Mathematics Rising

My post appeared on the Scientific American Guest Blog this morning.  Here’s the link:

Quantum Mechanical Words and Mathematical Organisms

My time this week is again taken up with work on a few writing projects that I’m trying to wrap up (not to mention end of the term grading).  But I should be back on track with my regular blogs next week.

In the meantime, an article on scientificamerican.com caught my attention, being about the mathematics of juggling!  But the article originated elsewhere, at Simons Science News.  And this is how I became aware of the Simons Foundation website.  It’s worth exploring.

The Simons Foundation’s mission is to advance the frontiers of research in mathematics and the basic sciences. We sponsor a range of programs that aim to promote a deeper understanding of our world.

It is a private foundation that provides some interesting grant opportunities.

The site provides some video interviews with mathematicians organized under the heading Science Lives.  These interviews include ones with John Nash, Michael Atiyah and Cathleen Morawetz from my own alma mater – The Courant Institute of Mathematical Sciences.   There’s also an interesting article on new observations in biology that are consistent with a mathematical idea of Turing’s proposed in 1952.

Now, more than 60 years later, biologists are uncovering evidence of the patterning mechanisms that Turing proposed in his paper, prompting a resurgence of interest in them, with the potential to shed light on such developmental questions as how genes ultimately make a hand.

I’m short on time today and working on a guest blog which I hope to be able to provide a link to shortly.  But I did begin exploring a website that has short video interviews with some of my favorite thinkers.  I found among a list of participants on the website Closer To Truth, Gregory Chaitin and David Deutsch.  They have each participated in a number of video interviews that seem always short, but always interesting.

I listened to one of Deutsch’s this evening called What is Ultimate Reality?  In the interview he described the four fundamental, interdependent aspects of reality, as  presented in his book, The Fabric of Reality.  These are, quantum physics, the theory of evolution, the theory of computation and the theory of knowledge.  Briefly,  he says that quantum physics constrains the kinds of theories that one can express and that evolution is the theory of emergent properties that cannot be expressed in terms of atoms.  Computation is about the processes in nature that are independent or that transcend the substance in which they are embodied.  And knowledge is the kind of information that can do things.  Knowledge is embodied in DNA, brains, books, computers, etc.  But Deutsch makes the point that what moves things, what changes things or creates things is information, not the substances in which the information is embodied.

For me, the most refreshing and important thought was his observation that all of these aspects of reality have been underestimated by being accepted as the right explanation in their own field.  But they have not become integrated.  The depth of these specialized fields has made it increasingly more difficult to consider their interdependence.

In another interview, Gregory Chaitin explores his own Platonism.

I recommend listening.

I was expecting to write about a paper I found recently by Oran Magal, a post doc at McGill University, On the mathematical nature of logic. I was attracted to the paper because the title was followed by the phrase Featuring P. Bernays and K. Gödel

I’m often intrigued by disputes over whether mathematics can be reduced to logic or whether logic is, in fact, mathematics, because these disputes often remind me of questions addressed by cognitive science, questions related to how the mind uses abstraction to build meaning. This particular paper acknowledges, in the end, that its purpose is two-fold. It makes the philosophical argument that an examination of the interrelationship between mathematics and logic shows that “a central characteristic of each has an essential role within the other” But the paper is also a historical reconstruction and analysis of the ideas presented by Bernays, Hilbert and Gödel (the detail of which is not particularly relevant to my concerns). It was Bernays’ perspective that I was most interested in pursuing.

Magal begins with the observation that

the relationship between logic and mathematics is especially close, closer than between logic and any other discipline, since the very language of logic is arguably designed to capture the conceptual structure of what we express and prove in mathematics.

While some have seen logic as more general than mathematics, there has also been the view that mathematics is more general than logic. It is here that Magal introduces Bernays’ idea that logic and mathematics are equally abstract but in different directions. And so they cannot be derived one from the other but must be developed side-by-side. When logic is stripped of content it becomes the study of inference, of things like negation and implication. But while logical abstraction leaves the logical terms constant, according to Bernays, mathematical abstraction leaves structural properties constant. These structural properties do seem to be the content of mathematics, and what makes mathematics so powerful.

Magal describes how Bernays understands Hilbert’s axiomatic treatment of geometry. Here, the purely mathematical part of knowledge is separated from geometry (where geometry is thought of as the science of spatial figures) and is then investigated directly.

The spatial relationships are, as it were, mapped into the sphere of the abstract mathematical in which the structure of their interconnections appears as an object of pure mathematical thought. This structure is subjected to a mode of investigation that concentrates only on the logical relations and is indifferent to the question of the factual truth, that is, the question whether the geometrical connections determined by the axioms are found in reality (or even in our spatial intuition). (Bernays, 1922a, p. 192) (emphasis added)

Magal then uses abstract algebra to illustrate the point:

To understand Bernays’ point, that this is a structural direction of abstraction, and the sense in which this is a mathematical treatment of logic, it is useful to compare this to abstract algebra. The algebra familiar to everyone from our school days abstracts away from particular calculations, and discusses the rules that hold generally (the invariants, in mathematical terminology) while the variable letters are allowed to stand for any numbers whatsoever. Abstract algebra goes further, and ‘forgets’ not just which number the variables stand for, but also what the basic operations standardly mean. The sign ‘+’ need not necessarily stand for addition. Rather, the sign ‘+’ stands for anything which obeys a few rules; for example, the rule that a+ b= b+ a, that a+ 0= a, and so on. Remember that the symbol ‘a’ need not stand for a number, and the numeral ‘0’ need not stand for the number zero, merely for something that plays the same role with respect to the symbol ‘+ ’ that zero plays with respect to addition. By following this sort of reasoning, one arrives at an abstract algebra; a mathematical study of what happens when the formal rules are held invariant, but the meaning of the signs is deliberately ‘forgotten’. This leads to the study of general structures such as groups, rings, and fields, with immensely broad applicability in mathematics, not restricted to operations on numbers.

Again the key to the discussion is the question of content. When mathematics is viewed as a variant of logic it could easily be judged to have no specific content. The various arguments presented are complex, and not everyone writes with respect to the same logic. But the consistency of Bernays’ argument is most interesting to me. He is very clear on the question of content in mathematics. And reading this sent me back to another of his essays, where he is responding to Wittgenstein’s thoughts on the foundations of mathematics is 1959. Here he challenges Wittgenstein’s view with the nothingness of color.

Where, however, does the initial conviction of Wittgenstein’s arise that in the region of mathematics there is no proper knowledge about objects, but that everything here can only be techniques, standards and customary attitudes, He certainly reasons: `There is nothing here at all to which knowing could refer.’  That is bound up, as already mentioned, with the circumstance that he does not recognize any kind of phenomenology. What probably induces his opposition here are such phrases as the one which refers to the `essence’ of a colour; here the word `essence’ evokes the idea of hidden properties of the color, whereas colors as such are nothing other than what is evident in their manifest properties and relations. But this does not prevent such properties and relations from being the content of objective statements; colors are not just a nothing….That in the region of colors and sounds the phenomenological investigation is still in its beginnings, is certainly bound up with the fact that it has no great importance for theoretical physics, since in physics we are induced, at an early stage, to eliminate colors and sounds as qualities. Mathematics, however, can be regarded as the theoretical phenomenology of structures. In fact, what contrasts phenomenologically with the qualitative is not the quantitative, as is taught by traditional philosophy, but the structural, i.e. the forms of being aside and after, and of being composite, etc., with all the concepts and laws that relate to them. (emphasis added)

Near the end of the essay he makes a reference to the Leibnizian conception of the characteristica universalis which, Bernays says was intended “to establish a concept-world which would make possible an understanding of all connections existing in reality. This dream of Leibniz’s (which it seems Gödel thought feasible) is probably the subject of another blog. But in closing I would make the following remarks:

Cognitive scientists have found that abstraction is fundamental to how the body builds meaning or brings structure to its world. This is true in visual processes where we find cells in the visual system that respond only to things like verticality, and it is seen in studies that show that a child’s maturing awareness seems to begin with simple abstractions. Mathematics is the powerful enigma that it is because it cuts right into the heart of how we see and how we find meaning.

I’d like today to stay on the topic of mathematics from the cognitive science perspective, and in particular, to make available another set of interesting studies summarized by C. R. Gallistel, Rochel Gelman and Sara Cordes. The studies are described in their contribution to the book Evolution and Culture (edited by Stephen C. Levinson and Pierre Jaisson and published by MIT press) and entitled: The Cultural and Evolutionary History of the Real Numbers. A pdf of this selection can be found here. These are provocative ideas that don’t seem to be getting a lot of attention yet.

Their premise:

that a system for arithmetic reasoning with real numbers evolved before language evolved. When language evolved, it picked out from the real numbers only the integers, thereby making the integers the foundation of the cultural history of the number.

Observations of the conceptual need for the real numbers, as well as their sometimes unwelcome presence, is peppered throughout the history of mathematics. But they were only formerly defined in the 19^th century. The authors clarify that this real number system – a continuous, uncountable set of rational and irrational numbers

is used by modern humans to represent many distinct systems of continuous quantity–duration, length, area, volume, density, rate, intensity, and so on. Because the system of real numbers is isomorphic to a system of magnitudes, the terms real number and magnitude are used interchangeably. Thus, when we refer to “mental magnitudes” we are referring to a real number system in the brain. Like the culturally specified real number system, the real number system in the brain is used to represent both continuous quantity and numerosity.

I liked their summary of the observed weakness of the rationals.

The geometric failing of the integers and their offspring the rational numbers arises when we attempt to use proportions between integers to represent proportions between continuous quantities, as, for example when we say that one person is half again as tall as another, or one farmer has only a tenth as much land as another. These locutions show the seemingly natural expansion of the integers to the rational numbers, numbers that represent proportions. This expansion seemed so natural and unproblematic to the Pythagoreans that they believed that the natural numbers and the proportions between them (the rational numbers) were the inner essence of reality, the carriers of all the deep truths about the world. They were, therefore, greatly unsettled when they discovered that there were geometric proportions that could not be represented by a rational number, for example, the proportion between the diagonal and the side of a square. The Greeks proved that no matter how fine you made your unit of length, it would never go an integer number of times into both the side and the diagonal. Put another way, they proved that the square root of two is an irrational number, an entity that cannot be constructed from the natural numbers by a finite procedure.

And this is the heart of my interest:

Our thesis is that this cultural creation of the real numbers was a Platonic rediscovery of the underlying non-verbal system of arithmetic reasoning. The cultural history of the number concept is the history of our learning to talk coherently about a system of reasoning with real numbers that predates our ability to talk, both phylogenetically and ontogenetically.

What I find provocative about the history of mathematics is while it may look like mathematics is just the conscious organization of practical symbols, over time it is inevitably discovered that these symbols contain more than was put into them. They grow deeper, become more entwined and produce unanticipated new possibilities. This has always suggested to me that every formalized idea emerges from a well-spring of possibilities to which the mathematician keeps gaining proximity. This alone is full of implications about the nature of abstract ideas, what they accomplish, and what moves the development of human culture. Recent papers, like this one on the evolutionary history of the real numbers, consistently encourage me to keep thinking along these lines.

The way these investigators identify the presence of this primitive use of continuous mental magnitudes is interesting. Some of the first studies cited involve pigeons, rats and monkeys, where their memory of ‘duration’ is observed by exploiting one of the difficulties with continuous measurements. The difference between nearby numbers is difficult to discern, for example, numerosities are represented by voltage levels, because of the noise in voltage levels. This is contrasted with numerosities represented by digital computers. Experiments were designed to identify one of these creature’s subjective judgment of durations, by using the behavior of the animals, as the indicator of their memory of duration. The variability in these judgments (called scalar variability) increases as the remembered durations get longer. It is believed that this is because the noise in a magnitude is proportional to the size of the magnitude. Their observations are fairly precise, and even extended to allow the observation of non-verbal animals doing arithmetic with these continuous magnitudes. Other studies designed to produce non-verbal counting in humans produced the same results. These mental magnitudes were also seen mediating judgments of the numerical ordering of symbolically presented integers.

I expect this kind of evidence will continue to grow.

For now I’ll leave you with their summaries:

In summary, research with vertebrates, some of which have not shared a common ancestor with man since before the rise of the dinosaurs, implies that they represent both countable and uncountable quantity by means of mental magnitudes (real numbers). The system of arithmetic reasoning with these mental magnitudes is closed under the basic operations of arithmetic, that is, mental magnitudes may be mentally added, subtracted, multiplied, divided and ordered without restriction.

In short, we suggest that the integers are picked out by language because they are the magnitudes that represent countable quantity. Countable quantity is the only kind of quantity that can readily be represented by a system founded on discrete symbols, as language is. It is language that makes us think that God made the integers, because the learning of the integers is the beginning of linguistically mediated mathematical thinking about both countable and uncountable quantity.

I am increasingly fascinated by the mathematics of fundamental cognitive processes – like creatures finding their way to and from significant locations, or foraging for food, or foraging with the eyes, or comprehending the duration of an event. I’m excited by the fact that there are cognitive neuroscientists that have become focused on the architecture of these processes in particular. Their work seems to always suggest that our formal mathematical systems are growing out of these very same processes.

I read today Charles Gallistel’s contribution to Dehaene and Brannon’s Space, Time and Number in the Brain. Gallistel has a link to the pdf version on the Rutgers website.

Gallistel is concerned with the abstractions of space, time, number, rate and probability that have been experimentally studied and found to be playing a fundamental role in the lives of nonverbal animals and preverbal humans. His premise is this:

the brain’s ability to represent these foundational abstractions depends on a still more basic ability, the ability to store, retrieve and arithmetically manipulate signed magnitudes.

He makes a point of distinguishing between magnitude and our symbolic numbers. Magnitudes are what he calls ‘computable numbers’ a quantity that “can be subjected to arithmetic manipulation in a physically realized system.”

Being a bit pressed for time, I’ll just reproduce some of his observations.

The representation of space, he says:

requires summing successive displacements in an allocentric (other-centered) framework, a framework in which the coordinates of locations other than that of the animal do not change as the animal moves. By summing successive small displacements (small changes in its location), the animal maintains a representation of its location in the allocentric framework. This representation makes it possible to record locations of places and objects of interest as it encounters them, thereby constructing a cognitive map of its experienced environment. Computational considerations make it likely that this representation is Cartesian and allocentric.

But in order to have a directive function, these representations of experienced locations must be vectors – ordered sets of magnitudes. And the organism accomplishes arithmetic with them.

A fundamental operation in navigation is computing courses to be run… Assuming that the vectors are Cartesian, the range and bearing are the modulus and angle of the difference between the destination vector and the current-location vector. This difference vector is the element-by-element differences between the two vectors. Thus, the representation of spatial location depends on the arithmetic processing of magnitudes.

Gallistel challenges the notion that time-interval experience is generated by an interval-timing mechanism, pointing out that

There is, however, a conceptual problem with this supposition: The ability to record the first occurrence of an interesting temporal interval would seem to require the starting of an infinite number of timers for each of the very large number of experienced events that might turn out to be “the start of something interesting”–or not

Instead, he proposes

that temporal intervals are derived from the representation of temporal locations, just as displacements (directed spatial intervals) are derived from differences in spatial locations. This, in turn leads to arithmetic operations on temporal vectors (see Gallistel, 1990, for details). Rats represent rates (numbers of events divided by the durations of the intervals over which they have been experienced) and combine them multiplicatively with reward magnitudes [9]. Both mice and adult human subjects represent the uncertainty in their estimates of elapsing durations (a probability distribution defined over a continuous variable) and discrete probability (the proportion between the number of trials of one kind and the number of trials of a different kind) can combine these two representations multiplicatively to estimate an optimal target time [1].

I found one of the most interesting parts of this discussion to be the one on closure.

Closure is an important constraint on the mechanism that implement arithmetic processing in the brain. Closure means that there are no inputs that crash the machine. Closure under subtraction requires that magnitudes have sign (direction), because otherwise entering a subtrahend greater than the minuend would crash the machine; it would not be able to produce a valid output. Rats learn directed (signed) temporal differences; they distinguish between whether the reward comes before or after the signal and they can integrate one directed difference with another [11].

I find this particularly interesting because it took us some time to find signed differences in our symbolic system of subtraction or even to recognize the significance of closure.

I’ll end this with his brief conclusion. Some of the details of these studies can be found in the linked pdf.

It seems likely that magnitudes (computable numbers) are used to represent the foundational abstractions of space, time, number, rate, and probability. The growing evidence for the arithmetic processing of the magnitudes in these different domains, together with the “unreasonable” efficacy of representations founded on arithmetic, suggests that there must be neural mechanisms that implement the arithmetic operations. Because the magnitudes in the different domains are interrelated–in for example, the representation of rate (numerosity divided by duration) or spatial density (numerosity divided by area)–it seems plausible to assume that the same mechanism is used to process the magnitudes underlying the representation of space, time and number. It should be possible to identify these neural mechanisms by their distinctive combinatorial signal processing in combination with the analytic constraint that numerosity 1 be represented by the multiplicative identity symbol is the system of symbols for representing magnitude.

A recent blog from Jennifer Ouellette (from the Scientific American Blog Network)  brought my attention once again to how mathematics is related to the structure-building functions of the brain. As I followed up on some of the references in her post, I found myself on a little journey through hallucinatory experiences that I really enjoyed.

Her post is generally about Turing models applied to patterns found in the characteristic features of animals.  But she got into territory that I find particularly provocative when she began to discuss the evidence for whether a Turing model can be applied to neurons in the brain.

If we really want to get into some interesting speculation, we can think about whether a Turing model can be applied to neurons in the brain, which could be “described mathematically as activators or inhibitors, encouraging or dampening the firing of other, nearby neurons in the brain.” And that could potentially explain why we see certain recurring patterns when we hallucinate.

I did a blog about how Turing insights appear to bridge otherwise disparate trends in science.  But the link to Turing in this context is drawn through hallucination patterns, categorized in the early 20^th century by University of Chicago neurologist Heinrich Kluever, into what he called the form constants: checkerboards, honeycombs, tunnels, spirals, and cobwebs.

Over seventy years later, another Chicago researcher, Jack Cowan – who holds dual appointments in mathematics and neurology – set out to reproduce those hallucinatory patterns mathematically, believing they could provide clues to the brain’s circuitry.

The paper is very technical and involves quite a lot of mathematics as well as neuroscience.  But the authors succeed in modeling a structure of the primary visual cortex whose activity would produce “certain basic types of geometric visual hallucinations.”  Bressloff and Cowam conclude:

…thus our new work provides a stronger link between the nature of hallucinatory patterns and the actual structure of cortex.  Indeed, we hypothesize that the symmetries and length-scales of these hallucinatory images are a direct consequence of the geometry of cortical interactions as well as the retino-cortical map.

Apparently they found that the patterns predicted by their calculations closely matched what people will see when under the influence of hallucinogenic drugs, and they suspect that these patterns could be emerging from a kind of Turing mechanism.

While the random fluctuations in brain activity might technically just be “noise,” the brain will take that noise and turn it into a pattern. Since there is no external input when the eyes are closed, that pattern should reflect the architecture of the brain, specifically the functional organization of the visual cortex.

Ouellette also spoke with neuroscientist Robin Carhart-Harris, who has done quite a lot of work on the brain mechanisms that lead to hallucinatory experiences, and how they might be used to help in the treatment of depression and addiction.  I very much enjoyed watching an interview with him, shot as part of a forthcoming documentary on consciousness. There he made some really nice observations of the integrative function of  ‘brain hubs’  that unify activity from different regions of the brain into coherent patterns or narratives.  The geometric patterns of hallucinations are perceptual errors but, he tells us, the error “is a function of how the perceptual system works.”  Carhart-Harris argues that the way we currently understand the action of the visual system,  “says, very strongly, that reality is a construction. Reality only becomes something as we piece it together.”

With respect to how hallucinatory patterns reflect brain activity, Carhart-Harris tells Ouellette:

You are not seeing the cells themselves, but the way they’re organized – as if the brain is revealing itself to itself.

This is, in fact, what I have thought about mathematics.

I saw an opinion piece by Stephen Ornes, in the March 16 issue of New Scientist which ties the ongoing debate about the nature of mathematical ideas, to a modern one about money and ownership.  Ornes argues that patentability is one of the most hotly contested issues in software development.  The problem, as many see it, is that not all software is patentable because of its dependence on mathematics.  Mathematics is understood as the exploration of abstract ideas, not the invention of new products.   Ornes referred to an essay by David Edwards (University of Georgia) in the April 2013 issue of Notices of the American Mathematical Society.  In the end, Edwards is calling for an update of the patent laws because the current laws do not promote the development of technological innovation.  I wasn’t very inspired by the discussion.  However, when I went to find the Edwards piece in the AMS Notices, I stumbled upon an essay, written in a completely different spirit, and published in January 2012.  Jason Scott Nicholson, then a Ph.D. candidate in mathematics at the University of Calgary, addressed Eugene Wigner’s consistently cited query into the “unreasonable effectiveness of mathematics.”  But Nicholson explores the puzzle of mathematics’ effectiveness using the structure of the ideas brought to life in the book Lila by Robert M. Persig, author of the widely read Zen and the Art of Motorcycle Maintenance.

Nicholson explains that, in Lila, reality is dual-aspected. One of these aspects is what Persig calls Static Quality, and the other, what he calls Dynamic Quality.  A very brief explanation of these ideas is this:

Dynamic Quality is understood as the creative urge, the constant stimulus to move, perhaps to something ‘better.’  Static Quality is what is given in the patterns reflecting the “realization” of the undefined Quality that is the world.  Static Quality is created in response to Dynamic Quality.   It exists on 4 discrete but related levels:

Inorganic                Biological                 Social                and                Intellectual

In this system, the biological builds on the inorganic, the social on the biological, and the intellectual on the social.  Nicholson tells us that Perig uses a computer analogy to illustrate this idea:

He describes the relationship between these levels as being analogous to the relationship of computer hardware to computer software—the software is run on the hardware, but has nothing, really, to do with it. The program that you run on your computer and write your article with has nothing to do with the computer hardware itself. Furthermore, the content of your article has nothing to do with the program you write it in. In this way the levels of static quality are related to each other: Biological is built on Inorganic, Social is built on Biological, and Intellectual is built on Social, but each level is independent of the other.

Persig’s Static Quality creates a relationship among manifold patterns – from the bonding of atoms, to the mating of animals, to the formation of nations, to the dogma of religions, and the intellectual patterns of art and science.  And this relatedness becomes the crux of Nicholson’s argument:

…since nature is simply inorganic and biological patterns of value that follow Dynamic Quality, it is not surprising that mathematics, a static intellectual pattern of quality that also follows Dynamic Quality, should arrive at the same conclusions. That is the reason that mathematics that is done in isolation ends up explaining nature so well—both are patterns of static quality created by following Dynamic Quality!

This configuration of Quality, Dynamic Quality and Static Quality is also used by Nicholson to describe the art/science character of mathematics:

Art is the realization of Dynamic Quality in a given medium—that is, Art is following Dynamic Quality, and the pattern of static quality which is a “work of art” is left in its wake, in whatever medium the artist chose. In this sense, mathematics, especially pure mathematics, is an art, as it is the realization of Dynamic Quality in the medium of mathematical definitions and their logical consequences.

But mathematics is also a science. It is commonly classified as such, being in the science faculty of most universities. More to the point, though, it is also generally seen as similar to empirical sciences in that it involves an objective, careful, and systematic study of an area of knowledge. It is, however, different because it verifies its knowledge using a priori rather than empirical methods. But, within the Metaphysics of Quality, its methods are totally empirical. In fact, it may be argued that from this perspective, it is even more empirical than the other sciences. Mathematics is following empirical reality (Quality) directly, whereas other sciences are one step removed from empirical reality (Quality): they follow nature, which, in turn, follows Quality. Thus mathematics is really both an art and a science and, in fact, can act as something of a bridge between the two.

The nature of Pirsig’s ‘Quality,’ and the use that Nicholson makes of it, reminded me of Leibniz again.  For both Pirsig and Leibniz, our perceived reality is the consequence of structure being brought to something we cannot see, something that isn’t even material in the way we understand material.  For Leibniz, this fundamental reality is the harmonious existence of monads. Leibniz’s monad is: 

Something that has no parts can’t be extended, can’t have a shape, and can’t be split up. So monads are the true atoms of Nature—the elements out of which everything is made.

The text of Leibniz’s Monadology is not easy reading.  It is a heavily logic-based analysis.  The Internet Encyclopedia of Philosophy is one of many philosophy sites that discusses the document. There the point is clarified that:

Leibniz thus distinguishes four types of monads: humans, animals, plants, and matter. All have perceptions, in the sense that they have internal properties that “express” external relations; the first three have substantial forms, and thus appetition; the first two have memory; but only the first has reason (see Monadology §§18-19 & 29).

There is no formal correspondence between the Persig and Leibniz.  But there are most certainly parallels.  Leibniz’s appetitions, for example, as explained by the Stanford Encyclopedia of Philosophy are:

“tendencies from one perception to another” (Principles of Nature and Grace, sec.2 (1714)). Thus, we represent the world in our perceptions, and these representations are linked with an internal principle of activity and change (Monadology, sec.15 (1714)) which, in its expression in appetitions, urges us ever onward in the constantly changing flow of mental life. More technically explained, the principle of action, that is, the primitive force which is our essence, expresses itself in momentary derivative forces involving two aspects: on the one hand, there is a representative aspect (perception), by which that the many without are expressed within the one, the simple substance; on the other, there is a dynamical aspect, a tendency or striving towards new perceptions, which inclines us to change our representative state, to move towards new perceptions.  (emphasis added)

I’ve been intrigued for some time by the view of reality Leibniz gave us and, to a large extent, because of its unmistakeable mathematical character. But I’ve also been captivated by how non-materialistic it is.  Also from The Stanford Encyclopedia of Philosophy is this about Leibniz’s philosophy of mind.

In short, Leibniz stands in a special position with respect to the history of views concerning thought and its relationship to matter. He rejects the materialist position that thought and consciousness can be captured by purely mechanical principles. But he also rejects the dualist position that the universe must therefore be bifurcated into two different kinds of substance, thinking substance, and material substance. Rather, it is his view that the world consists solely of one type of substance, though there are infinitely many substances of that type. These substances are partless, unextended entities, some of which are endowed with thought and consciousness, and others of which found the phenomenality of the corporeal world. The sum of these views secures Leibniz a distinctive position in the history of the philosophy of mind.

I thought it worthwhile to bring these ideas up again in the context of Jason Scott Nicholson’s response to Wigner.

A nice article, focused on the origins of creativity, appears in the March 13 issue of Scientific American. Author, Heather Pringle, surveys research that seems to indicate that the human talent for innovation actually emerged over hundreds of thousands of years ago, before homo sapiens left Africa.  This is contrary to the view held previously that a genetic mutation ignited sudden cognitive advances in homo sapiens already living on the European continent some 40,000 years ago.

Pringle describes the archeological evidence that motivates this revision in the story of our evolution.  She also fills the story in with new insights into the evolution of the modern human brain. While mathematics is never mentioned specifically, questions about the emergence of symbol in our experience are inevitably relevant to questions about the source of mathematical creativity.

I followed up on one of Pringle’s examples, finds at an archeological site on the very tip of Africa.

The hunter-gatherers who inhabited Blombos Cave between 100,000 and 72,000 years ago, for example, engraved patterns on chunks of ocher; fashioned bone awls, perhaps for tailoring hide clothing; adorned themselves with strands of shimmering shell beads; and created an artists’ studio where they ground red ocher and stored it in the earliest known containers, made from abalone shells.

I have given most of my attention to the engraved patterns on ochre.  Professor Christopher Henshilwood led the early investigation at this site.  His work continues more recently through the TRACSYMBOLS Project.  A wealth of information related to the project is easily accessed on their website.   Their home page has two videos and, in one of them, Ian Tattersall, Curator of the Museum of Natural History, introduces the work with a description of symbolic thought that certainly calls mathematics to mind (at least it does to my mind).

“The one thing that makes us feel unique is our extraordinary symbolic mental capacity.  We disassemble the world around us; we break it down into a mass of symbols. Then we recombine those symbols to remake the world in our heads.  The human brain is a product of 350 millions years of vertebrate evolution.”

This last statement is indicative of the trend in cognitive neuroscience to put us back into the larger history of our lives.

Science News had an article on the finds at Blombos Cave in June, 2009.  Here, author Bruce Bower, refers to the ochre engravings found in Blombos Cave as ‘meaningful geometric designs.’ The article can be accessed on this website.

“What makes the Blombos engravings different is that some of them appear to represent a deliberate will to produce a complex abstract design,” Henshilwood says. “We have not before seen well-dated and unambiguous traces of this kind of behavior at 100,000 years ago.”

An earlier (2007) paper in the Journal of Archeological Science focuses on another African excavation site on the western cape. But in that paper Alex Mackay and Aara Welz define the terms of the debate, namely, what we consider ‘design,’ and how we understand ‘symbolic.’  I thought these observations worth including.

The question, ‘‘What, in archaeological terms, constitutes a symbol?’’, remains anything but clear (‘‘What isn’t a symbol?’’ even less so).

Indeed, the formation of lines through a series of actions strongly implies an element of design, regardless of whether it was expediently formulated or realised over multiple stages. By design we require only that the artisan(s) undertook the act(s) of scoring in order to give physical manifestation to a mental concept.

Whether or not engraved ochre necessarily carries any symbolic significance is a different matter. In order to be symbolic, it is necessary that the design has a cognitively constructed and conventionally maintained relationship with some other thing, either physical or conceptual (Chase, 1991; Noble and Davidson, 1991). Clearly, no such relationship can be demonstrated on the basis of the available evidence. Of course, the same argument can be made with regard to the engraved ochre from Blombos Cave. Though there is almost a self-evident sense of meaningfulness to the Blombos piece, this is not, in truth, sufficient to make any argument for its symbolic significance in the sense above.

Our suspicion is that the KKH ochre, the finds from Blombos, and the various other shell beads all had symbolic significance to their makers, and that some MSA people thus had the capacity to create and deploy symbols, and to store information externally. However, we must also accept the possibility that the motivations for engraving and breaking this particular piece were far more mundane…              (emphasis added)

Pringle completes her Scientific American piece with references to anthropological studies of the brain and the effect that the size of a population has on the likelihood that new ideas will emerge from with it and connect with other ideas that enhance their utility.  Regarding the brain,

At the University of California, San Diego, physical anthropologist Katerina Semendeferi has been studying a part of the brain known as the prefrontal cortex, which appears to orchestrate thought and action to accomplish goals. Examining this region in modern humans and in both chimpanzees and bonobos, Semendeferi and her colleagues discovered that several key subareas underwent a major reorganization during hominin evolution. Brodmann area 10, for example—which is implicated in bringing plans to fruition and organizing sensory input—nearly doubled in volume after chimpanzees and bonobos branched off from our human lineage. Moreover, the horizontal spaces between neurons in this subarea widened by nearly 50 percent, creating more room for axons and dendrites.

Researchers have imagined that it is this bigger brain that led to our ability to free-associate, and also to encode finer-grained memory.  But free association needs analytic thought (referred to as the default mode) if we are to make something of freely associated connections. The body’s somehow learning to regulate subtly altering concentrations of dopamine (and other neurotransmitters) in order to switch smoothly from one mode to another, may be one of the keys to our idea-driven modern lives. And this mechanism, they say, could have taken tens of thousands of years to fine tune.  These ideas are now being tested on an artificial neural network. Pringle concludes:

Once that final piece of the biological puzzle fell into place—perhaps a little more than 100,000 years ago—the ancestral mind was a virtual tinder box, awaiting the right social circumstances to burst into flame.

I enjoyed this view of our ancestors.  And if our talent for free association is as critical to creativity as it would seem, one might wonder about how this electrochemical move from thought to thought actually translates into a useful idea.  I think this question runs parallel to many of our questions about how surprisingly effective mathematics can be in the exploration of our worlds.

I read another New Scientist article today. The article was written by Brian Greene. While it didn’t give me a lot of new information, it made an interesting point about what it means (and when is it particularly effective) to take our mathematics seriously.  He talked about Einstein’s insight regarding the speed of light.  It was in the late 1800s, he explains, when Maxwell’s equations gave it the value of 300,000 kilometers per second (close to experimental measurements).  But the equations didn’t say anything about the standard of rest that gave this speed meaning.  Greene reminds us of the postulated invisible medium for transmitting light (the ether) which he calls a makeshift resolution to the problem. He then goes on to highlight a particular aspect of Einstein’s insight.

It was Einstein who in the early 20th century argued that scientists needed to take Maxwell’s equations more seriously. If Maxwell’s equations did not refer to a standard of rest, then there was no need for a standard of rest. Light’s speed, Einstein forcefully declared, is 300,000 kilometers per second relative to anything. The details are of historical interest, but I’m describing this episode for a larger point: everyone had access to Maxwell’s mathematics, but it took the genius of Einstein to embrace it fully. His assumption of light’s absolute speed allowed him to break through first to the special theory of relativity – overturning centuries of thought regarding space, time, matter and energy – and eventually to the general theory of relativity, the theory of gravity that is still the basis for our working model of the cosmos. (emphases added)

This is a detail about Einstein’s thinking that I hadn’t understood in quite that way.  It’s a provocative idea. Mathematical necessity overrides the expectations created by our physical intuition.  If the equation doesn’t depend on a standard of rest, than neither does the speed of light.  Mathematics, here, is acting much like a human sense, a mode of perception.

After reading this, my own thoughts went down a number of different paths, which I can’t recall well enough to repeat here.  But the precedence that mathematics has taken in physical theories, eventually led me to look at discussions centered around whether reality was fundamentally made of material or meaning.  One of the schools of thought that reflects this question finds information to be more fundamental to reality than material.  Paul Davies and Niels Henrik Gregersen compiled a collection of essays that address this issue in the book Information and the Nature of Reality. In his introduction, Davies describes Einstein’s theory of special relativity and general relativity as the first blow to our confidence in the idea of ‘matter.’

By stating the principle of an equivalence of mass and energy, the field character of matter came into focus, and philosophers of science began to discuss to what extent relativity theory implied a ‘de-materialization’ of the concept of matter.

Later, of course, quantum physics not only amplified this question, but also raised other yet unanswered questions about the significance of the observer.  Again from Davies:

A wave function is an encapsulation of all that is known about a quantum system. When an observation is made, and that encapsulated knowledge changes, so does the wave function, and hence the subsequent quantum evolution of the system. Moreover, informational structures also play an undeniable causal role in material constellations, as we see in, for example, the physical phenomenon of resonance, or in biological systems such as DNA sequences.

In an interview for the radio show To The Best of Our Knowledge Davies said this about the view of reality that quantum theory may be expressing:

…when we human beings, make observation of the world, we are interrogating nature, we are getting yes/no answers in the most primitive way. Every scientific experiment consists of doing exactly that. Come back to the simple example I gave where it is obviously true that the electron bounds to the left or the right. You get a yes/no answer. In the world of Quantum Physics, we get into another subtlety here. Which is the possibility of the super position. Now, you toss a coin it is heads or tails. But in Quantum Physics, if you toss a quantum coin, and this might be like the spin of particle or something, you can have a little of heads and another tails. Or a little bit of tails, but a lot of heads. You can have any mixture of the two.  In other words, an atom can be in the head and tails state, or in both states, at once. So in this sense the theory can take us to a God’s eye view, not a human view. Whenever human beings make observations, they get definite yes/no answers. But, if we could look to the world through these God-like eyes, and see the superposition, we would see that there is more than just yes and no or one and zero.

Certainly this opens the door to theological discussions, which the book does include. But just as interesting is the more fundamental question:  What is the mode of perception that mathematics provides?  Our visual systems structure the data that floods the retina.  To what does mathematics give structure?