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By Joselle, on March 21st, 2011 I’ve been spending a lot of time reading about the significance of Riemann’s Habilitation Dissertation and, today, a little bit of looking into the pervasive human desire to generalize led me yet again to Plato. I keep thinking that a closer look at what Plato actually said is consistent with even the most brain-based thoughts on how we come to know anything. The words that first got me going today were from William Kingdon Clifford’s translation of Riemann’s lecture (#20 on this list of Riemann papers has a pdf of the translation):
Researches starting from general notions, like the investigation we have just made, can only be useful in preventing this work from being hampered by too narrow
views, and progress in knowledge of the interdependence of things from being checked by traditional prejudices.
The key words for me are general notions, knowledge of the interdependence of things, and traditional prejudices. Riemann’s thoughts are carefully considered generalizations, directed at the notions of space, geometry and measurement, that greatly affected the course of modern mathematics. And I will likely address them again in future posts. But my point for this post centers more on epistemological ideas and hence another reference to Plato.
In Book 6 of the Republic, Plato describes what has been called the simile of the sun. In the dialogue he writes, he makes the following statements:
The old story, that there is many a beautiful and many a good, and so of other things which we describe and define; to all of them the term “many” is implied.
And there is an absolute beauty and an absolute good, and of other things to which the term “many” is applied there is an absolute; for they may be brought under a single idea, which is called the essence of each.
The many, as we say, are seen but not known, and the ideas are known but not seen.
He goes on to talk about sight which he says, unlike the other senses, is bonded to something else – visibility, or light, or the sun itself. While each of the following is formed as a question, he makes these observations:
And the power which the eye possesses is a sort of effluence which is dispensed from the sun
Then the sun is not sight, but the author of sight who is recognized by sight
And the soul is like the eye: when resting upon that on which truth and being shine, the soul perceives and understands, and is radiant with intelligence; but when turned toward the twilight of becoming and perishing, then she has opinion only, and goes blinking about, and is first of one opinion and then of another, and seems to have no intelligence
About students of geometry and arithmetic and the “kindred sciences” he says (or asks!):
And do you not know also that although they make use of the visible forms and reason about them, they are thinking not of these, but of the ideals which they resemble; not of the figures which they draw, but of the absolute square and the absolute diameter, and so on — the forms which they draw or make, and which have shadows and reflections in water of their own, are converted by them into images, but they are really seeking to behold the things themselves, which can only be seen with the eye of the mind
Among many there is a reluctance to accept as ‘a world,’ the world of forms. There is material that gives rise to ‘the mind,’ and it is the body alive in its world but what the mind gives rise to seems to be material-less. Yet clearly there is structure and meaning to ideas. These structures can be explored as vigorously as one might explore any number of materials. And, in fact, there will always be some relationship between thought and material. The prejudice we have is believing that ideas are ‘contained’ in us while physical things are external to us. But it is worth remembering that the images constructed by what neuroscientists call the visual brain are largely internal events. I think Plato’s simile of the sun contains more than just an interesting metaphor. I think it is taking note of how human worlds (both physical ones and conceptual ones) can only be brought into existence in relationship, leaving us to wonder again about the source of the conceptual ones. Our habitual distinction between internal and external experience may yet give way.
Mathematics, I believe, uniquely demonstrates where an extraordinarily careful consideration of concepts can lead and assigns indisputable value to the research of form.
By way of an ode to Riemann I’ll end with a quote from Labyrinth of Thought by Jose Ferreiros Dominguez:
…Riemann transgressed the limits of the traditional conception of mathematics, turning it into a discipline of unlimited extent and applicability, since it embraced all possible objects.
By Joselle, on March 15th, 2011 I would like to follow up on Alain Connes’ statement in my last blog. The weave of mathematical thought is tight. The seeds of mathematics are found in early explorations of number relationships and in observations of what we call space. But symbol, stripped of content, has led to heightened powers of thought and discernment. A simple illustration of what happens is this quick look at the weave of algebraic and geometric ideas.
One of the things that may not be clear to individuals with very little math experience is that the algebraic way of thinking about something and the geometric way of thinking about something are often two ways of thinking about the same thing. Everyone who has taken an algebra class knows that a function, say defined by y = 2x assigns a y value to every x value. We can draw this relationship using Cartesian coordinates. It is a line passing through the origin (the point 0,0) with a particular positive incline. You can use Pythagoras’ observation about right triangles to calculate the distance between any two points on the line or any two points on the infinitely extended Cartesian plane. The space here is the plane and the objects in it are the points. We identify a particular collection of points, determined by an algebraic function, when the line is drawn.
But the distance between “points” with any number of coordinates is defined using the Pythagorean idea. So we can calculate the distance between two points in three dimensions, or four, or infinitely many. Even more impressive is the extension of this idea of distance, to the more general idea of a metric which is the distance between any two mathematical objects in a mathematical space that has a metric. For a any metric to be valid (including the distance formulas in more than three dimensions), just three things about it must be true (inherited from the original):
-the distance must always be positive (or 0 if we measure from one object to itself).
-we get the same result if we measure from say object A to object B or from object B to object A.
-the distance from A to B added to the distance from B to say C is greater than or equal to the distance from A to C. (mimicking a triangle)
The metric we all know is the Euclidean metric, or the distance formula we started with. But every non-Euclidean space has a defined metric and, if properly established, we can talk about the distance between two objects in any collection of any number of mathematical objects. Functions themselves, like the one we started with, can be the objects in a collection. Then the distance between two functions can be defined. In the theory of fractals, this kind of thinking can describe very complicated shapes as the limit of simpler ones (like the numerical limit in calculus).
The fruits of this kind of interweaving are everywhere in mathematics and, as Alain Connes observes, the weave cannot be separated without threatening the integrity of the whole body of thought. But my own mind wanders in two directions when I think about this kind of mathematical development. I wonder first how it is that we successfully identify the essence of an idea or, more specifically, what gives us confidence in our three properties of distance. This confidence can only be justified mathematically. It is the very thing that makes research in mathematics so difficult. When we can no longer see the distance we wish to calculate, the mind has to hold onto something internal to itself. Somehow the mathematician learns, through purely mathematical experience, the conditions for invention. Related to this, but looking in the other direction is the miracle that it works. Physics and engineering are built on these relationships. And what follows is a book review in Evolutionary Psychology, where the author comments on mathematics applied to observations of evolutionary progression:
……we turn to the kinds of mathematical arenas, invented between the time of Newton and the time of Einstein, used to represent this evolutionary change. By and large these arenas are sets of elements, each element representing or “marking” a possible state of the evolving system. Usually the elements are labeled with numbers, or strings of numbers, that demarcate them quantitatively. The equations of motion specify the rates (or something similar) at which any one state gives way to others accessible from it, and so on through each moment of time in the evolutionary progression. The set of elements often has a metric, or natural measure of distance, associated with it, which allows us to say when a state has changed a little or a lot, and by how much. When suitably posed, the metric can be read as equipping the set of elements with geometric properties.
These properties are intrinsic to the evolutionary change and can be freed from the arbitrary manner in which we might map, or link, the state elements to their numerical indices. It is then natural to think of such a set, equipped with a natural geometry, as a “space” of “points,” each point marking a state, and of the evolutionary change as tracing out a path or trajectory though this space. So, for example, a state element or point represents a population in which the frequency of a gene variant has a specific value and that of a meme variant also has a specific value. Points with slightly different frequency values are nearby in the space. The equations of motion connect the points in an axiomatic game of “join-the-dots,” to predict which point will follow which as the population evolves. A glance in any textbook about mathematical population genetics, ecology, or neural network theory, for instance, will reveal endless content based on this general point of view.
By Joselle, on March 7th, 2011 Here is an excerpt from a piece by Alain Connes in The Princeton Companion to Mathematics:
It might be tempting at first to regard mathematics as a collection of separate branches, such as geometry, algebra, analysis, number theory, etc., where the first is dominated by the attempt to understand the concept of “space,” the second by the art of manipulating symbols, and the third by access to “infinity” and the “continuum,” and so on.
This, however, does not do justice to one of the most important features of the mathematical world, namely that it is virtually impossible to isolate any of the above parts from the others without depriving them of their essence. In this way the corpus of mathematics resembles a biological entity, which can only survive as a whole and which would perish if separated into disjoint pieces.
The scientific life of mathematicians can be pictured as an exploration of the geography of the “mathematical reality” which they unveil gradually in their own private mental frame.
It is this about the nature of mathematics that has led me to wonder about the way it reflects cognition itself. Our nervous system integrates the work of delicate sensory devices, which often overlap in function, bringing discernible form to the flux around us. In this way we find our fundamental orientation in the world. The human development of music, language, art, and science are likely extensions of this. Perhaps mathematics draws on some of these internal mechanisms, ones that extract form from flux, and then somehow formalizes the possibilities for relationship among the things we experience.
Connes later says:
Where things get really interesting is when unexpected bridges emerge between parts of the mathematical world that were remote from each other in the mental picture that had been developed by previous generation of mathematicians. When this happens, one gets the feeling that a sudden wind has blown away the fog that was hiding parts of a beautiful landscape.
It is with mathematics that we find an unexpected link between the imagination and reality. Our confidence that we can rely on the consequences of a particular kind of reasoning leads to startling new possibilities. It makes sense that there was some concern, particularly in the eighteenth century, that we could violate the demands of this reasoning with ill-conceived generalities and non-constructive proofs.
As Donal O’Shea says nicely in The Poincare Conjecture:
Popular accounts of mathematics often stress the discipline’s obsession with certainty, with proof. And mathematicians often tell jokes poking fun at their own insistence on precision. However, the quest for precision is far more than an end in itself. Precision allows one to reason sensibly about objects outside ordinary experience. It is a tool for exploring possibility; about what might be as well as what is.
By Joselle, on March 2nd, 2011 For me, one of the more intriguing things that happened in mathematics is what is called the arithmetization of the Calculus. This is not because it contributes to my understanding of fundamental concepts (because it doesn’t). Nor is it because the ideas are exotic (they’re not). I’m captivated, instead, by what it may demonstrate about the way we see – the way we come to understand or discern meaning. How does one separate calculus ideas from the intuitive notions of space and motion that gave rise to it? That this was accomplished is usually irrelevant to students of mathematics unless, of course, they aspire to be mathematicians.
The work of Karl Weierstrass (building on ideas developed by Cauchy) established what many call the soundness of calculus. He removed, from the limit process, the idea of motion or of changing values of a variable. The Princeton Companion to Mathematics tells us how he began:
He attempted a unified approach to the definition of rational and irrational numbers which involved unit fractions and decimal expansions and now seems somewhat murky. Weierstrass’s definition of the real numbers appears unsatisfactory to modern eyes, but the general path of arithmetization of analysis was established by this approach.
Then a bit later:
The former notions of variables tending to given values were replaced by quantified statements about linked inequalities.
These linked inequalities are the epsilon/delta definitions of limits and continuity. In the book, Where Mathematics Comes From (Lakoff/Nunez), which I’ve referenced in other blogs, the authors try to characterize what Weierstrass had set out to do:
He had to eliminate one of most basic concepts – the natural continuity of a trajectory of motion – and replace it with a concept that involved no motion (just logical conditions), no continuous space (just discrete entities), no points (just numbers), with functions that are not curves in a plane (but, rather, sets of ordered pairs of numbers).
And here, the very thing that I first loved about calculus is being taken away – that the things we were looking at weren’t static and discrete, but seemed to be moving, and often across endless distances. It was my first calculus class that brought me to attention and opened my eyes to the depth of these creative thoughts. Of course, this meaning isn’t really taken away and that I find equally provocative. Nunez wrote an interesting paper (Nunez 2008 in his list of publications) about the gestural components of these abstractions.
In another essay in the Princeton Companion to Mathematics, Jose Ferreiros draws attention to the disagreement between Weierstrass and some of contemporaries, namely Riemann, Dedekind, and Cantor, who favored conceptual meaning over calculated or formula driven results. Dedekind’s definition of the real numbers (the Dedekind cut) is judged more successful than Weierstrass’s. It is grounded in the logic of order, the concept of collections, and the association of numbers with points on a line. He starts by considering the system of rational numbers as a whole, an actual infinity. It is a very abstract description of ordered elements which, nonetheless, obey the laws of arithmetic.
While Weierstrass may have been at odds with some of his contemporaries (whose work would have far-reaching consequences), he set something big in motion. He is credited with bringing the required precision to calculus ideas, grounding them in relationships that could be clearly stated and repeatedly applied. Newton’s definition of a limit had been given in this way:
Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer to each other than by any given difference, become ultimately equal. (Principia Mathematica, Book 1)
And this one from Cauchy was an improvement:
When the values successively attributed to a variable approach indefinitely to a fixed value, in a manner so as to end by differing from it by as little as one wishes, this last is called the limit of all the others. (Cours d’Analyse)
Weierstrass’s definition is strictly arithmetic and gave calculus concepts indisputable validity. It has been said that Weierstrass’s limit merely hides the more intuitive, geometric and motion-based idea, and that Dedekind’s definition of a real number is embedded in the everyday meanings of cutting and separating things. (See the paper Visualism in Mathematics).
But it is clear that symbol itself opens the door to the perception of more detail (even in its worldly application), which I find most interesting. And maybe here is my real point. Somehow the mind (or the body) takes its experience of number, which it uses to categorize quantities of things and to compare distances, and extends it. The experience becomes externalized in symbol. The notation we currently use was found to be best for calculation purposes. Infinite decimal expansions and incommensurable quantities opened other questions, which were addressed from many directions. And in the 19th century, the simplest of relations (less than or greater than), would provide a completely abstract definition of number, the Dedekind cut, one of a few that now meet the mathematician’s needs. In the set of real numbers, something new is expressed.
The approach of mathematicians like Dedekind, Riemann and Cantor eventually won out, and confidence in the precision and power of conceptual meaning bore the fruit of modern mathematics. That confidence comes from an intimacy with the subject, where one can see what is correct and what is not. What it is that the mathematician is getting closer to is yet to be understood.
By Joselle, on February 21st, 2011 Quite a lot of work is being produced by cognitive scientists about metaphor – what they are -what they do, how they shape thought – and I find it all interesting and provocative. The way in which metaphor shapes the way we see the world is the subject of James Geary’s book I Is an Other. Metaphor, as it is now understood, is said to transfer physical experience to psychological experience. Some quick writing from Geary can be found on his blog.
Metaphor has also been used to account for the complex weave of ideas in mathematics, as seen in the book Where Mathematics Comes From by George Lakoff and Raphael Nunez. It’s a way to see the purely conceptual being drawn out of the physical. A useful critique of the book appeared in 2001 in the Notices of the AMS.
But I can’t help but think that, given the ubiquitousness of metaphor in our experience, there is something of a ‘chicken or the egg’ problem here. Which comes first, the concept warmth or all warm things in our experience? The answer may not be as obvious as it seems. Abstraction does happen at the cellular level. There are specialized brain cells that respond preferentially to straight lines at a particular angle. There are cells that synthesize multiple views of an object into a “view-invariant” image. It is possible that the brain and the nervous system are primarily concept driven, or that their functions rely heavily on what we would think of as form rather than content, all being played out before anything even hits our awareness. This would certainly contribute to a discussion of what mathematicians find, when they find it ‘intuitively.’
Geary also brought up Watson, the Jeopardy champion, in one of his blogs.
What does this have to do with metaphor? Well, the kinds of things Watson and his human opponents will be parsing tomorrow are the same kinds of things that go into metaphors: loose associations, punning relationships, sidelong and sidereal correlations. Until now, computers have not been very good at making these kinds of intuitive connections, as the wealth of useless information thrown up by the simplest Google search demonstrates. If Watson can do it, though, that is one giant leap for computerkind…
But a recent post on Mashable gives us some inside info from a journalist who had access to Watson’s development. I grabbed this piece:
Q: Can you give us an example of a concept that was deceptively hard to teach Watson — something the researchers assumed would be straightforward but ended up being a challenge for the AI?
A: You mentioned the killer word in your question: concept. Watson and other machines don’t master concepts. A four-year-old child somehow figures out that a chihuahua and a great dane are both dogs. Conceptually, the two very different animals have something in common. Computers have to be taught such things. It’s a laborious process. (Watson, I should point out, hasn’t been taught such facts and relationships. It comes up with its responses by studying and analyzing data, most of it written English.)
While we may only be able to approximate the origin of what we call concept with our conscious minds, I’m sure it is a useful enterprise. In defense of the idea that concept is not derived, I’ll defer again to Poincare. This is from his discussion of Space and Geometry in his Science and Hypothesis:
It is seen that experiment plays a considerable role in the genesis of geometry; but it would be a mistake to conclude from that that geometry is, even in part, an experimental science. If it were experimental, it would only be approximative and provisory. And what a rough approximation it would be! Geometry would be only the study of the movements of solid bodies; but, in reality, it is not concerned with natural solids: its object is certain ideal solids, absolutely invariable, which are but a greatly simplified and very remote image of them. The concept of these ideal bodies is entirely mental, and experiment is but the opportunity which enables us to reach the idea.
By Joselle, on February 14th, 2011 I had the opportunity to listen to Paul Churchland when he gave a talk last Friday, on Cognitive Enhancement, at the University of Texas at Dallas. He used the time to address, not enhancement drugs or exercises, but the enhancement effects of language and symbol. I poked around today to find more more on the ideas that he only had time to reference and I stumbled upon Dudley Shapere’s critique of Churchland’s book The Engine of Reason, the Seat of the Soul.
I was most interested in Churchland’s conviction that cognition is not based in “sentence crunching” or logic-like processes, but is grounded more in small neuronal pattern adjustments of synaptic connections. This is consistent with my sense that mathematics is born more of pattern and image than logic. But language as the externalization of learning, makes the transference of cognitive acquisitions to another individual possible and also presents explorers with new ground to ponder. In his book, Churchland asks some interesting questions:
How is it that human cognition manages to reach behind appearances? How do we discover, for example, that light consists of submicroscopic waves?…..How then does a neural network such as a human scientist – a network doomed, after all, to be trained on a uniform diet of observable phenomena – ever manage to form concepts or prototypes of unobservable phenomena?
The answer, to a first approximation is that we learn all of our prototypes solely within the domain of observable things.
Shapere argues strongly that recycling prototypes is not enough to account for how the world gets opened up by the sciences. To describe some of his difficulty with this notion, he uses the idea of curvature in mathematics, of which, he says, the commonsense view is in terms of space. In other words, the curvature of a one-dimensional line is generally imagined by seeing it on a two-dimensional surface, and the curvature of a two dimensional surface is imagined by seeing it in a three dimensional space. He continues:
In this commonsense view, it makes no sense to speak of the curvature of our three-dimensional space, because we cannot access a four-dimensional containing space with respect to which we could say it curved.
He goes on to say:
However, Gauss’s theory of surfaces, generalized by
Riemann to spaces of any dimensionality, introduced a notion of curvature capturing the important features of the commonsense view except for being intrinsic to the space (line, surface, etc.), determinable by measurements
entirely within the space itself, without appeal to anything about any higher-dimensional containing space, even its existence.
This is the idea Einstein would use. “Such transformations,” Shapere argues, “are uncapturable in today’s neural network models.”
I want to agree, although I, myself, am captivated by the relationship between the original, spatial notion of curvature and the later, very powerful, transformation of it. And I’m convinced that neuroscience will make a significant contribution to how we understand what we do, particularly when it reveals unexpected relationships, like the fact I learned today that the visual cortex is recruited by blind individuals when they learn to read Braille. In these individuals, the visual cortex is processing information derived from touch. We could say the brain is being efficient since the visual cortex in these individuals is not being used. But I think it’s also important to note that Braille is the touch version of the visual word.
The symbol is most important to Shapere’s understanding of scientific investigation.
Neither the character nor the methods of science can be understood in terms of individual thought-processes, with their attendant prejudices, myths, and misunderstandings; they must be understood in terms of an objective body of
doctrines and evidence developing sometimes in radical directions. Herein lies the importance of language, logic and, I would add, mathematics, which gives us our only means of conceptualizing that which lies well beyond our middlesized
experience.
It’s not the individual, he says,
For a primary attribute of the modern scientific
knowledge-seeking process, and of the knowledge that results from it, is its independence of the individual.
I’ll say it again, because I like it so much, from last week’s post:
The symbol jumps over the blind spot of cognition with a flash of unexpected light.
By Joselle, on February 7th, 2011 I’ve thought that one of the reasons it’s difficult to resolve questions about the nature of mathematical reality is that we’re not exactly clear on what it means to ‘perceive’ something. Trying to establish whether or not even the data of our senses is somehow independently ‘real,’ has fueled centuries of philosophical debate. I found a paper today with the title Visualism in Mathematics by Fernando Flores of Lund University. The paper applies the visualism of Professor Don Ihde to mathematics (from his book Expanding Hermeneutics, Visualism in Science). It’s an interesting read in that it also addresses how the formalism of modern mathematics is, in itself, visual and the rigor of logic based conceptualizations just hides its “direct connection to the intuition of everyday life.” One of the passages I liked was this one about Dedekind’s cuts, (used to give a firm logical foundation for the real number system):
There is a very important and unconscious manipulation of text-depicting gestalts in Dedekind‘s construction, the praxis of cutting and separating, the praxis of finding things spread around in suitable successions make this proof a master piece of art rather than a scientific result. That talks a lot about the nature of mathematical knowledge, which is in fact deeply rooted in the everyday world.
Flores also suggests a sameness between mental and physical space and early in the paper says the following:
A common denominator of all these visual constructions is to represent a certain type of ‘logic visual reality’, which could be illustrated by John Venn‘s (1834 –1923) configurations of circles. The geometric constructions in logic, works generally as analogies but they are more than that, they are parallel phenomenological worlds.
The idea that abstract constructions are objects in a perceived world makes the old questions, like what’s really there and what did we invent, harder to separate.
For me, these kinds of thoughts inevitably connect to the work of cognitive scientists like Mark Turner and Gilles Fauconnier who look closely at the details of how conceptual worlds are built with metaphor-based mental processes. Some of this can be found in their paper Rethinking Metaphor.
But the conceptual is pushed right back into nature in work known as biosemiotic mimesis, where metaphor is, essentially, the way the organism perceives. This view of cognition is discussed in a paper by Andreas Weber from Hamburg, Germany called Mimesis and Metaphor. This work has its own history (some of which you can see in the paper’s references). I’ll save this discussion for another post, but I will take from this paper what is consistent with the theme of this one, that the power of this thing we call metaphor is that it also brings us something new, something that wasn’t there before and can now be perceived. By Weber it is described in this way:
The deciding moment in the symbolic (or poetic) achievement, the so called “keen metaphor” (Haverkamp 1995), is that it does not only arouse a vital import through the synaesthetic enactement of feeling: it produces something entirely new, something never heard of, which becomes an opaque part of the world itself.
And also:
The symbol jumps over the blind spot of cognition with a flash of unexpected light. It knows more than what it was borrowed for. Its wisdom stems from a double source, always merging body and culture.
By Joselle, on January 31st, 2011 I started today by taking a look at what might be the latest on what cognitive scientists were saying about mathematics. The broad scope of cognitive science includes the investigation of what Mark Turner calls (in the title of one of his books) “the riddle of human creativity.” When exploring the origins of conceptual systems, the rise and effectiveness of mathematical conceptual systems is often one of the more mysterious, despite recent attempts to demystify it (like the work of Rafael Nunez). There are some papers to be found at The Cognitive Science Network. Research in these areas provides valuable insights into how our image-filled worlds are formed. I am particularly interested in the relationship between gesture and thought and will follow this up in future posts. But today I was a little side-tracked by an essay I found by Paul Bernays written in 1959. He is responding to Ludwig Wittgenstein’s Remarks on the Foundations of Mathematics.
I decided to write a bit about this essay because I found the mathematician’s response to Wittgenstein’s critique of mathematics contains important insights into what cognitive scientists explore or have yet to explore.
Early on, Bernays makes the following observation (without characterizing Wittgenstein a behaviorist). He says:
Two problematic tendencies, however, appear throughout. The one is to dispute away the proper role of thinking – reflective intending – in a behavioristic manner.
Bernays remarks that Wittgenstein allows only one kind of ‘factuality,’ namely concrete reality. Quoting Wittgenstein:
I can calculate in the imagination, but not experiment.
To this Bernays adds:
An engineer or technician has doubtless just as lively an image of materials as a mathematician has of geometrical curves, and the image which any one of us may have of a thick iron rod is no doubt such as to make it clear that the rod could not be bent by a light pressure of the hands.
Bernays makes the argument that “concept-formations” are “an extract from experience.” and observes that Wittgenstein points to this when he says “Imagination teaches us it.” But Bernays then makes an observation of the role of intuition in mathematics that crystallizes one of its most provocative characteristics. While mathematics can always be shown to follow reason, or to make sense, it can lead to a fact that seems unreasonable, yet is true. Here’s what Bernays says:
In considering geometrical thinking in particular it is difficult to distinguish clearly the share of intuition from that of conceptuality, since we find here a formation of concepts guided so to speak by intuition, which in the sharpness of its intentions goes beyond what is in a proper sense intuitively evident, but which separated from intuition has not its proper content.
In other words, mathematics brings thought beyond what is immediately apparent to the intuition, but which only has meaning because of it. This is the hallmark of the discipline. And, in this way, mathematics liberates us from the limits of our perceptions. Again from Bernays:
The strictness of the logical and the exact does not limit our freedom. Our very freedom enables us to achieve precision through thought in a perceptive world of indistinctness and inexactness.
Despite its history of foundational uncertainties and philosophical disputes, the mathematician’s intimacy with his or her subject reveals its nature, however difficult it may be to justify it. The indistinctness of perception is resolved by a disciplined use of the imagination. This is what the body manages with mathematics. And this is what I hope will continue to inform research efforts in cognitive science.
By Joselle, on January 24th, 2011 In a recent post I said that one of the things that dissuades us from accepting the existence of a truly Platonic mathematical world, or believing in the timeless existence of its forms independent of human minds, is the habit we have of distinguishing ourselves from the rest of nature, despite all the evidence we’ve seen of nature’s tightly knit oneness. This is why Plato came to mind for me again, when I read an article about how fish can count.
The article describes some of the findings published in an animal cognition paper. Researchers found that angelfish can successfully distinguish between various sized angelfish groups, provided the groups differ by at least a 2:1 ratio. Data has also been collected that suggests that these fish can, more precisely, distinguish between the quantities 2 and 3. Since the 2:1 ratio doesn’t apply here, it is believed that this talent arises from a different neural mechanism. Researchers say that there is no reason the fish could not get beyond 3, but rather that there has been no particular advantage to them doing so.
A good deal of work has been done on the presence of some number sense in babies and other animals, often mammals. But these fish findings got my attention because fish, in our understanding of evolution, show up a few hundred million years earlier than mammals. And Robert Gerlai, who conducted much of the research in the article, played right into that observation when he said that given how widespread the basic math-related skills are throughout the animal kingdom, it’s possible that the action of counting, “could have arisen in one ancestral species from which all species with this ability evolved.”
What all of this suggests to me is a kind of biological underpinning of mathematics that can’t be disassociated from the world itself, or the universe for that matter. Everything neurological has its source in the signaling talent of individual cells, ultimately tied to the substances of stars. If counting grows out of this as does, breathing, digestion, movement and later memory and imagination, then it may be that Plato intuited that mathematics, as a thoughtful pursuit, was tapping into something very fundamental, something not quite ours, whose truth value was greater than the particulars to which we applied it. It seems inevitable that we will find that mental processes are as exploratory as the work of the senses.
I’ll close with a quote from Alain Connes:
One of the essential things a mathematician does is recognize the internal coherence and generative character belonging to certain concepts. It happens that very simple concepts can suggest all sorts of ideas or models. Investigating these, one truly has the impression of exploring a world step by step — and of connecting up the steps so well, so coherently, that one knows it has been entirely explored. How could one not feel that such a world has an independent existence?
–Conversations on Mind, Matter, and Mathematics
By Joselle, on January 17th, 2011 I just flipped back and forth between reading about 18th and 19th century developments in mathematics (analysis in particular) and 18th and 19th century transitions in art. The language of art history and the language of math history is very different. It does feel a little like going from color to black and white, or from something sunny, to something just starkly lit. But because so much of my recent reading has included the phrase modern mathematics, I couldn’t help but spend some time thinking about what people say about modern art.
What I find common to both are thoughtful shifts in the human questions: What’s really there? How can we talk about it? How can we express it?
For the sake of simplicity I’ll refer only to Paul Cezanne whose work is often thought of as the bridge between impressionist painting and modern art. Impressionist painting seems most centered on how we see, how our visual experience changes continuously, in time, with light. Cezanne, however, also took note of the geometric forms of nature, and further explored the effects of binocular vision, wanting to capture, if one could, the truth of his perception.
Painting must be motivated, to some extent, by a desire to reveal something about what is contained in what we see. And, while I’m taking a big leap here, mathematics can be said to be the exploration of what is contained in the whole of what we see and what we can subsequently imagine. As it is characterized by precision, a good deal of strictly thoughtful analysis must be assumed.
While mathematics can only be known with symbolic representation, its roots are in experience. It manages to extract forms and generalities from sensation and then it explores the value and significance of those forms. The transition to what we call modern mathematics can be seen as a shift away from continuing to play with the observations made (with respect to things like functions, limits, derivatives and integrals, series, alternative geometries) toward finding the simplest generalities that successfully characterize these things – essentially pinning down their truth. It required a new detachment, not unlike the modern artist, in order to find the forms rather than all of the examples of the forms. (I will write more specifically about some of these changes in another post).
For now I would like to quote Cezanne:
“May I repeat what I told you here: treat nature by means of the cylinder, the sphere, the cone, everything brought into proper perspective so that each side of an object or a plane is directed towards a central point. Lines parallel to the horizon give breadth… lines perpendicular to this horizon give depth. But nature for us men is more depth than surface, whence the need to introduce into our light vibrations, represented by the reds and yellows, a sufficient amount of blueness to give the feel of air.”
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