![\"\"](\"http:\/\/aristotle2digital.blogwyrm.com\/wp-content\/uploads\/2017\/11\/throwing_darts.png\")
<\/a><\/p>\nHow does the person learn to throw accurately?\u00a0 The equations governing the flight of the dart are nonlinear and subject to a variety of errors.\u00a0 Clearly practice and feedback from eye to brain to hand are required but how exactly does the training take hold?<\/p>\n
While some believe that they know the answer, an honest weighing of the facts suggest that we don\u2019t have more than an inkling into the mechanism by which intelligence and repetition interact to produce a specific skill.\u00a0 In Aristotelian terms, the mystery lies in how a human moves from having the potentiality to score a bullseye into being able to do it dependably.<\/p>\n
What does seem to be clear is that this human mechanism, whatever it actually is, is quite different from any of the mathematical processes that are used in machine applications.\u00a0 To appreciate just how different, let\u2019s look at one specific mathematical targeting method: differential correction.<\/p>\n
Differential correction is the umbrella term that covers a variety of related numerical techniques, all of them based on Newton\u2019s method.\u00a0 Related terms are the secant method, Newton-Raphson, the shooting method, and forward targeting.\u00a0 Generally speaking, these tools are used to teach a computer how to solve a set of nonlinear equations (e.g., the flight of a dart, the motion of a car, the trajectory of a rocket, etc.) that symbolically can be written as<\/p>\n
\\[ \\Phi: \\bar V \\rightarrow \\bar G \\; ,\\]<\/p>\n
where $\\bar V$ are the variables that we are free to manipulate and $\\bar G$ are the goals we are trying to achieve.\u00a0 The variables are expressed in an $n$-dimensional space that, at least locally, looks like $\\mathbb{R}^n$.\u00a0 Likewise, the goals live in an $m$-dimensional space that is also locally $\\mathbb{R}^m$.\u00a0 The primary situation where these spaces deviate from being purely Cartesian is when some subset of the variables and\/or goals are defined on a compact interval, the most typical case being when a variable is a heading angle.<\/p>\n
These fine details and others, such as how to solve the equations when $m \\neq n$, are important in practice but are not essential for this argument.\u00a0 The key point is that the mapping $\\Phi$ only goes one way.\u00a0 We know how to throw the dart at the board and then see where it lands.\u00a0 We don\u2019t know how to start at the desired landing point (i.e., the bullseye) and work backwards to determine how to throw.\u00a0\u00a0 Likewise, in the mathematical setting, the equations linking the variables to the goals can\u2019t be easily inverted.<\/p>\n
Nonetheless, both the natural and mathematical targeting processes allow for a local inversion in the following way.\u00a0 Once a trial throw is made and the resulting landing point seen and understood, the initial way the dart was thrown can be modified to try to come closer.\u00a0 This subsequent trial informs further throws and often (although not always) the bullseye can be eventually reached.<\/p>\n
To illustrate this in the differential correction scheme, suppose the flight of the dart depends on two variables $x$ and $y$ (the two heading angles right-to-left and up-to-down) and consider just how to solve these two equations describing where the dart lands on the board (the values of $f$ and $g$):<\/p>\n
\\[\u00a0 x^2 y = f \\; \\]<\/p>\n
and<\/p>\n
\\[ 2 x + \\cos(y) = g \\; . \\]<\/p>\n
To strike the bullseye means to solve these equations for a specified set of goals $f^*$ and $g^*$, which are the values of the desired target.\u00a0 There may be a clever way to solve these particular equations for $x$ and $y$ in terms of $f$ and $g$, but one can always construct so-called transcendental equations where no such analytic solution can ever be obtained (e.g. Kepler\u2019s equation<\/a>), so let\u2019s tackle them as if no analytic form of the solution exists.<\/p>\n Assuming that we can find a pair of starting values $x_0$ and $y_0$ that get us close to the unknown solutions $x^*$ and $y^*$, we can \u2018walk\u2019 this first guess over to the desired initial conditions by exploiting calculus.\u00a0 Take the first differential of the equations<\/p>\n \\[ 2 x y dx + x^2 dy = df \\]<\/p>\n and<\/p>\n \\[ 2 dx + \\sin(y) dy = dg \\; .\\]<\/p>\n Next relax the idea of a differential to a finite difference \u2013 a sort of reversal of the usual limiting process by which we arrive at the calculus in the first place \u2013 and consider the differentials of the goals as the finite deviations of the target values from those achieved on the first throw:\u00a0 $df = f_0 – f^*$ and $dg = g_0 – g^*$.\u00a0 Likewise, the differential $dx = x_0 – x^*$ and $dy = y_0 – x^*$.\u00a0\u00a0 Since we now have two linear equations in the two unknowns $x^*$ and $y^*$, we can invert to find a solution.\u00a0 Formally, this approach is made more obvious by combining the system into a single matrix equation<\/p>\n \\[ \\left [ \\begin{array}{cc} 2 x y & x^2 \\\\ 2 & \\sin(y) \\end{array} \\right] \\left[ \\begin{array}{c} x_0 – x^* \\\\ y_0 – y^* \\end{array} \\right] = \\left[ \\begin{array}{c} f_0 – f^* \\\\ g_0 – g^* \\end{array} \\right] \\; .\\]<\/p>\n The solution of this equation is then<\/p>\n \\[ \\left[ \\begin{array}{c} x^* \\\\ y^* \\end{array} \\right] = \\left[ \\begin{array}{c} x_0 \\\\ y_0 \\end{array} \\right] – \\left [ \\begin{array}{cc} 2 x y & x^2 \\\\ 2 & \\sin(y) \\end{array} \\right]^{-1} \\left[ \\begin{array}{c} f_0 – f^* \\\\ g_0 – g^* \\end{array} \\right] \\; . \\]<\/p>\n Since the original equations are non-linear, this solution is only an estimate of the actual values and, by taking these estimates as the improved guess, $x_1$ and $y_1$, an iteration scheme can be used to refine the solutions to the desired accuracy.<\/p>\n Everything rests on two major assumptions:\u00a0 1) a sufficiently close first guess can be obtained to seed the process and 2) the equations are smooth and can be differentiated.\u00a0 By examining each of these assumptions, we will arrive at a keen distinction between machine and human thought.<\/p>\n First the idea of a first guess is so natural to us that it is often hard to even notice how crucial it is.\u00a0 Just how do we orient ourselves to the dartboard.\u00a0 The obvious answer is that we look around to find it or we ask someone else where it is.\u00a0 Once in sight, we then lock our attention on it and finally we throw the dart.\u00a0 But this description isn\u2019t really a description at all.\u00a0 It lacks the depth needed to explain it to a blind person let alone explain it to the machine.\u00a0 In some fashion, we integrate our senses into the process in a high non-trivial way before even the first dart is thrown.<\/p>\n Returning to the mathematical model, consider how to find $x_0$ and $y_0$ when $f^* = 2.5$ and $g^* = 4.0$.\u00a0 The traditional way is to make a plot of the mapping $\\Phi$, which may be considered as taking the usual Cartesian grid in $x$ and $y$ and mapping it to a set of level curves.\u00a0 A bit of playing with the equations leads one to the plot<\/p>\n where level curves of $f$ are shown in black and those for $g$ are in blue.\u00a0 The trained eye can guess a starting point at or around (1.5,1.0) but, for the sake of this argument, let\u2019s pick (1.0,0.5) (the green dot on the figure).\u00a0 In just a handful of iterations, the differential correction process hones in on the precise solution of (1.57829170833, 1.00361110653) (the black dot on the plot). \u00a0But the mystery lies in the training of the eye and not in the calculus that follows after the first guess has been chosen.\u00a0 What is it about the human capacity that makes looking at the plot such an intuitive thing?\u00a0 Imagine trying to teach a machine to accomplish this.\u00a0 The machine vision algorithms needed to look at the plot and infer the first guess still don\u2019t exist, even though the differential correction algorithm is centuries old.<\/p>\n Even more astonishing, once realized, is that the human experience neither deals with symbolically specified equations nor are the day-to-day physical processes for throwing darts, driving cars, or any of the myriad other things people do strictly differentiable.\u00a0 All of these activities are subjected to degrees of knowledge and control errors.\u00a0 Inexactness and noise pervade everything and yet wetware is able to adapt to extend in a way the software can\u2019t.\u00a0 Somehow the human mind deals with the complexity in an emergent way that is far removed from the formal structure of calculus.<\/p>\n So, while it is true that the human and the machine both can target and that targeting entails trial-and-error and iteration, they do it in completely different ways.\u00a0 The machine process is analytic and local and limited.\u00a0 The human process is emergent, global, and heuristic.\u00a0 In addition, we can teach ourselves and others how to do it and even invent the mathematics needed to teach a pale imitation of it to computers, but we really don\u2019t understand how.<\/p>\n","protected":false},"excerpt":{"rendered":" Riding a bike, catching a ball, driving a car, throwing a dart \u2013 these and countless other similar, everyday activities involve the very amazing human capacity to target.\u00a0 This capacity… Read more 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