As a mathematician and scientist who engages in cross-disciplinary research, I have had to spend a lot of time thinking about how various disciplines differ and how they fit together. My interest in this subject first began, however, in college and graduate school where I was regularly exposed to the quarrels and prejudices between academic disciplines and sub-disciplines. Distaste often develops along the lines of “pure vs. applied”. For example, some pure mathematicians disdain applications and consider the methods of physics to be “hand-waving”. I vividly remember my own shock and outrage as a young math major when a physics professor, with a cavalier flourish, canceled π with 3, as in . Some physicists roll their eyes at mathematicians’ fanatical precision and devotion to proof. Many biologists take an even dimmer view of mathematicians. Mathematicians, they note, have no interest in the “real world”; as long as their theorems follow from their axioms they are happy in their little playhouse. Similar disdains arise between the “hard and soft sciences”, between the sciences and humanities, and between science and religion or theology.

At this point let me assure non-academic readers that most academics get along wonderfully well and are patrons of the liberal arts. Indeed, the scientist who swoons over Mozart and the historian who takes a keen interest in quantum physics are the heart and soul of the intellectual enterprise. The prejudices appear mostly in good-natured jokes or the occasional raised eyebrow, and the academy would be an insipid place without these colorful rivalries. When prejudices do become rancorous, it is usually because funding or other resources are scarce. Interdisciplinary rancor also can occur when one discipline feels that its intellectual domain is being threatened by another. Witness, for example, the struggle between Mormon theologians and archaeologists as continuing research by non-Mormon and Mormon scholars alike turns up no evidence for the historicity of the Book of Mormon.

When disciplines become antagonistic, the academy becomes dysfunctional. Often it becomes clear during debate that practitioners of one discipline do not understand the epistemological methods or scope of the other discipline. Such misunderstandings can prevent mutual respect, especially in the context of science and religion.

While some have promoted conflict by exerting the primacy of one discipline over the other (Richard Dawkins comes to mind), many others have promoted dialogue. The late Stephen J. Gould famously suggested that science and religion are “non-overlapping magisteria”. This model suggests that science has authority in the realm of “what is”, and religion has authority in the realm of “what it means”. I am humbled by Gould’s great scientific stature and I appreciate his influential efforts to defuse conflict; however, I do disagree in a sense. I think all disciplines are concerned with “what is”. I also think that the truth about reality, although many-faceted and multi-layered, is of one piece. I think all disciplines are searching for that truth, but with different ways of knowing (epistemologies), different standards of “proof”, and different scopes of questions that can be addressed.

In contrast to the model of non-overlapping magisteria, let us consider a model of nested epistemologies (Fig. 1), using only three circles for the sake of simplicity. The inner circle represents mathematics, the middle circle science, and the outer circle the humanities, including religion, philosophy, and theology. The circles represent groupings of epistemologies, not divisions of facts or layers of reality, and they are nested to indicate regions of overlap, as in a Venn diagram.
*Figure 1. A model of nested epistemologies. The circles represent groupings of epistemologies, not divisions of facts or layers of reality.*

*Mathematics*

The inner circle is mathematics. Most people do not realize that mathematics is pure deduction, codified into symbols for the purpose of efficiency and precision. Deduction is reasoning from the general to the specific. The following syllogism is a “cartoon” that illustrates deduction:

All Andrews University students own cars. Ben is an Andrews University student. Therefore, Ben owns a car.

Mathematics begins with unproven axioms and deductively draws conclusions. The conclusions are called “theorems”, and the finite list of deductive steps that lead from a collection of axioms and/or theorems to another theorem is called a “proof”. Mathematics is the only discipline in which arguments are 100% conclusive; theorems are always true if the axioms are true. And here lies the limitation of the deductive method: at some point one must begin with unproven axioms. Mathematicians care only that their axioms are logically consistent; they do not concern themselves with the truth of the axioms in the real world. Thus, although mathematical arguments are 100% conclusive, pure mathematics does not in itself produce new knowledge about reality except in the sense that it teases out information already contained (but hidden) in the axioms(1).

*Science*

Scientists appreciate and utilize the power of the deductive method, but unlike mathematicians they care whether the axioms are true in the real world. They call axioms “hypotheses” and test them rigorously with data. To do this, the scientific method requires another type of reasoning in addition to deduction.

Induction is reasoning from the specific to the general. A cartoon example is:

All of the Andrews University students I’ve taught this year own cars. Therefore, all Andrews University students own cars.

Clearly induction is not 100% conclusive unless all data points are observed. However, induction is the crucial connection between logic and the real world.

The scientific method is an alternating cycle of induction and deduction (Fig. 2).
*Figure 2. The scientific method, an alternating cycle of induction and deduction.*

Patterns in data are inductively generalized into hypotheses, from which conclusions (predictions) are deductively drawn. The predictions are then tested against further data, and the hypotheses are inductively revised. This cycle continues until a high degree of correspondence between predictions and data is achieved (Fig. 3).
*Figure 3. The success of the scientific method is demonstrated by technological and medical advances.*

Surprising predictions borne out by new data lend the most weight in this process.

The addition of induction to deduction, which takes us from the inner circle to the middle circle, is both a strength and a weakness. It is a strength, because science, unlike pure mathematics, can address many types of questions about reality. It is a weakness because this broadening of scope comes at the cost of loss of certainty; scientific arguments, unlike mathematical proofs, are never 100% conclusive (2). Nevertheless, the scientific method works extremely well. If you have any doubts about this, consider your continual reliance upon modern technology and medicine (Fig. 3).
*
Humanities*

Although science can address some questions about reality, it cannot address many of the most important ones, including questions of meaning. This leads us to the largest circle. Humanities utilize both deduction and induction, as well as extra-rational ways of knowing. By “extra-rational” I do not mean “irrational” or “a-rational”, but rather “beyond rational”. The inclusion of extra-rational epistemologies allows the humanities to address questions about reality that science cannot address. Necessarily this comes, however, at the cost of a further reduction in certainty. At this point I need to clarify what I mean by “certainty”, because the introduction of extra-rational ways of knowing allows the term to be used legitimately in a more subjective way (“I am certain in my own heart that…”). But here I am using “certainty” to mean “conclusivity” in the same objective sense I was using for mathematics and science.

*Trade-off between Conclusivity and Scope*

By “conclusivity”, I mean the property of an argument that convinces other people of what the arguer thinks or knows internally. By “scope”, I mean the range of questions about reality that can be addressed by a type of argument. In the nested model, there is a trade-off between conclusivity and scope (Fig. 4).
*Figure 4. The trade-off between conclusivity and scope.*

Mathematical argument is 100% conclusive, scientific argument is fairly convincing, and arguments in the humanities have relatively little conclusivity. At the same time, pure mathematics can address essentially no questions about reality, science can address some questions about reality, and the humanities can address the most questions about reality, including questions of meaning.

Boundary Permeability How permeable are the boundaries of the circles in Fig. 1? Are there legitimate boundary crossings?

The larger circles contain springs of creativity that nourish the smaller circles. For example, working mathematicians look for patterns and induce “conjectures” (candidates for theoremhood), much as scientists propose hypotheses through induction. The final mathematical research product, however, does not use these inductive steps, known to mathematicians as “scratch-work”. The final product is purely deductive, containing only theorems and proofs. Mathematical inspiration can come from the humanities as well. I find that the music of J. S. Bach helps me do mathematics. There have been several times of desperation in my career when I prayed for help in solving a mathematical problem. Many mathematicians have had the experience of solving problems in dreams, and occasionally the approaches actually work in the light of day. In no case, however, would music, prayers, or dreams be mentioned in the resulting research paper. Mathematics is pure deduction, and anything else, no matter how enlightening, helpful, or true, is something other than mathematics.

In the same way, creativity can flow across the boundary from the humanities into science. For example, a belief in the inspiration of the Adventist “health message” may lead a medical scientist to propose certain hypotheses that are then tested according to the scientific method. The final product, however, must contain only rigorous applications of data collection, induction, and deduction. Good scientists and the process of good scientific peer review work as hard as possible to eliminate subjective bias. Science is a cycle of induction and deduction, and anything else, no matter how enlightening, helpful, or true, is something other than science.
*
Boundary Violations*

Some boundary crossings are illegitimate. There have been attempts by some (for example, Richard Dawkins and Daniel Dennett) to impose scientific materialism on the humanities. There have been attempts by others to inject supernatural considerations into the methods of science.

Postmodern deconstruction of science is a boundary violation. Although it is true that all scientists have bias, it is also true that the methods of science are designed precisely in order to reduce bias as far as possible. The fact that all systems of thought have bias does not imply that they are equally valid. Astronomy works better than astrology. Chemistry works better than alchemy. Psychology works better than phrenology. Clearly the reduction of subjectivity is extremely effective.

One of the more bizarre examples of boundary violation by deconstructionism is the attempt by some feminist scholars in the 1980’s to “feminize” mathematics. They thought mathematics should be warmer and fuzzier and that the notion of conclusive proof was inherently masculine. This was an insult to female mathematicians, and now that women are taking higher mathematics courses in equal numbers with men, it looks patently absurd.

A more recent controversial boundary crossing has been the attempt to present Intelligent Design (ID) as an alternate scientific hypothesis in public school classrooms. I think this is a boundary violation, certainly not because there is anything wrong with discussing ID, but simply because ID is not truly a testable hypothesis from a scientific point of view. The hypothesis that there exist “irreducible complexities” is not a well-posed scientific hypothesis because it is extremely difficult, if not impossible, to “prove a negative”. I hope the discussion of ID will live long and prosper in philosophy of science classes everywhere. I simply think that ID, along with many other important topics, belongs in the outer circle and should not be confused with science in the minds of students. *Indeed, trying to reduce any pursuit in the outer circle to the domain of science violates those very qualities that allow the pursuit to transcend science in the first place.*

The last boundary violation I want to mention is what I call “religion masquerading as pure deduction”. In this case, religious believers present their scriptures as the axioms in a purely deductive system. For example, a “True Blue Mormon” might say, “I don’t care what archaeologists or DNA researchers find. I *know* there were advanced civilizations of Jewish descent in North America 1600 years ago who had steel, glass, cattle, and horses, because the Book of Mormon says so.” The problem with this approach can be illuminated by recalling that in a purely deductive system, the truth or falsity of the axioms is irrelevant. Such a religious believer, however, is extremely invested in the truth of his or her axioms. If you ask why the axioms (scriptures) are true, he or she begins to cite evidence of one kind or another (induction) and it becomes clear that the system is not purely deductive after all. If you push a bit further, you may find that the argument reduces completely to the person’s “testimony”—the “burning in the bosom”—which means the argument is squarely in the outside circle and is in no-wise completely deductive. There is in fact no way to compress religion into the deductive circle without destroying the very aspects of religion that make it meaningful.
*
Conclusion*

Today’s canonical disciplines did not begin with well-defined epistemological boundaries. For example, early mathematicians often were scientists or philosophers or even religious leaders. Over time, the modern disciplines developed naturally into epistemological categories, which resulted in great advances. Today, interdisciplinary work is once again taking the lead, but this time its synergy is leveraged in a powerful way by the clear-eyed recognition of epistemological categories.

I think that all disciplines are searching for truth and reality, but with different ways of knowing (epistemologies), different standards of “proof”, and different scopes of questions that can be addressed. The outer circles (Fig. 1) use more types of epistemologies and can address more types of questions. The inner circles provide more warranty to convince someone else of what one knows or thinks but can address fewer types of questions. Larger circles include all smaller circles, and hence cannot logically contradict them. There may be complex paradoxes, certainly, but not logical contradictions. Outer circles nourish and inspire the inner circles but cannot be included in the final arguments or products of the inner circles. Inner circles provide an internal structural support for outer circles that allows them to grow and flourish.

The nested model in Fig. 1 is exactly that—a model. A good model is an accessible mock-up that captures the main points of a system while remaining much simpler than the original. In this account I have fought the urge to use jargon or refine ideas and terms to a fine-scale resolution. I have not tried to counter every possible objection or make every important point. A model is useful whenever it provides a simple framework for organizing ideas and clarifying discussion. If the model presented here is helpful, then readers will refine it, revise it, and fill in the missing blanks.

*Shandelle M. Henson is Professor of Mathematics at Andrews University. Her master’s research specialty was mathematical logic and her doctoral and current research specialty is mathematical ecology. She gave this paper as an invited oral presentation to the St. Albert the Great Forum on Theology and Science at the University of Arizona, the Berrien Springs, Michigan chapter of the Adventist Forum, and the Andrews University Seventh-day Adventist Theological Seminary.*

1. There is another type of limitation in the mathematical method. Gödel’s Incompleteness Theorem shows that no mathematical system at least as complicated as arithmetic can be finitely axiomatized. That is, the axiomatic method cannot completely generate all the truths that exist in such a system. This ended efforts to reduce the whole of philosophy to symbolic logic.

2. There is also a theoretical limitation to the scientific method that is akin to Gödel’s Incompleteness Theorem. The incompleteness arises from problems of self-referentiality. Science cannot completely explain human consciousness, because we are using the object of study (the mind) to explain itself.

3. The mathematician in me cringes at the phrase “boundaries of the circles”. In fact, the circles are the boundaries of the disks. But I am using “circle” to mean the whole disk, following common parlance.

This is a companion discussion topic for the original entry at http://spectrummagazine.org/node/1722