Yes, but the axioms of set theory are independent of the mind who’s come up with them – different mathematical communities might come up with different axiomatic systems, but the mathematical truths are one and the same.

Models in natural science are, however, dependent on the context in which they’re created – the technology available at the time, the “paradigms” of science (in Kuhn’s sense), what’s considered acceptable for scientific study, etc. Pure mathematics lies at one level above this and I don’t see how someone can reasonably claim that it is invented.

]]>Informal surveys suggest most working mathematicians are Platonists, at least with respect to mathematics itself, but there are many mathematicians and philosophers of mathematics who are not and there are philosophical approaches to mathematics that do not require the assumption that mathematical results are true in and of themselves (e.g. it could be that mathematical truths are constructed rather than discovered). At best this is still an open question.

]]>It goes so *well* with this gentleman in a top hat, this flower, this… err… woman pulling her hair?, and this dancer!

You can’t tell me that those perfect match-ups are just coincidence!

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]]>What counts as a round number is as arbitrarily defined as how many licks it takes to get to the center of a tootsie pop.

We just happen to almost always use base 10. It’s our arbitrarily chosen number system, and it’s not special. We just happen to have ten fingers, so it only seemed natural to base our numbers off that.

Edit:

Pardon me, φ in base φ is 10.

That’s when n++ crashed. which is hilarious. It can search through a 10000000+ character file to find and snip out formatting, but looking for a six-character string crashes it.

Anyhow, threw the file into Vim and had a look,

Very clever, the sequence actually repeats four times “24242424″ and starts at the 242423rd character including “3.”

Unless I’m totally wrong. In that case please correct me.

]]>At least in terms of some meanings of the words ”size” and “set.”

]]>]]>Phi is sometimes called “the most irrational number,” meaning that it is the hardest to approximate with a ratio of rational numbers.

“Out of all of the infinitude of numbers in the world” (which you said, basically)

“Out of the infinity of numbers in the world” (which is accurate but sounds a little clunky to me)

“Out of all the countless numbers in the world” (I like that one because it invokes uncountability)

“Out of all of the infinite number of numbers in the world” (which has a nice rhythm to it)

I might even recast it to remove the leading “Out”.

So yeah, I was probably being overly critical to call it an “error”. A “poor phrasing” is better. I blame the copyeditor.

]]>Euclid had a construction for the A:B::B:A+B ratio ratio pretty early in The Elements. Comes up again in the double-angle isosceles triangle, which is the basis for constructing the pentagon. The diagonals of a pentagon cross each other in a golden ratio. The ratio also comes up in the perfect solids a few times.

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