What are the Origins of Math?
Four scientists debate ideas on whether math is an inherent part of our reality, or merely something our brains use to cope with and explain our environment.
THE RECENT DISCOVERY OF THE HIGGS BOSON PARTICLE, which was predicted by mathematical formulas, shows the power of math to describe and predict the world around us--from the helical structure of DNA and the spirals of galaxies, to how rapidly epidemics spread and our universe is expanding. But is that because everything in our world is inherently mathematical and follows precise rules? Or do we tend to see mathematical patterns everywhere because of the way our brains embroider an orderly overlay over what we experience?
The origins of math has become a hot topic of debate as neuroscientists continue to uncover mathematical abilities we seem to be born with, and have pinpointed regions in the brain responsible for mathematical thinking. Other scientists are finding that certain math capabilities vary by culture and depend on how we interact with the world. Both types of findings suggest math is a human construct rather than a phenomenon that determines how the cosmos is constructed.
To explore this debate on the origins of math and why it matters, The Kavli Foundation led a roundtable discussion with two physicists, a neuroscientist, and a cognitive scientist:
- Brian Butterworth—Emeritus Professor of Cognitive Neuropsychology at the Institute of Cognitive Neuroscience at University College London;
- Simeon Hellerman—Associate Professor at the Kavli Institute for Physics and Mathematics of the Universe at the University of Tokyo, Japan;
- Rafael Núñez—Professor of Cognitive Science at the University of California, San Diego and member of UCSD’s Kavli Institute for Brain and Mind;
- Max Tegmark—Professor of Physics at the Massachusetts Institute of Technology, and member of MIT’s Kavli Institute for Astrophysics and Space Research.
The following is an edited transcript of the discussion.
The Kavli Foundation (TKF): Let us begin by having Dr. Tegmark discuss his hypothesis that the universe is inherently mathematical in the way it is constructed and doesn’t know any other way to behave other than according to mathematical rules.
MAX TEGMARK: It’s actually an old idea. Even Galileo exclaimed that the universe is a grand book written in the language of mathematics because he was wowed by all the astronomic regularities discovered in his time, for instance the precise circular or elliptical orbits of planets. Long afterwards, a whole set of subatomic particles were predicted by mathematical principles and then discovered, the Higgs boson being the most recent example. Even the possibility of curved shape of outer space was predicted centuries earlier by non-Euclidian geometry. So nature is clearly giving us hints that the universe is mathematical. I’ve taken it to the extreme by proposing that our entire physical reality isn’t just described by math, but that it is a mathematical structure, having no properties besides mathematical properties.
TKF: How can that be?
Author of the soon to be published book, Our Mathematical Universe, Max Tegmark developed the Mathematical Universe Hypothesis that our universe is a mathematical structure. Tegmark was elected Fellow of the American Physical Society in 2012 for his contributions to cosmology, which is the study of the large-scale structure and history of the universe.
MAX TEGMARK: Someone might say that their cat has the properties of being cute, black and slightly neurotic. But a cat is actually a very elaborate arrangement of particles such as electrons, which are purely mathematical objects in the sense that they have no properties other than mathematical properties—numbers like one-half, minus one, and one, for example, that we physicists have named spin, electric charge and lepton number. If we zoom out from this microscopic scale to the largest scale of our cosmos, we realize that space itself is also purely mathematical, with no properties other than mathematical properties as dimensionality, curvature and topology.
TKF: As a fellow physicist Dr. Hellerman, do you agree with Dr. Tegmark’s mathematical universe hypothesis?
SIMEON HELLERMAN: I think many physicists, including myself, agree that there should be some complete description of the universe and the laws of nature. Implicit in that assumption is the universe is intrinsically mathematical. The discovery of the Higgs boson, which completes what physicists call the Standard Model, and others have called “the theory of everything” is a great example of this idea that there should exist some complete and mathematically consistent description of nature. Although the principles in the Standard Model hold true for atomic or subatomic particles—at the level of quantum physics--there may be inconsistencies in how they relate to fundamental physics, such as the law of gravity. But increasingly there is compelling evidence for bringing gravity into the model and still having a single internally consistent mathematical framework.
"I think many physicists, including myself, agree that there should be some complete description of the universe and the laws of nature. Implicit in that assumption is the universe is intrinsically mathematical." – Simeon Hellerman
MAX TEGMARK: Generally we feel we are taking a step forward when we get more out of a model than is put into it. The Higgs boson is a beautiful example of something that came out that we did not have to put in. Many mathematicians feel that they don’t invent mathematical structures, they just discover them—that these mathematical structures exist independently of humans.
Coauthor of the book, The Mathematical Brain, Butterworth’s research explores the neuropsychology of numbers and arithmetic, and models neural networks for arithmetic and reading.
BRIAN BUTTERWORTH: We’ve pinpointed an area of the human brain where there’s a specialist neural network that responds to counting the number of objects in a set. This area of the brain can recognize numbers across modalities. In other words, it can recognize three cats, three tones or three wishes. A similar area in the brain of the monkey does the same job. We even discovered the guppy — a small fish with a tiny brain — has one system for detecting small sets of up to four objects and one for larger sets. My argument is that we evolved a brain-based system for detecting and comparing the number of items in groups. Humans have developed symbolism for these numbers and elaborated on them to create the kinds of mathematics that Max and Simeon need to describe the universe. Numbers are not necessarily a property of the universe, but rather a very powerful way of describing some aspects of the universe.
TKF: So you disagree with Dr. Tegmark’s notion that electrons are merely numbers?
BRIAN BUTTERWORTH: Yes, because in order to have a physical explanation for phenomena, you have to have a cause for it. But how can a number be a cause? It’s true that you can use numbers to describe electron properties, but that doesn’t mean those numbers are actually a property of that physical object. Twoness is a property of a set of objects, such as two cups, or two electrons. But it is independent of the kinds of objects that are in the set for which it is a property. A set of two cups is different from a set of two electrons so twoness can’t have the same causal property for cups and electrons.
TKF: Dr. Núñez, what is your response to these hypotheses, given that your research has detected cultural differences in math abilities and suggests many mathematical principles are learned from our interactions with the world?
Coauthor of the book Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, Núñez conducts field research, psychological experiments, and neuroimaging studies to understand the human nature of mathematics and its foundations.
RAFAEL NÚÑEZ: I agree with Brian that numbers are not properties of the universe, but rather that they reflect the biological grounding for how people make sense of the world. Math is a form of human imagination that is not only brain-based but also culturally shaped— and this is crucial. It’s true that without a brain we can’t do math, but it’s also true that we need a brain to play the piano or tennis or go snowboarding. And none of these actions are genetically determined. We need a brain for all of them, but we also need a sophisticated cultural apparatus that shapes how those basic brain functions are recruited and expressed. Brain areas support the invention of mathematical principles, but these principles don’t come straight out of a particular area of the brain.
TKF: Can you give an example that supports the notion that math can be culturally shaped?
RAFAEL NÚÑEZ: Take the mathematical notion that ‘0 factorial = 1’. This ‘truth’ doesn’t exist anywhere in the universe, and it doesn’t come out straight from brain activity. But in the culture of mathematical practice, certain mathematicians realized that they needed this ‘truth’ for certain things to work out, and adopted it. In modern mathematics this is routinely done via formal definitions and axioms. These are outcomes of cultural practice — not merely conventional, but highly constrained cultural practices. In the domain of numbers, I’ve done research in remote areas of the world, such as Papua New Guinea, and in the highlands of the Andes. Some cultures operate with precise numbers concepts and others don’t have the concepts for, say, the numbers 8 or 11—their languages don’t have words that discriminate those numbers from something like 9 or 10. When you investigate these niches of cultural practices, you see some fundamental notions of number that are not present, such as precision, for example.
BRIAN BUTTERWORTH: Are you saying that math is a cultural invention, which is sort of arbitrary?
RAFAEL NÚÑEZ: No, because culture is not arbitrary. Cultural practices are constrained, among others, by the biology of the individuals that make up the culture. Speech accents, for example, are related to cultural (linguistic) practices that are not genetically determined— nothing in my genes says that my native language is Spanish and that I speak English with a Spanish accent. And humans can’t just produce any arbitrary sound they want in any frequency—because they are highly constrained biologically. So it’s not purely arbitrary.
BRIAN BUTTERWORTH: You said that if you don’t have the word for nine, you’re not going to have the concept of nine. But John Locke, the British philosopher of the 17th century, reported talking to Amazonian Indians that had no number words beyond 5. Yet, if he asked them to explain to him about larger numbers, these Indians would hold up their fingers as well as the fingers of other people present in order to show what these larger numbers were. So they had a concept of all these numbers, even though they didn’t have any words for them. Our own research in Australian cultures that have no counting words shows that if you present in a culturally appropriate way, you’ll find these children have the same concepts of numbers and arithmetic that kids brought up speaking English do.
RAFAEL NÚÑEZ: I agree that we could have an idea of a regular polygon with 103 sides, even though we don’t have a name for it. But I don’t think that this is the essence of the question. In fact, I don’t think that the origin of math is ultimately about numbers. Instead, it is much more about logical constraints, postulates and axioms, inferential mechanisms, and so on. A good accountant who does a lot of number crunching does not make for a good mathematician. Number may play a role, but is not necessarily the cornerstone of mathematics. And we have lots of different logical principles or axioms from which to choose, each of which may be internally consistent but inconsistent with others. So you can’t just say, for example, that a particular statement about infinity is true in the universe because its truth status will depend on the axioms you start with and those are concocted from the human imagination, which is mediated by language and culturally shaped. There is no inherent single form of logic in the universe. Humans operate with different types of logics in different contexts and for different purposes.
"Many mathematicians feel that they don’t invent mathematical structures, they just discover them—that these mathematical structures exist independently of humans." – Max Tegmark
SIMEON HELLERMAN: But we know that given the usual rules of logical inference, it’s possible to construct all operations involving numbers. So we can agree that whole numbers and the laws of all forms of geometries are consistent and universal, whether or not they may be realized in nature.
BRIAN BUTTERWORTH: It’s not clear that you can derive the properties of numbers from logic alone or that having an arithmetic technology is necessary for logic. It may make doing complicated shades of logical reasoning easier. In any case, formal logic is not going to turn out to be sufficient to give you any of the kinds of mathematics that we are interested in, even the relatively simple arithmetic we’re familiar with. I think formal reasoning stems from our frontal lobes of the brain and there are some axioms about numbers that come from the parietal lobes of the brain. The frontal lobe operates on these numerical concepts in order to give us what we understand as the rest of mathematics.
MAX TEGMARK: When different cultures evolve, they aren’t all going to come up with the concepts and words for all the different mathematical structures, but I think they will all come up with some of the most useful concepts. All cultures find it useful to distinguish between one and two, so they can know if they left one kid behind in the forest—ducks are really good at keeping track of how many babies they have swimming after them--whereas studying abstract algebra may not be something important to all cultures.
This 'baby picture' of our universe represents the spherical region of space from which light has had time to reach us during the 13.8 billion years since our Big Bang. Some physicists, such as Max Tegmark, think our universe is inherently mathematical and follows precise rules. (Credit: Max Tegmark and Planck Collaboration)
RAFAEL NÚÑEZ: That’s right. Beginning with Galileo’s time, the math that was created and developed became intimately intertwined with physics so it fit the phenomena humans observed in nature. For centuries now, we have been cherry-picking the math that has been useful and discarded the math that hasn’t. At this point, contemporary physics can no longer exist without the mathematics that goes with it. You ascribe the number properties as if they are in the universe, but in fact in mathematics there are all kinds of choices that have been made beforehand for that very mathematics to be what it is. For example, set theory says that the empty set is a subset of every set, even though we don’t see that fact physically materialized anywhere in the universe. Yet, we now realize that such ‘truth’ is ‘needed’ and therefore we make it true. This kind of cherry-picking has happened all over the history of math, essentially after the 19th century with the invention of non-Euclidian geometry, which altered certain postulates and axioms set previously, and with the creation of modern new logical systems.
MAX TEGMARK: The fantastic twist of this is that non-Euclidian geometry was invented almost 200 years ago when physicists thought it didn’t describe our own physical space, which they thought was flat, not curved, so two parallel lines could never cross. Then Einstein came along and after studying non-Euclidean geometry supposed space was curved and that this suggested light would bend around the Sun, which it does, and that there could be black holes, which were later found. Don’t you think it’s surprising that such mathematics could predict things in nature that we later found?
RAFAEL NÚÑEZ: Yes, at first glance it seems surprising, but when you dig in a little more you realize that not all the tools that mathematicians have invented have been useful in physics in finding new things. We humans are pretty good at trying to make sense of things and excel at developing new tools for such purposes. You are giving examples for cases in which mathematics does work apparently in nature. But, how about all those cases for which it doesn’t, including for making precise weather predictions? The saga of mathematics in science has been to invent new mathematical tools that help make testable predictions and to keep those that work, while discarding those that aren’t useful. But there are tons of other things in pure mathematics that aren’t testable or useful in empirical science proper.
"Numbers are not necessarily a property of the universe, but rather a very powerful way of describing some aspects of the universe." – Brian Butterworth
BRIAN BUTTERWORTH: What about things that can only be described using probability, such the position of an electron at any point in time. How does that fit in with your hypothesis Max?
MAX TEGMARK: Quantum mechanics famously threw that monkey wrench into the old idea of causality when it turned out there are certain experiments where you can’t say for sure what’s going to happen. But you can take a purely mathematical description, known as the Schrödinger equation, and say that it always applies to everything, so there is no random or indeterminate thing about that. It just means that the actual full reality is bigger than the reality that we can see.
TKF: Are you saying that to us it feels subjective and random, but above it all there is this order that we just can’t perceive?
MAX TEGMARK: Yes. It’s like if they put one clone of you in a room labeled A and the original you in a room labeled B. When you come out the next morning and look at your room label, you can’t predict whether you are going to see A or B because you have no way of knowing whether you’re the clone. So it’s going to seem subjectively random to you whether you come out of room A or room B. But someone who is observing both you and your clone will be able to predict that if your clone comes out of room A, than your original version will come out of room B.
BRIAN BUTTERWORTH: Understanding the origins of math is important for education. If we have an innate system that underlies much of our mathematical abilities, then things can go wrong with it’s genetic transmission in the brain, so there will be some people who aren’t going to be able to learn this arithmetic in the usual way. You have to find different ways to teach these people, just like you have to find different ways to teach dyslexics to read.
MAX TEGMARK: If math is inherent out in the universe, then mathematics can give us hints for solving future problems in physics. If we really believe that nature is fundamentally mathematical, then we should look for mathematical patterns and regularities when we come across phenomena that we don’t understand. This problem-solving approach has been at the heart of physics’ success for the past 500 years.
SIMEON HELLERMAN: I agree with Max and want to add that, in the physical sciences, the gold standard of a theory is that it predicts qualitatively new phenomena. If we thought math was so culture bound and flexible that it could describe whatever you observe—maybe there’s a Higgs boson, maybe not, and math can describe either situation on a democratic basis—then there would be a great deal in physics we wouldn’t bother doing and we never would have had the successes that we’ve had.
"Math is a form of human imagination that is not only brain-based but that is also culturally shaped— and this is crucial." – Rafael Núñez
RAFAEL NÚÑEZ: I agree with Brian that understanding the origins of math has a tremendous impact on what education could or should be. It also has implications for understanding other cultures’ beliefs and logics. Many wars are due to not understanding another culture’s logic. Logical systems embody mathematical principles that are incorporated in our legal systems and religions, both of which prescribe behavior. Understanding the origins of math will help us understand human nature better.
MAX TEGMARK: I’ve really enjoyed this interdisciplinary conversation. Perhaps the reason why Simeon and I are more gung-ho about nature being mathematical than the neuroscientists is that it’s much easier to study and mathematically describe one little electron than to study the zillions of electrons that comprise the human brain. There is beautiful complexity there and we have a lot of work cut out for us, even if nature is ultimately mathematical at the root.
BRIAN BUTTERWORTH: There are still some unanswered questions. For example, would the Higgs boson exist if there wasn't the mathematics to describe it? Perhaps this is a question best solved after a few drinks.
— Summer, 2013
Writer: Margie Patlak