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By Joselle, on October 30th, 2013 New Scientist published an article by Amanda Gefter in their August 15 issue which describes how and why the notion of infinity has come into question again. The distinction between a potential infinity (the process of something happening without end), and an actual infinity (represented, for example, by the set of real numbers) was disputed among mathematicians for a long time until Cantor brought new meaning to the nature of infinities in the set theory he created. Now infinities are part of the living tissue of mathematics. But infinities have thwarted the success of some physical theories and continue to do so. The survey of challenges from mathematicians and physicists alike make for an interesting read. One of the more reasonable claims comes from Max Tegmark of MIT.
When quantum mechanics was discovered, we realised that classical mechanics was just an approximation,” he says. “I think another revolution is going to take place, and we’ll see that continuous quantum mechanics is itself just an approximation to some deeper theory, which is totally finite.
I believe Riemann himself had not decided whether space was continuous or discrete.
One of the links within this piece was to another New Scientist article by Gefter, published in October of last year. The older piece had the title: Reality: Is everything made of numbers? In it, Amanda Gefter surveys some of emerging perspectives in physics that equate, in one way or another, mathematical reality with physical reality. She recalls Einstein’s fix for the equations that described an expanding universe, years before there was clear evidence that the equations were correct.
How did Einstein’s equations “know” that the universe was expanding when he did not? If mathematics is nothing more than a language we use to describe the world, an invention of the human brain, how can it possibly churn out anything beyond what we put in? “It is difficult to avoid the impression that a miracle confronts us here,” wrote physicist Eugene Wigner in his classic 1960 paper “The unreasonable effectiveness of mathematics in the natural sciences”
With respect to current physics investigations she says:
The prescience of mathematics seems no less miraculous today. At the Large Hadron Collider at CERN, near Geneva, Switzerland, physicists recently observed the fingerprints of a particle that was arguably discovered 48 years ago lurking in the equations of particle physics.
Gefter then tells us about how some prominent physicists, like Brian Greene and Max Tegmark, are tackling the riddle. She has more to say about Tegmark’s view than Greene’s, and Tegmark’s view is fairly extreme. Gefter quotes him:
I believe that physical existence and mathematical existence are the same, so any structure that exists mathematically is also real.
I’ve always liked this kind of talk. While the content of a remark like this might be difficult to specify, I suspect that it’s full of insight. Tegmark goes on to suggest that the mathematical structures that have no physical application in this universe correspond to other universes. But his ideas rest, in part, on his judgment that mathematical structures don’t exist in space and time. ” Space and time themselves,” he says, “are contained within larger mathematical structures.”
Now, one might argue that brains and computers exist in space and time, and without them there is no mathematics, or at least not the kind we used to thinking about. But I don’t want to move here into a difficult philosophical debate. Instead, I’d like to suggest that if we consider that everything that exists is known only in relationship, some new light might be shed on the question of how mathematics can do what it does. Physicists like Tegmark can seem to be replacing the physical subjects of their investigations with mathematical ones, putting the mathematics ‘out there.’ Cognitive scientists seem to be finding mathematical notions mirrored in sensory and learning processes, suggesting that mathematics is part of how the body learns about its world. But the body’s perceiving mechanisms are built entirely in relation to its surroundings, or more specifically, to the properties of things like air and light, or even gravity. If we can see mathematics as one of the body’s actions, action that stretches the reach of its perceiving and learning mechanisms, then perhaps we can imagine that it develops not ‘for’ the world we live in, or from it, but ‘with’ it. Like color, perhaps.
By Joselle, on October 7th, 2013 Understanding the neural functions that contribute to the birth of mathematical structure and meaning is an active subject of research in cognitive science. A significant amount of work has been done to identify an innate ability we share with other creatures, namely the ability to perceive quantity. This is sometimes called our approximate number sense. Finding a path from this talent to how we come to use the symbolic representations of mathematics is certainly challenging. And a recent study published in Science contributes what is, perhaps, an unexpected observation. Researchers participating in the study observed that numerosity (defined as the set size of a group of items), while processed by the association cortex, mirrors the properties of primary sensory processing. Specifically, the neurons tuned to small numerosities follow topographical principles. These are principles associated with topographic representations that occur in vision, for example. In other words, neighboring points on the retina correspond to neighboring neural responses as the retina’s data is mapped to the cortex. The mapping is smooth, perhaps continuous in the mathematical sense. It is true that formal mathematical abilities, the ones we are trained to do, rely on different cognitive processes, but studies have found that individual differences in an innate number sense and other mathematical abilities are correlated. The Science paper not only suggests that topographic mappings may be active in higher-order cognitive processes, but also that number sense shares something with the primary senses.
Our results demonstrate that topographic representations, common in the sensory and motor cortices, can emerge within the brain to represent abstract features such as numerosity. Similarities in cortical organization suggest that the computational benefits of topographic representations, for example efficiency in wiring, apply to higher-order cognitive functions and sensory-motor functions alike. As such, topographic organization may be common in higher cognitive functions. On the other hand, topographic organization supports the view that numerosity perception resembles a primary sense. These views are not mutually exclusive, but both challenge the established distinction between primary topographic representations and abstracted representations of higher cognitive functions.
It seems to me that the brain is a remarkable system that integrates everything that the body detects in sensation. And that the paths of this integration are continuous, from the most fundamental to the most complex. I recently had a conversation with my daughter about reading. For whatever reason, my daughter resists the idea that words on the page are symbolic representations of sounds. And this resistance inhibits her ability to read new words. There was a time when she needed to read out loud in order to fully comprehend what she was reading. And now, while she’s made significant progress reading to herself, it will happen that the words on the page aren’t making sense. But if she hears them, meaning is immediately comprehended. As we talked about it, I told her that her brain was doing an amazing thing when it translated something it was seeing into the meaning that was carried in sound. This action is another kind of integration of various cognitive processes. A systematic approach, contained in our biology, allows the body to build meaning over and over again. This conversation came to mind again today, when I read today about language in a recent piece on scientificamerican.com by Joshua Hartshorne. Hartshorne makes the following observation:
The second striking feature of language is that when you consider the space of possible languages, most languages are clustered in a few tiny bands. That is, most languages are much, much more similar to one another than random variation would have predicted.
This design-constraint is interesting if you consider that there is likely a design-constraint on every aspect of cognition, on everything that we think we know. Perhaps this is a consequence of the way cognitive processes are consistently integrated. And, I would argue, that while mathematics is full of its own design constraints, the magic of what it does is found in the extent to which it can provide eccentric possibilities, or let more into the constrained system, by providing a uniquely thorough investigation of its own constraints.
By Joselle, on September 25th, 2013 The most recent issue of New Scientist has an article called Thoughts: The inside story. In it, philosopher Tim Bayne begins with a survey of all of the things we mean by the word ‘thought’ – the mental activity that accompanies perceptions, problem solving, the integration of various perceptions, the uncontrolled associative train of connected concepts, the organization of these associations, possibilities that are not perceived but fully imagined, and so on. Early on, Bayne makes the following remark:
Although the distinction between perception and thought is intuitive, no one has been able to characterise it unequivocally.
I believe that this particular observation is worthy of more attention than it was given. And mathematics, given its role in the development of science, may be in a good position to contribute to the discussion. Mathematics is, in some sense, the conscious application of thought to perception. So how does the body make thoughts and what is it doing?
Bayne spends some time outlining why a “physicalist conception of thought” is more easily defended than an immaterial ‘mind’ or ‘soul’ conception, largely because of our observations of the significance of the brain’s role in thought. But, I believe even more to the point, he argues that “the materialist account of thought does justice to the continuity of nature,” and our relationship to other living creatures. The significance of the role that language plays in the presence of thought is somewhat undermined by studies that demonstrate thoughtful activity in non-human species – the chimpanzees’ ability to compare quantities and grasp simple fractions, or the baboon’s awareness of social hierarchies or the monkey’s ability to assess the difficulty of a task . While he didn’t mention it, the ability to discern quantity has been observed in fish as well. We are only beginning to grasp the complexity of non-human lives.
Language and symbol are here understood as facilitators of thought, perhaps providing for the unique sophistication of human thought, and the shared or social nature of our cognitive breakthroughs. Bayne quotes philosopher Andy Clark:
As the philosopher Andy Clark has remarked, “experience with external tags and labels thus enables the brain itself… to solve problems whose level of complexity and abstraction would otherwise leave us baffled.”
But Bayne seems to accept the usual bias about mathematical thoughts:
Sometimes thought is controlled by the application of a rule. Mathematical and logical operations, for example, are rule-based, and philosophers have invented many other systematic “thinking tools” to help them think more clearly (see “Tools for thought”). But this is an unusual kind of activity, and most episodes of thinking involve no rule.
It is certainly true that mathematics is used to organize thoughts as well as perceptions, but mathematics doesn’t emerge as a ‘tool,’ it emerges as a ‘thought.’ And, I would argue that the birth of many mathematical thoughts is an action of the body, and not solely directed by our conscious will. Perhaps mathematics itself is an act of perception.
As the foundation of many of the sciences, it would seem to be in a unique position to reveal something about the relationship between our thoughts and the world in which we live.
Bayne also makes this observation:
Until recently, undirected thought was seen as a useless and wasteful aspect of our internal mental lives. But research now suggests that it is a normal and even necessary aspect of thought. Brain activity during mind-wandering is reminiscent of that seen when people are deliberately engaged in creative thinking. It may be that, paradoxically, undirected thought is when we get our best thinking done.
The great pleasure that mathematicians feel about their discipline may be due to the fact that the source of mathematical thoughts, lies deep within the layers of our perceptive and cognitive processes, and their rise to consciousness is, in fact, not willed. Yet they invite us to make more of them, with very carefully directed reflection.
I appreciated Bayne’s consideration of the limits of thoughts.
Given that the machinery of human thought is part of our biology, there is every reason to suspect that it suffers from the kinds of bugs and blind spots that constrain other biological systems. It is doubtful whether chimpanzees possess the ability to think about quantum mechanics, for example. Perhaps that is one of the limitations of lacking language. But if there are parts of reality that are inaccessible to other thinking species, why should we assume that no part is inaccessible to us?
The role that mathematics plays in science demonstrates the extent to which our images and ideas can exceed the limits of our perceptive abilities. While mathematics opens the door to, what still appear to be, limitless imagined structures, it also opens the door to the inaccessible reaches of our physical world. So what is thought, as an action of the body, designed to accomplish? And what might mathematics tell us about that?
By Joselle, on September 10th, 2013 One of the reasons that the nature of mathematics has been such an enigma, is that we associate it with thought, and we tend to distinguish thought from the physical world. We do find mathematics in natural structures – some of these beautifully represented in a film you may have seen called Nature by the Numbers. We’re somewhat familiar with the efficiency of honeycomb structures built by the honey bee. Their geometry maximizes robustness while minimizing weight. Honeybees execute the building of these honeycombs with great precision. Researchers have also seen mathematics in the behavior of insects. Ants calculate the path to food which takes the least amount of time rather than the one that is the shortest distance. Huffington Post reported on a study this past April.
En route to their roach banquet, the ants did not follow the most direct travel path, the study found. Rather, they followed an angled path, traveling over more of the smoother material in order to reach the food morsels in the shortest amount of time. The findings demonstrate that Fermat’s principle of light travel also applies to living creatures, the researchers conclude.
In The Math Instinct, Keith Devlin made a nice survey of the mathematics in nature – the navigation feats of migratory birds, the logarithmic spiral of a falcon catching its prey and the mathematics of locomotion are some of his examples. Generally speaking, these observations highlight the pragmatic value of the non-symbolic mathematics. But pragmatic expectations might actually obscure the path to a new insight. Much of this blog is devoted to investigating the ubiquitous presence of mathematics, in living and non-living things, in order to raise some novel questions about the roots as well as the significance of mathematics in our human experience.
A recent story in a National Geographic blog describes the mathematical behavior of a fish, but the pragmatic motivation for the circular objects these fish create is not so obvious. Large (6.5-foot-wide) structures on the seafloor were once a mysterious feature of the underwater landscape. But these decorated circles have now been attributed to 5-inch long male pufferfish. The fish use their bodies to construct and decorate these nests, within which accepting females will lay their eggs.
The scientists aren’t sure exactly what the females are looking for when they judge a male’s nest. It could be the central patterns made of fine sand, the decorations on the outside, or the nest’s size or symmetry.
The mail fish remains in the nest to supervise the eggs’ hatching. He then looks for a new site where he starts the nest making process again. It takes a significant amount of time for this small fish to build these large geometric structures, swimming toward the center of the circle in a straight line then around the center in a circular motion. But the pragmatic value of this geometry is not obvious. It reminded me of the bowerbird. David Rothenberg described the creativity of the bowerbird’s courtship structures in his book Survival of the Beautiful.
Bowerbirds, say biologists are unique. There is perhaps no other species besides human beings that is known to create things so beautiful beyond their function, structures that we have a hard time calling anything else but art, the arrangement of objects that please us. Bowers are built to attract females, but they are far from the simplest solution to such a problem. Yet a male won’t get a female without one. And somehow, evolution has led them to build them in exact and precise ways.
I enjoy both of them because they just don’t quite fit into the functionality-driven ideas that we use to describe our world. They seem to allow the possibility that life creates for no reason. Creations become tied to what look like reasons, but the creations are not fully captured by the reasons. This is the way I sometimes think of mathematics. One of the things that gets in the way of an informed appreciation of the human development of mathematics is pragmatism.
By Joselle, on August 27th, 2013 David Deutsch is proposing a very interesting conceptual shift in how we understand the nature of a physical theory. It’s an idea he has for “generalizing the quantum theory of computation to cover not just computation but all physical processes.” The theory in question he calls Constructor Theory. A video of Deutsch outlining the ideas (as well as the accompanying written text) were provided by Edge.org last October.
Constructor Theory rests on the nature of information is understood. According to Deutsch,
There’s a notorious problem with defining information within physics, namely that on the one hand information is purely abstract, and the original theory of computation as developed by Alan Turing and others regarded computers and the information they manipulate purely abstractly as mathematical objects. Many mathematicians to this day don’t realize that information is physical and that there is no such thing as an abstract computer. Only a physical object can compute things…information has to be a physical quantity. And yet, information is independent of the physical object that it resides in.
That information is physical, yet independent of the physical object in which it resides, is one of the cornerstones of the theory. Deutsch makes the point:
I’m speaking to you now: Information starts as some kind of electrochemical signals in my brain, and then it gets converted into other signals in my nerves and then into sound waves and then into the vibrations of a microphone, mechanical vibrations, then into electricity and so on, and presumably will eventually go on the Internet. This something has been instantiated in radically different physical objects that obey different laws of physics. Yet in order to describe this process you have to refer to the thing that has remained unchanged throughout the process, which is only the information rather than any obviously physical thing like energy or momentum.
The way to get this substrate independence of information is to refer it to a level of physics that is below and more fundamental than things like laws of motion, that we have been used thinking of as near the lowest, most fundamental level of physics. Constructor theory is that deeper level of physics, physical laws and physical systems, more fundamental than the existing prevailing conception of what physics is (namely particles and waves and space and time and an initial state and laws of motion that describe the evolution of that initial state).
The theory works on yet another level of abstraction, (and a familiar mathematical idea) transformations.
The new thing, which I think is the key to the fact that constructor theory delivers new content, was that the laws of constructor theory are not about an initial state, laws of motion, final state or anything like that. They are just about which transformations are possible and which are impossible…When they’re possible, you’ll be able to do them in lots of different ways usually. When they’re impossible, that will always be because some law of physics forbids them, and that is why, as Karl Popper said, the content of a physical theory, of any scientific theory, is in what it forbids and also in how it explains what it forbids.
The laws of Constructor Theory are expressed with an algebra.
The first item on the agenda then is to set up a constructor theoretic algebra that’s an algebra in which you can do two things. One is to express any other scientific theory in terms of what transformations can or cannot be performed. The analog in the prevailing formulation of physics would be something like differential equations, but in constructor theory it will be an algebra. And then to use that algebra also to express the laws of constructor theory, which won’t be expressed in terms of subsidiary theories. They will just make assertions about subsidiary theories.
Chiara Marletto (a student I’m working with) and I are working on that algebra. It’s a conceptual jolt to think in terms of it rather than in the terms that have been traditional in physics for the last few decades. We try and think what it means, find contradictions between different strands of thought about what it means, realize that the algebra and the expressions that we write in the algebra doesn’t quite make sense, change the algebra, see what that means and so on. It’s a process of doing math, doing algebra by working out things, interleaved with trying to understand what those things mean. This rather mirrors how the pioneers of quantum theory developed their theory too. (emphasis added)
…one of the formulations of quantum theory, namely matrix mechanics as invented by Heisenberg and others, isn’t based on the differential equation paradigm but is more algebraic and it, in fact, is another thing that can be seen as a precursor of constructor theory.
Deutsch acknowledges that, while promising, the effort is new and if it turns out to be wrong, “Then we would have to learn the lesson of how it turned out to be wrong.” At the end of last year, Deutsch authored a paper that may be found here.
One of the things I most enjoy about David Deutsch is how he expresses his own captivation with how we have come to know things that lie far outside the range of our experience. In this particular talk he says the following:
One of the central philosophical motivations for why I do fundamental physics is that I’m interested in what the world is like; that is, not just the world of our observations, what we see, but the invisible world, the invisible processes and objects that bring about the visible. Because the visible is only the tiny, superficial and parochial sheen on top of the real reality, and the amazing thing about the world and our place in it is that we can discover the real reality.
We can discover what is at the center of stars even though we’ve never been there. We can find out that those cold, tiny objects in the sky that we call stars are actually million-kilometer, white, hot, gaseous spheres. They don’t look like that. They look like cold dots, but we know different. We know that the invisible reality is there giving rise to our visible perceptions. (emphasis, again, my own)
I think the significance of this was failed to be appreciated by Arnold Trehub, a psychologist at the University of Massachusetts, Amherst and author of The Cognitive Brain. He contributed this critique of Deutsch’s ideas:
Deutsch is to be commended for pointing out that information necessarily involves electrochemical signals in his brain. But the greatest volume of information signaling in his brain is not a part of his conscious experience. It is only after unconsciously selected mini-patterns of preconscious neuronal activity are projected in proper spatio-temporal register into his brain’s perspectival representation of his surrounding space that they become part of his conscious experience of the world around him as well as his conceptions of possible worlds. The physics that he knows and strives to advance exists in this domain of neuronal activity.
We are each born with a system of brain mechanisms that constitute the full scope of our occurrent phenomenal universe. The neuronal structure and dynamics of this cognitive brain system actually construct the world of our experience. But the brain mechanisms that give us this world are evolutionary adaptations selected to better enable us to survive and thrive on our uncertain earth. When David Deutsch regards science as an enterprise for discovering what the world is really like, he seems to be seeking ultimate truths. How can we possibly know the ultimate nature of the world when the cognitive brain, the best tool we have, is a pragmatic and opportunist organ? It seems to me that science, like the brain that conceives it, is a pragmatic enterprise. We may seek to know what the world is really like, but what the world is really like, within the purview of science, is always provisional.
The provisional nature of knowledge is understood. But the constructive action of the brain is by no means being ignored in Deutsch’s ideas. The projection of that preconscious neuronal activity may be one of the things Deutsch’s method is meant to capture. That “volume of information signaling” in the brain, that is not a part of his conscious experience, may be the very thing for which mathematics compensates. Science (particularly the mathematical side of science) at the very least , suggests that the brain is terribly underestimated when it is characterized as “a pragmatic and opportunist organ.” The more interesting question would be how is mathematics and science contributing to the brain’s construction of its world.
David Deutsch does a remarkable job of integrating specialized disciplines. In The Fabric of Reality there are four fundamental aspects of reality represented by quantum physics, the theory of evolution, the theory of computation and the theory of knowledge. This new effort follows the path he seems to want to clear.
By Joselle, on August 14th, 2013 As I read more discussions of the relationship between mathematics and physics, I find that what mathematics might reveal about how physical science progresses becomes an increasingly interesting question.
I recently found the text of a lecture given by Paul Dirac in 1939. It was reproduced on the occasion of the Dirac Centennial Celebration organized by the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge in 2002. The title of the lecture is “The Relation between Mathematics and Physics.” Dirac remarks, right away, that “there is no logical reason” why mathematical reasoning should succeed as one of the two methods used by the physicist to study natural phenomena (the other being experiment and observation). And, early in the talk, he says the following:
One might describe the mathematical quality in Nature by saying that the universe is so constituted that mathematics is a useful tool in its description. However, recent advances in physical science show that this statement of the case is too trivial. The connection between mathematics and the description of the universe goes far deeper than this, and one can get an appreciation of it only from a thorough examination of the various facts that make it up.
He goes on to survey the conceptual shifts that have happened in physics – from the equations that represent the laws of motion in Newtonian physics, to the geometry of Einstein’s space-time, and the non-commutative algebra of quantum mechanics. He made what I thought was an unexpected distinction between classical laws, governed by “a principle of simplicity,” and the mathematical beauty that makes the theory of relativity so compelling. About this Dirac says the following:
The research worker, in his efforts to express the fundamental laws of Nature in mathematical form, should strive mainly for mathematical beauty…It often happens that the requirements of simplicity and of beauty are the same, but where they clash the latter must take precedence.
In this light he proposes that a powerful method of research for physicists may very well be to first choose a promising branch of mathematics “influenced very much in this choice by considerations of mathematical beauty,” and then proceed to develop it, keeping an eye on the way it lends itself to physical interpretation. String theorists are among those who seem to have chosen this route. And here’s an interesting statement:
One may describe the situation by saying that the mathematician plays a game in which he himself invents the rules while the physicist plays a game in which the rules are provided by Nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen.
‘The rules,’ of course, are the consequences of various observed relationships among the concepts captured in mathematical symbol. These may be relationships among numbers, or among operations and transformations, or among spatial properties and geometric structures. While there continues to be disagreement about whether the rules are invented or discovered , what Dirac may be emphasizing about mathematics is that the mathematician looks only at the mathematics, requiring no other validation of what mathematician Richard Courant once called ‘verifiable fact.’
Yet mathematics is an equal partner in the design of physical theories, what we consider a purely empirical science. This fact must say something about the nuances of what we mean by ’empirical,’ which are often reflected in disputes between rationalists and empiricists. For some, the senses are like detectors to which a rational device of the mind is applied. But this perspective has been consistently challenged by research in cognitive science. These studies indicate strongly that the senses are not easily distinguished from that rational device. And perhaps it is this that is reflected in the growing blend of mathematics and physics.
Stephen Hawking was one of the speakers at the Dirac Centennial Celebration. The title given his talk was “Gödel and the end of physics.” But he wasn’t predicting the end of physics. The view he presents is, to some extent, a critique of the standard positivist approach to science, where the mathematical treatment of sensory data is the only source of knowledge. In this light, mathematics is not considered a product of the mind. Intuition and introspection, after all, play no role in the acquisition of knowledge. But how could one divorce intuition and introspection from mathematics? Physics’ increasing reliance on mathematics must be pointing to a relationship between mathematics and perception. And Hawking sees another problem. Physical theories, or mathematical models of physical systems, are self-referencing.
…we are not angels, who view the universe from the outside. Instead, we and our models, are both part of the universe we are describing. Thus a physical theory, is self referencing, like in Gödel’s theorem. One might therefore expect it to be either inconsistent, or incomplete. The theories we have so far, are both inconsistent, and incomplete.
Like Gregory Chaitin, Hawking seems to find this incompleteness promising.
Some people will be very disappointed if there is not an ultimate theory, that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind. I’m now glad that our search for understanding will never come to an end, and that we will always have the challenge of new discovery.
By Joselle, on August 1st, 2013 In the August issue of Scientific American, Meinard Kuhlmann addresses, yet again, the conceptual difficulties inherent in the interpretations of experimental data of modern physics.
…the particle interpretation of quantum physics, as well as the field interpretation, stretches our conventional notions of “particle” and “field” to such an extent that ever more people think the world might be made of something else entirely.
Kuhlmann is currently a philosophy professor at Bielefeld University in Germany and has dual degrees in physics and philosophy. I was happy to see that he is firmly committed to the idea that the task of understanding the physical world requires both disciplines.
The two disciplines are complementary. Metaphysics supplies various competing frameworks for the ontology of the material world, although beyond questions of internal consistency, it cannot decide among them. Physics, for its part, lacks a coherent account of fundamental issues, such as the definition of objects, the role of individuality, the status of properties, the relation of things and properties, and the significance of space and time.
Kuhlman takes the time to describe, in fairly simple terms, the content of the Standard Model which consists of groups of elementary particles and the forces that mediate their interaction. He describes how the particles blur into fields while, at the same time, the fields are quantized rather than continuous. His discussion of how the particles are not really particles and the fields are not really fields leads him to his point:
If the mental images conjured up by the words “particle” and “field” do not match what the theory says, physicists and philosophers must figure out what to put in their place.
Kuhlman then takes his article in two interesting directions. The first is to focus on the notion of structure.
A growing number of people think that what really matters are not things but the relations in which those things stand…We may never know the real nature of things but only how they are related to one another…New theories may overturn our conception of the basic building blocks of the world, but they tend to preserve the structures. That is how scientists can make progress.
I was immediately reminded of a passage in the Courant/Robbins classic What is Mathematics? When I first read the book, I was impressed with implications of this observation which appears early in the text.
The “ether” was invented as a hypothetical medium capable of not entirely explained mechanical motions that appear to us as light or electricity. Slowly it was realized that the ether is of necessity unobservable; that it belongs to metaphysics and not to physics. With sorrow in some quarters, with relief in others, the mechanical explanations of light and electricity, and with them the ether, were finally abandoned.
A similar situation, even more accentuated, exists in mathematics. Throughout the ages mathematicians have considered their objects, such as numbers, points, etc., as substantial things in themselves. Since these entities had always defied attempts at an adequate description, it slowly dawned on the mathematicians of the nineteenth century that the question of the meaning of these objects as substantial things does not make sense within mathematics, if at all. The only relevant assertions concerning them do not refer to substantial reality; they state only the interrelations between mathematically “undefined objects” and the rules governing operations with them. 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. A clear insight into the necessity of a dissubstantiation of elementary mathematical concepts has been one of the most important and fruitful results of the modern postulational development.
Fortunately, creative minds forget dogmatic philosophical beliefs whenever adherence to them would impede constructive achievement.
In the context of the Courant book, this is an important observation about the development of mathematics. But I have always thought that it can be seen as an important observation of a more general intellectual maturity. And this, I think, leads to Kuhlmann’s second alternative for interpreting the meaning of quantum physics which chooses ‘properties’ rather than ‘objects’ as having an existence.
What we commonly call a thing may be just a bundle of properties: color, shape, consistency, and so on.
This idea is consistent not only with current theories in cognition, but also has roots in 19th century philosophy and science (in the work of Hermann von Helmholtz and Johann Friedrich Herbart, for example). Kuhlmann rightly argues that our first experiences are of properties.
As infants, when we see and experience a ball for the first time, we do not actually perceive a ball, strictly speaking. What we perceive is a round shape, some shade of red, with a certain elastic touch. Only later we do associate this bundle of perceptions with a coherent object of a certain kind – namely, a ball. Next time we see a ball, we essentially say, “Look, a ball,” and forget how much conceptual apparatus is involved in this seemingly immediate perception.
With respect to physics,Kuhlmann explains
theory predicts that elementary particles can pop in and out of existence quickly. The behavior of the vacuum in quantum field theory is particularly mind-boggling: the average value of the number of particles is zero, yet the vacuum seethes with activity…A particle is what you get when those properties bundle themselves together in a certain way.
The forgetting of ‘conceptual apparatus’ to which Kuhlmann refers is the very thing that I always hope (and expect) that mathematics will remind us of – in one way or another.
By Joselle, on July 17th, 2013 I’ve referred to category theory on more than one occasion (particularly with respect to physicist Bob Coecke’s graphical language). Not too long ago, Ronald Brown, at Bangor University, brought my attention to the work that he and colleagues have been doing to investigate the kind of mathematics that could be used to model the complexity of neural activity (and the products of this activity). Their success would suggest to me again that there is a correspondence between mathematics and how we perceive or come to know anything. I looked at some of his papers and have chosen some excerpts from one of them because they serve, I believe, to illustrate some of the ideas and their motivation. Brown co-authored the paper with Timothy Porter (also at Bangor). It has the title: Category Theory and Higher Dimensional Algebra: potential descriptive tools in neuroscience Here is an early observation:
One may presume that the power of abstraction, in some sense of making maps, must be deeply encoded in evolutionary history as a technique for encouraging survival, since a map gives a small and manipulable model of the environment.
The idea of ‘mapping,’ or making a correspondence, is one of the most fundamental ideas in mathematics. Mathematics could not proceed without it. But it has also been used to describe how the brain functions. For example, the brain performs what have been called computations to construct a visual image from visual input. The computations involve actions referred to as ‘mappings’ (in the mathematical sense) of aspects of the world onto elements of the brain. It makes sense that one would expect this kind of neural activity to be deeply encoded in our evolutionary history. And, with this in mind, one could argue that mathematics may be built on underlying, and mostly unconscious, biological action.
Approaching it from the other direction.
A study of why and how mathematics works could be useful for making models for neurological functions involving maps of the environment. Mathematics may also possibly provide a comprehensible case study of the evolution of complex interacting structures, and so may yield analogies helpful for developing and evaluating models of brain activity, in order to derive better models, and so better understanding. We expect to need a new language, a new mathematics, for describing brain activity. To see what is involved in this search, it is reasonable to study the evolution of mathematics, and of particular branches such as category theory.
This look at the evolution of abstractions is an interesting way to proceed. The authors provide a nice analogy for understanding the value of the idea of a colimit in category theory.
To focus on a common example, consider the process of sending an email document, call it E. To send this we need a server S, which breaks down the document E into many parts Ei for i in some indexing set I, and labels each part Ei so that it becomes E′ i. The labelled parts E′i are then sent to various servers Si which then send these as messages E′′ i to a server SC for the receiver C. The server SC combines the E′′i to produce the received message ME at C. Notice also that there is an arbitrariness in breaking the message down, and in how to route through the servers Si, but the system is designed so that the received message ME is independent of all the choices that have been made at each stage of the process. A description of the email system as a colimit may be difficult to realise precisely, but this analogy does suggest the emphasis on the amalgamation of many individual parts to give a working whole, which yields exact final output from initial input, despite choices at intermediate stages.
One question for neuroscientists is: does the brain use analogous processes for communication between its various structures? What we can say is that this general colimit notion represents a general mathematical process which is of fundamental importance in describing and calculating with many algebraic and other structures.
…Thus a conjecture as far as biological processes are concerned, is that this notion of colimit may give useful analogies to the way complex systems operate. More generally, it seems possible that this particular concept in category theory, seeing how a big object is built up of smaller related pieces, may be useful for the mathematics of processes.
The paper goes on to describe the value of higher dimensional algebra in putting the pieces together:
Information is often ‘subdivided’ by the sensory organs and is reintegrated by the brain. To enable different parts of that information to be integrated, there must be some ‘glue’, some inter-relational information available. If we are given arrows a, b, c, d with no information on where they start or end, then we could form combinations
which make no geometric sense. The colimit/composition process makes sense only where the inter-relations are also such as to enable the ‘integration’ to be well defined. Higher dimensional algebra allows more complex notions of ‘well formed composition’, and ones more adapted to geometry.
The authors do a nice job of describing the strength higher dimensional algebra and I recommend taking a look.
By Joselle, on July 11th, 2013 My piece on Riemann and cognition was published this week in +Plus. Here’s the link.
By Joselle, on July 3rd, 2013 Step by step, our ideas about the nature of our reality have moved far from the sensory constructions of space and time that define our immediate experience. And once fully outside the knowledge brought with sensation, we lose our footing. It’s difficult to manage ‘what can’t be sensed.’ But our conceptual difficulties with quantum mechanics are very reasonable if we imagine that, however abstract, the mathematics that got us there is rooted in our sensory experience – our sense of space and duration, our perception of quantity, and perhaps the cognitive mechanisms that manage these. The remarkable refinement of mathematical ideas has forced a reconsideration of what we think we see, and the conceptual possibilities that mathematics provides may indicate that we’ve enhanced our sensory apparatus in such a way that it has been made sensitive enough to ‘reach’ the edge of what is ‘sensible’ making us aware of the reality that escapes us. It is beginning to look like vast parts of our reality are not sensible.
These thoughts came to mind after reading a few discussions concerning the reconciliation of quantum mechanical strangeness. A New Scientist article addresses, specifically, questions about the meaning of space and time and their place (or lack of it) in modern physics. Space and time are constructed by the body and further explored by mathematics. But questions about whether or how they are real are very old. As Anil Ananthaswamy says,
…arguments about the nature of space and time swirl on. Are both basic constituents of reality, or neither – or does one perhaps emerge from the other in some way? We are yet to reach a conclusive answer, but it is becoming clear that if we wish to make further progress in physics, we must. The route to a truly powerful theory of reality passes through an intimate understanding of space and time.
I like the phrase ‘intimate understanding.’ It suggests getting very close to their source. The difficulty in physics, more specifically, is this:
A quantum object’s state is described by a wave function, a mathematical object living in an abstract space, known as Hilbert space, that encompasses all the possible states of the object. We can tell how the wave function evolves in time, moving from one state in its Hilbert space to another, using the Schrödinger equation. In this picture, time is itself not part of the Hilbert space where everything else physical sits, but somehow lives outside it…As for space, its status depends on what you are measuring. The wave function of an electron orbiting the atomic nucleus will include properties of physical space such as the electron’s distance from the nucleus. But the wave function describing the quantum spin of an isolated electron has no mention of space: according to the mathematics, the picture we often paint of an electron physically
rotating is meaningless.
Although a Hilbert space is a vector space whose structure is completely abstract, it rests on the meaning that it borrows from the relationships among vectors that we imagine in two and three dimensional Euclidean space. And so it is grounded in a familiar spacial idea. But mathematics has grown in such a way that the vectors of a Hilbert space can be used to represent possible states of a quantum mechanical system.
In a recent Scientific American blog, George Musser gave University of Maryland philosopher Ruth Kastner the opportunity to discuss her ideas about to resolve the difficulties with what is called the transactional interpretation of quantum mechanics. She begins with a reference to physicist and writer Han Christian von Baeyer’s article in the June issue of Scientific American. She says about his article that in response to the deep questions about the meaning of quantum theory von Baeyer “discusses one proposal – a denial that the theory describes anything objectively real …”
This is a very misleading summary of what von Baeyer discusses. I also wrote a few weeks ago about this article. It’s the wave function that is thought to have no objective reality from the point of view that von Baeyer discusses. This perspective (called Qbism) combines quantum theory and probability theory and sees the wave function as a powerful mathematical tool that provides the observer a way to make decisions about the surrounding quantum world. I took note, in my post, of the fact that Bayesian statistics (the ones used in Qbism) are also used by cognitive scientists to model how we build our very immediate expectations of our world from sensory data. And I quoted physicist Christopher Fuchs, a prominent proponent of Qbism, who said:
…even if quantum theory is purely a theory for apportioning and structuring degrees of belief, the question of “Why the quantum?” is nonetheless a question of what it is about the actual, real, objective character of the world that compels us to use this framework for reasoning rather than another.”
This doesn’t sound like a denial that the theory describes anything real.
The transactional Interpretation that Kastner discusses in her blog was first proposed, she tells us, by physicist John Cramer in the 1980s and can be traced back to physicists John Wheeler and Richard Feynman. She says the following:
My development of the Transactional Interpretation makes use of an important idea of Werner Heisenberg: “Atoms and the elementary particles themselves … form a world of potentialities or possibilities rather than things of the facts.” This world of potentialities is not contained within space and time; it is a higher-dimensional world whose structure is described by the mathematics of quantum theory. (emphasis added)
I wasn’t able to get a very clear picture of the history of the idea, nor her development of it from the blog, but the punch line is clear enough. Her transactional picture assumes that “there is more to reality than what can be contained within space-time.” It is somehow with the encounter of “potential events” (not contained within space-time) that “real energy may be conveyed within spacetime…” and “delivered in the normal future direction.”
She goes on to say:
The transactional picture is conceptually challenging because the underlying processes are so different from what we are used to in our classical world of experience, and we must allow for the startling idea that there is more to reality than what can be contained within spacetime.
I had the impulse to bring these ideas together because of what they have in common, specifically, that what we see (in the most abstract sense of that word) consistently indicates how much we don’t see. And, perhaps that there is even a strain on our mathematical ways because mathematics itself may have its roots in how the body ‘perceives.’ This doesn’t mean that progress can’t be made. Bringing back a little bit from Christopher Fuchs:
For the Qbist, the lesson that the structure of quantum theory calls out to be interpreted in only this way is that the world is an unimaginably rich one in comparison to the reductionist dream. It says that the world has excitement, risk, and adventure at its very core.
Considering how our mathematics brings us closer to this core is equally exciting.
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