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 metaphorically in terms of sets as . 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 , 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 at this “foundational” level? Well, . What is the set containing 1 and 2 as numbers (i.e. ). Well, exactly the same thing: . 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.