Last Book Club

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 \mathbb{R}. 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 f_n(x)= n semicircles of perimeter \frac{\pi}{2^n} (where the first one starts at (0,0) and the last one ends at (1,0). Now each f_n has arclength \frac{\pi}{2}, but the sequence of functions converges to [0,1]. There is an apparent contradiction since the arclengths of the sequence converges to \frac{\pi}{2} and thus $[0,1]$ has “length” \frac{\pi}{2}.

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 d(f,g)=sup_x\big| f(x)-g(x)\big| +\int_0^1\left(\big|f'(x)-g'(x)\big|\right)dx. 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.


Book Club V

This chapter was about what the authors call BMI or Basic Metaphor of Infinity. Apparently most of the rest of the book is about examples of this. I don’t really get what is so revolutionary about it. Most mathematicians that I know already think about infinity in this way, so there is not much for me to write about. I actually found this chapter rather dull and tedious with all the examples.

BMI just says that we can embody infinity despite being finite by metaphorically considering infinity to be iterations of finite things without end. Oh, so looking through this chapter again right now, there are also lots of terms from cog sci and phil language that are defined, but also rather obvious. Like some terms naturally have endings (like jump inherently implies a landing) and some do not (like leaving does not have a termination, it is an open ended verb).

So there are two types of infinity, both subcases of BMI. The one that is conceptualized as something without and, and the one that is the “number” infinity conceptualized as a number greater than any other. Then there are examples…

I will be out of town for the next week and will not be able to update, but hopefully I’ll be done with the book by then and we’ll be able to move on to something a little more exciting.

Math is not formalistic

Technically this is the next book club, but I think it deserves a title in its own right since I was planning on tackling it at some point. I have a friend that is a die hard formalist. He says that once we lay down the formal system of mathematics, everything is completely determined as true or false (ignoring obvious Godelian problems which he acknowledges). We don’t invent math precisely because we have to discover whether or not propositions are true in the formal system.

This argument is extremely difficult to refute. I can’t claim that we can invent a proposition that is false, but can be deduced through proof to be true (what he interprets as my “invention” standpoint). My argument essentially has been that mathematicians could care less about the truth of a proposition. You are missing the point of math if all you want to know is whether or not the Riemann Hypothesis is true. Mathematicians care about the ideas used in proof (invented things), and the method used (also invented). Thus mathematics is invented. Still, I had no hard way of convincing him. He still says that it is discovered since we couldn’t change the result no matter how it was done.

Finally! Lakoff and Nunez provide a good example as to why math cannot be considered purely formal. Basically chapter 7 is about our metaphorical constructions of sets. They basically talk about the fact that most confusion in math is the same term being used in two different ways (does this sound Wittgensteinian to you? All problems are actually just problems in language…). Philosophers call this the fallacy of equivocation. Now Lakoff and Nunez don’t really claim that this is an equivocation, instead they say that there is a definition and there are metaphorical ways to interpret that definition.

Lots of interesting examples in this chapter I wish I could write about, but here is the actual one. We can interpret an ordered pair (a,b) metaphorically in terms of sets as \{\{a\}, \{a,b\}\}. Check for yourself that this is a well-defined metaphor. Also, it is very common to interpret at a foundational level numbers as sets (look this up), but counting starting at zero would be \emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}, etc. We confuse these metaphors to be actual definitions. This is not the case. We actually lose conceptually what we are talking about if we look at it that way.

E.g. What is the ordered pair of numbers (0,1) at this “foundational” level? Well, \{\{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}. What is the set containing 1 and 2 as numbers (i.e. \{1,2\}). Well, exactly the same thing: \{\{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}. These conceptual distinctions exist at a mathematical level (the ordered pair and the unordered set are not the same), but when we take the formalistic metaphor to be the definition we lose that distinction. All formalism is is one way of interpreting things. It is purely a metaphor. We lose concepts (as any metaphor by definition does not preserve everything). Thus, mathematics is not literally reducible to set theory.

Any counters to this? I believe this is my best weapon against formalists, and “discoverers” now.

Book Club IV

So. I didn’t get through as much reading as I thought I would. Two chapters isn’t bad, but nothing more than I would have done. There are a couple of interesting claims here. I’m still completely confused about how they are rejecting Platonism. I think that they are quasi-Platonists at best, maybe full out Platonists at worst. They beginning of chapter 5 talks about forms and essence. I kind of want to skip to the end and read their summary first just to get the big picture.

This is about the essence of algebra. They claim there is a general algebraic essence metaphor. The most interesting part of these two chapters is the claim that algebraic structure is not inherently in what we use like we naively think. The integers under addition does not have the group structure in it. We impose that metaphor onto the set in order to make more sense of it. This seems to make sense. It is just nothing I would have ever guessed or thought of on my own.

Next comes a partial answer to my Platonism question. I don’t feel like explaining the term folk theory (since I’ve used it so much in 20th Century Philosophy). We have a western folk theory of essence in mathematics. This is the reason we feel that we should try to characterize all of mathematics in the fewest number of independent axioms possible. We believe these axioms to describe the forms of mathematics. Then when we do math we are just discovering things about these essences. (Look up these terms as well if you are fuzzy on “form” and “essence”).

It turns out that the folk theory of essences does not describe mathematical cognition. When we talk about rotations of the triangle (cyclic group of order 3), the rotations and group structure are conceptualized independently. The thing that gets us from one to the other is the idea of “linking metaphors” (as opposed to the previously encountered grounding metaphors).

Hmm…I don’t remember coming across anything particularly interesting in chapter 6, which is about Boolean logic. It talks about the Classes as Containers metaphor as a basis for Boolean logic. This was already touched upon though. It just goes one step further to the symbolic-logic mapping that mathematicians use. This is how we get the feeling of “blind manipulation.” All of this traces back to the embodied container schema, though. The closing thought is that no matter how advanced our system of logic becomes, it will never be able to capture true human reason. Thus if we solely use logic for reasoning we will miss some things.

Book Club III

Chapter 4 has lots of repeats. It basically just gives more evidence that the four grounding metaphors are true plus some examples of them. There have been some really interesting new points brought up, though.

1. ERF’s (i.e. Equivalent Result Frames). This is a set of three things: a desired result, essential actions and entities, and a list of alternative ways of performing those actions with those entities to achieve the result. From the embodied mind standpoint, ERF’s are how we get associativity of arithmetic. Thus, according to this theory, associativity is not an axiom, but a consequence. I sort of like this. We’ll see how this stands up later (e.g. when we throw out associativity in non-associative algebras or loops).

2. The four grounding metaphors are isomorphic as mappings over the natural numbers. Well, I’d hope so. Also, the “proof” of isomorphism is a little sketchy. Also, as structures they aren’t isomorphic, since some of them produce things like fractions, irrationals, and negatives while others don’t. This isomorphism is only with respect to how they map to the naturals.

3. Uh. Why does Reuben Hersh endorse this book again? Numbers are things in the world, thus “Since things in the world have an existence independent of human minds, numbers have an existence independent of human minds. Hence, they are not creations of human beings and they and their properties can be ‘discovered.'” Well, I guess that settles the discovered vs invented question. Hersh is a hard-core invented proclaimer, though. Also, I’m not sure how much I buy the “things in the world have an existence independent of human minds.” I guess that depends on how you interpret the necessity of an observer in quantum mechanics.

Also, is this really what embodied mind is saying? Embodied mind is the idea that how we interact with these objects is how we perceive them and interpret them. Isn’t it true that if another species of intelligent beings formed somewhere, and they interacted with objects, in say a non-associative way, associativity would not be a consequence? In the same section we have, “There is only one true arithmetic, since things in the world have determinate properties.” I disagree precisely because of what I just said. There are many true arithmetics. In fact, in physics non-associative structures are rearing their ugly heads. If we had developed such that we saw and interacted with those structures before interacting with macro-structures, non-associativity could be the norm.

4. They are heavily pushing the metaphor idea. Good idea, but do all these metaphors they are referencing exist in all languages? This seems to be an English-centric argument. Maybe a certain person who has studied the cultural influence of mathematics can tell us if there are cultures that didn’t use, say, the motion metaphor. (i.e. 5 is close to 6 since we interpret motion from one natural number to the next as a short path).

5. Extension of subitizing to closure. I don’t remember if I mentioned this. The way we tell the difference between small numbers of things (up to about 4) is to subitize. We know whether something has 1,2,3, or 4 objects instantly. Groups of larger size get harder. We need to group into these smaller groups that we know to figure out how many are there. So this is not a closed operation. 2+3=5. We can subitize 2 and 3, but not necessarily 5. Why is it natural that we extend this to a closed operation? It seems unnatural actually, since the natural innate thing is not closed.

6. Probably the best thing about this chapter is the idea that (I can’t actually find it right now) symbols are important in the sense that the symbol representing the number (or whatever) is not the thing itself. A proper choice of notation is necessary to make calculating and manipulating as easy as possible. This has been my biggest soapbox argument for the past year. Let’s get rid of some of this outdated crap. String diagrams are amazing, box notation is amazing for algebra, categorification is amazingly compact, etc.

Now, don’t get me wrong. I love what these people are trying to do, but I think they aren’t even interpreting their own results properly. Maybe I’ll take my most cogent points from these posts and write a more well-founded and thought out essay to send to them.

Book Club II

Today I’ll do chapters 2 and 3 since neither one had anything completely revolutionary. Chapter 2 reminded me why I love philosophy of language. It talked about how metaphors are subconsciously built into our cognition. This is interesting because we know that we all use metaphors, but this claim is that even at the most basic level we think in metaphors. This is seen in many examples that I won’t go into. It definitely has to do with language, though.

Chapter 2 also talked a little about the neuroscience of math. What parts of the brain are active when people are doing math. Unsurprisingly, when people do actual math (e.g. solve the Riemann Hypothesis) a different part is active than when people are doing arithmetic. This would not be surprising at to professional mathematicians, since most Ph.D. mathematicians would have trouble with 54+72. This cognitive science approach now gives the reason why.

Chapter 3 starts to build the metaphor approach. The idea, I can already tell, is that mathematics is just one gigantic metaphor. So they talk about what they call “grounding metaphors.” These are the metaphors on which mathematics is founded upon. These are opposed to what they call “linking metaphors.” Linking metaphors are what allows us to abstract past the grounding ones. These were not discussed in this chapter.

The four grounding metaphors for arithmetic (and hence not math yet) are “arithmetic as object collection,” “arithmetic as object construction,” “…as a measuring stick,” “…as motion.”

Object collection: Numbers are just cardinalities of physical objects. Cannot be extended easily, but most basic. Can get zero by the empty collection.

Object construction: The collection of two objects is made up of one object and one object. Every number can be broken apart or constructed. We now can get fractions.

Measuring stick: Numbers as lengths. Now we can get irrational numbers (do Pythagorean theorem with a triangle with leg lengths of 1).

Motion: Numbers as distance. Since we can move backward we can get negative numbers.

So basically they talk about how each interpretation yields different things. How each metaphor interprets addition, subtraction, multiplication, and division as separate things. How we naturally combine various versions of the metaphors to get different ways of doing things. How each interprets the number one for example differently. Interesting, but not earth-shattering.

The one thing I didn’t like about these two chapters is that it sort of skimmed over the question that they said they would answer: Is mathematics invented or discovered? The talked about the foundations of mathematics, but sort of miss the actual foundational question. I’ll try to think about which I think they mean based on their “foundation” and post my answer tomorrow.