|
|
By Joselle, on November 1st, 2010 I’ve become a bit preoccupied recently with the world of early 20th century mathematicians, partly because of a book I’m working on, but also because of how late 19th and early 20th century thinking largely defines the mathematics students learn today. In this light I found a book of selected writings of Hermann Weyl, a prominent 20th century mathematician. The book, Mind and Nature, was edited by Peter Pesic. One of the Weyl selections from 1932 is called The Open World: Three Lectures on the Metaphysical Implications of Science.
Weyl’s preface to the lectures begins like this:
One common thought holds together the following three lectures: Modern science, insofar as I am familiar with it through my own scientific work, mathematics and physics make the world appear more and more as an open one, as a world not closed but pointing beyond itself……It remains to be added that science can do no more than show us this open horizon….
He follows this with an analysis of perspectives that touches on the thoughts of individuals from Plato to W. Pauli and includes references to the work of some of his contemporaries. Weyl’s lectures and papers are never easy reading, but they are an opportunity. Rarely do mathematicians write about what they do.
I would like to point today to just a few of Weyl’s thoughts as they correspond to ideas I’m trying to build.
About the very nature of reality, Weyl says that it is “an error of idealism” to assume that the mind or the ego’s mental images guarantee a reality to the ego which is more certain than the reality of the external world. What he says next reminded me of the kinds of ideas in the notion of embodied cognition.
….in the transition from consciousness to reality the ego, the thou and the world rise into existence indissolubly connected and, as it were, at one stroke.
Weyl distinguishes two domains in human life, creation (or construction) and reflection (or cognition). He warns that constructive activity that is not aided by reflection may depart from meaning. Passive reflection, however, “may lead to incomprehensible talking about things which paralyzes the creative power of man.” Truth, he says, must be attained by action, “..we do not perceive it if we merely open our eyes wide…” And science cannot be done with intuitive cognition alone. He uses this idea to conclude that Bouwer’s restriction of mathematics to “intuitively cognizable truths” would not help us in the sciences. The action, to which he refers, is the mathematics that begins with axioms followed by what Weyl calls “the practical rules of conclusion.”
With respect to the infinite, the subject of much disagreement among late 19th century mathematicians, Weyl tries to put down “the experiences which mathematics has gained in the course of its history by an investigation of the infinite.” His words again:
The infinite is accessible to the mind intuitively in the form of the field of possibilities open into infinity, analogous to the sequence of numbers which can be continued indefinitely; but the completed, the actual infinite as a closed realm of absolute existence is not within its reach. Yet the demand for totality and the metaphysical belief in reality inevitably compel the mind to represent the infinite as closed being by symbolic construction.
Mathematicians who would not allow the existence of an actual infinity rejected the set theoretic work of Georg Cantor which distinguished different kinds of infinities, established equivalence relations among them, and now defines the the structure of much of modern mathematics.
Weyl makes a final point when he rejects “the categorical finiteness of man.” in both its atheistic and religious forms. He concludes that mind is “freedom within the limitations of existence” and is “open toward the infinite.” Further, “God as the completed infinite cannot and will not be comprehended by it.”
By Joselle, on October 25th, 2010 I noticed an irrepressible smile on my face as I listened to Peter Sarnak’s lecture in a complex analysis class I took in graduate school. I tried to account for the pleasure I felt about ideas I could just barely comprehend. It seemed I understood the significance of the exploration, that these abstractions were discerning realities not accessible to the senses.
This weekend I picked up The Lightness of Being by physicist Frank Wilczek. About the title Wilczek says:
A central theme of this book is that the ancient contrast between celestial light and earthy matter has been transcended. In modern Physics, there’s only one thing, and that thing is more like the traditional idea of light than the traditional idea of matter.
This statement is grounded in work done in particle physics, which the book explores. In the first few pages he makes the following remark:
But the ultimate sense-enhancing device is a thinking mind. Thinking minds allow us to realize that the world contains much more, and is in many ways a different thing, than meets the eye.
Seasonal repetitions, the motion of celestial objects, hidden regularities in harmonies, or in material in general (both organic and inorganic), are not captured by the fundamental action of the senses, but only by the thoughtful observation of pattern. And mathematics is the way we do this.
I’m convinced that my smiling in that complex analysis class was a consequence of my recognizing how far-reaching and liberating the observation of pattern and relationship can be. We do it in other things. In painting, music, poetry and fiction, we produce maps to what we cannot see directly by finding relationships in our experiences. We create what one might call emotionally defined aggregates, of the things we see and feel, and organize them with the words or images that best capture the sensations . And the precision of mathematically defined classes and equivalence relationships is the creativity that builds what we call science.
It is our nature to find our reality in relationship and not in ‘the thing itself.’ Our bodies are built for it. Mathematics helps us get behind misleading appearances by relying exclusively on discerned relationship. It admits no other kind of judgment. As a thorough investigation of relationship, it ultimately breaks through stubborn prejudices (like the work of Riemann described in my previous post). Richard Courant points to the necessary dissubstantiation of mathematically defined objects in What is Mathematics? when he says:
What points, lines, numbers “actually” are cannot and need not be discussed in mathematical science. What matters and what corresponds to “verifiable” fact is structure and relationship….
This dissubstantiation of mathematical objects is the prerequisite for finding our way. On this rests The Lightness of Being…..
By Joselle, on October 20th, 2010 My attention was brought today to a very pretty short film visualizing displays of the Fibonacci Sequence in nature on YouTube. The sequence was introduced to western mathematics in 1202. Descriptions of the film’s displays can be found here. A more complete description of the mathematical properties and applications of the sequence can be found here.
By Joselle, on October 18th, 2010 Even students of mathematics rarely have the opportunity to explore the kind of thinking that leads to ground-breaking achievements in their discipline. I was struck, very recently, by how students in my calculus class would not likely reflect on how it was possible that the tedious arithmetic they were doing (solving equations involving clumsy fractions and roots), which could appear to have nothing to do with anything, led them to a conclusion about how much human labor and capital investment would maximize the production of a particular product. Their small calculations were somehow giving them information about very worldly activity modeled by solids and curves on a grid. But this is not directly related to Riemann (perhaps Descarte and certainly LaGrange).
Yet I have often been excited by the kind of thoughtfulness that lead Bernhard Riemann to peel apart ideas that, before his approach, had no parts. We first learn about the coordinates of a space as being the way to locate a point in that space. On the plane, given the location of some central point, we can locate any other point by knowing its horizontal and vertical distance from that center. The Cartesian coordinate system definitively connects the geometric properties of objects drawn on the plane with algebraic equations that represent those objects. The relentless exploration of this mathematical space and the physical spaces of our lives led to increasingly generalized notions of space, object and dimension. There is no one line of thinking one can follow. But Riemann broke one of our most stubborn habits.
In my general searching for accounts of imaginative departures from the consensus, I found a few publications by a historian of math and science, Jose Ferreiro. In one paper in particular he says:
Many of the investigations about geometry in the 19th century, and especially on non-Euclidean geometry, were of a foundational character. Not so with Riemann: his main aim was not to axiomatize, nor to understand the new ideas on the basis of established geometrical knowledge (say, projective geometry), nor to analyze questions of independency or consistency – rather, he aimed to open new avenues for physical thought.
He is making a distinction here between the investigation of axioms in geometry like whether Euclid’s fifth postulate was independent of the others or, as had been determined, whether a different set of axioms could establish a valid non-Euclidean geometry. Instead, one might say that what Riemann did was distill geometry into its component parts.
Riemann began by criticizing traditional geometry, grounded in axioms, finding the definitions of space and constructions in space and their relations “in darkness.” He demonstrated that one could embed physical space into a more general concept which is now referred to as a manifold. This is a mathematical space that can be made to resemble a Euclidean space of a particular dimension but, in its most general form, opens the door to less restricted ideas and rich mathematical landscapes. The generality is a manifold extended by any number of components, the dimension of which is that number minus one. Euclidean examples are the line and the circle, each a 1-dimensional manifold embedded in a 2-dimensional space (the Cartesian plane) and the plane and the sphere, each 2-dimensional manifolds in a 3-dimensional space. But one can imagine a purely abstract space determined by all possible values a variable might take within certain constraints.
About these Riemann says:
It will follow from this that a [n-dimensional manifold] is capable of different measure-relations, and consequently that space is only a particular case of a triply extended manifolds. But hence flows as a necessary consequence that the propositions of geometry cannot be derived from general notions …, but that the properties which distinguish space from other conceivable triply extended manifolds are only to be deduced from experience.
It was an extraordinary insight with far-reaching consequences. My interest here is to draw attention to the unlikely yet amazingly productive impulse to extrapolate the notions of position and measure or magnitude from their geometric origin (which has its own roots in spatial experience) and in this way plant the seeds for the revolutionary ideas in both math and physics (Relativity being one among others).
A translation of Riemann’s famous address can be found here. A comprehensive look at Riemann can be found in a book by Detlef Laugwitz. I’ll end with a statement from another paper by Ferreiros:
Riemann consciously avoided the image of Reason as the a priori source of knowledge. In his view, all knowledge arises from the interplay of experience broadly conceived (Erfahrung) and ìre ectionî in the sense of reconceiving and rethinking (Nachdenken); it begins in everyday experiences and proceeds to propose conceptual systems which aim to clarify experience going beyond the surface of appearances. Reason in the old sense is found nowhere…
By Joselle, on October 11th, 2010 My attention was brought again today to cognitive scientists working in what has come to be called embodied cognition. My initiation into these ideas happened when I read the book Where Mathematics Comes From by George Lakoff and Rafael Núñez. The book explores mathematical ideas from the perspective that our bodies, living in their world, structure concepts and reason, including those found in mathematics. Or, to put it another way, the physical interaction between the body and the world largely constitutes the cognitive processes that emerge.
Recognizing the extent to which we are not conscious of what the body is doing (how we see, how we breath, why we laugh, why we sleep – to name just a few) raised the possibility, for me, that the things we think we are directing might also be driven by processes outside of our awareness. I find this prospect invigorating, a perspective that could fling open doors to understanding ourselves better and crack some of the masks we hide behind.
Only recently did I realize the extent to which this perspective is growing in cognitive science. It has roots in the phenomenological philosophy of Merleau-Ponty where the body is said to stand between subject and object, somehow existing in both. But more and more cognitive scientists are developing research to support their view. A new book by Lawrence Shapiro was brought to my attention today. The NYTimes had a related story last February. Disputes are also growing. I read an interesting post responding to a paper challenging the ideas.
Francisco Varela was one of the pioneers of the perspective and wrote about what he called autopoietic systems (derived from the words self and creation). The cell is such a system in that the production of its components is based on an external flow of molecules and energy. Then the components continue to maintain the organized bounded structure that gives rise to them. It is said that autopoietic systems are structurally coupled with their medium. This continuous dynamic in any living system is thought of as, at least, a rudimentary form of knowledge and cognition. A nice review of one of his book Embodied Mind can be found here.
Rafael E. Núñez continues to work on the study of ideas in mathematics through this lens. You can get access to some of his publications here. He makes an interesting statement in one of them:
I will argue that the dynamic component of many
mathematical ideas is constitutive of fundamental
mathematical ideas such as limits, continuity, and infinite
series, providing essential inferential organization for them.
The formal versions of these concepts, however, neither
generalize nor fully formalize the inferential organization of
these mathematical ideas (i.e., epsilon/delta definition of limits and continuity of functions as framed by the arithmetization
program in the 19th century). I suggest that these deep
cognitive incompatibilities between dynamic-wholistic
entities and static-discrete ones explain important
dimensions of the great difficulties encountered by students.
I will inevitably return to all of this. For now I would just like to add that the whole perspective is, in my opinion, compatible with mystical insights as well as analytic ones, when you consider that you can’t quite see outside your life but can become cognizant of your place within it.
By Joselle, on October 4th, 2010 I wrote not too long ago about the recording of the aftermath of particle collisions in ongoing high energy physics experiments. The post took note of the imaginative management of uncertainties (quantum mechanical uncertainties, measurement uncertainties and statistical errors). This hotbed of uncertainties is disentangled with the mathematics of probability. Mathematics here is being used to hold onto things, so that we can get a look at an extraordinarily unstable situation. It gives us a way to see events too numerous and fast for the body to sense or record.
But today, my attention was drawn to the fact that the very idea of a detector grows out of some generalization of how we see. And when a significant ‘generalization’ is effective, it will inevitably bring me back to mathematics.
There is a very nice website put together by the Particle Data Group at Lawrence Berkeley National Laboratory to communicate particle physics ideas and methods. To pave the way for the generalization of ‘seeing,’ they describe the effects of light in this way:
What we think of as “light” is really made up of billions and trillions of particles called “photons.” Photons, like all particles, also have wave characteristics. For this reason, a photon carries information about the physical world because it interacts with what it hit.
For example, imagine that there is a light bulb behind you, and a tennis ball in front of you. Photons travel from the light bulb (source), bounce off the tennis ball (target), and when these photons hit your eye (detector), you infer from the direction the photons came from that there is a round object in front of you. Moreover, you can tell by the different photon wavelengths that the object is green and tan.
Our brain analyzes the information, and creates the sense of a “tennis ball” in our mind. Our mental model of the tennis ball helps to describe the reality around us.
Note the care taken in the words “mental model.” Greater detail about how the electrons in the atoms and molecules of material interact with light waves can be found on this physics website. Whether the energy from a light wave will be absorbed, reflected or transmitted will depend on how it compares to the vibrational frequencies of the electrons in the material it hits.
It’s clear that detectors are a variant on eyes. Since the wavelength of visible light is too wide to analyze anything smaller than a cell, particles of shorter wavelengths must be used as probes. These particles are made to hit a target or other particles and, through various detector processes, the aftermath of their interactions is recorded and analyzed. The body has somehow imagined a way to mimic its eyes and see what it can’t see.
Physics has developed with the extension of conceptual landscapes (generalizations of number, space, dimension, give us calculus, non-Euclidean geometries, vector spaces and topology). And the key to the existence of mathematical objects is structure and relationship. As Courant says concisely in the book What is Mathematics?
What points, lines, numbers “actually” are cannot and need not be discussed in mathematical science. What matters and what corresponds to “verifiable” fact is structure and relationship, that two points determine a line, that numbers combine according to certain rules to form other numbers, etc
The high energy experiment’s ability to extend the range of empirical data (both what we see and how we see it) happens in much the same way. Sight in these complex configurations of material and electronics is defined by structure and relationship. Thought is driving sensation. The body’s ability to find ways to give shape to its world may be inexhaustible.
By Joselle, on September 27th, 2010 It isn’t often that the human experience of mathematics is explored in the arts, but it does happen. A Beautiful Mind and Proof are two recent examples of math-related dramas. But it seems that, by many accounts, A Disappearing Number has succeeded in weaving mathematics itself into the mystery of human lives. The play premiered in England in 2007 but was performed again this past July at Lincoln Center. I was intrigued by the reviews I read and the clips and interviews one can find on youtube. The images and words of this particular story suggest the real confluence of mathematics, the world, desire, and creativity, as the body lives. NPR did a spot about it on Morning Edition and there it indicates that a video version of the play will be shown in select theaters beginning in October.
The story being told is one which has been told before. It is about the working and personal relationship between G.H. Hardy and Srinivas Ramanujan. When I was first told about how this collaboration happened it enlivened my own love affair with mathematics because it revealed something about the power of intuition in mathematical quests. Ramanujan, an Indian clerk without working ties to the mathematics community of his time, investigated theorems from first principles and communicated some of his results in letters to Hardy, a prominent Cambridge mathematician at the time. Hardy was not the only mathematician Ramanujan reached out to, but he was the only one who recognized the significance of Ramanujan’s results despite his unconventional notation and presentation. Hardy brought Ramanujan to Cambridge and they began working together in 1914. Later, when Hardy was asked what he thought his greatest contribution to mathematics was, he said it was the discovery of Ramanujan.
The play was conceived and directed by Simon McBurney who was intent on not leaving the mathematics out of the story. It’s in the music, the dialog and the visuals. It lives in the individuals and in their relationships, a unique and refreshing perspective. About the necessity of this approach, McBurney has been quoted as saying, “When the brain gets lost, it doesn’t stop working. It tries to make sense of things. It begins to speculate and guess, and that’s when things open up. That’s exciting.”
I completely agree.
By Joselle, on September 20th, 2010 I decided to write today a little more directly about mathematics. A book I’m working on led me to review a story I like very much, the incubation and birthing of the complex number. The story has been told many times (Dantzig’s Number and Nahin’s An Imaginary Tale to name just two). But most of my students have not been taught anything about how this other kind of number came to be, or they’ve been told the wrong story.
In the book Number, Tobias Dantzig takes note of an essay written in the late sixteenth century by Franciscus Viete in which he proposes the use of vowels to represent unknown quantities in an algebraic expression, and consonants to represent the given magnitudes (Descarte later revised the idea so that x,y,z are used for the unknowns and a,b,c for the givens). Merely writing down these numberless expressions, allowing them to be, Dantzig suggests, freed the algebraist of their prejudice for only natural number solutions to problems. Without any forsight, the path to the refinement of the number concept was opened.
But more revealing of the kind of blind searching we never associate with mathematics is the story of the complex number and the imaginary unit. Before the logical foundation of the real number was even close (a 19th century development), a path to the complex number was opened. A very short version of the story is this: The 16th century mathematician, Scipione del Ferro found a formula for the solution to a depressed cubic equation (one with no x squared term). He could use the formula (which involved a number of roots) to find the one real solution to the cubic. A short time later, Girolamo Cardano (known as Cardan) found a way to extend this idea to the solution of all cubics. Unlike del Ferro, when Cardan worked with the square roots in the formulas, he allowed himself to work with negative numbers inside the root. He is quoted in Rudin as saying, “Putting aside the mental tortures involved,” everything works out. And later, “So progresses arithmetic subtlety the end of which, as is said, is as refined as it is useless.” The mathematician Rafael Bombelli (also 16th century) took note of the role complex conjugates were playing in finding roots, and proceeded to develop rules for operations on these yet to be understood thoughts. One could say he found the complex number domain, a territory that would remain uncharted for some time (a bit more than 200 years). In 1673 John Wallis began a geometric interpretation of the numbers but it was not until 1797 that Caspar Wessel sees clearly what they are.
Dantzig says the following about the development of mathematical ideas:
Distant outposts were acquired before the intermediate territory had been explored, often even before the explorers were aware that there was an intermediate territory. It was the function of intuition to create new forms; it was the acknowledged right of logic to accept or reject these forms, in whose birth it had no part. But the decisions of the judge were slow in coming, and in the meantime the children had to live, so while waiting for logic to sanctify their existence, they throve and multiplied.
By Joselle, on September 13th, 2010 The hoopla about Hawking’s new book made it frustratingly clear that there’s a real impasse in the centuries-old debate over whether science, and its intrinsic rationality, can or should definitively dispute religious ideas. The impasse is, I believe, a consequence of our not seeing the elephant in the room, namely ourselves. The debate proceeds, as if we can make a simple objective evaluation of thousands of years of human experience without ever reflecting back on our own cognitive and cultural development.
Religious systems do contain creation stories and interpretations of worldly events, but the intent of these stories is to structure our relationship to everything which is ‘not us’ or to guide our interaction with ‘all that is.’ The nervous system guides our day-to-day existence: breathing, movement, finding nourishment, digestion, protection. Yet components of this system have evolved, and cognitive systems have emerged that fill us with lively internal images, language and story. The body builds these mental images from the same stuff that finds it food and shelter and, I believe, it may be doing this to become more aware of what and where it is.
In the recent history of our species, new mental images emerged in science and mathematics. It’s like we grew more eyes and they can see other galaxies, earlier times in the universe. The creature that we are keeps looking. Why? What is it looking for? I think this question deserves very careful and perhaps fresh consideration. It is more relevant to a discussion of God than is the completeness or lack thereof of physic’s analysis of material.
The body has extended its sensory limits with physic’s and mathematics, by allowing the development of complex cognitive mechanisms, which create new conceptual possibilities, that transport us out of our time and space. These neural structures have grown outside of our awareness and are also working to discern our relationship to the whole (to which as creatures, we must intuitively connect). The fruits of these new visions should not be used to indicate that our quest for relationship is immature.
We still know very little about how material and awareness are related, or how material gives rise to our conceptualization of it. We can’t see yet how consciousness grows out of the organization of atoms which we deem to be not consciousness. But we need to wonder about it, reflect on it. And perhaps we can correct some of the mistakes of judgment we may still be making about how objective we can be or what it means to know.
By Joselle, on September 6th, 2010 I was impressed when my husband, who participates in one of the experiments at the Large Hadron Collider in Geneva, first recounted the discovery of the top quark at the Tevatron Particle Accelerator in Batavia, Illinois. He told me about it shortly after we met, in the summer of 1998. I had already read some popular books about physics, but none of them had given me a look at what physicists actually mean when they say, with confidence, that they’ve seen the particle they were hunting down. Seeing the top quark required, among other things, a consensus among hundreds of experimentalists about how to sort out the electronic buzz from a highly sensitive 4-story tall detector. The buzz is created by millions of particle collisions per second. Software and hardware systems have been designed to permanently record the electronic effects of 200 of those collisions per second. In fact, the sighting of an elusive particle relies on this enormous accumulation of data because, in the end, it is an analysis of various probabilities that determines whether or not something has been observed. Distributions from data can best be compared to expectations when the numbers are high (a few million flips of a coin will get you heads 50% of the time). Then there are the collaborations and disputes over how to do the analysis of the data, how to identify the presence of a particular particle, how to draw a conclusion from what they know about the probabilities.
There are probabilities everywhere. They begin, perhaps, with the fact that in quantum mechanics, the very appearance of a particle is probabilistic. The quantum world is not deterministic. There is no unique outcome to some given set of initial conditions. Particle interactions can only happen probabilistically. But there are also the probabilities that data will conform to a particular distribution or that the energy recorded is just noise from the detector itself and not from the collision of particles. Data plots are the way measurements are communicated. Simulated collisions (aptly called Monte Carlos) are also created to establish probabilities.
There can be disagreement over a myriad of considerations, how quickly a particular analysis can produce a publishable result, will the method itself produce errors or are the errors mostly statistical errors, how reliable are the simulations.
But, after all, we’re trying to get a glimpse of the moments after the Big Bang. “What we’re trying to do is almost impossible,” my husband once said concisely. And inside the little bit of space provided by that almost, experimentalists have managed to design an experiment, that can somehow hold steady what is known in order to isolate and test what is not known, even when that happens to be the conditions in the early universe. They manage it with extraordinary control over material, the material of the detector, the material of their computers, and partner this with a powerful conceptual development – probability and statistics.
The Tevatron and the Large Hadron Collider (LHC) are now in the race to find the Higgs particle. This is the big one. It’s the particle that would account for the mass of truly fundamental particles. It is the only standard model particle that has not been observed and, finding that it isn’t there, would mean abandoning the model. I plan to follow up with more on the Higgs and more on probabilities.
|
|
Recent Comments