[Fis] Fluid Foundations

I’d like to address Aleks’ third point about mathematics having drifted into
senescence because nobody is questioning this move towards greater and
greater discretization. I believe this is no longer true. There is a
fledgling field within cognitive psychology called “mathematical idea
analysis,” outlined by George Lakoff and Ralph Nunez in their 2002 book,
“Where Mathematics Comes From: How the Embodied Mind Brings Mathematics Into
Being.” The authors claim that all abstract theory is embodied in origin and
so begins concretely as sensorimotor experience during infancy that is
systematically bootlegged into abstraction with maturation, both
individually and collectively.

Within mathematics, not even so basic a concept as number is immune. At
least three entirely different metaphors for number exist --“objects in a
set,” “points on a line,” and “units of measurement.” Lakoff and Nunez show
how entailments from each metaphor, plus blends between them, pave the way
for entirely different branches of study. What is particularly interesting
is how different foundational metaphors can result in INCOMPATIBLE notions
of similar concepts, such as that of infinity. On the one hand, Newton’s
method of limits in calculus and entailments by Cantor lead to the notion of
transfinite numbers, including different sizes of infinity. On the other
hand, metaphors available through Leibniz’s alternative method of calculus
lead to the notion of infinitesimals. Entailments extending the metaphor to
granular numbers, posit a system of immeasurably “huge” numbers
incommensurate with the transfinites.

It seems to me that such an approach allows for a less reified, more fluid
view, than one which takes for granted this relentless enterprise of
breaking the universe up into smaller and smaller discrete objects. Rather
than applying the agenda of discretization to the continuity of things,
perhaps this notion of conscilience that we are discussing relates to a
figure/ground reversal, by applying the agenda of making more continuous
that which has previously viewed as discrete.

Cheers,
Terry Marks-Tarlow

>Some random thoughts:
>
>1. Independent reinvention: another path to consilience
>
>Two years ago, I have been trying to solve a particular problem in machine
>learning, my field of research. Looking around, I found a particular
>generalization of mutual information, which was constructed by my
>colleague.
>I performed numerous experiments, and it performed well. Then I tried to
>check if anyone thought of that before, and, empowered by Google, I was
>surprised to find virtually the same formula (re)invented independently
>with
>different names over the past 50 years in: biology (Quastler), psychology
>(McGill), information theory (Han, Yeung), physics (Kikuchi,Cerf&Adami
>arXiv:quant-ph/9605002), neuroscience (Brenner), robotics (Yairi et al),
>and
>closely related notions in game theory (Grabisch & Roubens), complexity
>theory (Gell-Mann), statistics (Darroch), and chemistry (Kirkwood). I could
>go on, but I wonder how many I already missed. Most researchers, but not
>all, arrived at it by employing the inclusion-exclusion principle outside
>of
>strict context of set theory. That was my encounter with consilience: but
>it
>was not a discovery of applicability of a different method in an unexpected
>domain, but the independent discovery of the same hypothesis (actually a
>tool) in a number of different domains: isn't this the same.
>
>
>2. Hierarchies of sciences
>
>Perhaps that is demonstration enough that vertical integration is
>beneficial
>if not necessary. And here come in Pedro's diagrams of sciences. I do not
>see a need for politically correct circularity. I also agree with the
>optimistic viewpoint that there needs to be only a single vertical field
>that provides the cognitive "tools" (which are, however, psychologically
>feasible) to all the horizontal fields. Role models already exist: logic,
>mathematics and statistics (and philosophy that studies their foundations).
>
>As a side note, I found Pedro's reference to circle of knowledge very
>interesting. There, religion and philosophy are in the center. Where is
>religion today? While philosophy sets the framework for what and how to
>think, religion answers the Why?. While some people could involve in the
>infinite regress of why why why, most stop recursing at a certain level and
>move forward. Religion is a formal setting of this direction, combining
>with
>the informal herding behavior. Ultimately the quality of this direction is
>judged by survival and growth of both the religion and its followers.
>
>
>3. Objects reified
>
>Mathematics and logic, for one, have been cast in stone. Nobody dares
>question the fundamentals anymore, it would be unimaginable throwing all
>that away. This seems like a proof of Stan's senescence. Mathematics and
>logic are all discrete, based on objects and their behavior. Probability
>theory, too, is discrete, with the division of the universe into discrete
>events. And without probability theory, there is no statistical notions of
>information or entropy. If you now look at the natural world, you don't see
>objects: you see the fuzz of interlocking leaves, the swaying fur of grass,
>the smooth gusting of wind, swarming of flies, the gritty mess of mud, soil
>and pebbles on the ground, the swirling stream, the foamy foam.
>
>Then you turn and look at a urban landscape: objects (discrete buildings),
>objects (discrete chair), objects (discrete road-side signs), objects
>(discrete potted plants), objects (discrete buttons), objects (discrete
>humans), objects (discrete molecules of air), objects (discrete plant
>species), objects (discrete knots of wind speed), objects (centimeters of
>leaf surface area), objects (zeros and ones encoding the music on a CD),
>pummeling of photons, cascading of electrons. We don't know what there is
>inside an atom, but surely it's objects, and objects within them. Computers
>are engineered so that everything is fully deterministic and predictable,
>and there is nothing that is not an unnested discrete binary object, all
>equal, all egalitarian.
>
>We're trying to fit every aspect of nature to objects and to relationships
>between them. The problems we have with concepts like continuity (what is
>infinity? what is zero? what is divide-by-zero?), forces, fields,
>relationships and interactions (are two particles interacting at a
>distance,
>or are they really a single object? is a molecule of my body an independent
>molecule or is it my body?) are a kludge retrofitted on objects that tries
>to capture everything that objects alone miss. Indeed, our representations
>have senesced, and all we will ever do will be in the context of objects.
>*Can* we think in some other way than by reifying objects, or has our
>cognition senesced a long time ago? If computers don't, do we still have
>the
>freedom to think in non-objects? Fuzzy logic is not an answer: every fuzzy
>variable is still an object: either it's hot or it is not, two values, two
>objects. Just as in logic, a hypothesis is either perfectly true or
>perfectly false.
>
>You might wonder for a few seconds, perhaps admit it, but you would
>certainly not want to work in this direction. Life's too short and that's
>senescence.
>
>Aleks
>
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Received on Fri Oct 8 12:16:43 2004