Mathematics Rising

I wanted to take a closer look at the Lakoff/Nuñez book Where Mathematics Comes From and its relationship to what has come to be called the embodied mind.  It seems to me that the biologists who pioneered embodiment had a more radical view of cognition than many of the cognitive scientists who use the paradigm.  For example, in one of his papers, Humberto Maturana Romesin, an early explorer of a very broadly defined notion of cognition, has said:

In my view the central theme of cognition is the explanation of experience, not reality, because reality is an explanatory notion invented to explain experience.

This might sound vexingly circular, but this is only because we are so locked into our usual point of view.  One of the things that came to mind when I read it, was the difficulty we have with many of the ideas in modern physics  – like, for example, the idea that there is no place and time before the big bang, that there is no such thing as nothing, or that virtual particles can pop in and out of existence.  These realities are the explanatory notions invented to explain a very broadened experience, the experience that physics provides.

The difficulty with the view presented in Where Mathematics Comes From is that, while it associates mathematics with some very fundamental human experiences, it fails to fully appreciate the way living systems are understood as totalities by many of the biologists whose work inspired embodiment ideas.  In particular, it fails to fully appreciate the way living systems exist in relation to their medium and are, in no way, independent of it. As Maturana says:

the behavior that appears is not a feature of the organism, but a condition of its existence in the relational space in which it is a totality, and in which behavior as a relational dynamic involves both the organism and the medium in which it exists.

He goes on to say that one of the reasons we have difficulty with this view:

arises from our cultural training that leads us to think in terms of external causes to explain the occurrence of any phenomenon. This attitude blinds us to the spontaneous nature of all processes in the molecular domain in which we exist. All molecular processes occur spontaneously following a path that arises moment after moment according to the structural dynamics of the different molecules involved.

Where Mathematics Comes From approaches mathematics in a very tool-like way, with an emphasis on how we might be building it.  And this is certainly a useful perspective.  But it likely misses something because it only minimally addresses the spontaneity of cognitive processes.

In a review of the book back in 2001, James Madden takes note of other deficiencies in the Lakoff/Nuñex perspective:

If I think about the portrayal of mathematics in the book as a whole, I find myself disappointed by the pale picture the authors have drawn. In the book, people formulate ideas and reason mathematically, realize things, extend ideas, infer, understand, symbolize, calculate, and, most frequently of all, conceptualize. These plain vanilla words scarcely exhaust the kinds of things that go on when people do mathematics. They explore, search for patterns, organize data, keep track of information, make and refine conjectures, monitor their own thinking, develop and execute strategies (or modify or abandon them), check their reasoning, write and rewrite proofs, look for and recognize errors, seek alternate descriptions, look for analogies, consult one another, share ideas, encourage one another, change points of view, learn new theories, translate problems from one language into another, become obsessed, bang their heads against walls, despair, and find light. Any one of these activities is itself enormously complex cognitively—and in social, cultural, and historical dimensions as well. In all this, what role do metaphors play?

With a critique more centered on mathematics education, Martin Schiralli and Nathalie Sinclair make a related observation:

WMCF is right about the sensory-motor basis of abstract concepts, but their reduction of abstract concepts to more concrete ones through metaphor fails to explain the fundamental processes involved in acts of abstraction. The very phrases ‘abstract thought’ and ‘abstract concept’ are misleading. The expression that needs to be analysed is ‘thinking abstractly’.

An intuitive framework (more Kantian naturalism than Platonic idealism) might be a given, but later ‘intuitions’ might be the result of the ideational being (the thinker) tapping into the embodied ordering principles and categories that the visceral being (the organism) has been subliminally and experientially processing.

I believe a more careful look at this biological view of mathematics has been explored by Yehuda Rav.  Here are some excerpts from his essay on mathematics as seen in the light of evolutionary epistemology.

Thus, Maturna (1980, p. 13) writes: “Living systems are cognitive systems, and living as a process is a process of cognition”. What I wish to stress here is that there is a continuum of cognitive mechanisms, from molecular cognition to cognitive acts of organisms, and that some of these fittings have become genetically fixed and are transmitted from generation to generation. Cognition is not a passive act on the part of an organism, but a dynamic process realized in and through action.

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. From a different perspective, Maturana and Varela (1980, p. 131) express it this way:  “all states of the nervous system are internal states, and the nervous system cannot make a distinction in its process of transformations between its internally and externally generated changes.”

Thus, the logical schemes in hypothetical representations are the same as the logical schemes in coordination of actions, schemes which have been tested through eons of evolution and which by now are genetically fixed.

As it is a fundamental property of the nervous system to function through recursive loops, any hypothetical representation which we form is dealt with by the same ‘logic’ of coordination as in dealing with real life situations. Starting from the elementary logico-mathematical schemes, a hierarchy is established. Under the impetus of socio-cultural factors, new mathematical concepts are progressively introduced, and each new layer fuses with the previous layers.  In structuring new layers, the same cognitive mechanisms operate with respect to the previous layers as they operate with respect to an environmental input. …..The sense of reality which one experiences in dealing with mathematical concepts stems in part from the fact that in all our hypothetical reasonings, the object of our reasoning is treated by the nervous system by means of cognitive mechanisms which have evolved through interactions with external reality.

Mathematics is a singularly rich cognition pool of mankind from which schemes can be drawn for formulating theories which deal with phenomena which lie outside the range of daily experience, and hence for which ordinary language is inadequate.

I would like to add one further note.  Near the beginning of his essay, Rav says this:

mathematics and objective reality are related, but the relationship is extremely complex and no magic formula can replace patient epistemological analysis.

And later:

The nervous system is foremost a steering device for internal and external coordination of activities.

For me the most provocative thing about mathematics is what it may be telling us about the connectedness of the internal and the external – experiences that we often have great difficulty reconciling.

I listened to three short talks today and found that they had something nice in common – they each show us how sensory experience (often vision) gives rise to mathematics that provides access to what cannot be seen, and clarifies what is seen.

The first of these talks was called Symmetry, reality’s riddle presented by Marcus du Sautoy.

It begins with a description of Galois’ famous death in a duel. Galois stayed up the whole night before the duel writing letters and outlining his mathematical ideas.  Marcus du Sautoy explains how Galois had found a new language, the language of symmetry, and goes on to take note of the ubiquitous presence of symmetry, showing it to us in, among other things, molecular structure, particle physics and art.  Du Sautoy explains how Galois saw that it is not just the symmetries we see in an object that characterize the object, but also the way the symmetries of an object interact with each other. With this insight, he develops a language that can identify the substance of the things unseen.  The symmetry that underlies a physical object is captured by a number.  And du Sautoy makes the point that this language, expressed in number, now makes it possible for him to create symmetrical objects in high dimensional spaces.  This is the mathematical idea of group theory.

The second was a talk given by David Deutsch: A new way to explain explanation. While I didn’t exactly enjoy his critique of pre-scientific ideas, he made some clarifying points about empiricism and the scientific method. He observes that science is originally distinguished by the conviction that all knowledge is derived from the senses.  While, he says, this helped by promoting observation and experiment, “it was obvious that there was something horribly wrong with it.”

Empiricism is inadequate because, well, scientific theories explain the seen in terms of the unseen.  And the unseen, you have to admit, doesn’t come to us through the senses.  We don’t see the origin of species.  We don’t see the curvature of space-time, and other universes.  But we know about those things.  How?

What we see, in all these cases, bears no resemblance to the reality that we conclude is responsible – only a long chain of theoretical reasoning and interpretation connects them

Deutsch makes the argument that testability is only valuable when the theory to be tested is “hard to vary, because every detail plays a functional role.”  By way of an example, he identifies all of the functional relationships in the modern explanation of the seasons – “that surfaces tilted away from radiant heat are heated less, that a spinning sphere, in space, points in a constant direction, that tilt also explains the sun’s angle of elevation at different times of year, and that it predicts that the seasons will be out of phase in the two hemispheres.”

Finally, it was Benoit Mandelbrot’s talk: Fractals and the art of roughness that I most enjoyed.

Mandelbrot also found a number that characterized an observation, a number that denoted the roughness of a surface.  In an engaging informal style he just tells us stories.

Humanity had to learn about measuring roughness…..Very few things are very smooth…What’s the length of the coastline, which seems to be so natural because it’s given in many cases, is, in fact a complete fallacy; there’s no such thing. You must do it differently.

What good it that, to know these things?  Well, surprisingly enough, it’s good in many ways…Now a lung is something very strange…The volume of a lung is very small, but what about the area of a lung?…Anatomists were arguing very much about that. Some say that a normal male’s lung has an area of the inside of a basketball court. And others say, no, five basketball courts…The bronchi branch, branch, branch and they stop branching, not because of any matter of principle, but because of physical consideration: the mucus, which is in the lung. So what happens is that in a way you have a much bigger lung, but it branches and branches down to distances about the same for a whale, or a man and or a little rodent….And I think that my mathematics, surprisingly enough, has been of great help to the surgeons studying lung illnesses and also kidney illnesses, all of these branching systems, for which there is no geometry.   So I found myself, in other words, constructing a geometry, a geometry of things which had no geometry.

Mandelbrot very gently makes clear that the objects conjured up by 19^th century mathematicians, functions that have been called monstrous and used to demonstrate the break between mathematics and visible reality, are now used to describe some aspects of nature’s complexity. He shows us the appearance of a “fractal-to-be” in an 18^th century Japanese painting and notes the intuitive grasp of fractals in the work of engineer Gustave Eiffel and in the tower that bears his name.  He tells us that halfway through his career he decided to test himself.  “Could I just look at something which everybody had been looking at for a long time and find something dramatically new?”   So he looked at “these things called Brownian motion,”

Then I was telling my assistant, “I don’t see anything.  Can you paint it?”  So he painted it, which means he put inside everything.  “Well, this thing came out…,” he said. And I said, “Stop! Stop! Stop! I see; it’s an island.”

Mandelbrot could see that this island had a fractal dimension but it was his friends who, 20 years later, won the Fields medal when they proved it.

Here, again, a sensory idea (roughness) is characterized by a number which takes some things away (like the length of a coastline) and brings other things into view (like a refined understanding of the lungs).

This talk is very pleasant to listen to and I recommend it if you haven’t seen it. Mandelbrot gives a sketchy description of the Mandebrot set but successfully describes the simplicity of its starting point. And he concludes with these very nice words:

Bottomless wonders spring from simple rules, which are repeated without end.

Unfortunately for us, philosophies of science and mathematics are rarely brought to the attention of individuals who are not engaged in these efforts.  Yet, while difficult to access, the views of the world provided by mathematics and science are pregnant with meaningful implications for all of us.   I have always been struck by the depth of reflection in Hermann Weyl’s writing. These are thoughts that come clearly from a person, an individual working to reconcile all of the images of himself and the world that have taken shape in his mind’s eye.

At the end of his essay The Unity of Knowledge he summarizes that at the basis of knowledge is (1) intuition “the mind’s originary act of seeing what is given to him,”  (2) expression, “the active counterpart of passive understanding” and (3) Thinking the possible, or imagination (he then apologizes for some of his indecision of mind)  For me, these three things are mathematics.  And seeing mathematics in this way tells us something about ourselves and the extent of our potential.

I’ve pulled out other excerpts that I found particularly provocative from a collection of selected Weyl pieces, edited by Peter Pesic, entitled Mind and Nature.

The first is from The Open World published in 1932.  It is a reflection on the non-causal aspect of quantum theory, which Weyl sees as bridging organic and inorganic nature.

According to vitalism the living organism reacts as a whole; its functions are not additive. The manner in which its structure is preserved throughout growth, in spite of a outside influences and perturbations, is not to be explained by small scale causal reactions between the elementary parts of the organism.  Now we see that according to quantum physics the same applied even to inorganic nature and is not peculiar to organic processes.  It is out of the question to derive the state of the whole from the state of its parts.  This leads to conditions which may most plainly if not most correctly be interpreted as a peculiar non-causal “understanding” between the elementary particles, that is prior to and independent of the control exercised by differential laws which regulate probabilities…It seems therefore that the quantum theory is called upon to bridge the gap between inorganic and organic nature; to join them in the sense of placing the origin of those phenomena which confront us in the fully developed organism as Life, Soul and Will back in the same original order of nature to which atoms and electrons also are subject.  So today less than ever do we need to doubt the objective unity of the whole of nature…..

Within a lengthy discussion of the concept of infinity in mathematics he makes the following remarks:

Mathematics is not the rigid and uninspiring schematism which the layman is so apt to see in it; on the contrary, we stand in mathematics precisely at the point of intersection of limitation and freedom which is the essence of man himself.

Something entirely new takes place when I embed the actually occurring number symbols in the sequence of all possible numbers…The given is embedded in the ordered manifold of the possible, not on the basis of descriptive characteristics, but on the basis of certain mental or physical operations and reactions to be performed on it – as, for example, counting.

Consider the definition “n is an even or an odd number according as there exists or does not exist a number x for which n = 2x.  For one who accepts this with its appeal to the infinite totality of numbers x as having a meaning, the sequence of numbers open into infinity has transformed itself into a closed aggregate of objects existing in themselves, a realm of absolute existence which “is not of this world,” and of which the eye of our consciousness perceives but reflected gleams.

We reject the categorical finiteness of man…..On the contrary, mind is freedom within the limitations of existence; it is open toward the infinite…The completed infinite we can only represent in symbols.  From this relationship every creative act of man receives its deep consecration and dignity.  But only in mathematics and physics, as far as I can see, has symbolical-theoretical construction acquired sufficient solidity to be convincing for everyone whose mind is open to these sciences.

The last one is from the piece called Mind and Nature (1934).

The impossibility of designing a picture of reality other than on the background of possibility appears to be founded on the circumstance that existence is a penetration of the what and the how, and consequently arises from a contact of object and subject, of pure factuality and freedom.

I was particularly taken by the words, “existence is a penetration of the what and the how, and consequently arises from a contact of object and subject”  Penetration is not a word one might expect to use to characterize our existence, but I think it leads to the right images – like probe, permeate, immerse, sink into, as well as fathom and understand.

In his introduction Pesic makes the point that Weyl’s praise of the symbol “includes the mathematical no less than the literary, artistic, and poetic.” He tells us that

Weyl gained perspective, insight, and altitude by thinking back along the ever-unfolding past and studying its great thinkers, whom he used to help him soar, like a bird feeling the air under its wings.

Hearing about visual processes, from neuroscientists and artists alike, consistently brings mathematical thoughts to mind for me – like Samir Zeki’s descriptions of how visual images are constructed, or the Impressionist painters’ attention to the sensations in the eye rather than the subject of the painting, and, of course, Poincaré’s suggestion that visual space has more than three dimensions.

The relationship between vision and mathematics (or sensation and symbolic thought) came up again today, when I followed the links in a December 15 Vision Help Blog. The blog begins with a reference to an Oliver Sacks story about a blind individual given sight, and concludes with a paper pointing to the psychological significance of the reluctance of organ donors to donate corneas.

The Sacks vignette tells us a lot about vision, and explains why freeing the cornea of obstruction doesn’t just give us sight. We have to learn to see.  We have to learn how the many appearances of an object belong to the one object.

We achieve perceptual constancy — the correlation of all the different appearances, the transforms of objects — very early, in the first months of life. It constitutes a huge learning task, but is achieved so smoothly, so unconsciously, that its enormous complexity is scarcely realized (though it is an achievement that even the largest supercomputers cannot begin to match).

It is for this reason that the subject of Sacks’ story who, at age 50, was given back the sight he had lost as a very young child, could not integrate the various pieces of visual stimuli into a coherent image.

…with half a century of forgetting whatever visual engrams he had constructed, the learning, or relearning, of these transforms required hours of conscious and systematic exploration each day. This first month, then, saw a systematic exploration, by sight and touch, of all the smaller things in the house: fruit, vegetables, bottles, cans, cutlery, flowers, the knickknacks on the mantelpiece — turning them round and round, holding them close to him, then at arm’s length, trying to synthesize their varying appearances into a sense of unitary objecthood. [...]”  (italics added)

While I cannot make any precise correspondence, I do see the visual processing mechanisms that this individual had not developed as ‘the mathematics’ of sight, as the way to capture the many facets of any sensation into a recognizable whole, one that the mind can manage.  Even some of the words used in the story call to mind things like algebra and geometry:

An infant merely learns. This is a huge, never-ending task, but it is not one charged with irresoluble conflict. A newly sighted adult, by contrast, has to make a radical switch from a sequential to a visual-spatial mode, and such a switch flies in the face of the experience of an entire lifetime.  (italics added)

But there is also a detail in the story that points to what I see as the body’s sheer tenacity, the tenacity to be, however that is accomplished.  Everyone was surprised when one of the first things Virgil could recognize was a symbolic thing – it was the alphabet:

Virgil’s first formal recognitions when the bandages were taken off had been of letters on the ophthalmologist’s eye chart…How was it that he had so much difficulty recognizing faces, or the cat, and so much difficulty with shapes generally, and with size and distance, and yet so little difficulty, relatively, recognizing letters? When I asked Virgil about this, he told me that he had learned the alphabet by touch at school, where they had used letter blocks, or cutout letters, for teaching the blind. I was struck by this and reminded of [R.L.] Gregory’s patient S.B.: “much to our surprise, he could even tell the time by means of a large clock on the wall. We were so surprised at this that we did not at first believe that he could have been in any sense blind before the operation.” But in his blind days S.B. had used a large hunter watch with no glass, telling the time by touching the hands, and he had apparently made an instant “cross-modal” transfer, to use Gregory’s term, from touch to vision. Virgil too, it seemed, must have been making just such a transfer. […]

I remembered reading about a blind mathematician, and today found more about him in a 2002 AMS article called The World of Blind Mathematicians.  In this story, representations created by touch are also noted:

A visitor to the Paris apartment of the blind geometer Bernard Morin finds much to see. On the wall in the hallway is a poster showing a computergenerated picture, created by Morin’s student François Apéry….The surface plays a role in Morin’s most famous work, his visualization of how to turn a sphere inside out. Although he cannot see the poster, Morin is happy to point out details in the picture that the visitor must not miss.

Later Morin brings out clay models that he made in the 1960s and 1970s to represent shapes that occur in intermediate stages of his sphere eversion.

The models were used to help a sighted colleague draw pictures on the blackboard…It is startling to consider that such a precise, symmetrical model was made by touch alone. The purpose is to communicate to the sighted what Bernard Morin sees so clearly in his mind’s eye.

And this:

Morin recalled that, when a sighted colleague proofread Morin’s thesis, the colleague had to do a long calculation involving determinants to check on a sign. The colleague asked Morin how he had computed the sign. Morin said he replied: “I don’t know—by feeling the weight of the thing, by pondering it.”

Morin believes there are two kinds of mathematical imagination. One kind, which he calls “time-like”, deals with information by proceeding through a series of steps. This is the kind of imagination that allows one to carry out long computations. “I was never good at computing,” Morin remarked, and his blindness deepened this deficit. What he excels at is the other kind of imagination, which he calls “space-like” and which allows one to comprehend information all at once.

Space-like imagination is tactile as well as visual:

One thing that is difficult about visualizing geometric objects is that one tends to see only the outside of the objects, not the inside, which might be very complicated. By thinking carefully about two things at once, Morin has developed the ability to pass from outside to inside, or from one “room” to another. This kind of spatial imagination seems to be less dependent on visual experiences than on tactile ones. “Our spatial imagination is framed by manipulating objects,” Morin said. “You act on objects with your hands, not with your eyes. So being outside or inside is something that is really connected with your actions on objects.” Because he is so accustomed to tactile information, Morin can, after manipulating a hand-held model for a couple of hours, retain the memory of its shape for years afterward.

Another anecdote in the article refers to mathematician Norberto Salinas

In a contribution to a Historia-Mathematica online discussion group about blind mathematicians, Eduardo Ortiz of Imperial College, London, recalled examining Salinas in an analysis course at University of Buenos Aires. Salinas communicated graphical information by drawing pictures on the palm of Ortiz’s hand, a technique that Ortiz himself later used when teaching blind students…

I particularly like this suggestion:

…blind people often have an affinity for the imaginative, Platonic realm of mathematics. For example, Morin remarked that sighted students are usually taught in such a way that, when they think about two intersecting planes, they see the planes as two-dimensional pictures drawn on a sheet of paper. “For them, the geometry is these pictures,” he said. “They have no idea of the planes existing in their natural space.

Happy New Year!

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!

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.

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.

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.

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 19^th century developments in mathematics (the ones that lay the groundwork for mathematics as it is understood today) to see references to the late 18^th 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.

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.