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By Joselle, on December 21st, 2011 I see mathematics as associated with a searching, instinctual will, whose direction is shaped by our biology. I find some of its roots in the way our visual system constructs what we see, or in the way grid cells (neurons lit by location) tell a rat where it is, or the way ants can find their way home with a kind of internal vector analysis. In the past, I’ve thought of mathematics as somehow making unconscious processes like these, available to the needs of a conscious will. But I also see the emergence of mathematics as one of the body’s actions, something the organism just began to do, like language. Why would our nervous system begin to build something like mathematics?
One can find a meaty collection of essays on mathematics in the book: 18 Unconventional Essays on The Nature of Mathematics edited by Reuben Hersh. In one of the essays (Philosophical Problems of Mathematics in the Light of Evolutionary Epistemology), Yehuda Rav says that we should not just accept the mystery of the effectiveness of mathematics in the natural sciences. It “is the task of any epistemology” to furnish some explanation for it. And Rav proposes something consistent with my own ideas:
The core element, the depth structure of mathematics, incorporates cognitive mechanisms, which have evolved like other biological mechanisms, by confrontation with reality and which have become genetically fixed in the course of evolution. I shall refer to this core structure as the logico-operational component of mathematics. Upon this scaffold grew and continues to grow the thematic component of mathematics, which consists of the specific content of mathematics.
Rav makes the observation that
When we form a representation for possible action, the nervous system apparently treats this representation as if it were a sensory input, hence processes it by the same logico-operational schemes as when dealing with an environmental situation.
He quotes biologists Humberto Maturana and Francisco Varela who said:
all states of the nervous system are internal states, and the nervous system cannot make a distinction in its processes of transformations between its internally and externally generated changes.
These observations support the idea that the nervous system treats mathematical objects the same way it treats objects of the senses. And with this idea we can understand why it feels like mathematics is being discovered.
But Maturana and Varela also made the following statement
Living systems are cognitive systems, and living as a process is a process of cognition. This statement is valid for all organisms, with or without a nervous system.
Clearly Maturana and Varela defined cognition far more broadly. Cognition, in this sense, is like interaction or adaptation. This school of thought associates cognition with life in the most fundamental way. But, as Rav considers, this has implications for how we characterize knowledge itself.
This equivalence between what we mean by cognition and what we mean by living systems calls to mind (for me) Schopenhauer’s will – to which I should devote another blog. Shopenhauer’s Will is fundamental and ubiquitous, like Maturana’s cognition. I will just quote from translation of The World as Will and Representation (first published in 1819) that can be found here.
Here we already see that we can never get at the inner nature of things from without. However much we may investigate, we obtain nothing but images and names
The act of will and the action of the body are not two different states objectively known connected by the bond of causality; they do not stand in the relation of cause and effect, but are one and the same thing, though given in two entirely different ways, first quite directly, and then in perception for the understanding. The action of the body is nothing but the act of will objectified, i.e., translated into perception. Later on we shall see that this applies to every movement of the body, not merely to movement following on motives, but also to involuntary movement following on mere stimuli; indeed, that the whole body is nothing but the objectified will, i.e., will that has become representation.
I would also like to note that Rav proposes that the fact that the nervous system responds to representations in the same way it responds to sensory input can account for what he calls the Platonic illusion. But, as I see it, there is reason to wonder about this neutrality of the nervous system. I still think Plato’s intuition was correct. There is something behind or beneath the objects of our experience, even if it is the pure commonality of everything. And mathematics helps us see that.
……Best wishes for the upcoming holidays!
By Joselle, on December 13th, 2011 This blog is motivated in part by my conviction that life itself is far more mysterious than we are yet able to ponder. And it is mathematics that has often redirected my attention back to that mystery as its wealth of conceptual possibilities shows me more of what we don’t understand. David Deutsch very nicely articulated the source of my own perplexity at a TED conference in 2005. Near the end of his talk he made the following remarks regarding human structure and universal structure or our physical system and the physical system of which we are a part.
The one physical system, the brain, contains an accurate working model of the other not just a superficial image of it (though it contains that as well) but an explanatory model embodying the same mathematical relationships and the same causal structure.
The faithfulness with which the one structure resembles the other is increasing with time
Its structure contains, with ever-increasing precision, the structure of everything.
This was brought back to me today when I read an October Discover article by Carl Zimmer.
The article reviews recent work investigating the FOXP2 gene, a gene that appears to be related to the development of language in our species and hence has been dubbed the language gene.
Modern ideas about our cultural evolution can be pragmatic and dull – the purpose of language is communication, communication enhances survival prospects, life is directed by survival needs. In fact, the single most frequent complaint I hear from students in required math classes is “But when will I ever need this?”
Alternatively, a candid look at investigations of the language gene suggests that small, apparently random changes work within themselves to offer new expressions of life itself. Zimmer explains that this language gene produces a protein that seems especially active when human embryos are developing. It switches on in neurons within particular regions of the brain and then latches onto other genes in developing neurons and switches them on or off as well. And so it has the quality of being able to regulate the activity of other genes.
Humans are not the only species to benefit from FOXP2. Researchers have shown that the gene is associated with vocal learning in young songbirds, which produce higher levels of FOXP2 protein when they need to learn new songs. If their version of FOXP2 is impaired, they make singing mistakes. Other vocal-learning species, such as whales, bats, elephants, and seals, may also rely on the gene.
These findings hint at what happened to FOXP2 in our ancestors. It may have started out hundreds of millions of years ago as a gene that helped regulate the learning of body movements, such as those involved in running, calling, and biting. Later mutations in the gene spurred more neural growth in certain areas of the brain, including the basal ganglia, creating the connections essential for learning and mastering complicated sounds and, eventually, full-blown language.
FOXP2 didn’t give us language all on its own. In our brains, it acts more like a foreman, handing out instructions to at least 84 target genes in the developing basal ganglia. Even this full crew of genes explains language only in part, because the ability to form words is just the beginning. Then comes the higher level of complexity: combining words according to rules of grammar to give them meaning.
There is reason to believe that mirror neurons, those motor neurons that are activated when we are watching action as well as when we’re doing it, play a role in the development of language (there being a strong motor component to spoken language). It’s interesting, however, that these neurons don’t mirror all action, but do mirror action that seems to be characterized by some intentionality.
It is the interactive nature of neurons that got my attention again. The area of the brain more recently found to be important for language processing is located at the junction where auditory and visual sense data are processed as well as stimuli from the skin and internal organs. And these neurons are multimodal, meaning that they can process different kinds of stimuli (auditory, visual, sensorimotor, etc.) This combination of traits makes this area (the inferior parietal lobule) ideal for grasping the different properties of spoken and written words: their sound, their appearance, their function, etc. It is also thought that this area may enable the brain to classify and label things, leading to the formation of concepts and thinking abstractly.
There’s a nice multi-level description of language with information about its social, psychological, neurological, cellular and molecular aspects on The Brain From Top to Bottom. The levels you can look through are on the top right of the page.
Mathematics is not the same as language, but its development has relied on the development of language. And the great mystery reveals itself to me again when I see that David Deutsch’s observation rests on the action of a protein.
By Joselle, on December 5th, 2011 Mathematics is usually thought of as a tool that quantifies things in our lives and there is good reason for this. Early in our experience, it is presented to us as a counting and measuring device, not as a way to see something. But this characterization of mathematics is misleading. Quantification alone would not get us very far. The true value of numbers is that they give us a way to perceive order and relationship, and these produce the images and forms in mathematics that have become so powerful. Despite the ubiquitous presence of these forms in living things and social phenomena, we still tend to associate mathematical ideas with physics, or the forces that structure material. Yet mathematics itself emerges, is brought to life, from our own biology. How or why we find it is as mysterious today as it ever was.
It is for this reason that every indication of its living presence is interesting to me, like the instinctual vector analysis that ants seem to manage to find their way home. Or what I found today – the living phase transitions of starling flocks. The video posted on Wired Science is definitely worth a look. The text of the article describes the unexpected character of the patterns displayed by the flock.
What makes possible the uncanny coordination of these murmurations, as starling flocks are so beautifully known? Until recently, it was hard to say. Scientists had to wait for the tools of high-powered video analysis and computational modeling. And when these were finally applied to starlings, they revealed patterns known less from biology than cutting-edge physics.
Starling flocks, it turns out, are best described with equations of “critical transitions” — systems that are poised to tip, to be almost instantly and completely transformed, like metals becoming magnetized or liquid turning to gas. Each starling in a flock is connected to every other. When a flock turns in unison, it’s a phase transition.
Another great video is featured here.
In the abstract of a paper on biological criticality the authors explain:
Many of life’s most fascinating phenomena emerge from interactions among many elements–many amino acids determine the structure of a single protein, many genes determine the fate of a cell, many neurons are involved in shaping our thoughts and memories. Physicists have long hoped that these collective behaviors could be described using the ideas and methods of statistical mechanics. In the past few years, new, larger scale experiments have made it possible to construct statistical mechanics models of biological systems directly from real data. We review the surprising successes of this “inverse” approach, using examples form families of proteins, networks of neurons, and flocks of birds. Remarkably, in all these cases the models that emerge from the data are poised at a very special point in their parameter space–a critical point. This suggests there may be some deeper theoretical principle behind the behavior of these diverse systems.
It’s not just interesting that these events can be modeled using mathematics. What’s noteworthy is that the living actions themselves seem to contain mathematics. They manifest the mathematical forms we investigate as plainly as do the organic structures in this very pretty film about the Fibanocci numbers.
What this suggests to me is that mathematics is opening our awareness to something truly fundamental about our reality by giving it conceptual shape. And this window we’ve created should ultimately tell us something about ourselves.
A quote I like from Blaise Pascal (that I found on MAA Mathematical Sciences digital library) is this one:
Nature is an infinite sphere of which the center is everywhere and the circumference nowhere.
By Joselle, on November 28th, 2011 My husband is one of the experimental physicists participating in the ATLAS experiment at the LHC at CERN. He left this morning on a trip to Geneva to visit CERN and that may be why I clicked on Kelly Oakes blog at the Scientific American blog network: Why the Higgs Boson Matters.
The stuff that has the attention of particle physicists may seem far removed from what appears to be the real world, but in the words that Oakes borrowed from Carl Sagan I find the kind of thought that keeps me interested.
everyone you love, everyone you know, everyone you ever heard of, every human being who ever was, lived out their lives on the pale blue dot we know as Earth — and none of it would have ever existed without the Higgs boson.
The history of the Higgs boson, the idea that a particular, yet unobserved phenomenon is responsible for all of the mass contained in the universe, is nicely told in a 2007 article in The Guardian.
But when all of the words associated with this quest in physics become attached to people and places, they can come alive. A very nice effort to do just that is produced at a website called Colliding Particles. There one can find short films that bring you into some of the day-to-day of researchers and their reflections. In the first of this series, Gavin Salem says nicely that the fundamental motivation for this now highly technical work is that “We want to know how we got here.” And, he continues “part of how we got here is knowing why we’re made up of the things we’re made up of, what the structure of the world is.” The work, he explains, is essentially born of fundamental human curiosity. In the same segment, Jonathan Butterworth explains that physics is not just observation, although it starts with observation, but is more about fitting ideas, that have predictive value, to observations. Jon’s student explains that the Higgs particle is a mathematical way of introducing mass into the universe. This idea fits, and it can be tested.
Ideas are laboriously explored mathematically, in a reality that has been quantified for just this purpose. Mathematics is the only thing that can structure an idea. In a another episode called Problems Gavin tells us that one of the difficulties in physics is figuring out which problem to solve. And his student volunteers that when he tries for days to solve something, and doesn’t, he almost wants to “give up everything.” Gavin adds that part of solving the kinds of problems he and his students work on is “believing you can solve it and having the persistence to think, and wait, and think, and come back to it, until you find a solution.” And, he tells us, “knowing that that whole process does actually go somewhere– the belief that you can get there, is an important part of solving problems.”
Physicists are often asked to do what the physicists in these films are doing, that is to bring words and pictures to the ideas they explore so that they can be comprehended by the rest of us. The mathematics is left out because, I would argue, the mathematics is the thinking. It is the mechanism that, like our cognitive mechanisms, actually shapes the observations that are made. It seems very likely to me that mathematics is some conscious extension of the body’s inherent way of learning from its world. And the body is built to acquire knowledge that has predictive value. The cause and effect reasoning studied in babies shows us some of our most basic wiring.
I can imagine that the mathematical description of a fundamental particle is like a neurological description of a tree.
By Joselle, on November 21st, 2011 To give shape to this blog, I’ve been jumping around quite a lot through the fields of mathematics, physics, and the neurological and cognitive sciences. I decided today to let more of my weight drop into philosophy.
It’s not unusual when reading about 19th century developments in mathematics (the ones that lay the groundwork for mathematics as it is understood today) to see references to the late 18th century work of Immanuel Kant. Often the references describe how the discovery of non-Euclidean geometries contradicted him. This is primarily because Euclidean geometry was the only one possible for him. But Kant was at work reconciling the conflict between the rationalists, who saw knowledge as a product of the intellect, and the empiricists, who saw it as a product of the senses. I don’t believe he would have found the development of non-Euclidean geometries inconsistent with his perspective that knowledge emerged from the interplay of sensibility and reason.
Philosophers have influenced developments in physics, but today, the implications of research are mostly discussed by the researchers. There are interdisciplinary impulses, like the physicists and cosmologists from the Foundational Questions Institute, who organized a multidisciplinary conference on the perception of time. Yet I don’t think I’ve read any truly philosophical critiques of string theories, for example. The controversies surrounding string theories stem largely from our inability to test the theories, to make an observation that anchors their mathematical meaning to some clear physical meaning as well. The dispute over the value of this research is not a philosophical one. It’s a pragmatic one. So today I wondered if the rationalist/empiricist argument is worth remembering in the context of some of the conceptual difficulties in physics. I pulled this excerpt of Kant from a paper written by David Kaiser .
Without sensibility no object would be given to us, without understanding no object would be thought. Thoughts without content are empty, intuition without concepts blind. It is, therefore, just as necessary to make our concepts sensible, that is, to add the object to them in intuition, as to make our intuitions intelligible, that is, to bring them under concepts. These two powers or capacities cannot exchange their functions. The understanding can intuit nothing, the senses can think nothing. Only through their union can knowledge arise.
Kant used mathematics (and in particular Euclidean geometry) as a demonstration of knowledge that doesn’t come from experience. You can find an outline of his thinking on Philosophy Pages. About his view of mathematics they say:
Understanding mathematics in this way makes it possible to rise above an old controversy between rationalists and empiricists regarding the very nature of space and time. Leibniz had maintained that space and time are not intrinsic features of the world itself, but merely a product of our minds. Newton, on the other hand, had insisted that space and time are absolute, not merely a set of spatial and temporal relations. Kant now declares that both of them were correct! Space and time are absolute, and they do derive from our minds. As synthetic a priori judgments, the truths of mathematics are both informative and necessary.
I found a website that described itself as being devoted to tackling age-old philosophical questions with the help of cybernetic theories and technologies. There is an article there (written more than 15 years ago) by Valentin F. Turchin with the title: From Kant to Schopenhauer.
In it Turchin says:
In classical mechanics we use much more of our neuronal world models.
In other words, classical mechanics talks about a world consistent with the one the body organizes with sense data.
There is a three-dimensional space; there is time; there are the concepts of continuity, a material body, of cause and effect, and more.
Mach and Einstein would be, probably, impossible without Kant. They used the Kantian principle of separating elementary facts of sensations and organizing these facts into a conceptual scheme. But the physicists went further. Einstein moved from the intuitive space-time picture given by the classical mechanics down to the level of separate measurements, and reorganized the measurements into a different space, the four-dimensional space-time of the relativity theory. This space-time is now as counterintuitive as it was in 1905, even though we have accustomed to it.
In quantum mechanics, the physicists went even further. They rejected the idea of a material body located in the space-time continuum. The space-time continuum is left as a mathematical construct, and this construct serves the purposes of relating micro and macro-phenomena, where it has the familiar classical interpretation. But material bodies lost their tangible character. … In the relativity theory observations (measurements) at least belonged to the same universe as the basic conceptual scheme: the space-time continuum. In quantum mechanics, on the contrary, there is a gap between what we believe to really exist, i.e. quantum particles and fields, and what we take as the basic observable phenomena, which are all expressed in macroscopical concepts: space, time and causality.
Kant elevated abstract knowledge to the same level of significance as experience or sensation, a perspective that sets the stage for the counter-intuitive conceptual schemes that show us the world we don’t otherwise see. But quantum mechanics introduces a deeper conceptual difficulty.
Turin goes on to argue that if we want to construct a theory that describes the ultimate reality of physics, one that can, step by step, construct the observables, we need a philosophical basis that goes further than Kant. He explains:
We must go further down in the hierarchy of neuronal concepts, and take them for a basis. Space and time must not be put in the basis of the theory. They must be constructed and explained in terms of really existing things.
Kant’s metaphysics, Turin explains, needs to be pushed further. More has to be understood about the relationship between observables and conceptual schemes. These ‘really existing things’ Turin describes as “the most essential, pervasive, primordial elements of experience.” And so Turin moved on to Schopenhauer who, finds the essence of reality in action more than substance.
Let us examine the way in which we come to know anything about the world. It starts with sensations. Sensations are not things. They do not have reality as things. Their reality is that of an event, an action. Sensation is an interaction between the subject and the object, a physical phenomenon. Then the signals resulting from that interaction start their long path through the nervous system and the brain. The brain is tremendously complex system, created for a very narrow goal: to survive, to sustain the life of the individual creature, and to reproduce the species. It is for this purpose and from this angle that the brain processes information from sense organs and forms its representation of the world. Experiments with high energy elementary particles were certainly not included into the goals for which the brain was created by evolution. Thus it should be no surprise that our space-time intuition is found to be a very poor conceptual frame for elementary particles.
We must take from our experience only the most fundamental aspects, in an expectation that all further organization of sensations may be radically changed. These most elementary aspects are: the will, the representation, and the action, which links the two: action is a manifestation of the will that changes representation.
Why not see this as an indication that action should have a higher existential status than space, time, matter?”
The Kantian idea that conceptual knowledge is a necessary partner to sensory knowledge lines up with many of the opinions of mathematics expressed in posts I’ve authored. But Schopenhauer’s view that the universe is apprehensible through introspection approaches the heart of much of how I see things. I will want to speak to this soon.
By Joselle, on November 14th, 2011 In another blogging heads interview (and in a related blog), John Horgan explores with David Rothenberg the significance of beauty in scientific thinking. Rothenberg’s new book Survival of the Beautiful, is the subject of much of their discussion. While the conversation centers on questions of beauty (how biology does or does not take it into account) for me, the value of work like this lies in how it promotes the view that the many of the facets of human culture are aspects of nature itself. Drawing a similar portrait of mathematics is one of the motivations for this blog. And so I find work like Rothenberg’s exciting. He’s a musician, and a philosopher with very productive curiosity.
One of the points made in the interview is that a purely functional view of creative activity can miss the significance of what may be happening. In other words, if one takes the view that bird song functions to defend territory or to attract mates, there is little reason to consider the wide variety among different bird songs, from short calls to complex melodies. They become equivalent by virtue of our decision to think in terms of function. Rothenberg also makes the point that we may be missing an opportunity to better appreciate the significance of human creativity and culture by failing to see its counterparts in the rest of the natural world.
The discussion is interesting and Rothenberg’s approach to the art of nature’s creatures is enlightening. It points, again, to the limits of purely functional, reductionist approaches and raises the question of how science might profit from, even a subtle, reorientation of what it expects to see.
For me, the heart of the matter rests in the idea that the essence of all life is to occupy its world by engaging it, expressing it and creating it. As John O’Donohue said so well in Anam Cara “Essentially, we belong to nature. The body knows this belonging and desires it.”
There were two questions Horgan raised, that were interesting to me, but I don’t think Rothenberg adequately considered them (at least not in the interview). The first was given the tendency in biology to overlook the significance of the beauty of things (like peacock tails and bird songs) and our habit of viewing them functionally (as mating strategies) Horgan wondered about the predisposition of physicists to use beauty and elegance as guides in their investigations. As Wilczek once said, symmetry is not so much an aesthetic choice as it is a strategy. Rothenberg addressed the contrast by first taking note of the way biological models of life contain an inherent arbitrariness, inconsistent with the more predictive nature of physics. (I think that’s what he was saying). But he also went on to characterize the patterns and forms for which (or with which) physicists search as “basic forms” and “rules” that could prejudice our view, because life can actually be very messy. I think this view of physics is common and mistaken. A more interesting way to look at it is that physicists trust something that they don’t necessarily understand. They let something that has no particular function, something whose value is that it pleases, something one might call instinctive, direct them.
Horgan also asked him about Semir Zeki’s thoughts on art. And Rothenberg suggested that Zeki was engaged more in looking at what was happening in the brain when we looked at a work of art. But Zeki has suggested some things that are much more interesting than what Rothenberg described. In particular, Zeki has suggested that the visual arts may be extending the function of the visual brain – that is to see, to get information about the world, to find the essence of things. This is a fresh way of bringing art and science together. I have a blog discussing both Wilczek and Zeki here.
I think Zeki’s ideas about art and Horgan’s hunch about physics are related. They both address what may be less than conscious, but fairly complex strategies. I am of the mind that mathematics itself can be viewed in this way, as a living part of us, doing what living things do – engaging, playing, expressing and creating. I see it as one of the actions of our searching, instinctual will.
While I may have been hoping to hear Rothenberg say something more about math, physics and Zeki, his own work with music, art and biology most certainly opens up some very important doors and windows, and I look forward to reading it.
By Joselle, on November 8th, 2011 As I talked about in a recent post, string theories, and the multiverse models they imply, have been widely criticized for their lack of testability. Some physicists argue that the problem is that the theory is more mathematics than it is physics. Is the distinction becoming fuzzier? And why isn’t that discussed? Why not bring the role mathematics plays in this and other ideas into the foreground? Does mathematics make things? The following statements are from a transcription of Alan Boyle’s interview with Brian Greene from Cosmic Log on msnbc.com.
While Greene consistently acknowledges that mathematics is leading the way, the debate never seems to include questions about why or how the mathematics may be leading the way. Here’s an excerpt:
Brian Greene: Well, when we are doing mathematical investigations in physics, we as theorists allow the math to take us where it will go. We have seen, time and again, that math is a very potent guide to revealing the true nature of reality. That’s what the past couple of hundred years have established. So all we’re doing is following the same kinds of procedures that we always have. And as we follow the procedures, as we push the mathematics forward, the math is clearly suggesting that there may be other universes out there.
Q: You make the point that it’s very difficult to have any sort of direct contact with other universes. The differences are just so great. The only way to conceptualize other universes, I suppose, is through mathematics and the bits of evidence that can be gleaned from particle collisions or the cosmic microwave background radiation. Is there any possible avenue to get substantive information about the bigger picture, or are we pretty much stuck in our own little corner of the multiverse?
A: I think we’re certainly stuck physically. But I would not underestimate the power of mathematics to provide the kinds of insights you are referring to. We are definitely at a rudimentary state in our understanding of these multiverse proposals. But if we can refine that understanding, we could produce detailed “universe demographics.” We could gain a very detailed understanding of the percentage of universes that would have this or that quality.
Perhaps it’s a good thing that, despite the fact that mathematics plays so significant a role in driving physics theories, the theories are described in a non-mathematical way, or without any reference to the mathematics. This could mean that, in physics, mathematics is never fully distinguished from the actions that define observation itself. In other words, the brain (or the body) builds what we perceive from fairly disparate pieces of sensory data and now we accept that mathematics extends this nervous system action to the things the senses cannot access directly. We don’t talk about our eyes when we talk about seeing something.
But this warrants some discussion itself, since most people do not think about mathematics in this way. Even for many readers of popular science books, mathematics isn’t seen as building the reality that physicists encounter, but more, as something that provides a technical description of it. This misunderstanding is evident in another interview with Brian Greene on NPR’s Fresh Air when Terry Gross:
GROSS: Now, you said something that really baffles me. You said: When we study those universes in mathematical detail – what do you mean by that? I mean, we don’t even know those universes exist. So when you say when we study them in mathematical detail, what are you talking about?
GREENE: Well, that is a confusing idea, I think, for people who don’t actually engage in the kind of research that I’m talking about because what we do is we sit down with equations, equations that describe space and time, equations that describe how matter can move through space and time. And using those mathematical equations, we can get a sense of what it would be like to be in one of those other universes, even if we can’t actually visit or see or interact with that universe in any real sense. That’s the power of mathematics.
And I have to say, underlying everything that we’re talking about, in fact underlying everything I do with my entire life, pretty much, is a firm belief that mathematics is a sure-footed guide to how reality works. If that’s wrong, then all bets are off.
I should note here that one of the criticisms of the kind of research in which Brian Greene is engaged is that it is too completely characterized by mathematics. But researchers find the strength of this work in exactly this – how it puts both quantum mechanics and general relativity into a mathematically consistent framework, when nothing else that we’ve imagined can. Lawrence Krauss once said that string theories were, mathematically, the most intoxicating ideas around.
I plan to discuss the string theory debate again in another post. But I want to end this one with a transcription from a talk given by David Deutsch at a TED conference in 2006. The punch line of his talk was actually about what he considers to be the only productive response to the problem of global warming. But on his way to that thought, he painted this wonderful image of human understanding as a kind of construction, beautifully re-connecting us (and our mathematics) to the universe itself. I should clarify that referring to humanity as chemical scum points back to a description of our existence from Stephen Hawking.
Deutsch is describing the energy (that was pressed out by magnetic fields around a galaxy collapsing into a black hole) that shot out in jets (producing a quasar). His observation is that this happened:
in precisely such a way that billions of years later, on the other side of the universe, some bit of chemical scum could accurately describe and model and predict and explain what was happening there, in reality. The one physical system, the brain, contains an accurate working model of the other not just a superficial image of it (though it contains that as well) but an explanatory model embodying the same mathematical relationships and the same causal structure. Now that is knowledge. The faithfulness with which the one structure resembles the other is increasing with time. This chemical scum has universality. Its structure contains, with ever-increasing precision, the structure of everything. This place is a hub which contains within itself the structural and causal essence of the whole of the rest of physical reality. The fact that the laws of physics allow this or even mandate that this can happen is one of the most important things about the physical world.
By Joselle, on November 5th, 2011 Contrary to the by-line, this post is by Bob not Joselle. She wanted me to post an item that’s been of interest lately.
As a reader of Mac and Apple rumor sites over the years, I was surprised the night of October 5th when I went to cnn.com to show Joselle a news item which I have completely forgotten now. Instead, on the page was notice that Steve Jobs had died. Incomprehensibly to me, I became somewhat sad and quiet. Why? I never met him. I am a fan of many of the developments that came out of Apple this last decade, but I am under no illusion that they are all solely Jobs’s invention. I have read repeated accounts of his sometimes difficult behavior, and it is quite possible I would not even enjoy his company.
Perhaps partly to identify the reason for my reaction, since that day I have thought about several of the blogs and editorials about Jobs that have been published online since his passing. While interesting, none of them helped me understand my own reaction. Many of them seemed compelled to summarize what it was that Jobs had given us, what it was that caught our attention. Initially, many of the conclusions were fairly specific – the iPad is what he was after all along, or maybe the iPhone. Perhaps it was the new technology, like the touch or voice interfaces. Others claimed it was his attention to detail. A common theme was that he found a way to create interfaces and technologies that are close to us. This used to be called ‘the look and feel’ in the days of the Mac vs. PC battles. Some called attention to his evident marketing genius, or seeming creation of new industries. While it does seem to be the case that he accomplished much in these areas, I am not sure all credit can be given to him in each instance. Regardless, I don’t think any of these are the true gift of Steve Jobs.
Later postings have concluded that his greatest success was the company Apple itself, an organization they claim is uniquely able to match artistic and technology sense with the untapped desires of the public. Maybe it is a great result. Jobs appears to have concluded it was, according to one interview. The company seems to have achieved this during his tenure as CEO, and time will tell if it is able to deliver going forward. But I don’t think this is the important gift that he gave.
It was his sister, author Mona Simpson, who touched upon it in her eulogy, I think, when she mentioned the strength of his will during his illness. To me, the true gift he gave was the unique demonstration of will. Not so much his will in illness. Just will. It is perhaps most concisely called to attention by looking at Apple in late 1996, bleeding to death and with drums of doom pounding in almost every news item of its impending end. When he returned, it was rightly believed that it would be impossible for Apple to survive, let alone return to a place of prominence at the leading edge of technology. I dare say that for any other CEO, it would have been. But Jobs clearly saw this differently and had the will to make it happen. Looking in hindsight at the iMac, the cancellation of Newton, Mac OS X, the iPod, iTunes, iPhone, iOS and iPad, there was clearly tremendous focus and long-term planning of staging and testing developments in the marketplace. I am admittedly no expert, but I know of no other similar sequence in corporate accomplishments.
As an experimental scientist, I tend to rely on physical metaphors. The qualities of ferromagnetism come to mind. Magnetic domains in a ferromagnet become aligned with the application of a persistent and strong magnetic field. In the case of Apple, the magnetic field was the will of Steve Jobs, and the resulting ferromagnetism is the consistent output of his company. The course of Apple was clearly an external manifestation of the internal will of its leader. And this brings me to the reason for posting this in Joselle’s blog.
As her blogs often point out, the world inside and the world outside are inextricably linked. The will is perhaps one path linking them. In the case of Jobs, his internal world was brought out for all to see in surprising levels of design and ease of use of what are usually temperamental and difficult to use technologies. His attention to his own intuition and consistency seemed to verge on the probing thinking demanded of many research mathematicians. Indeed, Simpson revealed that, had his life progressed differently, Jobs felt he might have been a mathematician. That might surprise some people given his ties to art and his subordination of technology to artistic impulses. But I’m not surprised.
Steve Jobs gave us the wonderful demonstration of the resurrection of Apple and the power of will to facilitate it. Whatever you think of him, such a story is as remarkable as a real-life Rocky, and worth the telling. But he also gave us a unique demonstration of an unusual connection of the exterior material technology we use to the carefully tended interior world of our thought. In that sense, what he was doing was similar to mathematicians and artists alike.
For that demonstration of will, I am thankful.
By Joselle, on October 31st, 2011 Back in July, David Castelvechhi blogged about a conversation between John Horgan and George Musser. I missed it when it was new, but I’m glad I didn’t miss it completely. Most of their discussion focuses on the value or viability of what has come to be known as string theory. It was a thoughtful debate that addressed the subtleties of considering the meaningfulness and testability of the theory.
One of the major difficulties with string theories is that there is no clear way to test them. High energy physics experiments have been enormously successful at finding the reasoning, and the equipment, that can make a view of the early universe possible. Particle physicists have identified, measured and recorded the fundamental constituents of matter. They are able to describe and predict the matter and energy states of physical systems very precisely. Various string theories have been proposed to unify the forces described in these quantum mechanical systems with those of general relativity. But it is striking that, within the experimental possibilities of particle physics, there is no way to see, or to detect, the fundamental constituents of nature that various string theories propose.
The particle detectors of physics experiments are like giant elaborations on the senses. They detect the presence of something when that thing hits the detector and interacts with it, the way visual components of the world are produced when light hits our eyes or, more generally, the way various aspects of the world hit sensory mechanisms of the entire body. If the elements of string theories are outside the range of even the sensory mechanisms of particle detectors, then the question arises, can a theory like this contribute to science? John Horgan is of the mind that it cannot. But George Musser defends the possibility that indirect evidence can, over time, develop into the pictures that string theories predict.
The elements of string theories have been considered to precede space and time, matter and energy. And so Horgan understandably asks,
What is a string then? If it’s not something that can be situated in space and time and if it’s not constituted of matter or energy, what the hell is it? Is it some kind of pure mathematical form?
Except for this one moment, mathematics doesn’t really come up, despite the fact that string theory is largely a mathematical idea. Musser described how space and time or energy and matter have been described as emergent properties of the elements of these theories. Talking about something that precedes space and time,” as Horgan says, seems to not make sense and we might even think that this is modern physics leading itself astray. But this is not just a modern idea. As Frank Wilczek says in The Lightness of Being,
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.
Plato was an idealist, but so was Leibniz. For Leibniz the world was not made of material. Anything that took up space, or had extension, was by definition complex, meaning that it could be divided. The truly fundamental elements of the universe would not be divisible and so were necessarily immaterial. The idea that the physical world emerges from immaterial elements is outside the range of most contemporary science-minded individuals (but not all).
I would like to suggest that what is being left out of this discussion is the lingering problem, identified by physicists themselves – the problem of objectivity or that there really is none. It is the way we perceive that builds the worlds we see. It could be said that some of the profound developments of 19th century mathematics were prodded along by insights into the illusion, produced by the senses, that the space we see around us is an objective space that contains us. And that Euclidean geometry describes it. But space is an organizing principle, not a thing. It shows up often, in things like the circular display of time, or the portrayal of color on a wheel. We talk about the years ahead of us or behind us, or the ups and downs of the market. Riemann’s insights into the foundation of geometry were inspired, in part, by insights into visual processes. And in his work are the seeds of the kind of topological thinking that is the groundwork for string theories.
The senses function largely as organizing processes and I have come to see that mathematics often reflects that. What’s missing from these discussions about the true nature of reality, or how far science can reach, are questions about how the mathematics we discover brings novel possibilities into our imagination and otherwise hidden aspects of nature into our awareness. If we find a universe in our mathematics that we can’t see yet, that doesn’t mean it isn’t there. Perhaps we just haven’t grasped our relationship to it yet, which we likely have, since it emerged from the symbolic systems that define physics ideas. Trying to see how it happens, how it is that mathematics can extend the range of our awareness is a difficult path to navigate. But there is no question that it is worth exploring.
By Joselle, on October 24th, 2011 I ventured down a series of paths today, no doubt related, but with no quick and easy way to tie them together. So I decided to invite you to look with me and let your mind play.
I started with a couple of talks at a recent at a recent Foundational Questions Institute conference on the Nature of Time.
I listened first to a panel made up of neuroscientist David Eagleman, physicist Paul Davies, philosopher Tim Maudlin, and mechanical engineer and computer scientist Raissa D’Souza who is currently most involved with models of complexity. It is an interesting discussion that begins with Paul Davies’ observation that there is nothing in the equations of physics that says that this moment, the present, is special. There is nothing that conveys the now of our experience. In modern physics, time is not specifically represented. Tim Maudlin took note of the fact that now it is physicists, not philosophers that say some of the strangest things about time, but that he will make the more straightforward claim that time exists, that it has a direction, that we are objectively getting older, and that our task is really to understand how things are related to each other. Raissa D’Souza is interested in the local ordering of emergent properties. For her, time is characterize by the increasing complexity of systems, or their increasing interdependence. Mathematics itself only came up once in this discussion and it was the philosopher who brought it up, making the argument (I think?) that we need to be careful to distinguish between mathematical possibilities and the properties of physical things. I didn’t hear enough from him to be sure, but I think he is of the mind that mathematics is a tool applied to the description of our reality rather than a strategy for unraveling some of the illusions of our experience.
David Eagleman, the neuroscientist, got my attention when he answered a question from the audience. The question was whether time, like solidity, might one day be understood as not being a fundamental property of the world, but rather as one that is built up in some way from more fundamental fragments. Eagleman found the question interesting given that our sense of time is thought of as a meta-sense. It is so named because different aspects of our notion of time – duration, temporal order, something called rate of flicker, and simultaneity – actually happen through different neural mechanisms. While they normally work in concert, and produce a unified experience, they can be teased out from each other and separated.
The group did address the fact that, in physics, words like entropy, force, energy or work have very precise meaning, distinct from the common use of these words, and that there was also some difference in how they were used to describe macroscopic events versus quantum mechanical ones. Eagleman, wanting to say something about the macroscopic energy efficiency of the brain, near the end said something like ‘we’re mobile creatures and if you want to simulate the brain, you need to simulate the stomach because we’re driven by hunger..’ I would agree, and I am intrigued by the fact that hunger has brought us to some of the most enigmatic riddles of our experience. Eagleman is of the opinion that time itself was calculated internally with respect to how much energy the brain felt itself using.
I then listened to the talk given by Julian Barbour. He mentions early in his talk that Newton got it wrong and Leibniz got it right. I’ve recently spent time exploring the Leibniz view of the world, and today tried to search out what he said about time, specifically. I found a nice discussion on the Internet Encyclopedia of Philosophy, where the author uses the structure of virtual realities as a metaphor for the mathematician’s metaphysical ideas. I think it works nicely. For Leibniz, space and time were the physical representations of ideal relations, and not things in themselves.
Take the analogy of a virtual reality computer program. What one sees on the screen (or in a specially designed virtual reality headset) is the illusion of space and time. Within the computer’s memory are just numbers (and ultimately mere binary information) linked together. These numbers describe in an essentially non-spatial and temporal way a virtual space and time, within which things can “exist,” “move” and “do things.” For example, in the computer’s memory might be stored the number seven, corresponding to a bird. This, in turn, is linked to four further numbers representing three dimensions of space and one of time–that is, the bird’s position. Suppose further the computer contains also the number one, corresponding to the viewer and again linked to four further numbers for the viewer’s position, plus another three giving the direction in which the viewer’s virtual eyes are looking. The bird appears in the viewer’s headset, then, when the fourth number associated with the bird is the same as the viewer’s fourth number (they are together in time), and when the first three numbers of the bird (its position in virtual space) are in a certain algebraic relation to the number representing the viewer’s position and point of view. Space and time are reduced to non-spatial and non-temporal numbers. For Leibniz, God in this analogy apprehends these numbers as numbers, rather than through their translation into space and time.
This site also explains this about the Leibniz perspective:
Leibniz argues that it would be a great waste of possible perfection to only allow living beings to have bodies at that particular level of aggregation with which one is phenomenally familiar. (Perhaps Leibniz was understandably impressed by the different levels of magnitude being revealed by relatively recently invented instruments like the microscope and telescope.) Leibniz writes:
“Every portion of matter can be thought of as a garden full of plants, or as a pond full of fish. But every branch of the plant, every part of the animal, and every drop of its vital fluids, is another such garden, or another such pool. […] Thus there is no uncultivated ground in the universe; nothing barren, nothing dead.” (Monadology, §§67 & 69)
Barbour’s talk presents what he calls shape dynamics, where shape, identified as the only intrinsic property of something, leads the theoretical construction. I won’t try to capture it here. But it will likely be the subject of another post.
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