Jeez Louise, it appears no two mathematicians agree on the definition of mathematics. KM: In my view, mathematics is definitely *not* the study of thinking. There are huge areas in cognitive psychology for that and mathematics exists irrespective of one's level of thinking (yes, my philosophy is showing). I lean toward the view that math is the study of structure, patterns, and form that arise as a consequence of rule sets. The things that Warren listed that *seem* formless are not actually. Chaos has form; fluid turbulence has structure. Even truly random events & sets have structure that can be studied mathematically. --WDS: well then, is there *anything* you would call structureless? MS: Topology started with the bridges of Konigsberg, Euler's formula V-E+F=2, and Lhuilier's generalization to polyhedra with higher genus V-E+F=2-2g. --WDS: well, I would agree that numbers are very useful and almost unavoidable. But they are avoidable sometimes, and math sometimes proves statements that can be expressed without numbers. Thus proving: numbers are not the litmus test. JP: Graduate students in the humanities are learning how to think more intelligently in their chosen specialties. The same goes for lawyers. But they're not doing mathematics. --WDS: Well, as far as I've been able to tell, law does not involve thinking, it involves "lying" and "manipulating"; in particular, virtually everything you read about in a law book is false. Well, let me give you a few examples from my experiences in traffic court. I always fight traffic tickets. So far, I always lost the case. Anyhow, in court #1, the judge's first move was to tell me I wasn't allowed to wear what I was wearing, which was, jeans & a jacket. I mean, literally, his very first move, before trial even started. So, evidently, his personal fashion sensibilities trumped the very highest law of the land, the 1st amendment to the US constitution, which states that even if the US congress tried to make a law about what I was allowed to wear in court, that law would be invalid. Shortly thereafter, I presented two written motions to the judge, following the advice of a law book, which was to make your motions be written, citing verbatim law, etc etc. What happened then was not in my law book, which was: the judge refused to even read the motions. He refused even to look at them. Instead he told me to explain them orally, and immediately dismissed them after 1.5 sentences for motion #1, and 0.5 sentences for motion #2. In court #2, the judge opened ceremonies with a long long speech about the single most important principle, more important that any law in the world, even than the constitution (or so he said, for about 10 minutes on end) which was the principle that everybody is presumed innocent until proven guilty. So then after many many hours he got to my case. "How do you plead?" he said. "Innocent" I said. "You aren't allowed to plead innocent" said the judge. "If you wanted to do that, you had to do it in writing on a special form, not here at your trial." If there is any actual logic and fairness and thinking going on in the law, I do not believe it happens at a low budget level. I believe the written law is simply ignored until you pay at least $5000. It's basically an extortion scheme. But anyhow, re the humanities, the law, etc, I disagree with Propp in a more fundamental respect than that. Sure, people being educated in any field are learning how to think better. But, so what? That is not relevant: those fields do not help humanity as a whole to improve its thinking. I contend that humanity is presently much better at thinking than it was 1000 years ago. This is due to the development of math, science, technology. But is humanity any better at legal thinking than it was 1000 years ago? I would say: no. I think there has been zero improvement there. If you took a lawyer from today and time-transported him back 1000 years, would he be any better at it than a lawyer of that time? No. What about humanities generally? Well, we have more convenient access to information today, which helps, but if that factor is removed, then it seems to me that the only improvements in humanities thinking that happened during the last 1000 years, are (what I would call) mathematical. And as far as I can see, every mathematical contribution ever, has helped humanity to be better thinkers. (If not, then I claim it doesn't deserve to be called mathematics...) What does "better" mean? Means able to do more. Superset. And re JP's own suggestion that maths is the study of tautologies, how about if I do a computational study of approximation algorithm #1 vs #2, finding #1 seems to work better. I have no idea why, but 99% of the examples I try, it works better. Seems to me, that would be a perfectly legitimate and valuable mathematical contribution, and it would improve our thinking, but there is no theorem or obvious truth there at all. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)