Mathematics is often grouped together with the many other sciences, despite science supposedly being an empirical study of some domain; while math for centuries has been claimed by the rationalists. I argue however that math is indeed a kind of science, albeit a very strange one. While physics is the study of natural laws, chemistry the study of chemicals, biology the study of life, math it would seem, is the study of math. This is a little tongue and cheek, and when looked at more closely it becomes apparent that what math is, is the study of inherently meaningless axiomatic symbolic systems, as well as the relationships between and rules imposed upon, those symbols. It is however the goal of meta-mathematics to interpret these symbols and insert meaning from our world into them, and extract useful information out. But more on that later. What is more important now is understanding what math is and what math can tell us, if anything, about the world. This will be done in two parts, Part 1, Incompleteness. Part 2, Meta-mathematics and Meaning.
Toward the end of the 19th century David Hilbert, inarguably the most important mathematician of his time, proposed 23 of the most important math problems to be solved in the 20th century. The second of these problems was boring to most at the time, but wound up certainly being the most important as far as mathematics as a whole was concerned. Hilbert’s second problem was to show that second order arithmetic (the closest thing to a grounding of mathematics had at the time) was consistent, i.e. that these rules of mathematics could not result in a contradiction. Despite it being boring, the larger issue was that no one quite knew where to start. That is until mathematicians Frege and Dedekind came along with the logicism movement. Bertrand Russell and Alfred Whitehead, a friend of Russell’s who was also a prominent philosopher and mathematician at the time, were the spearheads of this movement, with the goal to ground all of mathematics in pure logic. Doing so would ensure that the axioms of mathematics were deemed true through their logical form alone, and not based on their content. Much progress was made with Russell and Whitehead publishing a “Principia Mathematica” famously taking up the first half to show that 1+1=2 falls out of a Hilbert system, which is a simple system of logical rules. Logicism was making great strides, but the movement had not fully succeeded, as a few axioms of mathematics were still not based in logic. With the foundation of mathematics being a proper field of study now, Hilbert’s second problem, the consistancy problem, had some grounding.
Little progress was made after period, and Logicism fell out of fashion along with its sister, Positivism. That is, until Austrian logician Kurt Gödel published a remarkable paper in response to both Hilbert’s second problem and Principia Mathematica. To get right to the point, Gödel published his two incompleteness theorems which delivered a shocking blow to the logicist movement, and a very lateral solution to Hilbert’s second problem. Gödel’s first incompleteness theorem states: “Any sufficiently expressive math system must be incomplete or inconsistent.” By sufficiently expressive, Gödel means the symbols could be mapped in order to do second order whole number arithmetic (effectively set theory with Peano arithmetic). Let us examine what Gödel’s first theorem means for mathematics as whole.
The first question you might have is, well clearly the math system we use is sufficiently expressive, so which is it? Is our math system incomplete, inconsistent, or both? Gödel offered an immediate answer to this: our most extensive mathematical system known today as Zermelo–Fraenkel set theory (ZF from here on), is definitely incomplete. There are statements which you can formulate in ZF that are impossible to answer in ZF. Another way of saying this is that these statements are independent of the axioms of ZF. (Some examples of these are: The continuum hypothesis (Hilbert’s first problem), that every surjective function has a right-inverse, every vector space has a basis). Each of these can be declared true or false then appended to the ZF axioms, and this new system will be consistent so long as ZF is consistent. The most famous example of this is the controversial “Axiom of choice”. However this does not help us get around completeness as with this new set of axioms we will have new unprovable statements, and so on. I would like to emphasize here specifically that Gödel’s work makes no reference to the truth or falsity of claims; only the provability or un-provability via the axioms.
So now we’re stuck with an incomplete system, that is, it is full of unprovable statements. At least it is consistent right? In fact it gets worse. This is Gödel’s second incompleteness theorem, which states: “Any sufficiently expressive, consistent math system cannot prove its own consistency.” This is doubly frustrating! It means firstly, that if we can prove our system is consistent, it must be inconsistent, and that if it is consistent (which we would like it to be), we cannot show it.
This is one of the many pits rationalists will avoid going when discussing math. In order to do mathematics in a way that is not entirely epistemologically pragmatic, one has to take its consistency on faith; entirely analogous to the assumption of the uniformity of nature in physics. The more math we do, and the longer we go without proving its own consistency or finding a contradiction, the more evidence we have that it is consistent, but if it actually is consistent, we can never show it.
This is how math is empirical, and not fundamentally rational. To drive the point home: in order to get anywhere, one must assert the axioms of their mathematics to be true, and one must also assume their system is consistent, as no decent rationalist would accept claims made from an inconsistent system. It follows then that the truth value of any (decidable) claim made in a math system is reliant upon tautology (through the axioms and deduction) and the pre-supposition of non-contradiction. Any derived mathematical statement is only true because we assume the axioms are true, and that the axioms do not lead to contradiction. This shows that when rationalists claim mathematics is a form of a priori truth, the truths they declare are at best tautological, circularly assumed to be true in the first place, and at worst contradictory. The reason we trust Zermelo–Fraenkel set theory is not because we find these axioms to be self evidently true, but because its’ meta-mathematics have been extensively studied and we have a strong empirical basis to assume its’ non-contradiction.
While this certainly discredits most of the a priori claims rationalists make about math, the rationalists and platonists have much to cling to when it comes to the ontology or objective existence of mathematical objects, such as numbers. In part 2, we will explore more thoroughly the success and failures of logicism, and the relationship between meaning, math and meta-mathematics.
The Third Triumvirate 