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.

May 27, 2008 at 3:38 pm

I asked my formalistic professor about that example, and he said:

“If you think = {1,2} is absurd I’d have thought that was more of a problem for the (von Neumann) set-theoretic reduction of numbers (and ordered pairs) than for formalism. There is a consistent formal system which consists of PA arithmetic (with quantifiers appropriately restricted to Nx for numbers) with numerical terms like zero and successor primitive combined with ZF set theory (quantifiers appropriately restricted to Sx for sets) plus the axiom that ~(Ex)(Sx & Nx) and that formal system does justice to intuitions that the above identity is false. For me it’s true in that system that, for example, 2 is distinct from {{ }, {{ }}},

and I recognise no mathematical truth independent of truth in a formal system.”

May 27, 2008 at 3:41 pm

…oops, I lost part of that cutting and pasting. It should have begun “If you think = {1, 2} is absurd…”

Personally, I liked your example, which does use the most popular formal definitions within mathematics. And I’m unconvinced that there wouldn’t be a similar problem for the system mentioned. At the very least, it would not be able to say why different elements of different systems were all called “2,” what the justification for so-calling them was, which would seem to be intimately connected to applications, to ordinary language.

May 27, 2008 at 3:45 pm

…double oops, I see the problem, I can’t use pointy brackets! So it should’ve begun “If you think (0, 1) = {1, 2} is absurd…” I had originally used round brackets, as they are standard for ordered pairs in mathematics (e.g. for Cartesian coordinates), but I guess that logicians use the pointy brackets. I’ve noticed that they are also used in algebra, for defining structures. I guess there is a close relationship between algebra and formalism. I’ve no idea why they don’t use the round brackets though. There is always a shortage of brackets for other collections, e.g. atomic mereological fusions.

May 27, 2008 at 3:57 pm

Incidentally (sorry for the FOUR comments) I wonder if there isn’t a regress problem for formalism. I mean, its meaningless symbols are just that, meaningless unless they are interpretted. Even when the formal system specifies the background logic, what makes those strings of symbols a background logic? At its most basic, what makes “x” the same as “x”? It is as you say, a problem in language.

Having said that, I’m finding it hard to tell the difference between Lakoff and formalism. We invent the formal structure, and then discover truths about it, given only it. The way our brains are guides us towards certain structures, e.g. Heyting arithmetic, or von Neumann’s. And those have to be specified as formally as possible, because that’s how our brains tell us to do mathematics. (Having said that, I should admit that I’m a theistic Platonist, just trying to understand the enemy!)

May 27, 2008 at 5:01 pm

Lakoff would definitely say that formalism is one metaphor for looking at problems and proofs, but it has limitations like all other metaphors. I think the statement “I recognize no truth independent of truth in a formal system” is interesting, because it sort of touches on a very fine point.

Mathematical truth vs the nature of mathematics. Prove that 1+1=2 in a purely formalistic way, and 90% of mathematicians will not recognize it as truth. This is because we are using the wrong metaphor for that instance. There are lots of different formal systems, why is that particular one the chosen one?

Try to convert the proof of Fermat’s Last Theorem or the Poincare conjecture to purely formalistic terms. I’m not sure it can be done (literally it would probably take more space than exists in the universe thus being impossible). Does this mean it is not a proof?

OK. I think I’m not creating a clear argument going back and forth on formalism here. The main point I guess is that Lakoff is definitely not a formalist. He believes that formal systems sometimes are a useful metaphor to use, but often times you lose the ideas or distinctions which are the important part.

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December 19, 2009 at 6:34 am

Heinz von Forester once said that it is a choice we make whether to understand the world as invented or discovered, and it is totally fine providing we discover inventors or invent discoverers (depending on which side we stand on).

I’d prefer to make a combination of the two views, something along the idea of “justified formal systems”. In that sense everything is true in a formal system of some sort, and the formal system is able to embody truth by it’s similarities to reality, how it maps onto it or something.

Something like that. Of course the minimal formal system is a single axiom, stating whatever you say is true!