"What is Mathematics ?
Update: Welcome Econmistsviewers ? I am amazed you are here. This is, I think, a once in a lifetime opportunity to type a joke about Hendrik Hertzberg and the Georgia State Legislature. Huh ?
the Georgia State Senate has adopted a resolution allowing the state to nullify any federal laws it thinks are unconstitutional. Hendrik Hertzberg actually read the resolution, and wrote a post that made me want to read it as well: he described it as "a Kompletely Krazy Kocktail of militia-minded moonshine and wacko white lightning ... [and that the] resolution is written in "a mock eighteenth-century style, ornate and pompous", I thought it was an unnervingly good imitation of eighteenth-century prose.
[skip]"That to this compact each State acceded as a State, and is an integral party, its co-States forming, as to itself, the other party:..."
I googled a distinctive phrase, and lo! it turns out that the Georgia resolution is a lightly modified version of Thomas Jefferson's Resolutions Related To The Alien And Sedition Acts.
Odd. I always thought that the tern "co-state" had more to do with Hamiltonian than with Jeffersonian constitutional exegesis.
Oh and Mr Hertzberg, I love you, but you have to learn to use the google.
Mark Kleiman comments
the "formalist" view of mathematics, as summed up by Bertrand Russell: "Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true." That remains the dominant view among philosphers of mathematics, but the disconnect between the frailty of the wand and its evident power lends some support to the alternative view of Lakoff and Núñez that mathematics starts from basic human experience.
I must confess that my own views remain primitively Platonic; that is, I belive that it was the case that a group consisting of two pairs of trees contained four trees, even before there were humans around to notice that fact.
I think that Kleiman (like Plato) confuses mathematics and physics. I have no doubt that there are laws of physics and that they existed before humans did. For example, I'm sure that two trees plus two trees made four trees back when our ancestors were climbing in those trees.
However, the claim that two plust two must always equal four in general and not just for trees can be false. Consider a taught string. Wiggling it at one point with a given force will make a wave. Wiggling it somewhere else will create a wave. Wiggling it both places may create 0 waves. It also may create a wave with power 4 times that of each of the single waves. With waves 1+1 makes somewhere from 0 to 4.
The claim that massive objects must be like trees and not like waves on a bowed string must be a claim in phyisics and not in mathematics, because it is false. Arithmetic is just the same now as it was in 1900, however, now it is clear that Kleiman's statement about trees is a falsifiable hypothesis which happens to be true and not a necessary truth.
The title of Kleiman's post is "Yet Bridges Stand," but not all bridges stand. For example, the Tacoma Narrows Bridge was blown down by the wind.
How odd, a gust of wind is very light and the bridge was very heavy and strong. One could hardly notice the effect of even a very powerful gust of wind. However, when it comes to bridges, one plus one doesn't always equal two, it can be more than 2 if the wind is in harmony with the bridge.
update: Bruce Webb has video of destructive harmony.
I agree with Russell on this one. Mathematics concerns axiom systems. Mathematical terms are defined in use. The Euclidean term "line" can be approximately illustrated by something drawn with a pencil and a straight edge. It can also be equally validly illustrated by quite different sets which would be called curves in plain English. Neither illustration is more correct than the other. The meaning of the term is abstract, there is nothing except for the axioms and that means not enough for us to imagine it. For mathematics to be valid, it is not at all necessary for any entity which can fit in our universe to correspond to the axioms.
The hypotheses that sets of axioms can be intrepreted so that they correspond to physical laws are often very fruitful. However, they belong to physics not mathematics.
No doubt Lakoff and Núñez are right that, in history, many axioms began as hypotheses. I don't doubt that Russell agrees. This does not mean that they remained hypotheses, or that mathematicians consider, or should consider, their work vulnerable to evidence. Critically not all axiom systems began as hypotheses. The key example, of course, is Lobachevskian geometry (Euclid's axioms except that given a line A and a point B there is more than one line through B parallel to A). This began as an effort to prove by contradiction that no such axiom system was possible.
What is the basis in human experience for p-adic analysis ?