Many perceive math as a discipline of universal truths, but this assumption overlooks the foundations of mathematics. Math is the discipline of truths in a certain context, a context built up from a set of assumptions, or axioms, which we take for granted to be true. A brief look at history will show that math is not quite the absolutism with which it is often associated. Examining the history of mathematics may challenge our personal standards of objective logic, and inform the way we approach our own life philosophies.

The math community went through heavy turmoil in the 1900s, when the foundations of mathematics were being built. The appeal of unifying most of mathematics under one theory motivated a movement to establish these foundations, and mathematicians thought set theory could be the answer. The idea was that any mathematical object -- numbers, functions, ordered pairs, anything -- could be defined in terms of sets.

But the development did not proceed smoothly, and several challenged it. Most famously, Gottlob Frege, a German mathematician, wrote two large volumes constructing naive set theory, which he hoped would serve as a foundation for much of mathematics. Just as the second volume was being published in 1903, Bertrand Russell constructed a paradoxical set (now called Russell’s Set) that derived a fundamental contradiction in Frege’s theory. Frege quickly admitted his mistake and sought to remedy his error, albeit without success.

Soon after this famous incident, Ernst Zermelo and Adolf Fraenkel proposed another set of axioms for set theory, now called ZF. Unlike naive set theory, ZF rigorously defines what can and cannot be a set. (In ZF, Russell’s Set is not actually a set.) A slight expansion of ZF (called ZFC) is the axiomatic system that most mathematicians work with today.

Does ZFC work? Kurt Gödel, a famous logician, published his two incompleteness theorems in 1931, which firmly limited the power of axiomatic systems like ZFC. Informally, his second incompleteness theorem states that any reasonable axiomatic system either is inconsistent or cannot be proven to be consistent within itself. So if ZFC is consistent, there is no way to show that it is consistent, given the axioms of ZFC. And if we are able to show that ZFC is consistent by only using ZFC, then it is inconsistent! A consequence of this theorem is that there is no way to confirm that ZFC is not paradoxical unless it is indeed paradoxical. The only thing mathematicians can do is have faith that ZFC is consistent.

Most of us completely trust the results of mathematical research to be true, as we should—math proceeds as logically as is humanly possible. But even math hinges on assumptions whose logical consistency is unknown. In this way, a strict definition of faith does not run counter to logic as we often think. In the development of mathematics, we see that faith—in assuming the truth and consistency of axioms—is actually necessary to begin logical deduction. Mathematicians have demonstrated that rigor can be built from uncertain axiomatic foundations by being aware of the limitations of their assumptions (see Gödel’s incompleteness theorems), careful to build every theory up from foundational and relatively universal axioms, and quick to discard inconsistent theories (see Frege).

Can we similarly rigorize our philosophies? Can we make our beliefs and philosophies consistent and self-aware in the spirit of the history of mathematics? Can we mathematize the mind?[1]

We all live with our own axioms inscribed in our hearts, but often, they are so ill-defined and convoluted that we can’t enumerate them clearly. Sometimes we have no proof, not even a bad one, for our supposed theorems. Yet, in their roots, our philosophies are analogous to mathematics: both start with assumptions and build up into theories. How would our philosophies be different if they were scrutinized in the same way that mathematics has been held to a decided set of axioms?

Math relies on logical deduction from assumptions to produce theorems. It is difficult to overlook a misstep in logic when consistently referring back to a transparently articulated theorem. If we mathematize our minds, logical inconsistencies reveal themselves willingly, perhaps too much so. In Frege’s case, a simple paradox, in three lines, revealed a clear inconsistency that tore down two volumes of work. Hidden contradictions are insidious— thinking our systems of intellectual and moral standards are consistent, we may continue to build upon a contradictory set of beliefs. Approaching personal philosophies and spiritual orientations with a critical rigor minimizes the risk of logical inconsistencies in our systems.

But what is left in our minds when we reorganize our philosophies in the manner above? Perhaps almost nothing. We may find that cleaning up our beliefs results in a trash can full of unfounded theorems and little else. We may find that we actually know nothing of what we thought we knew. Examining our personal philosophies in the conventionally “rational” framework of an axiomatic system sheds light, perhaps too much of it, on the limits of our knowledge and beliefs.

For persons of faith, sifting through philosophies in a rigorous manner will almost certainly reveal that there is very little we can know about what lies beyond the physical or conceivable. But doing so also strengthens the few inklings of conviction that do survive reorganization. You may think that after such a process, one would know less than before, but can we really qualify the unfounded, disposable theorems as knowledge? I would argue the meta-knowledge of what we do and do not know to be one of the most valuable bodies of knowledge one can explore. The few axioms and theorems that survive a rigorous sifting are, then, indisputably yours. You own them confidently, you know them inside and out. Through this perhaps painful process, we can reclaim our philosophies as truly our own.

I’ll conclude with a meta-analysis of the axioms I’ve assumed thus far. Perhaps the most important axiom that this article accepts is that not believing a false proposition is preferable to believing one. Is this a reasonable axiom to accept? We put considerable effort into showing that faith does not run counter to logic, but what if one disagrees with the basis upon which we define logic? Is it worth mathematizing the mind without first accepting the mainstream standard to which these life-defining philosophies will be held? I have no answers, and you may find yourself similarly lost at many points along your analyses. But note that we were only able to ask those questions by using the self-awareness and rigor exemplified in the history of mathematics. Let us continue to organize, rethink, and reclaim our philosophies in our surely incomplete but worthy quest of understanding religion, morality, and the likes.

Glossary:

AXIOM: an assumption taken to be true without proof

AXIOMATIC SYSTEM: a set of axioms with a language and deduction rules

GÖDELʼS SECOND INCOMPLETENESS THEOREM: for any reasonable axiomatic system that can describe the natural numbers, it is either consistent or it cannot be proven to be consistent within itself

NAIVE SET THEORY: a set theory that defines sets to be a collection of objects with a certain property, with little restrictions on what the property can be

RIGOR: the precautions taken in the methods of mathematical proof to ensure correctness

RUSSELLʼS SET: the “set” R of objects x such that x does not contain itself. It turns out that R simultaneously contains itself and does not contain itself -- it is too “large” to be a well-defined set

SET THEORY: the mathematical study of sets, or collections of objects

ZF: a set of 8 axioms that was one of the first attempts to resolve problems in set theory such as Russellʼs Set

ZFC: ZF in addition to the Axiom of Choice; underlies most of modern set theory

[1] The purpose of the following paragraphs is to draw analogies from math to apply to our personal philosophies. Note that personal philosophies are quite different from formal systems. Appeals to Godel and other math results are merely looking to mathematics for inspiration. Also note that references to technical terms such as “axiomatic systems”, “axioms”, “proof”, etc. are colloquial and meant to associate philosophies with formal systems, not to equate them.