reality, a response was often the unfalsifiable, “God works in mysterious ways.” Assertions about holy book predictions coming true are usually post-dictions (written after the event) or interpretations that they try to make say what they don’t say. While all of us are susceptible to confirmation bias, I think that’s particularly true in religion. One theologian claimed that the Bible had it right in ways that prominent scientists who believed in an eternal universe had it wrong. Genesis opens with “In the beginning,” which was alleged to be scientific evidence that the Bible described the big bang. I pointed out that Genesis goes on to say that God then created two lights, the greater to rule the day, and the lesser the night. Almost as an afterthought, God then made stars (which biblical writers did not know were other suns, many larger than our sun).
To describe a significant difference between mathematicians and theologians, I’ll close with a popular cartoon on the door of many mathematicians. It shows one mathematician explaining his complicated multi-step proof. Another mathematician interrupts, and says, “I think you should be more explicit here in step two.” Step 2 says, “Then a miracle occurs.”
Jacobsen: Thank you for the opportunity and your time, Dr. Silverman.
Scott Douglas Jacobsen founded In-Sight: Independent Interview-Based Journal and In-Sight Publishing. He authored/co-authored some e-books, free or low-cost. If you want to contact Scott: Scott.D.Jacobsen@Gmail.com.