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By Joselle, on May 30th, 2011 It happens many times in class that I say, “in mathematics when you see something you don’t know, you try to figure it out using something you do know. And, recently, in the context of thinking about the generalizations that blossomed in late 19th and early 20th century mathematics, I’ve also wondered how it is that we stay ‘on track’ so to speak. How is it, for example that Riemann’s foundation for geometry holds onto the working properties of the original ideas?
I think that both of these things are partially addressed by recent research in cognitive science. In this post, I’d like to bring your attention to the work of a team led by Josh Tenenbaum, at MIT. It was recently reported that this team made the observation that babies reason – that they will expect an outcome of a situation based on a few physical principles.
These experiments make use of a tool that has been developed to measure a baby’s surprise, because this provides a way to identify a baby’s expectations. Their surprise (or lack of it) is measured by the amount of time they look at something. In other words, the baby will look at something longer when an unexpected thing has happened. These times have been carefully and repeatedly recorded. As Tenenbaum has said, researchers have quantified surprise. The same kind of tool has been used to measure a baby’s number sense. But this particular study identifies reason based on principles, before there is language. According to Tenenbaum, the study
suggests infants reason by mentally simulating possible scenarios and figuring out which outcome is most likely based on a few physical principles.
The report made me want to look more at Tenebaum’s work. I found a link to his recent article in Science: How Minds Grow. The link is on his web page under the heading Representative reading and talks. The article begins with a question that I think can be applied directly to mathematics: How do our minds get so much from so little? But the article is a complex analysis of how we build our conceptual structures on probabilities. Early abstractions develop when we order the distinguishable features of a perceived object, and these develop with experience, opening vast territories of knowledge, with what Tenenbaum calls a hierarchical Bayesian framework. The body builds its world based on probabilities related to experience or evidence. And conceptual systems grow in tree-like structures. The full content of the article is beyond the scope of this post. But the work contributes to the current view that perception and understanding happen together, that their interaction is seamless. Abstraction is a fundamental aspect not only of vision but of learning and all aspects of the body’s interaction with its environment. Perhaps the body ‘knows’ how to use abstraction the way it knows how to use light, for example.
In the conclusion of the article, Tenenbaum says the following:
How can structured symbolic knowledge be acquired through statistical learning? The answers emerging suggest new ways to think about the development of a cognitive system. Powerful abstractions can be learned surprisingly quickly, together with or prior to learning the more concrete knowledge they constrain. Structured symbolic representations need not be rigid, static, hard-wired, or brittle. Embedded in a probabilistic framework, they can grow dynamically and robustly in response to the sparse, noisy data of experience.
These observations give me a way to think about how mathematics stays on track and why intuition can play so crucial a role. It may be that mathematics searches the paths taken by concepts (first rooted in the body’s managing physical experience with abstract principles) or just searches the possibilities for concepts, within the constraints (or principles) the body knows. The ‘intellect’ and the senses are here united in a provocative way.
By Joselle, on May 23rd, 2011 Some of George Berkeley’s fame comes from his vehement critique of Newton’s calculus. His criticism was harsh and inspired a number of responses from contemporaries who accepted the vanishing quantities Newton used to formulate his notion of fluxions or, in modern terms, his understanding of instantaneous rates of change. The discussion that followed Berkeley’s 1734 publication (entitled The Analyst; or a Discourse Addressed to an Infidel Mathematician) is worth looking at. An index of publications related to the controversy (including Berkekey’s Analyst can be found here).
It is generally accepted that Berkeley’s criticism actually contributed to mathematics by motivating later work that was devoted to giving rigorous meaning to calculus concepts. These efforts finally culminate in Weierstrass’s arithmetic definition of a limit (discussed in an earlier post). But today I want to look more at Berkeley. I find it interesting that he was willing to systematically dismantle objects into their sensory components, and imagine a world built from ideas, but at the same time was reluctant to reconsider his expectations of mathematics – ideas that can give the senses greater reach.
In Book 1 of his Principia Mathematica Newton makes the following claim:
Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer to each other than by any given difference, become ultimately equal.
This is his description of a limit, the fundamental notion that gives calculus meaning. An instantaneous rate of change or fluxion, as Newton called it, is an application of this idea.
And this is what Berkeley had to say about it:
These Expressions indeed are clear and distinct, and the Mind finds no difficulty in conceiving them to be continued beyond any assignable Bounds. But if we remove the Veil and look underneath, if laying aside the Expressions we set ourselves attentively to consider the things themselves, which are supposed to be expressed or marked thereby, we shall discover much Emptiness, Darkness, and Confusion; nay, if I mistake not, direct Impossibilities and Contradictions.
Berkeley was a prominent philosopher, and a religious man who relied heavily on reason. He trusted geometry. With geometry he says:
there is acquired a habit of reasoning, close and exact and methodical: which habit strengthens and sharpens the Mind, and being transferred to other Subjects, is of general use in the inquiry after Truth.
His difficulty with fluxions was mathematical not philosophical. He didn’t trust them. For Berkeley, fluxions overstepped the close and exact method of geometry and analysts like Newton were getting away with it. His disapproval is clear:
Whereas then it is supposed, that you apprehend more distinctly, consider more closely, infer more justly, conclude more accurately than other Men, and that you are therefore less religious because more judicious, I shall claim the privilege of a Free-Thinker; and take the Liberty to inquire into the Object, Principles, and Method of Demonstration admitted by the Mathematicians of the present Age, with the same freedom that you presume to treat the Principles and Mysteries of Religion; to the end, that all Men may see what right you have to lead, or what Encouragement others have to follow you.
He objected to the material-based views of the world (and the determinism they implied) that are often traced back to the success of classical mechanics (a view abandoned by modern physics).
But for Berkeley, science was not the problem. The problem was taking an abstraction too far, to the point where it seemed to contain nothing.
The further the Mind analyseth and pursueth these fugitive Ideas, the more it is lost and bewildered.
Yet his philosophical writing lines up well with some very modern developments in science – like his idea that objects are not actually things in themselves but more bundled sensations. He may have used this observation to justify the existence of God, but it is consistent even with current neurological perspectives. He makes the unexpected argument that the things in our physical experience are not made of material but rather ideas:
By touch I perceive, for example, hard and soft, heat and cold, motion and resistance, and of all these more and less either as to quantity or degree. Smelling furnishes me with odours; the palate with tastes, and hearing conveys sounds to the mind in all their variety of tone and composition. And as several of these are observed to accompany each other, they come to be marked by one name, and so to be reputed as one thing. Thus, for example, a certain colour, taste, smell, figure and consistence having been observed to go together, are accounted one distinct thing, signified by the name apple.
Berkeley’s psychological insights made him something of an idealist. The world comes to be with ideas, visual and tangible ideas, ideas of taste or smell, “imprinted on the senses,” and others formed with the help of memory and imagination. The physical world of objects was mind-dependent. Real objects were distinguished from imaginary or hallucinatory ones only by their regularity. And science investigated that regularity.
Yet the same inquiring mind rejected the mathematics that drove the development of very productive new thoughts. In time, mathematicians came to conclude that mathematical objects were not things in themselves, but existed only in structure and relationship. The deterministic view of the universe, encouraged by the success of calculus, has been completely abandoned. Physicists now see our physical reality as alive with spontaneous, unpredictable quantum activity that can only be understood in interaction. Perhaps it can be said that mathematics has just extended the range of visual and tangible ideas.
From physicist Frank Wilczek in The Lightness of Being:
We try to find mathematical structures that mirror reality so completely that no meaningful aspect escapes them. Solving equations tells us both what exists and how it behaves. By achieving such a correspondence, we put reality in a form we can manipulate with our minds.
By Joselle, on May 16th, 2011 I’m a little pressed for time this week so I thought I would try to provide some fun links.
Steven Strogatz, mathematician and writer, speaks on a radiolab broadcast about an early insight. It was in a high school math class where he says he was being taught how to use graph paper. The teacher gave them each a pendulum, whose length was adjustable. They were asked to choose a length, then swing the pendulum and plot the time it took for it to complete 10 swings. The students would then increase the length and plot the time again. As Strogatz plotted the times, he saw the parabola emerge and was wonderstruck. He tells the story on Radiolab’s The Wonder of Youth which also discusses the surprising order of the periodic table.
Radiolab produced a video of parabolic images inspired by Strogatz’s early experience that can be found here. The visual of the fingers, the clicks of the stop watch, the swinging pendulum, and penciled points on graph paper, help drive the point home. You see the parabola, the hidden image, emerging out of the order given to shapeless and distinct events.
Perhaps it was the direct experience of actually doing the distinct things – swinging, counting seconds, and plotting dots – that brought depth to Strogatz’s observation. He had already learned about parabolas in algebra, and now it seemed like this inanimate object produced one for him. It was then, he tells us, that he grasped the meaning of ‘a law of nature.’ And it was mathematics that could illuminate nature’s invisible structure. The simultaneous use of distinct ideas – ordered magnitudes, time, and defining the location of a point – all of them subjective, cast this light on the pendulum.
Another radiolab broadcast (Desperately Seeking Symmetry) centers around observations of mirror images (of which there are many in mathematics). It turns out that molecules have mirror images and, using our hands as a model, it is often said that there are left and right handed molecules. While not the same, these molecules are mirror images of each other, just like with our hands and so, not surprisingly, this property is often called the handedness of a molecule. All of the matter around us seems to contain a 50/50 split of right and left-handed molecules. Yet all living molecules are left-handed, an interesting mystery, and one perhaps anticipated by Alice when she wondered about the looking glass. She asked “is mirror milk any good to drink.” It likely wouldn’t be if it existed because changing the handedness of a molecule can lead to bad tastes and dangerous toxins (from another broadcast: Mirror Mirror) There are illustrations of the handedness of a molecule at this web site.
The symmetry of a reflected image is how light is vision. Yet, like the plotted times of the pendulum swings, a conceptual re-presentation of it gives us a way to perceive profound structure otherwise unavailable to the senses. Conceptual symmetry exists by virtue of experienced symmetries or (like an object and its mirror image) is one with its physical counterpart. Here’s a nice symmetry video, also from Radiolab.
By Joselle, on May 9th, 2011 I have spent some time pointing to milestones in the history of modern mathematics where a conceptual shift produces provocative new thought – as when Riemann gave a new foundation to geometry, or when Cantor brought precision to the notion of countability. Modern mathematics, partnered with physics, increasingly refines what the human mind can perceive. In this light it is easy to understand the nineteenthcentury German physicist Heinrich Hertz (first to detect electromagnetic radiation) when he said:
One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them.
Frank Wilczek found this worth repeating in The Lightness of Being when he discusses the potential implications of metric field mathematics in physics. He explains that general relativity can be described using either of two mathematically equivalent ideas: curved space-time or metric field. He refers to curved space-time as the geometric version of the field view. But the field view, Wilczek explains, makes Einstein’s theory of gravity look more like other theories of fundamental physics and the hope is that its use will help usher in the sought after unification of the four physical forces or interactions. But I have also thought that the role of something much simpler, something like trigonometry, can illustrate the way, as Wilczek puts it, “our concepts are not just our products but also our teachers.”
Trigonometry is grounded in the simple idea of ratios among lengths, in particular, of line segments forming right triangles. The properties of these relationships were documented early in our history, in ancient Greek mathematics and in early Hindu astronomy. Trigonometric thoughts often start with the circle and Greek mathematicians studied the properties of the angles opposite various chords of a circle. They proved theorems that are equivalent to modern trigonometric formulas. The modern sine function was also defined in an ancient Indian treatise using the shadows cast by a sundial.
Various facts and relationships among these ratios were calculated and widely used in the analysis of the ground we walked on, as well as distances in the sky. Most of us begin learning the fundamental ancient facts in school, using the sides of right triangles and those familiar words: opposite over hypotenuse, adjacent over hypotenuse, and opposite over adjacent.
But the growing idea of a function, a rule of correspondence between two values, which became more and more sharply defined in the seventeenth and eighteenth centuries, took hold and became the general way we looked at relations. Trigonometric ratios are now understood as functions of real numbers, where the radian measure of an angle is paired with the sine, cosine or tangent value of that angle. The idea is introduced in textbooks with the unit circle, where we imagine wrapping the real number line infinitely many times around the circumference of the circle. The x,y coordinates of the points on the circle, hit by the ray which describes the angle, correspond to the cosine and sine values of that angle. These become the y values of the sine and cosine curves when they are plotted as waves.
When the circle is drawn on the complex plane, the same kind of configuration describes the relationship between complex numbers and trigonometric ratios, and this exposes the relationship between the exponential function and trigonometric relationships. Circles, circles, and more circles. The fact that other functions could be represented by a trigonometric series (a sum where each term involves a trigonometric function), has also had a major effect on analysis and physics and found application in the pure mathematics of number theory. In physics, solutions to Schrodinger’s equation, an important differential equation describing non-relativistic quantum mechanics, often involve sines and cosines.
Trigonometric ideas clearly contained more than they were initially used to describe. Their relationship to the circle kept unfolding and their expression of periodicity, together with their analytic development, continues to support even the most modern ideas in physics. There can be no doubt that we got more out of them than we originally put into them. This is just a small example of the way a purely conceptual object may have its own nooks and crannies, ones that actually contain hidden jewels. It raises intriguing questions about the nature of mathematical thought, how it happens, and how it captures more than is immediately apparent even to its authors.
By Joselle, on May 2nd, 2011 I had the opportunity to attend a talk given by Frank Wilczek, Nobel laureate in physics and author of the book The Lightness of Being. During the Q and A after the talk he was asked if our aesthetic judgment of symmetry could be said to prejudice scientific inquiry. Wilczek first pointed to the rich benefits of our search for symmetries. He then added that he saw our taste for symmetry as a strategy more than a judgment. I enjoyed the use of the word strategy. He went on to say that Nature seems to have good taste or that Nature’s taste seems to agree with ours (I’m paraphrasing but I think I got it right). This choice of words triggered a series of associations, for me, concerning the weave of biology and culture, or of biology and the complex conceptual objects we now perceive.
One of these associations was with something neuroscientist Samir Zeki has said about visual art. Zeki understands the purpose of vision itself as the acquiring of information about the world. And he has concluded that since “a search for the constant, lasting, essential and enduring features of objects, surfaces, faces, situations, and so on,” is what allows us to acquire knowledge, it follows that one of the functions of art, (which is often characterized as finding the essential enduring features of a perceived world) “is an extention of the major function of the visual brain.” The full paper, Art and the Brain, can be found on the Neuroesthetics Institute web site.
Symmetry is a specific concept or property of objects. And I found a paper that explores the history of our awareness or use of it. In The First Appearance of Symmetry in the Human Lineage: Where Perception Meets Art (Derek Hodgson, Department of Archaeology, University of York) the author also refers to work that finds symmetry detection in nonhuman primates. He tells us, for example, that it has been demonstrated:
that monkeys, raccoons, and birds prefer symmetry to asymmetry and irregularity. This may be related to the fact that symmetry is an enduring aspect of the visual world that has been incorporated into the underlying neural capacities of the brain in terms of capturing the non-accidental properties of the world…..Thus, the preferred response of neurons may have originally derived from the fact that many of the objects that need to be detected are themselves symmetrical….. In addition, most biologically important objects are symmetrical and, in this regard, sensitivity to symmetry may have evolved because it is crucial for discriminating living organisms from inanimate objects. The existence of fractals in natural scenes also reflects the widespread existence of symmetry in nature. Symmetry therefore appears to provide a useful means by which the visual world can be encoded for the purpose of efficient recognition.
Mathematics, of course, brings its own characteristic precision to the definition of symmetry, but it remains tied to fundamental ideas of sameness, balance or regularity. Wilczek is very clear about the power of mathematical symmetry in one of the chapters of The Lightness of Being called Symmetry Incarnate:
If we know an object has symmetry, we can deduce some of its properties. If we know a set of objects has symmetry we can infer from our knowledge of one object the existence and properties of others. And if we know that the laws of the world have symmetry, we can infer from one object the existence, properties, and behavior of new objects…Thus symmetry can be a powerful idea, rich in consequences. It’s also an idea that Nature is very fond of.
Later in the book, Wilczek also describes physics theories (and their mathematics) as data compression.
The goal is to find the shortest possible message – ideally a single equation – that when unpacked produces a detailed accurate model of the physical world.
This is analogous to what he refers to as the compression of floods of sensory data out of which we construct “small representations of the world adequate to function in it.” The more compressed, profoundly simple equation we have, the more complex the calculations that unfold them and the “ever richer output that the world turns out to match.”
In general, these observations are telling us that in art, mathematics and science, we seem to be able to elaborate on what the body is built to do, making more (much more) available to the senses. Neural mechanisms that permit our functioning in the world are enhanced, somehow, and recruited to explore the world beyond our immediate needs. Again, according to Wilczek:
We try to find mathematical structures that mirror reality so completely that no meaningful aspect escapes them….By achieving such a correspondence, we put reality in a form we can manipulate with our minds.
Philosophical realists claim that matter is primary, brains (minds) are made from matter, and concepts emerge from brains. Idealists claim that concepts are primary, minds are conceptual machines, and conceptual machines create matter.
Finding mathematical structures that mirror reality completely make this two sides of the very same truth, and Wilczek agrees.
By Joselle, on April 25th, 2011 The essence of mathematics lies precisely in its freedom. This statement from Georg Cantor is quoted so very often, and perhaps this is because of the surprise coupling of the words mathematics and freedom, or because of the implications of the word essence, which calls to mind other words like intrinsic, inherent or something that is by nature. Various biographical accounts of Cantor will explain that, for him, the statement had meaning on more than one level. It does characterize his own experience with mathematics but it is also an admonition to traditionalists who might thwart the creativity of new ideas. I found a very nice discussion of Cantor’s battles with his contemporaries in a paper by Joseph W. Dauben on The Battle for Transfinite Set Theory. He says about Cantor’s statement:
This was not simply an academic or philosophical message to his colleagues, for it carried as well a hidden and deeply personal subtext. It was, as he later admitted to David Hilbert, a plea for objectivity and openness among mathematicians. This, he said, was directly inspired by the oppression and authoritarian closed-mindedness that he felt Kronecker represented, and worse, had wielded in a flagrant and damaging way against those he opposed.
There is no room in mathematics or an ‘authoritarian closed-mindedness.’ The history of this intellectual conflict, as Dauben among others presents it, is very interesting and shows us something about the hurdles mathematics had to clear before it could take us further. The philosophical message is consistent with my own early experience with mathematics, where I found the boundless reach of its conceptual possibilities invigorating and reassuring. I continue to wonder about how we find those useful generalities from particulars– like the real number from quantity, equivalence from equality, manifold from space, n-dimensional from 3-dimensional. The process can be very tedious and laborious, involving many minds and many centuries. The exploration of relations among ideas is enough to capture the full attention of the pure mathematician. As Alain Connes has said:
Investigating these, one truly has the impression of exporing a world step by step….
Their utility is the focus of the applied mathematician. Mathematics seems to carve out a path that leads from the world of substances (where thoughts find relationships among things we experience) to the world of purely thoughtful relationships (which often reveal something new about the things we experience).
Cantor’s observation about the difference between a countably infinite collection and a non-countable infinite collection is grounded in a very thoughtful analysis of something that is not found in our physical experience, yet is completely tied to the very concrete experience of counting. For many of his contemporaries his work was laughable or, worse, destructive. But his analysis of the infinite relies on something fundamental and at the same time submits to the requirements of logic. This is probably the source of its utility.
In the book Labyrinth of Thought, Jose Ferreiros Dominguez looks at the history of set theory and gives us this from Cantor:
By a manifold or a set I understand in general every Many that can be thought of as One, i.e., every collection of determinate elements which can be bound up into a whole through a law…
This is the notion of a class of things, an aggregate, that has penetrated and influenced many fields of mathematics. These collections of objects, also objects themselves, describe a new starting point and make an analysis of the infinite possible. The idea that the collection is bound up into a whole through a law refers, I think, to the logical necessities that give the theory of sets meaning.
The essence of mathematics is freedom perhaps because it formalizes thought processes with logic and deduction, while letting the intuition play freely with metaphors. This has the remarkable affect of allowing the imagination to perceive beyond the limits of the senses. This sounds much like something Hermann Weyl, one of Cantor’s critics, once said:
We stand in mathematics precisely at that point of intersection of limitation and freedom which is the essence of man himself.
By Joselle, on April 18th, 2011 I would like to go back today to Riemann, and the significance of his generalized notions of space and magnitude, but with an eye on what neuroscience may be adding to how mathematics gains its effectiveness.
In a recent post, I pointed to the influence the philosopher Herbart had on Riemann’s 1854 lecture in which Riemann proposed the most generalized notions of magnitude and space. I pulled the following excerpt from a book by David Cahan on the foundations of nineteenth century science:
Herbart argues that each modality of sense is capable of a spatial representation. Color could be represented as a triangle in terms of three primary colors, tone as a continuous line, the sense of touch as a manifold defined by muscle contractions and still other spaces as associations of hand-eye movements. “To be exact,” he wrote, “sensory space is not originally a single space. Rather, the eyes and the sense of touch independently from one another initiate the production of space; afterward both are melted together and further developed. We cannot warn often enough against the prejudice that there exists only one space, namely, phenomenal space.” Therefore, for Herbart, “space is the symbol of the possible community of things standing in a causal relationship.” He insisted that for empirical psychology space is not something real, a single container in which things are placed. Rather, it is a tool for representing the various modes of interaction with the world through our senses.
In a 2005 article Edward Hubbard, Manuela Piazza, Philippe Pinel and Stanislas Dehaene write on the interactions between number and space in the parietal cortex. The authors conclude:
In the more distant future, it might become possible to study whether more advanced mathematical concepts that also relate numbers and space, such as Cartesian coordinates or the complex plane, rely on similar parietal brain circuitry. Our hypothesis is that those concepts, although they appear by cultural invention, were selected as useful mental tools because they fit well in the pre-existing architecture of our primate cerebral representations. In a nutshell, our brain organization both shapes and is shaped by the cultures in which we live.
I did a little looking into the brain circuitry being talked about and found that a neuronal analysis of the relatedness of number and space in our experience strengthens Herbart’s perspective and could suggest something about the nature of Riemann’s insight. I mean to be careful here. I don’t want to be sloppy about my references to either the mathematics or the neuroscience, but I find it worth pointing to some interesting correspondences.
Quite a lot of work has been done on the spatial aspect of number and some of this is described in the 2005 article. In a 2009 article, Domenica Bueti and Vincent Walsh explore the parietal cortex and the representation of time, space, number and other magnitudes and suggest the following:
Imagine you are Darwin for a day and you are charged with granting a species the ability to count. Where would be the most efficient place in the brain for this discrete numerical system? The parietal cortex is already equipped with an analogue system for action that computes ‘more than–less than’, ‘faster–slower’,‘nearer–farther’, ‘bigger–smaller’, and it is on these abilities that discrete numerical abilities hitched an evolutionary ride.
The work of the parietal cortex is discussed in a 1995 issue of The Neuroscientist (by Charles Gross and Michael Graziano of Princeton). A number of things come together in the posterior parietal cortex:
In summary, the posterior parietal cortex is where vision, touch, and proprioception come together for the first time. It is the hub of a system for the processing of spatial information. This system includes not only several regions within the parietal cortex….but a widespread network of other cortical and subcortical areas, including the ventral premotor cortex, the putamen, the frontal eye fields, the superior colliculus, the hippocampus, and principal sulcus. These areas are specialized for a variety of different spatial functions, such as visuomotor guidance of limb, eye, and head movements, navigatinig in the external environment and holding recent memory about the location of objects in space. They appear to carry on, in specialized fashions, the processing of information about space that is begun in the parietal cortex.
What I find most striking in reading the literature is the extent to which the body creates structure out of multi-sensory interaction. And what we call magnitude, space and number emerge from these multi-sensory interactions. This is not, I think, dissimilar from Herbart’s insight. Brain circuitry handles the body’s need for orientation, the perception of distance, speed, size and quantity interactively, with both internal and external sensory systems. Much of this work means to understand the evolution of the conceptual side of our nature, our idea-driven experience. One of the conclusions of the Bueti/Walsh article is this one:
…it was suggested that different magnitudes originated from a single developmental algorithm for more than–less than distinctions of any kind of stuff in the external world. The development of magnitude processing proceeds by interactions with the environment and is therefore closely linked with the motor reaching, grasping and manipulating of objects. It was further suggested that the emergence of our ability to manipulate discrete quantities evolved from our abilities with continuous quantities.
Riemann was not engaged in a psychological analysis of the evolution of concepts, but he was interested in the essence of ideas which appeared to be experience driven. And the greatest generality would be found in the rendering of their most fundamental attributes. I would like to suggest that his careful analysis of what one could mean by magnitude or what one could mean by space holds within it remarkable insights into interactive chains of cognition itself. But these can be found in the purely abstract investigation of mathematical worlds.
By Joselle, on April 11th, 2011 I don’t think it’s actually possible to answer the question in the title of this post, but I still believe it’s worth asking. We’ve thought of things ‘hidden under a microscope,’ or obscured by great distances, but in mathematics when something is hidden, it’s because we haven’t been able to imagine it yet. And when we do, we often get there by following the implications of previously imagined possibilities. The way that mathematics can direct our imagination is certainly demonstrated in the history of the complex number. It is worth looking at how they were first imagined and then how far from this came their significant impact.
Let’s start at the end, so to speak, with what Riemann had to say about complex numbers:
The reason and the immediate purpose for the introduction of complex quantities into mathematics lie in the theory of uniform relations between variable quantities which are expressed by simple mathematical formulas. Using these relations in an extended sense, by giving complex values to the variable quantities involved, we discover in them a hidden harmony and regularity that would otherwise remain hidden.
This I took from the book Riemann, Topology and Physics by Michael I Monasyrsky.
Now lets go to the beginning. As a purely numerical manipulation, the square root of a negative number was first written down to demonstrate the impossibility of splitting the number 10 into two parts, the product of which was 40 (shown to be 5 + the square root of -15 and 5 – the square root of -15). However, after it was written down, Italians of the Renaissance found that they could actually use these strange expressions to find real solutions to cubic equations.
As late as the early 19th century, many mathematicians could not accept the use of these numbers. The general sense was that mathematics was not the study of conceptual truths but of real truths, ones that could be thought of as constructions of intuition. In this regard, Gauss defended the complex number by saying “the arithmetic of the complex number can be given a most intuitive representation.” (See, among other histories, The Shaping of Arithmetic by Catherine Goldstein).
There is a very nice animation of the way the value or meaning of the complex number can be hidden here.
Gauss proposed a more abstract theory of magnitudes where one measured not substances but relations among substances. Riemann’s study of functions of a complex variable led him to describe what we now call a Riemann Surface and likely motivated Riemann’s most general notion of a manifold. Riemann’s notion of manifold also gave new definition to the ideas of space, measurement, and geometry. Many would agree, his reworked foundations for geometry set the course for modern mathematics. And all of these realizations are clearly manifest in the significant role that the complex number plays in physics and engineering.
This is the story of the relentless unwrapping of an imagined object, and finding what was yet hidden – like the complex roots of a polynomial, or the properties of functions of a complex variable and their relationship to real valued functions, or the Riemann Surface and the shapes that space can take.
The relationship between the simple idea of number, that we all grasp quickly, and these unfamiliar but powerful conceptual worlds is completely unexpected. We are digging into a cognitive invention and finding more than we seem to have put there. Where was the hidden hidden? This must have something to say about the nature of what we call cognition.
By Joselle, on April 4th, 2011 Mathematics today can seem an isolated discipline, removed from the questions of life and questions of meaning. But even a brief look at some of the writing of individuals like Leibniz, Weyl, and Poincare demonstrates substantial interest on the part of the mathematician to reconcile mathematics with common human experience. I remember one of my teachers in graduate school telling me that he was grateful to be able to do mathematics because it gave him the opportunity to transcend. I can no longer place this remark in its context, but I have always remembered it because, for me, mathematics does lead beyond appearances.
Today, neuroscience is the place where questions of perception and consciousness are addressed. And, in the other direction, neuroscientists and cognitive scientists have addressed questions about how mathematics happens. But we have lost the inclination to look at what mathematics might be telling us about the world of the senses and qualities of thought. Yet it has been remarked that even in the relatively obscure philosophy of Leibniz, there are the seeds of modern approaches to cognition.
I have found in philosophical writing from the18th, 19th, and 20th centuries (in the works of Leibniz, Herbart, Poincare and Weyl), the distinct impulse to consider how perceived realities are somehow fused from a multiplicity of sensory stuff, and how this may actually be reflected in mathematical thought. There is a clear tendency to emphasize that the world comes to be in relationship or as Weyl put it, “The world does not exist independently but only for consciousness.” And this is relevant to the mathematician who is so often standing between the ideal and physical images of the ideal. Mathematics for many of these mathematicians has been a door to self-understanding and, I believe inevitably, it will be again.
In Science and Hypothesis (1905), Poincare spends a good deal of time making the point that geometric space is not the same as visual space. He does this by looking into the components of vision. Vision, he explains begins on a non-homogeneous 2-dimensional retina. The third dimension is created by “the effort of accommodation made by the convergence of the eyes.” And, since there are two aspects to the perception of distance (the sensation of the muscles and the eyes’ convergence) he argues that visual space could be said to have 4 independent variables and would therefore be 4-dimensional. He takes the time to argue that geometry is not the study of the space we think we see around us. Its ideal bodies are entirely mental and our experience just enables us to reach the idea.
In three 1932 lectures, collected under the heading The Open World, Hermann Weyl looks at Leibniz’s monad with respect to the problems associated with the infinite divisibility of a line (Zeno’s Paradox for example). Leibniz says that the conceptual difficulties with the idea of a continuum in mathematics arise because it is not understood that in the ideal or the continuum, the whole precedes the parts, but in substantial things, “the parts are given actually before the whole.” Ideals divide infinitely, substances only until there is no more. His perspective, grounded strongly in logic, holds that the ultimate constitutents of the world must be simple, indivisible, and therefore unextended, particles—dimensionless mathematical points. Extended matter is in reality constructed from simple immaterial substances, monads, or entelechies. And any monad can be said to represent the world as a whole. While the Leibniz worldview is drawn in a very unfamiliar way, this last detail about every individual monad somehow containing the whole is, I think, a kind of intuition that keeps showing itself, in math, science, and art.
A discussion of Leibniz on philosophy pages tells us this:
What is at work here again is Leibniz’s notion of complete individual substances, each of which mirrors every other. A monad not only contains all of its own past, present, and future features but also, by virtue of a complex web of spatio-temporal references, some representation of every other monad,. . . . In a universe of windowless mirrors, each reflects any other, along with its reflections of every other, and so on ad infinitum.
And this:
But Leibniz held that some monads—namely, the souls of animals and human beings—also have conscious apperception in the sense that they are capable of employing sensory ideas as representations of physical things outside themselves. And a very few monads—namely, spirits such as ourselves and god—possess the even greater capacity of self-consciousness, of which genuine knowledge is the finest example.
Weyl adds the following:
The application of mathematical construction to reality then ultimately rests on the double nature of reality, its subjective and objective aspects: that reality is not a thing in itself, but a thing appearing to a mental ego. If we assume Plato’s metaphysical doctrine and let the image appearing to consciousness result from the concurrence of a “motion” issuing partly from the ego and partly from the object, then extension, the perceptual form of space and time as the qualitatively undifferentiated field of free possibilities, must be placed on the side of the ego.
And with this observation Weyl concludes very nicely:
Mathematics is not the rigid and uninspiring schematism which the laymen is so apt to see in it; on the contrary, we stand in mathematics precisely at that point of intersection of limitation and freedom which is the essence of man himself.
By Joselle, on March 28th, 2011 When I looked recently at Riemann’s famous lecture On the Hypotheses which lie at the Bases of Geometry, I gave some attention to this remark:
besides some very short hints on the matter given by Privy Councillor Gauss in his second memoir on Biquadratic Residues, in the Göttingen Gelehrte Anzeige, and in his Jubilee-book, and some philosophical researches of Herbart, I could make use of no previous labours.
I’m most interested in the conviction these thinkers share that a kind of reasoning, a willingness to look more closely, can get past the appearances of things. This is what mathematics consistently accomplishes.
Johann Friedrich Herbart was a philosopher and would likely be referred to as an early 19th century pioneer in psychology-based theories of learning. But he wrote on the nature of many things and is said to have been significantly influenced by Leibniz’s philosophical writing (which I will get back to)
In her paper, An Overview of Riemann’s Life and Work, Rossana Tazzioli describes one of the directions that Herbart’s influence on Riemann took:
Riemann agreed “almost completely” with Herbart’s psychology, which inspired both Riemann’s model of the ether (the elastic fluid which was supposed to fill all the universe) and his principles of Naturphilosophie. According to Herbart, the “psychic act” (or “representation”) is anact of self-preservation with which the “ego” opposes the perturbations coming from the external world. A continuous flow of representations go from the ego to the conscious and back. Herbart studied the connections between different representations in mechanical terms as compositions of force.
In his “Neue mathematische Principien der Naturphilosophie” Riemann followed Herbart’s ideas and supposed that the universe is filled with a substance (Stoff)
flowing continually through atoms and there disappearing from the material world (Korperwelt). From this obscure assumption, Riemann tried to build a mathematical model of the space surrounding two interacting particles of substance…
But Herbart’s ideas about space or, more specfically, sensory space may have motivated some other generalizations. In a book about Hermann von Helmnoltz and nineteenth century science, David Cahan also talks about Herbart’s influence:
Herbart argues that each modality of sense is capable of a spatial representation. Color could be represented as a triangle in terms of three primary colors, tone as a continuous line, the sense of touch as a manifold defined by muscle contractions and still other spaces as associations of hand-eye movements. “To be exact,” he wrote, “sensory space is not originally a single space. Rather, the eyes and the sense of touch independently from one another initiate the production of space; afterward both are melted together and further developed. We cannot warn often enough against the prejudice that there exists only one space, namely, phenomenal space.” Therefore, for Herbart, “space is the symbol of the possible community of things standing in a causal relationship.” He insisted that for empirical psychology space is not something real, a single container in which things are placed. Rather, it is a tool for representing the various modes of interaction with the world through our senses.
Herbart treated a successful visual representation as a projected expectation or “hypothesis” formed through an unconscious inference……he treated vision as an experiment constantly performed by the brain with the aid of the eye as its measuring device.
Vision as an experiment would fit nicely with some very current ideas about the visual brain. And, while not an analysis of anything geometric, this perspective is provocative. It encourages the abandonment of static structures in favor of more interactive ones.
In my looking I became captivated by other ideas of Herbart’s. Some nice expressions of them are in An Introduction to the History of Psychology by B.R. Hergenhahn. Herbart agreed with empiricists that ideas were derived from sense experience. But they were, for him, independently active. From the text:
Herbart’s system has been referred to as pyschic mechanics because he believed that ideas had the power to attract or repel other ideas, depending on their compatibility.
According to Herbart, all ideas struggle to gain expression in consciousness, and they compete with each other to do so.
…and all ideas attempt to become as clear as possible…all ideas seek to be part of the conscious mind.
Herbart used the term self-preservation to describe an idea’s tendency to seek and maintain conscious expression.
Herbarts position represented a major departure from that of the empiricists because the empiricists believed that ideas, like Newtons particles of matter, were passively buffeted around by forces external to them….For Herbart, an idea was like an atom with energy and a consciousness of its own – a conception very much like Leibniz’s conception of the monad.
I need to take up Leibniz’s monad in another post. A simple statement from Philosophy Pages will suffice for now:
Leibniz concluded that the ultimate constituents of the world must be simple, indivisible, and therefore unextended, particles—dimensionless mathematical points. So the entire world of extended matter is in reality constructed from simple immaterial substances…
These are the monads. But there is life in these monads. And material does somehow come from the immaterial…
I did become particularly interested in the kind of analysis undertaken by Herbart and Leibniz. We inevitably learn the mathematics of the 19th century, but we’re not likely to spend time thinking about monads. It reminded me of how I felt when I read Thomas Mann’s Magic Mountain. Hans Castorp’s innocent and imaginative response to his first glimpse of a very young medical science reminded me that ideas can take us countless places, not just here.
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