Not having internet has been annoying since so much has been discussed in this book, and I wanted to get my ideas out before I forgot them. I tried jotting notes in the margins, but this post won’t do it justice. This will conclude the material of the book. There will still be a few more discussing implications that I’ve thought of while synthesizing the material. Here goes:
I’m still on the fence about the novelty of the idea of the BMI. This is the case even after many examples. It just seems that mathematicians already know they are doing this. Luckily, I am completely reverse of the reviews of this book. I think the first half was so-so, but it starts to kick out some great ideas in the latter half.
Chapter 9: Real Numbers and Limits. Some interesting commentary on why epsilon-delta definitions are difficult for students. Limits are conceptualized using the motion along a line metaphor for numbers, yet the definition is very static. It also accounts for ridiculous cases of epsilon (who cares about large epsilon? it is only the small epsilon that matter in limits). This is mathematically a necessity, but conceptually confusing. I agree. On to sums. Hmm…still not all that interesting. Just examples of the use of the BMI.
Chapter 10: Transfinite numbers. They talk about Cantor’s diagonal argument and some of the assumptions. The proof is usually taken to be formal, but in actuality it cannot be written down formally because you can’t express infinity as an actual entity. This means that the fact that there are more reals than rationals is inherently metaphorical. It is also discussed that Cantor’s one-to-one and onto definition for equivalent infinities is metaphorical and not absolute. It is one way to count infinities and see if they are the same. It is not the only way. We often lose sight of this, or not even realize it.
Note: Getting the picture yet? They are building this idea of metaphor to a pretty interesting climax.
Chapter 11: Infinitesimals. This was probably my favorite chapter because I had never seen a construction of the hyperreal numbers. They first builds what they call the granular numbers. This is essentially just the first “layer” of the hyperreals. You get the interesting result that I didn’t know about that there are number systems with no possible system of numerals (because it would need an infinite alphabet to express). This also brings up the concept that there are mathematical objects that are inherently metaphorical (since they can’t be expressed otherwise). I think that they think the most important part of this chapter is that “ignoring certain differences is absolutely vital to mathematics.” This refutes the idea of mathematics being perfect, exact, absolute, … , i.e. Platonistic. Yea! Finally, the big one comes out. This argument is much longer, but basically boils down to “calculus is defined by ignoring infinitely small differences.”
Chapter 12: Points and the Continuum. What to say about this… Basically it goes through the struggle of how to define a point. Do points on the real line touch? If they do, then by definition of having no length they are the same point. So they can’t touch. But the real line is continuous, i.e. there are no gaps. Thus points much touch. A paradox? Actually they break this down as a problem of blending two metaphors for talking about . This shows that when we talk about things as absolute truth, we may actually be referring to a metaphor which doesn’t exactly work in every situation. We must be careful what metaphor we are using and how it affects what we are talking about. Also the problem of attempting to discretize (write down mathematics in a precise and logical manner from axioms) the continuous is discussed. From a conceptual point of view this is impossible. In fact, it really hasn’t been insanely successful.
Chapter 13: Continuity for Numbers: The Triumph of Dedekind’s Metaphors. This talks about Dedekind cuts. Blah.
Chapter 14: Calculus Without Space or Motion: Weierstrass’s Metaphorical Masterpiece. This talks about how the geometric interpretation/metaphor for calculus had major limitations. There were functions that had nothing to do with motion. It talks about how Weierstrass extended calculus to work in these situations. Here again is the “choice of metaphor argument.”
Le Trou Normand. Here they give us the kicker. I’m going to do this in a more concise way. Construct the sequence of functions semicircles of perimeter (where the first one starts at and the last one ends at . Now each has arclength , but the sequence of functions converges to [0,1]. There is an apparent contradiction since the arclengths of the sequence converges to and thus $[0,1]$ has “length” .
The problem is the same as before. Our choice of metaphor is incorrect. We can’t say that the limit of the length of a pointwise convergent sequence of functions is the length of the limit under our current metaphor. But we can define a new metaphor in which this works. This is a common metaphor to use in functional analysis. Construct a function space in which our distance is . You can work this stuff out for yourself to see how it works.
Moral: Our choice of metaphor matters! Down with Platonism! We can’t treat functions as literally being curves in the plane or the motion of a particle. While these are useful metaphors at times, they should not be taken as literal objective representations that give us all the information and no excess incorrect information (careful on all the negatives I stuck in there).
Tomorrow: Philosophical implications.