*To what extent can mathematics, science, and spirituality answer questions about the world and about ourselves? How do these pursuits relate, conflict, and influence one another?* These are the questions which motivated the workshop I had the pleasure of hosting with BU GradHillel. They are decidedly big questions, which have been hotly debated within religious, scientific, and philosophical circles for centuries. But they are also deeply personal and individual, and have played a role in my journey as a budding scientist and logician, and in reconciling where Judaism and faith belong in my life.

My goals in hosting the workshop, and in writing this article, are humbler than definitively classifying the epistemological capabilities of math, science, and faith. If nothing else, I hope to share some fascinating ideas about mathematics and science, which may be unfamiliar to people outside of these disciplines. If we learn something about the nature of faith and spirituality along the way, then all the better.

###### The Limits of Mathematics

Truths in mathematics and logic are obtained through deductive proofs, which show step-by-step how to reach an exact conclusion, with certainty, from a set of premises. Many mathematical statements take the form of an *implication*: “if this, then that.” Consider the example “if *n* is a prime number, then *n* is either odd or 2.” A direct inference using this implication would be, for instance, to take the premise that 5 is prime, and deduce that 5 is either odd or 2 (and since 5 is not 2, we further deduce that 5 is odd). We can use this implication in reverse as well, called the *contrapositive*: with the premise that 6 is *not* either odd or 2, we deduce that 6 is *not* prime. Note, however, that we could not deduce that a number which *is* odd or 2 *is *prime; this would be a logical fallacy.

Another common type of mathematical statement is a *universal *claim, which asserts that something is true for *all* possible values, e.g. “*every* prime number is odd.” It can be difficult to prove that a universal statement holds, but is easier to disprove by providing a *counterexample*; in this case, 2 is prime but not odd. The complement to a universal claim is an *existential* claim, in which a property holds for at least one value, e.g. “*some* prime number is even.” An existential claim can be proven with an example, in this case 2. Disproving an existential claim is challenging, as it demands showing that no satisfying witness *ever* exists.

Although the facts and methods of deduction in mathematics are unambiguous, many statements are still tremendously hard to definitively prove or disprove. Some have literally taken hundreds of years to solve, while others still have no solution in sight. But the mathematical advances of the future may never be enough to solve these problems. It might be a surprise to learn that there are always problems in mathematics for which finding a solution is *impossible*. This astonishing fact (called Gödel’s incompleteness) rocked the world of logic a century ago, before which many believed that all mathematical problems should be solvable given the right tools.

Furthermore, every mathematical truth actually depends on assumptions. We derive all mathematical facts from previously established facts, and those facts are proven from other ones, up and up… until we reach *axioms*. Axioms are the facts and rules chosen as the basis for a mathematical system. A famous example of this is the “parallel postulate” that underpins Euclidean geometry, which declares that parallel lines never intersect. If you change this axiom, you get non-Euclidean geometries, which are just as valid as Euclidean geometry but yield very different results. In my work as a computer scientist studying formal methods, we frequently construct new systems by stating a set of axioms and inference rules, and then analyzing the resultant properties. *Every* system has axioms, and no fact can be derived without them. This means there is no single universally correct set of mathematical truths; rather, all truths are contextual to systems. The axioms we choose are not based on some absolute measure of correctness, but on their usefulness.

###### The Nature of Science

While mathematics in its purest sense is concerned with proving facts within logical systems, science has the arguably harder responsibility of establishing truths about the external world. Meaningful scientific results do not necessarily explain the true nature of the universe. What a good result does have is *explanatory power*: it describes a subject with accuracy, benefits from additional observations, and is not so specific as to be useless. Furthermore, a good theory is hard to vary, in that changing its details harms its accuracy. This is one way to discredit mythological explanations of natural phenomena. For example, claiming a volcano erupts whenever a deity is angry is no better an explanation than claiming it erupts when a deity is, say, overjoyed; thus, neither theory is especially useful, even if one of them might ultimately be true.

Characterizing exactly how the scientific process works, and what answers it can and cannot provide, has twisted philosophers into knots for centuries – far beyond what any working scientist or layperson need concern themselves with. Here I will address a few key ideas, to convey how science as a practice is fundamentally messy and imperfect. We, individually and collectively, have a responsibility to respect this imperfection.

To begin with, what qualifies as science? The reality is that no strict delineation exists between proper scientific inquiry, pseudoscience, or legitimate nonscientific pursuits – this is called the problem of *demarcation*. While we do learn a formal step-by-step scientific method in school, the actual process is often messier. Discoveries and the development of theories can span centuries across thousands of researchers, and plenty of incredible results have been achieved without the modern scientific method.

Next, consider that general scientific results are necessarily derived from finitely many observations, and furthermore, to utilize results we must assume that our universe in the future will generally resemble the present. This is the problem of *induction*. A consequence of these constraints is that, unlike in mathematics, it is usually impossible to definitively prove a universal scientific claim (something is *always* true about the world) or disprove an existential claim (something is *never *true in the world).

Finally, recall that in logic, all results reduce back to chosen axioms. Similarly, every scientific result must depend on countless assumptions, both explicit and implicit. Such assumptions might concern the calibration of instruments, the purity of samples and environment, the accuracy and impartiality of prior results, the reliability of one’s senses, the validity of underlying theories, and on and on. In any mathematical setting, we can identify exactly the assumptions being made, but in science they are utterly impossible to control for, let alone enumerate. This yields the problem of *infinite regress*: our assumptions in science lead back forever, never landing on unassailable self-evident truths. Clearly, this problem is not enough to prevent scientific progress, but has spawned centuries of philosophical debate. There are a few ways to reconcile this endless “why game.” One is *foundationalism*, in which we agree on certain self-evident facts about the universe, and derive all subsequent results from there. Another is *coherentism*, which prioritizes the consistency of scientific facts over their absolute truth. Ultimately, the practical reality is that science does not tell us what is true about the world; it tells us what is *probably* true, or at least what is true enough to be useful.

###### The Role of Faith

Why have I spent this time enumerating limits of mathematics and science? In truth, I am speaking to, more than anyone else, my younger self. He was cynical, weary of a vast and confusing world, and had condemned faith as intrinsically inferior to logic and science. He also had not yet been diagnosed with autism, which would eventually bring some welcome context to all of these things. To him, mathematics was an unwaveringly reliable friend in the face of chaos and irrationality, which could answer every question about the universe given enough time and human ingenuity.

The reality is that science is incapable of providing absolute truths about our world; and, while mathematics can provide certainty, it is confined to the axiomatic systems we design. So do we abandon these practices? Of course not. Rather, we ought to appreciate that each is equipped to ask limited kinds of questions, offer limited answers, and that these constraints are the nature of every human endeavor.

In these gaps left by the empirical and logical, we have other more subjective tools for studying the world and ourselves, such as the arts, leisure, and faith. The role of faith is of course a very personal matter. Some use faith to choose absolute truths upon which they can invariably rely (recall *foundationalism*), like the existence of a deity, the certainty of an afterlife, or a moral balance in the universe. Faith can offer solace in the face of things not yet understood by science, such as the ever-elusive true nature of consciousness. It can also explore what is meaningful, a notion which science ultimately cannot address. How individuals or groups come to derive and accept spiritual truths is its own matter.

Science, mathematics, and faith, when engaged with responsibly, each address distinct facets of our lives. Science is ineffective against metaphysical questions which, by definition, lie outside of our world – it cannot prove, disprove, or even meaningfully *ask* whether a deity, or souls, or the afterlife exist. Conversely, faith and spirituality are simply not effective tools for empirical analysis of the mechanisms of the universe. But these pursuits are not opposed to one another; they are complementary. This is what I did not appreciate when I was younger. I felt that I had excised faith from my life and “chosen” science. But faith is inescapable. I may not have been engaging specifically with religious doctrine – and I still seldom do. But we all use a form of faith every day when we hope that, for instance, gravity will continue to pull us to the ground as it always has. Science can model how gravity (apparently) works, but it literally cannot guarantee that gravity will be around tomorrow. To apply the approximately correct theories of science, or the contextual facts of mathematics, to the real world truly involves a leap of faith.

(*This post is part of Sinai and Synapses’ project Scientists in Synagogues, a grass-roots program to offer Jews opportunities to explore the most interesting and pressing questions surrounding Judaism and science. Jared Pincus is a PhD student at Boston University and wrote this piece based on Scientists in Synagogues programming at Boston University Hillel.)*

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