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First, Jeffs view of programming is too narrow. Secondly, and more importantly, so is his view of math. Being "mathematically inclined" does not mean being wel

• stonemetal 18y ago
  >> If you see someone sorting papers and try to figure out why they are doing it inefficiently, that is mathematical inclination. Not really, I would say it was a logical inclination not necessarily a mathematical one. And therein is the rub I see programming as logic and problem solving with little to no inherent math unless I am dealing with a mathematical problem. Kind of like the way writing books is not an inherently mathematical in nature even if quite a bit of math is involved in writing a mathematics text book.
  • lbrandy 18y ago
    This is bordering on semantics. What's the difference between mathematical inclination and logical inclination? The field of computer programming has made a separation in alot of people's mind that simply didn't exist before. People, especially programmers, view mathematics far too narrowly because they lack historical context. George Boole was a mathematician. The logic textbooks are full of words that betray their background, e.g. the "lambda calculus". All of these things have their roots in mathematics. And before there was computer science degrees, these things -were- mathematics. I doubt, very seriously, that Djikstra would draw much of a distinction between "logical inclination" and "mathematical inclination". So while you might, the original context of the quote in question needs to be considered.
    • bpyne 18y ago
      Agree fully with lbrandy's point. Just to add a little more, people seem to get confused between the expression of mathematical thoughts and having mathematical thoughts. Mathematical symbols are abbreviations for ideas that, if expressed in natural language, would become tedious. Mathematical thinking is something people do everyday. For example, parents have to use combinatorial optimization everyday to figure out a best strategy for running errands. Few parents are schooled to a formal expression of the mental processes they go through in determining a solution but they weigh different goals and optimize nonetheless. Oddly, a study that is all about exploration of ideas and abstraction has been narrowly defined in many people's minds. Hopefully mathematics educators can break these limited views in years to come.
    • potatolicious 18y ago
      Maybe it's my engineering background, but I don't feel like this is semantics - it's unfair to categorize logical thinking as "mathematical" thinking. There are plenty of us from different backgrounds who are big on logic, and IMHO make great programmers. Mathematicians are far from the only ones who can think logically, and the term is at least somewhat misleading.
      • donaq 18y ago
        Uh, logic is a branch of mathematics. From wikipedia: Logic is the study of the principles of valid demonstration and inference. Logic is a branch of philosophy, a part of the classical trivium, as well as a branch of mathematics.
        • carbon8 18y ago
          potatolicious has a point. In fact, in universities, logic (formal logic, with symbols) is usually taught by philosophy departments.
          • donaq 18y ago
            That raises an interesting question, doesn't it? I wonder how many mathematicians turn out to be good programmers vs how many philosophers turn out to be good programmers?
• akikuchi 18y ago
  "if we're going to talk about math, let's get out of the abstract and into the specific. Let's talk details. Examples. What could be more math-y than that?" (from the codinghorror post) This topic seems to come up relatively frequently, and I appreciate Jeff's attempt to add to the conversation with some interestingly selected quotes. Still, the above quote to me highlights how many people, including Jeff, vastly oversimplify the math side of the equation. To put lbrandy's comment slightly differently, knowing a large number of mathematical facts will not make you a good programmer. But to me a basic explanation of “mathematical inclination” could be: a strong understanding of how a mathematical system is governed by rules, and how a problem can be parsed, compartmentalized, and addressed in pieces. It seems fairly self-evident to me that those skills would be shared by anyone whol could be described as even a competent programmer.
• ConradHex 18y ago
  >What's the distance between a point and a line segment? I think I agree with you. I would argue that the mathematical ability needed to solve the question above is to realize how to phrase a query to google, find the resulting formula, and turn it into code. You don't need to be able to derive it, for the vast majority of jobs out there. Also: I've noticed that most programming is equivalent to what in grade school and high school they called a "word problem". Most of the class would groan when we had to do these; I loved them. I suspect if you struggled with word problems, you would have a hard time programming.
  • mechanical_fish 18y ago
    phrase a query to google, find the resulting formula, and turn it into code You left out the part where, months later, you pay a mathematically-inclined consultant hundreds of dollars to fix the bugs in the formula that you cut and pasted without understanding it. ;) Numerical methods was a very interesting class, even on day one. You'd be amazed how easy it is to screw up simple formulas by doing the math in the wrong order -- the finite precision of floating-point arithmetic means that you have to be constantly on your guard.
• jules 18y ago
  >What's the distance between a point and a line segment? Does anyone have an elegant solution?
  • ori_b 18y ago
    Draw a triangle; Do the trig. Between any arbitrary point on a line segment, the point on the line segment that contains the shortest point, and the point in question, you have a triangle. You can take the length between any point on the line segment and the point in space, find it's magnitude squared, and subtract the magnitude squared of it's projection along the line segment. This gives you the distance of between the point and line segment squared by pythagoras' theorem. x /| / | / | -a****----- a is the arbitrary point on the line, the *s are the projected segment, and x is the point you want to find the distance from.
    • jules 18y ago
      I'm not sure, but that sounds like the answer to a different question: what's the distance between a point and a line.
• jacoblyles 18y ago
  On the topic of the applications of mathematical thinking to programming, I immediately thought of this: http://www.zedshaw.com/essays/programmer_stats.html http://www.zedshaw.com/essays/programmer_stats.html
• palish 18y ago
  What's the distance between a point and a line segment? let A, B be the 2D start and end points of a line segment. Let P be a 2D point. We think about the problem and graph the problem like this: P . A ------------------------ B Then we realize that we can re-frame the problem in terms of a right triangle: P . / | / | / _| / | | A ------------------------- B Q Now for some definitions. Let's define the vector operations that we'll be working with. class Vec2( object ): def __init__( self, x, y ): self.x = x self.y = y This represents a 2D vector with properties 'x' and 'y'. def length( V ): return sqrt( V.x*V.x + V.y*V.y ) This computes the length of a 2D vector. def normalize( V ): V_len = length( V ) return Vec2( V.x / V_len, V.y / V_len ) This computes a new vector that points in the same direction as V, but is of unit length ( length( normalize( V ) ) == 1.0 ). def dot( V1, V2 ): return V1.x*V2.x + V1.y*V2.y This computes the "dot product" between two vectors. I'll clarify this operation in a moment. Finally, for clarity and succinctness: V1 + V2 represents Vec2( V1.x+V2.x, V1.y+V2.y ) V1 - V2 represents Vec2( V1.x-V2.x, V1.y-V2.y ) S * V represents Vec2( S*V.x, S*V.y ), which scales the vector V by a factor of S. So (2.0 * V) would result in a vector in the same direction as V, but twice as long. Goal: find the length of PQ First, we examine our graph above and write out the known and unknown quantities. AB = B - A AP = P - A AB_len = length( AB ) AP_len = length( AP ) Q = ? AQ = Q - A PQ = P - A It looks like our first step is to compute Q. A dot product trick The trick we'll use to solve this is via the following rule: For any line that passes through A and B, dot( normalize( B - A ), P - A ) is the distance between A and the closest point on the line to P. Huh? Let me explain this thoroughly, so you can add it to your own toolbox for your entire life. Building an intuitive understanding of vector math Imagine a line segment AB and a point P. Now picture point Q, which is the closest point to P on a line that passes through A and B, just like the graph above. You can use the dot product to find the distance between A and Q. First, compute the vector AB by computing (B - A). You can visualize this as follows. Picture a line segment from point A to point B. Now move it so that A is coincident with the origin (0,0). Since you moved it, you didn't change its length and you didn't change its direction, just its location. So the result of (B - A) could be intuitively described as "a line segment that begins at the origin (0,0) whose length and direction are equal to AB's". The next step in the dot product trick is to set the length of (B - A) to 1.0, which is "unit length". A few related trivia notes: - if a vector's length is exactly 1.0, then it is called a "unit vector". - when you set a vector's length to 1.0, you have just "normalized" the vector. - normalized vectors are a succinct way to represent a direction in space, whether it be 1D, 2D, 3D, or any other D. - remember the 2D line equation "y = mx + b"? 'm' is the line's slope. In 2D space, slope is just an alternate way of representing a vector's direction. You can compute a vector V's slope as "rise over run" (V.y / V.x). The only advantage of using slope to represent a direction in 2D is that it's efficient to compute and to use. There are two big disadvantages. First, it's not immediately clear how we'd compute slope in 3D. But more importantly, you can't represent vertical lines with a slope. A vertical line has no 'run', so (V.x / 0.0) == undefined. In programming terms, this could cause problems. So slope is generally avoided for those reasons. I explained it here because I am hoping that it helps you to understand vectors more intuitively. So, let's enumerate some features of unit length vectors: - they can be used to represent any direction in space. - the range of each component of a unit length vector is never outside of [-1.0 .. 1.0]. - if you compute the dot product between a unit length vector and a point, you have projected the point onto the vector. The result of the dot product is the distance between the origin and the closest point on the vector. -- dot product trick - the dot product of two unit length vectors is equal to the cosine of the angle between them. In other words, let V1 and V2 be unit length vectors. dot( V1, V2 ) == cos( angle between V1 and V2 ). Related facts: -- the dot product of two unit length vectors is always in the range [-1.0 .. 1.0]. -- if V1 points in the same direction as V2 (that is, V1 == V2), then the dot product is 1.0. -- if V1 is perpendicular to V2, then the dot product is 0.0. -- if V1 points in the opposite direction of V2 (that is, V1 == -V2), then the dot product is -1.0. -- And now, for something fun (and totally optional -- if you don't really understand, don't sweat it. But hopefully it will be interesting rather than confusing): Imagine a light bulb at point L. Now imagine a sphere at point S, being lit by that light bulb. Imagine a point on that sphere, P. You can compute the lightbulb's effect on the sphere at that point as follows: L = lightbulb position S = sphere position P = point on sphere Ldir = normalize( L - P ) surface_normal = normalize( P - S ) light_intensity = max( 0.0, dot( surface_normal, Ldir ) ) # that quanitity is called "NdotL" in computer graphics. # you would then multiply that quantity by the light's # "attenuation", which dims the light as it gets further # away. But that's outside the scope of this example. # If you're interested in computer graphics, # the book "Realtime Rendering" is fantastic. Solving the problem, finally. So, let's use our newfound ninja-guru knowledge of vectors and dot products to solve the original problem. After glancing at the above graph again, we remember we need to compute the length of the line segment PQ. One way to solve it is to compute the point Q, then length( Q - P ) is our answer. Getting down to business: def dist_point_to_line( A, B, P ): # represent the line segment AB as a vector. AB = B - A # determine the direction of B relative to A. AB_dir = normalize( AB ) # compute the distance between A and Q using the dot # product trick. The first argument is a unit length # vector. The second argument is a point *relative to # that vector*. AQ_len = dot( AB_dir, P - A ) # Now that we know the length of AQ, we can compute Q. # To do this, think of the following equation as "start # at A; move along the direction AB_dir by AQ_len units; # that position is Q." Q = A + AQ_len * AB_dir # return the length of PQ. return length( Q - P ) I hope the explanation was been illuminating, and not too confusing. If you have any questions, feel free to ask.