Willard Van Orman Quine, prominent American philosopher and inspiration to Bob Moses during his years at Harvard, said mathematics was a result of the regimentation of ordinary discourse: “A progressive sharpening and regimenting of ordinary idioms: this is what led to arithmetic, symbolic logic, and set theory, and this is mathematization.” Quine believed mathematics was developed by discursive innovations that made ordinary languages into more precise and formal languages over time. Mathematicians learn to speak conceptual and symbolic languages that no one speaks as a natural language. The immediate benefit of regimentation is that it defends against the pitfalls of ambiguity and vagueness characteristic of natural languages. The ultimate benefit of regimentation is that it bridges the gap between lived experiences and individual intuitions, on the one hand, and mathematical notations and collective concepts, on the other.

Madeline Muntersbjorn is an Associate Professor at the University of Toledo where she teaches symbolic logic and philosophy of science courses. She has a deep fascination with the history of mathematics, as well as artificial languages, or languages that, rather than occurring spontaneously, are developed to serve specific purposes.

“Feature talks or artificial languages are intermediaries between spontaneously spoken languages and mathematical formalism. There’s something about the process of mathematization that is artificial. But also strategic, right? There’s something about artificial languages as a class of languages in that they’re deliberately constructed rather than spontaneously occurring forms of expression.”

In the first three steps of the Five-Step Curricular Process, the only focus is on ordinary language. The kind you, me, and everyone else across the globe speaks. The abstract formalization of mathematization is intentionally postponed for a longer period than in a traditional math classroom. However, in early stages of “people talk” what is brought to the forefront and facilitated by teachers is the interpersonal negotiation of ideas between students.

In Step One: A Shared Physical Experience, students all partake in a shared, concrete event. It can be a bus ride, a walk, a trip on the subway, or anything else with meaningful mathematical features. They are instructed to note, to whatever degree makes sense to them, the details of the event. In Step Two: A Picture/Model, students model the event in photography, drawings, or 3D. At this juncture, it is not just important that they are now making internal connections about how to connect a physical experience to a diagram, but that they do so in a way that will be understood by their peers. In Step Three: People Talk, they discuss their findings in small groups, trial-and-error whether their descriptions and models are intelligible, and work together to create a conceptual framework that can describe their experience in a way that avoids confusion.

These steps are important in cultivating ideas, giving students ownership and agency over their learning, fostering collaboration and investment in each other’s growth, and creating a space that is comfortable and culturally responsive. But it serves another purpose as well: revealing how the languages of mathematics require a shared consensus of meaning and a shared regimentation of how to use those languages.

Madeline has written and presented on Step Four: Feature Talk. She, like Quine, believes mathematicians learn to speak specialized languages wherein Feature Talk is an on-ramp between ordinary dialog and fluency in the symbol systems of mathematics.

Not herself a math teacher, she differs from others versed in the Five-Step Curricular Process as she taught herself the material through written sources, rather than through Algebra Project Professional Development Specialists. In her exploration of the history of the philosophy of mathematics, making her way from Plato to Bob Moses, she developed a particular interest in Feature Talk.

She was a presenter at the July 2022 We the People – Math Literacy for All conference, co-hosted by the Algebra Project, where she presented on the Algebra Project as a philosophy, explicating the intellectual heritage of Bob Moses who encouraged students to approach mathematics through real-world experiences. In November 2022, she published The Algebra Project, Feature Talk, and the History of Mathematics which outlines the unique intermediary language of Feature Talk as it bridges between natural languages and symbolic representations. And this July, she is organizing a panel on Artificial Languages in the History of Science and Mathematics where she will be speaking about Feature Talk at the Congress on Logic, Methodology, and Philosophy of Science and Technology.

“It’s like the philosophy of science Olympics,” she tells me, as the Division of Logic, Methodology, and Philosophy of Science and Technology organizes a conference every four years, internationally and in collaboration with hundreds of presenters.

Talking about math for an extended period can be difficult for the average person. Talking about the history and philosophy of mathematics may be even harder. For Madeline, it’s a professional passion.

In describing what mathematics is, something contemporary mathematicians still debate, she says, “There are many ways to philosophically describe mathematics and some people want to say that math is discovered, like it’s somehow out there waiting for us to uncover it, and other people want to say no, mathematics is something we make up as we go along. And in my view, these two extremes – though they’ve battled it out, pro and con, for several centuries – are not exhaustive of the possibilities because, and this is where I think what Quine said is so interesting, there’s no discontinuity between our everyday experiences and our mathematical activities. Our mathematical activities result from the refinement or regimentation of our ordinary activities.”

A formal logic classroom, she points out, has many parallels to a math classroom. “In my assignments, I always say, here’s informal English, let’s put it into formal English. Then after formal English, put it into symbols, because there’s a continuity there. It’s important for students to learn that the logical symbols are abbreviations of the formal English, not the ordinary English. One of the examples I use that would’ve made sense to Quine because it’s in first-order predicate logic, his preferred language for modeling the natural sciences, is to say “all lobsters are red.” And then to put that into formal notation, you would first say something like, “for all X in our domain, if X is an L, where Lx is “x is lobster”, then X is also an R, where Rx is “x is red.” But that’s so much longer than saying “all lobsters are red.” And yet the formal English can be condensed into a symbolic package very neatly: (x)(Lx Rx). Eventually, students begin to see how conditional statements about one-place functions do the work that categorical statements did when we were conversing in ordinary language.”

This type of universal conditional statement is common in everyday language as well as abstract mathematics. Both, ‘For all students, if you study diligently, you will get good grades’ and ‘For all real numbers X and Y, if X is positive and Y is negative, then the product of X and Y is negative’ are both examples of universal conditionals.

Madeline believes this is an example of ordinary language made extraordinary: “We domesticate the wild relations around us by being more precise in our use of ordinary language. I sometimes think of Feature Talk as ordinary language being put to extraordinary uses, or ordinary language subject to additional constraints, rules, conventions, and stipulations.”

Madeline has her own go-to example of people talk versus feature talk, “In ordinary language, you take a word like large, it’s very vague. How large is large? Depends on what we’re counting, right? Even a small number can be large depending on what we’re counting. If we find that 1% of all people have some dreaded disease, that’s too many people, that’s a large number. Whereas if we discover something like 1% of learning permit holders need to re-take their driving exam, that’s a small number.

“It’s the same number. It’s 1% in both cases, but whether it’s large or small depends on what we’re counting. And so that is part of what the regimentation is supposed to do. It’s supposed to be more precise about terms like large, that are inherently vague or have fuzzy borders.”

The importance of the Feature Talk step is not only in the benefits of regimentation but also by regimenting intuitive language, it helps make the regimentation more accessible. In being explicit about what ordinary activity or understanding is being encoded in subsequent abbreviations, the abbreviations become a tool instead of a riddle.

Madeline explains how students often already come to the table with a certain implicit understanding of science and math concepts. “One of the more famous equations, f = ma or force equals mass times acceleration, is the symbolization of something people already kind of know. If you say to very young people, would you rather be hit with a pillow or a brick? Would you rather it fall from a short height or a tall height? Their answers reflect people’s intuitive understandings of force; even though it’s a technical and scientific concept, people have an implicit awareness of how much relative oomph there is in one kind of collision versus another.”

Feature Talk can serve as the bridge between the implicit understanding of how the world works shared by young people, and the regimentation of symbolic representation and procedural computation developed to describe and explore it. But this artificial language defined by its precision can become tediously lengthy. One must make sure they use precise words but also capture all the salient mathematical features of the event in such a way that there is no room left for ambiguity or vagueness. This often leads to sentences with multiple subjects and myriad predicates. And so, in order to retain that necessary precision but shorten the expression length, symbolic abbreviations are implemented.

“Symbolism is mostly a process of abbreviation, but one that we’re led to as a consequence of being more precise. Because when you try to be more precise, you end up being more prolix, but in order to fit big ideas into small spaces, we use symbols to pack those ideas into shorter expressions.”

The symbolic representation we’re used to seeing on the first day of Algebra I classes is the last step in the Five-Step Curricular Process for a reason.

“You can teach mathematics in a top-down, axiomatic way: Here are the abstract axioms, let’s see what follows. It’s just that when you adopt that pedagogy, you’re going to appeal to a smaller subset of the student population than if you had a pedagogy that starts from concrete experience and works its way towards the abstract notations. And I think that’s important because what it suggests is that there’s not only one unique right way to teach and learn mathematics, but there are more or less effective strategies that bring more people along with the process.”

Showing students the end product of an ordinary event spoken about in ordinary language, made more precise by discussing its features, and then abbreviating those features in symbols, often does not make sense unless they too follow that same trajectory. Even students who are gifted at manipulating and memorizing the computational formulas of mathematics frequently don’t understand the concepts involved.

Madeline recalls walking across campus one day when she overheard students returning from Calculus discussing delta x as the limit goes to zero. One of the friends was insisting on how important the limit was, how his friends needed to know about it, but also how they didn’t need to understand it in order to do the homework.

“I remember thinking, oh, I wish I could stop and interview him because he’s trying to get at something really important about the relationship between concepts and computation. It is possible to learn to follow algorithms or enact procedures without having a full grasp of the concepts that underpin the procedures.

“But when you learn math that way, as arbitrary abstract rules without conceptual understanding, it doesn’t stick. And that’s one reason why somebody can graduate from high school with the scores on the test they need to get into college. But then by the time they get to college, which might be months or years later, computational rules without conceptual understanding hasn’t stuck with them.”

The purpose of the Feature Talk step then is that by retaining the continuity between People Talk and Symbolic Representation, the conceptual understanding sticks with students.

For Madeline, “Feature Talk is more precise and rigid. It’s very wordy. And this then makes it more natural for students to embrace abbreviations. Like, do I have to use this entire sentence? And so I think that, especially in formal logic, what I notice is that students can’t grasp the symbolism because they’re looking for a direct match-up from ordinary language to the symbolism. And there’s not a direct match-up because the symbolism is not an abbreviation of ordinary language expressions. There’s this intermediary language. You have to learn to speak the intermediary language before the abbreviated symbols make any kind of sense. So for me, where a lot of logic textbooks will say, ‘Here’s the symbols, put it in natural English,’ I say, here’s some natural English. We must put the natural language into an artificial language before it can be put into symbols.”

In this way, the Five-Step Process emulates the way these symbols and formulas came into existence and does not just give students the end results with instructions to memorize them.

Madeline goes on, “By starting with a shared experience and then moving from Picture or Model to People Talk, to Feature Talk to Symbolic Representation, by the time you get to symbolism, you can do the computations and have the conceptual understanding that’ll make those algorithms and procedures stick around, right? Because they will become a part of your overall understanding of the everyday experiences of things like functions, relations, properties, individuals, and variables.”

These foundational aspects of Feature Talk, which are meant to be built upon and later subjected to greater regimentation via symbolic representation, are only the beginning. They open up an intuitive pathway to an exponentially greater level of complexity.

Returning to the red lobsters, “The lobster example is a fun one to use for logic class because, first of all, most American lobsters are brown and only turn red when cooked. Second, there are blue lobsters. They’re rare. They’re an exception to the rule. The rule does not universally hold. So then we can introduce the existential quantifier and say there exists at least one X such that X is a lobster and X is blue. And then you get to show them the rare blue lobster and everybody’s like, whoa, wow, I had no idea. Because most people have no idea. But then you can get at this idea of contradictory statements, only one of them can be true: Either it’s true that all lobsters are red, or it’s true that there’s a lobster that’s not red, but not both. And similarly, if all As are Bs, if that’s true, then it’s not true that there is an A that’s not a B. And if it’s true that there’s an A that’s not a B, then it’s false that all As are Bs.

“It looks like a simple example on the surface, but it’s not because the question for all natural kinds is, are we carving nature at the joint? Or is it just a fiction that things can be carved up in this way? In terms of color perception, we know certain wavelengths correspond to what we call red, but there are also different languages for different color words that divide the visible spectrum in different ways. But the advantage of wavelengths, of course, is that they allow people from different cultures to talk about the same phenomena, independent of their natural language, which might not share the same color names.”

This gets to the heart of the Feature Talk step. Despite cultural differences, differences in perception, and different languages, Feature Talk leads to a shared modality in which people can communicate ideas scientifically, across cultural divides, and which serves a great communicative purpose.