for example, via rakesh:
"Where do numbers come from? Do they exist outside human beings, or did humanity invent them? ...[snip]... Studies have found that rats, chimpanzees, and pigeons have a built-in "accumulator" that allows them to keep rough track of a limited number of objects, usually about three. "
this reminded me of a paper i read a while ago on the ability of the brain to perform what we'd call non-commutative mathematical operations.
1.) first, what is a (non-)commutative mathematical operation?
if someone gives you three cookies for sweeping floors and later two cookies for making your bed, you have 3 + 2 = 5 cookies.
if another scenario gives you two cookies for sweeping floors and later three for making your bed, you have 2 + 3 = 5 cookies, still.
pretty elementary, right? 2 + 3 = 3 + 2 = 5 is the commutative property of addition of numbers.
and if you need six 2x6's for some construction project, and one lumber yard sends 2x6's in sets of three's, and another in pairs, then one yard will deliver two sets of three 2x6's and the other will deliver three sets of two 2x6's, and you still get six 2x6's because 2 x 3 = 3 x 2 = 6.
still boring, eh?
well, there's more to things than addition and multiplication of numbers, right?
one of the first undergrad examples we teach of a non-commutative algebra is the mathematics of rotations in 3 dimensions.
take a book, imagine there are two axes attached to the book as you are looking at its cover: one along the binding (A), and one perpendicular to the binding and lying in the plane of the cover (B). (a chalk-board would be useful here).
rotate the book about the A axis by 90 degrees (A-90), then rotate it about the B axis by 90 degrees (B-90).
you wind up with one orientation for the book in space.
now reverse the order of the rotations. do B-90 followed by A-90.
what's that? different orientation of the book in space, eh?
a natural representation of rotation in 3 dimensions is matrix multiplication, which is non-commutative.
means: the 3x3 matrix representing the
rotation operation B-90
||||||
VVVVVV
matrix(A-90) * matrix(B-90) != matrix(B-90) * matrix(A-90)
^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^
do B-90 followed by A-90 != do A-90 followed by B-90
the symbol "!=" means "does not equal"
2.) here is the abstract and the first 3 andthe 6-th paragraphs of the paper:
*** Nature 399, 261 - 263 (1999) © Macmillan Publishers Ltd.
Non-commutativity in the brain
DOUGLAS B. TWEED*, THOMAS P. HASLWANTER?, VERA HAPPE? & MICHAEL FETTER?
* Departments of Physiology and Medicine, 1 King's College Circle, University of Toronto, Toronto M5S 1A8, Canada
? Department of Neurology, University Hospital, Frauenklinikstrasse 26, Zurich 8091, and Department of Physics, ETH Hoenggerberg, Zurich 8093, Switzerland
? Department of Neurology, University of Tbingen, Hoppe-Seyler-Strasse 3, Tbingen 72076, Germany
Correspondence and requests for materials should be addressed to D.B.T.
[Abstract] In non-commutative algebra, order makes a difference to multiplication, so that a x b != b x a (refs 1, 2). This feature is necessary for computing rotary motion, because order makes a difference to the combined effect of two rotations[3-6]. It has therefore been proposed that there are non-commutative operators in the brain circuits that deal with rotations, including motor circuits that steer the eyes, head and limbs[4,5,7-15], and sensory circuits that handle spatial information[12,15]. This idea is controversial[12,13,16-21]: studies of eye and head control have revealed behaviours that are consistent with non-commutativity in the brain[7-9,12-15], but none that clearly rules out all commutative models[17-20]. Here we demonstrate non-commutative computation in the vestibulo-ocular reflex. We show that subjects rotated in darkness can hold their gaze points stable in space, correctly computing different final eye-position commands when put through the same two rotations in different orders, in a way that is unattainable by any commutative system.
The vestibulo-ocular reflex (VOR) is perhaps the simplest of the many neural responses to rotary motion: sensors in the inner ear measure head velocity and send commands to the eye muscles, moving the eyes in the opposite direction when the head turns, so as to keep the eyeballs from rotating relative to space[6]. By stabilizing our retinal images, the VOR allows us to see while moving.
An ideal VOR must be capable of non-commutative operations. This point has been argued before[5,13], but it is perhaps more clearly illustrated by the thought experiment in Fig. 1, which shows that the VOR, if it is to keep the gaze on target, must compute different final eye-position commands when a person undergoes the same two rotations in different orders. Starting by looking at a space-fixed target 30° to the left, a subject who turns first 10° counterclockwise (CCW) and then 60° left must end up looking 30° right and 5° up to stay on target (Fig. 1a). A subject who turns first 60° left and then 10° CCW must end up looking to the right and down (Fig. 1b).
Surprisingly, this ideal behaviour is not predicted by most models of the neural circuitry underlying the VOR. For example, Fig. 2 shows a computer simulation of an influential theory[19]. This model elegantly explains many features of the VOR, but as the simulation shows, it fails to keep the eye on target when put through the rotations from Fig. 1. Regardless of the order of rotations, it leaves the eye in the same final position relative to the head. Relative to space, then, the eye is incorrectly positioned, pointing above or below the target.
...[snip]...
It has been proposed that non-commutative effects in ocular control need not imply non-commutative computations in the brain, because some of these effects can be explained by the geometry of the eye muscles, in particular the fact that the muscles run through pulleys in the orbital wall[22,23]. Pulleys do have major implications for eye control[12,19-23], but neither they nor any other muscle geometry can explain the non-commutativity in Fig. 3. In fact, the model in Fig. 2 incorporates pulleys, but it remains commutative. Whatever the muscle geometry, there is always a correspondence between final, steady-state eye position and motor-neuron firing: to hold different eye positions, you need different firing rates[6,24]. Even if the pulleys themselves are adjusted by a separate set of motor neurons, final eye position will still be determined by the total motor-neuron pool. The distinct final eye positions in Fig. 3 therefore imply that the final activities of the motor neurons differ, depending on the order of head rotations. There must be non-commutative processing between the inner ear and the motor endplate.
***
3.) full article for those interested in reading more.
les schaffer