This is a book well worth reading if you can cut the authors a little slack. Having read it, I can now for the first time understand Euler's famous equation, e to the pie-eyed = -1. Wow! Euler wasn't just lubricating!
A second or third edition could be very valuable as a textbook; what L&N have written is a great start, but there are a few rough spots.
To paraphrase Ogden Nash,
One thing that Cognitive Science would be greatly the better for
Would be a more restricted (and more appropriate) employment of simile and metaphor.*
Consider what is perhaps the metaphor most frequently quoted in WMCF, "Numbers Are Points on a Line."
Some subsets of the numbers, including some infinite subsets (e.g. the 'real' numbers, or the 'pure imaginary' numbers {0 + xi}) can be mapped one-to-one onto points of a Euclidian straight line in such a manner that the directed distance from the point designated as zero [call it p(0)] to the point designated as x [call it p(x)] is proportional to the number x, and such a mapping can be very useful. Fine, use it. But why let yourself get so confused as to think that numbers are points. Points are points and numbers are numbers; points and numbers are two different concepts, and 'all you are entitled to say at the very most'* is that a particular subset of the numbers is, in some very useful respects, like the points on a line, which is a simile, not a metaphor. In general, I think similes would be more appropriate than metaphors in math. Sometimes, I think even a simile is a stretch.
Near the top of p. 199: "The point of this is . . . to comprehend how we understand it."
Is there only one way that anyone can understand something? Does every human mind work the same way?
On p. 424: "-n and n are symmetrical points relative to the origin (zero)
"This symmetry is conceptualized in terms of mental rotation . . . . Cognitively, we visualize the relationship between the positive and negative numbers using a rotational transformation . . . ." When dealing with the complex plane, this is clearly the better way to do it, but when dealing with 'real' numbers only, I have always thought of it as sliding a sort of mental tape measure along the number line until the point on the tape that was at n is at zero, and then the point that was at zero will be at -n. And lo and behold, for the 'real number line' this works just as well as the rotation, no better, no worse.
On pp. 218-220 L&N discuss ordinal numbers. Another use besides cardinal and ordinal is nominal, using a number as simply a name for something, e.g. Joseph Heller's wonderful book
Catch-22
. There were no catches 1 thru 21; 22 was just part of the name for that particular catch.
A few other items I noticed:
Top of p. 86: "Moreover, we don't use binary notation, even though computers do, because our ten fingers make it easier for us to use base 10." Actually, the main reason we seldom use binary notation is because it is cumbersome. Our ten digits (8 fingers, 2 thumbs) led us rather naturally to base 10, but octal and hexadecimal are used quite a bit, and it is actually easier to keep from losing your place counting on your fingers in binary, in which 10 digits allow you to count from 0 to 31 on one hand, or to 1023 using both hands
p. 140: The SIMILE Classes Are Like Containers sort of breaks down for me with the Venn diagram of two overlapping circles for two classes that have some members in common but neither is a subclass of the other.
p. 331: "For example, we take the curvature at a point in a curve as being PART OF THE CURVE. (italics in original; I had to substitute CAPS because Amazon's text box doesn't permit italics.) I take the curvature to be a characteristic of part of the curve, not a part of the curve. And a tangent is most certainly not part of a curve.
p. 332: "A Function Is a Point in a Space" ???
p. 337: "What COMMONPLACE cognitive mechanisms do they use?" (emphasis added) You aren't even interested in any unusual cognitive functions they may use?
p. 347: ". . . every bit of thinking we do must be carried out by neural mechanisms of EXACTLY the right structure to carry out that form of thought." Our brains haven't the flexibility to 'make do' to the slightest extent?
p. 349: "The Pythagorean theorem hasn't changed in twenty-five hundred years and, we think, won't in the future." But what we know about it has changed considerably. I have discovered several facts that, so far as I have been able to find out, were not previously known, e.g. a set of 3 equations in r (row #) and k (column #) which define an infinity by infinity matrix of Pythagorean triangles, each row and each column of which is a family with a or b in arithmetic progression, and c - a or c - b constant:
a(r,k) = 4rk + 2k(k-1)
b(r,k) = 4r(r+k-1) - 2k + 1
c(r,k) = 4r(r+k-1) + 2k(k-1) + 1
p. 360: ". . . the form of arithmetic used in all computers." There is no such thing as THE form of arithmetic used in all computers, Many computers use two's complement binary arithmetic, but the Control Data 6600 and 7600 used one's complement binary arithmetic. The IBM 1401, 1440, and 1620 used binary coded decimal arithmetic. All computers I have ever programmed had integer arithmetic, and most, but not all, had floating point.
p. 361: "All of that arithmetic used floating-point, not standard arithmetic." Simply not true. See just above.
* Ogden Nash: Very Like a Whale.
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Top reviews from other countries
Brian Aicheler
I wish this book had been available when I was a graduate.
Reviewed in the United Kingdom on March 6, 2020Manuel Gonzalez
Brillante integración disciplinar
Reviewed in Spain on January 16, 2016
