Comprehensive learning aspires to integrate more and more of Humanity’s traditions of learning to better comprehend the world and how it works. As we learn more and more from these diverse ways of knowing, it becomes increasingly likely that the perspectives, hypotheses, and frames of reference we encounter will seem incompatible or even inconsistent with others. This resource suggests how to better accommodate these ambiguities as tools for our comprehensive practice.

This way of accommodating ambiguity in our learning is a central theme in the profound 2007 book “How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics” by William Byers. The book examines ambiguity while it surveys the folklore of mathematics culture and the philosophy of mathematics. It is easy to read and serves as a good introduction to mathematics for readers with weak math skills. It might be especially valuable for those wanting to expand the compass of their comprehensive learning with mathematics.

## Ambiguity, Contradiction, and Paradox

In the book “How Mathematicians Think”, William Byers argues that the power of mathematical ideas comes from their ambiguity. Let’s start with his definition of ambiguity:

Ambiguity involves a single situation or idea that is perceived in two self-consistent but mutually incompatible frames of reference.

— William Byers in “How Mathematicians Think”, p. 28.

Byers discusses how equations like “2+3=5” contain two separate frames of reference. On the left hand side “2+3” is a process, a verb, the addition of 3 to 2, an arithmetic combining of two numbers. On the right hand side is the number 5. Processes as verbs are different from objects as nouns. This seems to be a category error worse than combining apples and oranges. Yet, the equation, as we all know, is true. What are we to make of this? Does arithmetic reconcile ambiguous category errors?

Byers also looks at variables in algebraic equations such as 3x+2=8. Does the ‘x’ refer to any number, is it a variable that varies over all possibilities, or does it refer to the number 2? Byers explains: “the answer is both and neither. At the beginning x could be anything. At the end, x can only be 2”. Variables in mathematical discourse always entail this ambiguous status where they are general and specific depending on our frame of reference. My Synergetics colleague Tom Miller calls such situations the “both/neither” where two perspectives, hypotheses, or frames of reference are both true and, at the same time, each is not true. The both/neither is a kind of ambiguity.

Byers gives many more examples. He concludes:

The control in mathematics is provided by the logical structure, and the power and profundity of mathematics is a consequence of having deep ambiguity under the strictest logical control…. The ambiguity is ‘resolved’ by the creation of a larger meaning that contains the original meanings and reduces to them in special cases. This process requires a creative act of understanding or insight.

— William Byers in “How Mathematicians Think”, p. 77-8.

Is ambiguity at the heart of our understanding?

Contradictions are almost the same as ambiguities. The difference is that in contradictions the two “mutually incompatible frames of reference” cannot be resolved into a single situation or idea that is consistent. The two frames conflict so incompatibly that we call it inconsistent.

Byers’ favorite example of the power of contradictories in mathematics is zero. Byers characterizes zero with the title of Robert Kaplan’s 1999 book “The Nothing That Is”. The nothing that is is an apt description of zero’s inherently contradictory nature. How can we represent nothing? For students of mathematics, the task is simple: represent it with the word “zero” and the symbol “0”. We say that zero is a number, a definite object. But this object represents nothing! How can that be? Can we honestly say that the number zero is nothing? Byers writes, “Does ‘nothing’ exist, and if not how can we talk of it? … zero is a name for something that does not exist.”

The concept of zero was a crucial advance in the history of mathematics. Yet it seems inherently contradictory. Byers observes, “far from excising the contradiction from our thought, we have learned to use it in a constructive and creative manner.” He adds, that from zero “a concept that is inherently contradictory has been made ambiguous by creating out of zero ‘a single idea’.” Zero is the single idea of a number that stands for nothingness. We boldly ignore the inherent contradiction at the heart of the idea except, of course, when we divide by zero which is strictly verboten.

Byers adds, “Both coherence and contradiction are fundamental components of ambiguity.” The “both/neither” captures this principle by highlighting coherence (the both) and contradiction (the neither). Both/neither contrasts with both/and in our culture’s current zeitgeist. Both/neither is the logic of ambiguity. If ambiguity is as fundamental as Byers argues, then both/neither logic is more important than the both/and logic that many of us have been using.

Are contradictions essential to the both/neither logic of ambiguity?

Byers quotes the Encarta World English Dictionary for a definition of paradox: “a paradox ‘is a situation or proposition that seems to be absurd or contradictory but is or may be true.'” In contradiction, we know that the two frames of reference are inherently inconsistent. In paradox, there is a possibility of reconciliation, of resolving the absurdity. Byers writes,

Paradox is not something alien to life; it is the basic fabric out of which life is composed.

— William Byers in “How Mathematicians Think”, p. 112.

Is “zero” merely paradoxical or inherently contradictory? I think it depends on your spiritual beliefs about logic: if your spirituality of logic rejects contradictions, then to keep zero you would consider zero merely paradoxical. But if your spirituality tolerates truths in contradictions, you might prefer to call a spade a spade and say that “zero” is contradictory. Here is how Byers addresses this concern:

“zero” is paradoxical—it contains a contradictory aspect and yet [it] is a fundamental mathematical concept.

— William Byers in “How Mathematicians Think”, p. 113.

Does this suggest that if an idea is contradictory but sufficiently useful, we should set aside its contradictoriness so we can access the idea’s creative power? Does that also suggest that our culture’s overemphasis on the so-called law of noncontradiction which prohibits contradictions is misplaced?

Byers’ book goes on to explore the paradoxes of infinity in depth. He explains that contradictions and paradoxes are two aspects of ambiguity. So ambiguity, in all its manifestations, can be seen as the crucial idea.

Logic moves in … the direction of clarity, coherence, and structure. Ambiguity moves in the other direction, that of fluidity, openness, and release. Mathematics moves back and forth between these two poles.

— William Byers in “How Mathematicians Think”, p. 77-8.

Byers’ schema for ambiguity in learning accommodates these two poles or phases. The contracting or converging phase brings clarity, coherence, and structure by learning in depth. The expanding or diverging phase brings fluidity, openness, and release by learning in breadth. Control and logic work to make the system cohere in a meaningful structure, it focuses, contracts, and converges to produce depth. Learning in breadth with its context confronts us with ambiguities and their incompatible stories or data and frames of reference challenging our ability to find resonant wholes for meaning making as we diverge and expand our perspective.

Do ambiguities including contradictions and paradoxes organize and drive our learning as we navigate the essential poles of depth and breadth?

## Ideas, Great Ideas, and Truth

In the second section of his book, Byers builds on the above foundation of ambiguity to explore the nature and truth of ideas.

The creative in mathematics is expressed through the birth of new ideas. These ideas may consist of a new way of thinking about a familiar concept or they may involve the development of an entirely novel concept.

— William Byers in “How Mathematicians Think”, p. 191.

These quotes summarize Byers’ thinking about how ambiguity leads to ideas:

An idea emerges in response to the tension that results from the conflict inherent in ambiguity…. An idea is a principle that organizes experience…. [It] is an answer to the question, “What is going on here?” … The ambiguity does not limit the idea—the ambiguity is the very thing that flowers into the idea.

— William Byers in “How Mathematicians Think”, p. 191-203.

Byers asserts “great ideas come out of situations of great ambiguity”. He then develops the great idea of “one”:

“One” is an idea that goes beyond mathematics. As was the case with “zero” and “nothing,” the mathematical “one” is an idea that grows out of the human condition itself…. [N]otice that “one” and the connected idea of “oneness” are ambiguous, that is, they are used in two different and conflicting senses. In the first sense, “one” represents something that is a unit and distinct from all others—a unique individual in a world of other individuals. The second sense comes from the word “oneness” or “to be one with,” which means connected or part of a larger whole…. The first sense emphasizes the uniqueness and separateness of that which is designated as the “one,” the second the harmonious integration of parts.

— William Byers in “How Mathematicians Think”, p. 205-206.

Byers gives many examples of ideas in mathematics. He explores how “the idea is always wrong”. This reminds me of Jan Zwicky’s realization that our meaning making insights may not be true as we explored in the resource The Whole Shebang. It is the reason we have emphasized mistake mystique in several previous resources (see the resources Mistake Mystique in Learning and in Life and Redressing The Crises of Ignorance). It is also a reminder of the both/neither logic where the coexistence of coherence and contradiction are an inherent aspect of ambiguity. The both/neither logic is inherent in ideas too.

Byers summarizes:

We are beginning to see paradox, not as something to be avoided and eliminated, but as a potentially rich source of ideas.

— William Byers in “How Mathematicians Think”, p. 283.

Byers motivates an exploration of great ideas with a Simone Weil quote:

All true good carries with it conditions which are contradictory and as a consequence impossible. He who keeps his attention really fixed on this impossibility and acts will do what is good. In the same way all truth contains a contradiction.

— Simone Weil as quoted in “How Mathematicians Think”, p. 284.

Byers adapts her quote to say, “a great idea carries with it conditions which are contradictory and as a consequence impossible.” He emphasizes that “a ‘great idea’ begins with a gap—a gap that seems unbridgeable.” Logic then helps us formulate and organize the idea. Nonetheless, there are always limitations, some of the falseness remains. Byers clarifies, “it is not the idea so much as the universality of the idea that is false…. Great ideas are wrong but they are wrong in a brilliant and inspired way.”

I hear Byers characterizing truth as the insight of creative immediate certainty when the light is turned on. This seems analogous to “the experience of meaning” of Jan Zwicky in forming gestalt wholes (see the resource The Whole Shebang). Byers says “the truth cannot be completely objectified; it is not completely ‘out there'”. Truth is a kind of objective subjectivity or a subjective objectivity. It cannot be completely formalized or captured. It is like a rainbow: it is objective, but we can never find the end of the rainbow because it is not an object. The truth is the coherence of a great idea together with its falseness, its contradiction, its ambiguity, its inherent both/neither logic.

Does Byers’ thinking about ideas, great ideas, and truths as inherently ambiguous help us better understand how we learn?

## Ambiguity and Comprehensive Learning

What will we get if we incorporate William Byers’ thinking about the central role of ambiguity in ideas, truths, and learning in our developing story about comprehensive learning?

Comprehensive learning involves studying more and more sources from more and more of Humanity’s great traditions of inquiry and action. These explorations inevitably lead to expanding the number of potentially incompatible hypotheses, perspectives, and frames of reference that we consider. As a result we are sure to confront ambiguities. Byers’ approach sees these ambiguities as tools to help us organize our learning.

With Byers’ insights, we can appreciate the ambiguity between breadth and depth as the two poles of comprehensive learning. Breadth is the diverging, expanding, and contextualizing phase. Depth is the converging, contracting, and clarifying phase. Now, perhaps, we can better judge how to keep breadth and depth in balance in our learning.

Inspired by how mathematics has profitably organized the numbers one and zero, arithmetic equations, and algebra as powerful ideas, we too may apply various logics to see if we can ferret out ideas that integrate the different frames of reference and perspectives that we are studying. Even if we succeed, we know, from the examples of mathematics, that inevitably some ambiguity will remain to spur tomorrow’s learning efforts.

Byers’ discussion on seeing the light in the context of ambiguity may help us better understand the adequate wholeness we value for meaning making. We may now realize that our experiences of meaning, as Jan Zwicky calls them, inherently involve ambiguity, paradox, and even contradiction.

Byers’ insights reinforce our value of mistake mystique where we valorize looking for and identifying the gaps in our understandings, communications, and designs. Indeed inherent ambiguity explains why there must be such gaps! It also explains why the logic of the both/neither can help us better find the ambiguities in our ideas, principles, and truths to spur us to deeper thinking.

Ambiguity with its both/neither logic can also help us organize comprehensive comprehensions (see the resource Comprehensive Exploration, Comprehension, and Collaboration), our most integrated forms of knowing. Ambiguities naturally suggest questions that characterize what we do not understand, our ignorance. Our value for a thoroughly refined ignorance aspires toward a question structured survey of our learning (thoroughly refined ignorance was introduced in the resource Shifting Perspectives and Representing The Truth). This refined ignorance can now be seen as an outline of the ambiguities we have identified. If this way of integrating our knowledge facilitates effective judgment making, we will have understood the incisive aspects of our situation.

William Byers shows us that embracing ambiguity, contradiction, and paradox can reward us with access to the creative wellspring of great ideas and truths! As we try to integrate more and more of humanity’s traditions of inquiry and action we should consider engaging the ambiguities we may have shied away from previously.

How has a survey of William Byers’ thinking about learning in mathematics improved your understanding of comprehensive learning?

This essay was written to provide ideas in support of the 16 February 2022 session of “Comprehensivist Wednesdays” at 52 Living Ideas (crossposted at The Greater Philadelphia Thinking Society).

Addendum: 1h 41m video from the 16 February 2022 event:

### Read Other Resource Center Essays

- Intellectual Virtues for Comprehensive Practice
- The Standard for All Measurements: Our Judgment
- Measuring Beliefs
- The Measurements of Life (Tools for Comprehensivity)
- What Is Comprehensive Learning?
- Articulating Comprehensivity: The Comprehensive Design of Our Lives
- Comprehensive Exploration, Comprehension, and Collaboration
- The Ethics of Learning from Experience
- Dante’s Comedìa and Our Comprehensivity
- Chronofiles: Data Mining Your Life for Comprehensive Thinking
- The Whole Shebang: “to understand all and put everything together”
- Shifting Perspectives and Representing The Truth

Is ambiguity at the heart of our understanding?

Great insight! That’s like necessity is the mother of invention but proximity is its father!