cf. https://en.wikipedia.org/wiki/Divine_Proportions:_Rational_T...
An ultrafinitist is still allowed to call that 'i'.
After all, the Cayley-Dickson construction is not an infinite affair.
But it's because the sine of 60 degrees is said by modern tables to be equal to sqrt(3) / 2, which Wildberger doesn't "believe in", he prefers to state that the square of the sine is actually 3 / 4 and that this is "more accurate".
The actual paper is at [1]:
Personally I don't believe in either value. I prefer to state that the sine of 60 degrees is 2.7773. I believe that is more accurate.
The news from this paper (thanks for the link!) is that evidently the Babylonians preferred that, too. Surely Pythagoras would have.
But how do you actually do anything useful with this ratio ¾? Like, calculating the height of a ziggurat of a given size whose sides are 60° above the horizontal? Well, that one in particular is pretty obvious: it's just the Pythagorean theorem, which lets you do the math precisely, without any error, and then at the end you can approximate a linear result by looking up the square root of the "quadrance" in a table of square roots, which the Babylonians are already known for tabulating.
For more elaborate problems, well, Wildberger wrote the book on that. Presumably the Babylonians had books on it too.
Re: rationals, I mean there's an infinite number of rationals available arbitrarily near any other rational, that has to mean they are good enough for all practical purposes, right?
https://scispace.com/pdf/words-and-pictures-new-light-on-pli...
Robson's argument is that it isn't a trig table in the modern sense and was probably constructed as a teacher's aide for completing-the-square problems that show up in Babylonian mathematics. Other examples of teaching-related tablets are known to exist.
On a quick scan, it looks like the Wildberger paper cites Robson's and accepts the relation to the completing-the-square problem, but argues that the tablet's numbers are too complex to have been practical for teaching.
To defend Wildberger a bit (because I am an ultrafinitist) I'd like to state first that Wildberger has poor personal PR ability.
Now, as programmers here, you are all natural ultrafinitists as you work with finite quantities (computer systems) and use numerical methods to accurately approximate real numbers.
An ultrafinitist says that that's really all there is to it. The extra axiomatic fluff about infinities existing are logically unnecessary to do all the heavy lifting of the math that we are familiar with. Wildberger's point (and the point of all ultrafinitist claims) is that it's an intellectual and pedagogical disservice to teach and speak of, e.g. Real Numbers, as if they're actually involving infinite quantities that you can never fully specify. We are always going to have to confront the numerical methods part, so it's better to make teaching about numbers methodologically aligned with how we actually measure and use them.
I have personally been working on building various finite equivalents to familiar math. I recommend anyone to read Radically Elementary Probability Theory by Nelson to get a better sense of how to do finite math, at least at the theoretical level. Once again, on a practical level to do with directly computing quantities, we've only ever done finite math.
We use numbers in compact decimal approximations for convenience. Repeated rational series are cumbersome without an electronic computer and useless for everyday life.
The point is to not confuse the notational convenience with the underlying concept that makes such numbers comprehensible in the first place.
I can tell you that it is the output of a function, not a distinct entity that exists on its own independently of the computation.
The whole point is that as a theory for the foundations of mathematics, you do not need to assume numbers with infinitely long decimal expansions in order to do math.
As long as someone isn't a crank (e.g. they aren't creating false proofs) I enjoy the occasional outsider.
A little off-topic, but as a non native English speaker this sentence in the article made me look up whether there’s scientific consensus that Noah’s Ark has been found and I’d just never heard about it. Turns out there isn’t, and the end of the sentence actually refers to the tablet. Was still a fun rabbit hole to go down.
nyc111•6h ago