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A review by stephenmeansme
Math Without Numbers by Milo Beckman
2.0
Milo Beckman set out to show the reading public what real math is, without using numbers. Did he succeed? In some sense, I'm an unreliable reviewer, because I have a university degree in mathematics, so I already have an idea of what real math is. I know very well what Beckman means when he says that at some point, mathematicians stop worrying about or even using "numbers" (that is, quantities represented in Arabic numerals): it's both a joke and also a fact that real mathematicians hate doing arithmetic with numbers greater than, say, 10. And I appreciate the goal of trying to get this across using words and pictures rather than lots of scary notation.
The book is divided into sections for broad areas of the real-mathematics landscape: Topology, Analysis, Algebra, Foundations, and Modeling. Except for Foundations, each section has a few sub-sections. Most of the good stuff is in the first few sections, IMO: Beckman does a nice job of explaining topology's focus on "shape" and leading that into the example of classifying manifolds; he also sketches some good intuition about how the continuum (real line) is used in calculus. The Algebra section starts to suffer from disunity, but the first part about abstraction was pretty good. Modeling has its moments, but no cohesion.
My problems with the book are twofold, possibly describable as "choice of audience" and "choice of mathematics."
First, I started to become unsure who Beckman's target audience was. The selection of mathematics (see below) may or may not be interesting to a high school student, but sufficiently-interested students would probably want a meatier book anyway. He often affects a certain "mystery box" presentation ("There's X, but I can't tell you about X yet") that I found distracting and potentially condescending. It was mildly mysterious.
Second, the choice of what mathematics to present was a bit odd. While the continuum bits of the Analysis section were good, the section overall seemed a bit too trivia-oriented (types of infinities!); similarly with Algebra and Foundations. If the task had fallen to me, I might have gone for more of a themes or tools approach: linearity, invariance, and classification, for example, are three themes that show up across a wide range of mathematical topics. On the tools side, there might be simulation (including probabilistic methods!), induction, and abstraction, these last two being in the book already. These cross-threads could give the book a better sense of unity, and properly conveyed, could give close readers a sense of accomplishment for spotting the connections early.
As a minor point, the Foundations section seemed the most gratuitous, mostly because most "real" mathematicians don't care about foundations, but also because it's structured as a Hofstadterian dialog about the nature of mathematical truth. It spends some pages on a back-and-forth about the history of mathematics as a knowledge enterprise and its association for a few centuries with European imperialism. I'm not going to freak out like some other reviews and say "warrgarble wokeism," but to bring it up, only as shallowly as it was done here, seems more like flag-waving than enlightening or educating. (The other payoff is passing mention of Goedel's theorems, for which GOEDEL, ESCHER, BACH exists but also is something I think is better left to experts.) The Modeling section features other odd digressions, as when Beckman speculates about physics being like an automata (other examples: Conway's "game of life"), which is either naive-realist or Tegmarkian and I don't like either option. Science can use math as a description language but science isn't math!
2.5 stars rounded down. Nice try?
The book is divided into sections for broad areas of the real-mathematics landscape: Topology, Analysis, Algebra, Foundations, and Modeling. Except for Foundations, each section has a few sub-sections. Most of the good stuff is in the first few sections, IMO: Beckman does a nice job of explaining topology's focus on "shape" and leading that into the example of classifying manifolds; he also sketches some good intuition about how the continuum (real line) is used in calculus. The Algebra section starts to suffer from disunity, but the first part about abstraction was pretty good. Modeling has its moments, but no cohesion.
My problems with the book are twofold, possibly describable as "choice of audience" and "choice of mathematics."
First, I started to become unsure who Beckman's target audience was. The selection of mathematics (see below) may or may not be interesting to a high school student, but sufficiently-interested students would probably want a meatier book anyway. He often affects a certain "mystery box" presentation ("There's X, but I can't tell you about X yet") that I found distracting and potentially condescending. It was mildly mysterious.
Second, the choice of what mathematics to present was a bit odd. While the continuum bits of the Analysis section were good, the section overall seemed a bit too trivia-oriented (types of infinities!); similarly with Algebra and Foundations. If the task had fallen to me, I might have gone for more of a themes or tools approach: linearity, invariance, and classification, for example, are three themes that show up across a wide range of mathematical topics. On the tools side, there might be simulation (including probabilistic methods!), induction, and abstraction, these last two being in the book already. These cross-threads could give the book a better sense of unity, and properly conveyed, could give close readers a sense of accomplishment for spotting the connections early.
As a minor point, the Foundations section seemed the most gratuitous, mostly because most "real" mathematicians don't care about foundations, but also because it's structured as a Hofstadterian dialog about the nature of mathematical truth. It spends some pages on a back-and-forth about the history of mathematics as a knowledge enterprise and its association for a few centuries with European imperialism. I'm not going to freak out like some other reviews and say "warrgarble wokeism," but to bring it up, only as shallowly as it was done here, seems more like flag-waving than enlightening or educating. (The other payoff is passing mention of Goedel's theorems, for which GOEDEL, ESCHER, BACH exists but also is something I think is better left to experts.) The Modeling section features other odd digressions, as when Beckman speculates about physics being like an automata (other examples: Conway's "game of life"), which is either naive-realist or Tegmarkian and I don't like either option. Science can use math as a description language but science isn't math!
2.5 stars rounded down. Nice try?