That's false. Property used to mean a set of rights that gives legal control over valuable things, not limited to simply "physical objects", has been around for thousands of years. Ancients used it for future payments, interest (which could be traded), and much more.

Ancient Syrians (600BC) gave exclusive rights for breadmakers to make certain breads for a year window, and these were property rights, tradeable, sellable, had futures, etc. Ancient Greeks had a patent system for "a new refinement in luxury" that were property rights. Athenaeus (200AD) describes the system in place then where inventors could own their inventions and be the only one to profit for some time.

These are all property rights - something owned by a person, sellable, tradeable, has value, exclusive use. That you (and too many others) seem to think property can only be a "physical object" is as short-sighted as some who claim property can only be land.

... by repeating lots of the flaws that led to IEEE754, requiring extra accumulators (the "quire") and hardware to do basic ops since the posit format alone fails, and making numerical analysis a complete mess, breaking the ability to write correct numerical algorithms.

They lose precision over large dynamic ranges, making algorithms fail on many inputs, without extreme care (and loss of accuracy over such ranges), lack of NaN/inf makes them fail on lots of other issues (and there are algorithms requiring NaN and inf behavior under IEEE754 for performance - I'll list one I recently made below...), this lack makes it harder to debug where algorithms broke, costing development time, ....

The algo I recently developed needed to find extrema of cubics over a finite range. This requires solving a quadratic. A quadratic root solver can have /0 = inf and sqrt(-) = NaN cases, which are often fiddled with using branches.

In my case I knew I'd be doing these in batches, and wanted C/C++ code to auto vectorize and do them in SIMD, and did not want to pay the cost for branches. This speed up the flow by about 8x on almost all larger processors, at the cost of some slots having NaN or inf. Those with NaN or inf had underlying cubics I could discard. So by using the IEEE754 aware multi parallel root finder (written in strd c++), I could check that the roots were in my interval (also parallelized) as a <= root <= b, which fails for root being NaN or inf. This check is also parallelzied.

All in standard C++, no hint of parallelization intrinsics, handled by modern compilers perfectly, and getting massive speed gains.

This is but one place NaN and inf are extremely useful. This type of use appears all over in scientific computing, graphics, pysics sims, etc.

Posits cannot handle this type of stuff.

No, those are also stupidly small samples. Look at the papers I listed.

> That 2014 study you linked is noting that it does happen with other kinds of blindness

Yes, from 1950 till 2014, as more and more kinds of blindness were found with schizophrenia, the type of blindness has been dwindling to smaller and smaller classes, ensuring there is not enough predictive power in the claims. Again, look at the papers I listed. ALL of this is covered.

> Isn't that saying the same thing the article does.

The article says lots of nonsense, like the most likely outcome of data is somehow the best evidence for an unproven claim. It implies the Australia study says a thing it DOES NOT SAY. Is this not enough bad reporting to question the accuracy of the article?

> What am I missing?

Simply look at the papers I posted. They are right there for you to read. The article is click bait trying to claim there is some surprising scientific claim that HAS NO SCIENTIFIC PROOF.

Making debt of that form illegal would kill any company that needed money to stay afloat, such as during some emergency, or war, or COVID, or tons of events that companies regularly survive.

Laws that are enforced and more importantly are enforceable have a much higher rate of making a difference. The same works here.

1870/500,000 * 66 = 0.247

Not a single blind child getting it is the most likely outcome, and this is called "the most rigorous evidence"?

It didn't protect rats in a study https://www.sciencedirect.com/science/article/abs/pii/S09209...

There sure is a lot of reported cases of all sorts of blindness with schizophrenia, constantly shrinking the pool of types of the two, making this conjecture constantly shrinking https://pmc.ncbi.nlm.nih.gov/articles/PMC4246684/

It also seems the Australia study does not quite say what the article claims it does - a follow-up study: https://jmsgr.tamhsc.edu/the-lack-of-comorbidity-between-ear...

Too bad this article simply doesn't mention all this. Of course the article will get less clicks with a less wild title.

It absolutely does matter. The first, and most important reason, is one needs to know the guarantees of every operation in order to design numerical algorithms that meet some guarantee. Without knowing that the components provide, it's impossible to design algorithms on top with some guarantee. And this is needed in a massive amount of applications, from CAD, simulation, medical and financial items, control items, aerospace, and on and on.

And once one has a guarantee, making the lower components tighter allows higher components to do less work. This is a very low level component, so putting the guarantees there reduces work for tons of downstream work.

All this is precisely what drove IEEE 754 to become a thing and to become the standard in modern hardware.

> the problem is that languages traditionally rely on an OS provided libm leading to cross architecture differences

No, they don't not things like sqrt and atanh and related. They've relied on compiler provided libs since, well, as long as there have been languages. And the higher level libs, like BLAS, are built on specific compilers that provide guarantees by, again, libs the compiler used. I've not seen OS level calls describing the accuracy of the floating point items, but a lot of languages do, including C/C++ which underlies a lot of this code.

Then, subtraction is (x#y)#0 = x-y. Reciprocal is x#0 = 1/x. Addition follows from x+y=x-((x-x)-y). This used the additive identity 0.

Multiplication follows from

x^2= x-1/(1/x + 1/(1-x)), so we can square things. Then -2xy = (x-y)^2 -x^2 - y^2 is constructible. Then we can divide by -2 via x/-2 = 1/((0-1/x)-1/x), and there’s multiplication. In terms of #, this expression only needed the constant 1, which is the multiplicative identity.

Now mult and reciprocal give x * 1/y = x/y, division.

Any nontrivial ring needs additive and multiplicative identities 1!=0, which are the only constants needed above. If you assume this is Q or R or C, it may be possible to derive one from the other, not sure. But if you’re in these fields, you know 0 and 1 exist.

Then any element of Q is a finite set of ops. R can be constructed in whatever way you want: Dedekind cuts, Cauchy sequences, whatever usual constructions. Or assume R exists, and compute in it via the f(x,y).

This also works over finite fields (eml does not), division rings, even infinite fields of positive characteristic, function fields (think elements are ratio of polynomials), basically any algebraic object with the 4 ops.

I just looked through many of the best known real analysis texts, and not a single one defines them this way. This list included the texts by

Royden, Terence Tao, Rudin, Spivak, Bartle & Sherbert, Pugh, and a few others....

Can you cite a single text book that has this definition you claim is in every real analysis course? I find all evidence points to the opposite.

A limit process is a definition. Try computing with it. You’ll end up with an infinite sequence, or an approximation.

An iterative process is an infinite series. They’re equivalent.

Newtons method is the same. Completely equivalent to an infinite series as you increase precision.

And both require constants, infinitely precise. So you’re still not doing anything the 1/(x-y) operation cannot do, and to do those series you’ll compute using things amenable to being done via ops easy to do by hand or machine via the 1/(x-y) op.

If we’re making sloppy approximations to a tiny range of exp, then I too can do it with a few terms.

And the basic monomial basis is not a single binary operation capable of reproducing the set of basic arithmetic ops. If you want trivial and basic, pick Peano postulates. But that’s not what this thread was about.

Go ahead and show how to compute exp or ln without an infinite series without circular reasoning. You can’t, since they’re transcendental.

There are infinitely many ways to make these binary operators. Picking extremely high compute cost ones really doesn’t make a good basis for computation.