It's probably worth adding the context that Wildberger's agenda is to ground mathematics in integers and rational numbers, eliminating those pesky irrationals Euclid introduced, because reasoning about them invariably involves infinities or universal quantifiers, which everyone agrees are tricky and error-prone, even if they don't agree with Wildberger's radical variety of finitism. So he was delighted to find a kindred spirit millennia ago in the Plimpton 322 scribe and, presumably, the entire Babylonian mathematical tradition.
Thanks for the context - I was baffled at first how the Guardian would run with the tagline "a trignometric table more accurate than any".
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".
Thank you for this expansion. I was about to rabbit hole on how it could be that ratio-based trig (and what is that?) is more accurate than modern calculations.
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?
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.
I like to imagine at the end of the human race when the sun explodes or whatever, some angelic being will tally up all the numbers ever used by humans and confirm that there are only finitely many of them. Then they'll chalk a tally on the scoreboard in favor of the ultrafinitists.
As long as someone isn't a crank (e.g. they aren't creating false proofs) I enjoy the occasional outsider.
An eventual output of a calculation has to be a finite result, but the concepts that we use to get there are often not.
The standard way of setting up calculus involves continous magnitudes, hence irrational quantities, and obviously that's used all over physics and there doesn't seem to be a problem with it.
I think to make a compelling case for a finitist foundation for maths you would at the least have to construct all of the physically useful maths on a finitist basis.
Even if you did that, you should show somehwere this finitist foundation disagrees with the results obtained by the standard foundation, otherwise there's no reason to think the standard foundation is in error.
Topology, i.e. the analysis of connectivity, is built upon the notion of continuity and infinite divisibility, which seems to be difficult to handle in an ultrafinitist way.
Topology is an exceedingly important branch of mathematics, not only theoretically (I consider some of the results of topology as very beautiful) but also practically, as a great part of the engineering design work is for solving problems where only the topology matters, not the geometry, e.g. in electronic schematics design work.
So I would consider any framework for mathematics that does not handle well topology as incomplete and unusable.
Ultrafinitist theories may be interesting to study as an alternative, but the reality is that infinitesimal calculus in its modern rigorous form does not need any alternatives, because it works well enough and until now I have not seen alternatives that are simpler, but only alternatives that are more complicated, without benefits sufficient to justify that.
I also wonder what ultrafinitists do about projective geometry and inversive geometry.
I consider projective geometry as one of the most beautiful parts of mathematics. When I encountered it for the first time when very young, it was quite a revelation, due to the unification that it allows for various concepts that are distinct in classic geometry. The projective geometry is based on completing the affine spaces with various kinds of subspaces located at an "infinite" distance.
Without handling infinities, and without visualizing how various curves located at infinity look like (as parts of surfaces that can be seen at finite distances), projective geometry would become very hard to understand, even if one would duplicate its algorithms while avoiding the names related to "infinity".
Similarly for inversive geometry, where the affine spaces are completed with points located at "inifinity".
Such geometries are beautiful and very useful, so I would not consider as usable a variant of mathematics where they are not included.
> numbers methodologically aligned with how we actually measure and use them.
We use numbers in compact decimal approximations for convenience. Repeated rational series are cumbersome without an electronic computer and useless for everyday life.
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.
>”He bought it from Edgar Banks, a diplomat, antiquities dealer and flamboyant amateur archaeologist said to have inspired the character of Indiana Jones – his feats included climbing Mount Ararat in an unsuccessful attempt to find Noah’s Ark – who had excavated it in southern Iraq in the early 20th century.”
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.
kragen|6 months ago
cf. https://en.wikipedia.org/wiki/Divine_Proportions:_Rational_T...
LeifCarrotson|6 months ago
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]:
[1] https://doi.org/10.1016/j.hm.2017.08.001
vessenes|6 months ago
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?
7thaccount|6 months ago
numpy-thagoras|6 months ago
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.
ants_everywhere|6 months ago
As long as someone isn't a crank (e.g. they aren't creating false proofs) I enjoy the occasional outsider.
griffzhowl|6 months ago
The standard way of setting up calculus involves continous magnitudes, hence irrational quantities, and obviously that's used all over physics and there doesn't seem to be a problem with it.
I think to make a compelling case for a finitist foundation for maths you would at the least have to construct all of the physically useful maths on a finitist basis.
Even if you did that, you should show somehwere this finitist foundation disagrees with the results obtained by the standard foundation, otherwise there's no reason to think the standard foundation is in error.
dr_dshiv|6 months ago
adrian_b|6 months ago
Topology, i.e. the analysis of connectivity, is built upon the notion of continuity and infinite divisibility, which seems to be difficult to handle in an ultrafinitist way.
Topology is an exceedingly important branch of mathematics, not only theoretically (I consider some of the results of topology as very beautiful) but also practically, as a great part of the engineering design work is for solving problems where only the topology matters, not the geometry, e.g. in electronic schematics design work.
So I would consider any framework for mathematics that does not handle well topology as incomplete and unusable.
Ultrafinitist theories may be interesting to study as an alternative, but the reality is that infinitesimal calculus in its modern rigorous form does not need any alternatives, because it works well enough and until now I have not seen alternatives that are simpler, but only alternatives that are more complicated, without benefits sufficient to justify that.
I also wonder what ultrafinitists do about projective geometry and inversive geometry.
I consider projective geometry as one of the most beautiful parts of mathematics. When I encountered it for the first time when very young, it was quite a revelation, due to the unification that it allows for various concepts that are distinct in classic geometry. The projective geometry is based on completing the affine spaces with various kinds of subspaces located at an "infinite" distance.
Without handling infinities, and without visualizing how various curves located at infinity look like (as parts of surfaces that can be seen at finite distances), projective geometry would become very hard to understand, even if one would duplicate its algorithms while avoiding the names related to "infinity".
Similarly for inversive geometry, where the affine spaces are completed with points located at "inifinity".
Such geometries are beautiful and very useful, so I would not consider as usable a variant of mathematics where they are not included.
kevin_thibedeau|6 months ago
We use numbers in compact decimal approximations for convenience. Repeated rational series are cumbersome without an electronic computer and useless for everyday life.
dansmyers|6 months ago
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.
nyc111|6 months ago
rubycollect4812|6 months ago
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.
1970-01-01|6 months ago
https://www.cnbc.com/2019/04/10/toddler-locks-ipad-for-48-ye...
kbelder|6 months ago
curtisszmania|6 months ago
[deleted]