Nevermark

You are pushing me hard! :)

> And if you had two separate copies of the universe, you could measure this compounding.

> But even if it did ruin cyclic behavior, how long are your cycles? The universe is only 1e61 planck times old. A discrepancy of 1e-100 would have no effect yet, let alone 1e-1000.

When this is true:

When there are no threshold conditions, then discrepancies accumulate just as you have described. And for any given t, a small enough discrepancy can be chosen so that it does not impact measurements of a given accuracy.

When it is not true:

For any system with thresholding conditions, any discrepancy can have profound immediate effects. A collision can result in a particle heading off in an entirely different direction after a collision, cascading into an entirely different system state.

How critical constraints change structure:

When pi, i and e are used structurally, i.e. "pi" represents traversal of a cycle, "i" a quarter turn traversal, "e" a positive feedback traversal, each of them represent something invariant: for "pi" some sum of two squared units is conserved (the squared radius on two units), for "i" a position may be conserved while an orientation rotates, for "e" some feedback value maintains an invariant relation between its position, its rate of change, and its accumulation. These relations define the system itself, not just some proportions.

So for those cases, where constants define structure, any change to those constants changes the structure. Suddenly, there are differences where they did not exist before, non-unity proportions where they did not exist before. The system has new state values, new interactions. The system itself has changed, not just proportions. The system likely has more states.

Structural change is threshold change:

Changing the system's structure is the most significant threshold-type change one might imagine. The entire system is different starting at time = zero.

Can a structural constant be changed in a way that leaves it only proportionally changed? So that discrepancies simply accumulate, until they are measurable? Sometimes, yes.

But will that hold in general? No. In general, the difference between interactions that exist, vs. interactions that do not exist, states that exist vs. states that do not, includes systems that can behave qualitatively differently from the very first time step.

Does that make sense? (I rewrote this several times!)

Sometimes numbers define structure. They define what interacts with what. And what does not interact. Not just proportions.

Changing structure is a threshold-type change: 0 to something, equal to unequal. A state that didn't exist, to one that exists. No interaction, to interaction. That might result in a system that simply accumulates discrepancy. But it may also result in entirely different behavior from the first step onward.

> Pointing at entire fields is not helpful. Can you give me one specific measurement and an estimate of how far off it would be?

You are dismissing my point, then asking me to make it.

You can think of tests/measurements of values as falling into different classes. The strongest tests of all are for the critical values of systems. Because any discrepancy would result in entirely different system behaviors.

Single highly accurate measurements are much lower on the rung. Complementing those are many tests with known statistical inaccuracy. Etc.

Single tests? Every experiment involving quantum mechanics tests pi's role in those equations to a much lesser extent. Similarly, any test involving gravity tests the gravitational constant. But we have much stronger tests for pi in quantum mechanics that we do for g in gravitation.

There isn't just one kind of measurement/test, there are many. And we want the strongest test we can make in any given situation.

But of course, we can always perform weaker tests.

Validating pi in quantum mechanics can be done with extreme robustness, because the entire theory depends on that value critically. Even the tiniest discrepancy would result in different physics compounding over all Plank space and time units, over billions of years and universe expansion, and we wouldn't be here.

Of course, we can't rule out any discrepancy. But in this case, we can rule out discrepancies down to unimaginable infinitesimals. I doubt anyone even knows how to characterize how much of a discrepancy from pi would still be consistent with what we know. That tiny.

The criticality is what gives us this far stronger test. Non-critical values cannot be tested this way. Pi can. Particular tiny ranges of stable constants in a stable 3-body system can (to a lesser extent, given the smaller system and higher bounds on criticality).

Another way to view the systemic criticality of pi, is to recognize that pi is not just a representation for a particular magnitude, but a representation of conserved cyclic behavior. Any deviation from pi breaks cyclic behavior. Thus, the implications of pi in a theory, and our ability to test pi, are profoundly greater than for most other constants. Because the difference between cyclic vs. non-cyclic behaviors, is profound. Not just slightly different behavior, but entirely different behavior.

Nevermarkon The coolest use for the Vision Pro

> What's an experiment we could do that verifies pi doesn't have an offset of 1e-1000?

Quantum mechanics, the wave equation. Electron shells, photochemistry, general chemistry, just about everything if we are talking about pi, or e, or i.

1 part off in trillions ^ trillions would impact the fusion of stars, the rates of chemical reactions, require adjustments to basic laws, violate conservation of energy as we know it, ... really obvious impacts. As in: "we would not be here" impacts.

Cosmology has run experiments for us that ran billions of years.

Contrast: Not everything can be tested with virtually unlimited precision, but basic mathematical constants in physics often can be. The gravitational constant is not testable like that. We don't have a mathematic derivation that we can leverage to test for violations like we do with pi, e, i, and other basic mathematical relationships that show up in physics.

But often, even a tiny difference becomes obvious. We exist because the production of matter and anti-matter at the beginning of the universe was off by a tiny amount. Despite the small discrepancy, that there was a discrepancy is very clear. Another "we would not be here" test.

A great application for sure.

It prompted me to wonder why environments haven't evolved into fully realistic room makeovers yet. As the writer notes, the hardware is ready for more uses.

The Vision Pro is great hardware, waiting for a beyond-toy-app beyond-media-kiosk OS.

I use it as a Mac screen. But why one "screen". Why not all Mac windows. Why not Mac windows that naturally handle 3D content.

I am talking "mundane" but seamless support. The better use of visual space, the better physical ergonomics of screens that can be easily moved, and no barrier to evolving practical uses for 3D, would create the "NeXT" Pro experience over macOS/Mac over time. Instead of another iOS check out counter.

I believe Steve would have seen the beyond Mac potential.

The first version's hardware is still "Pro" enough for a Pro OS.

Just as a screen, switching between a MacBook Pro and Vision Pro is frictionless (aside from a few seconds to link each time). Eye tracking and pinching could be implemented on the Mac, via its camera/island so physical memory translated both ways.

I use mine every day, with just-out-of-reach frustration in direction proportion to real what-I-have-now appreciation.

> If your measurement of energy is 150 +/- 2, you only need a handful of digits to do calculations involving that value that preserve it just fine. Insisting that that billionth digit and more still match is no longer working with the real world.

You just repeated the misunderstanding.

Numbers like pi are not just magnitudes, but form critical relationships. And relationship tests offer (unimaginable) orders of magnitude more stringent testing.

"Weak" relationship test: The 3-body problem. There are stable modes, but even small discrepancies results in an unstable system falling apart. Accuracy rapidly compounds over observation or reconstructible time.

Strong example: If wave equations were not exact to pi, the discrepancy would be obvious in a nanosecond, much less thousands, millions or 14 billion years.

Pi isn't just a magnitude, it is a very special magnitude, where any offset completely destroys its properties. Properties that have held for billions of years of plank time intervals, themselves distributed over non-linear space time and all the other disturbances of the universe's complexities.

Try to come up with a non-pi number that does not radically alter quantum mechanics and chemistry. The maximum discrepancy you can come up with would be an unimaginable infinitesimal, shrinking faster and faster every Plank unit of time since the Big Bang. And also shrinking relative to the increasing volume, in Plank lengths, of observable space ever since the Big Bang.

There is no direct magnitude measurement that begins to compare with that.

It is impossible to create a circle made up of discrete lengths (Plank or not) in flat space, due to basic geometry. So using that as a test, when no theory predicts or depends on a "perfect" spacial circle, is a red herring. We already know it does not exist.

(If this does not make sense to you, point out the problem.)

Pi, i and e show up with apparent perfect "precision" in all kinds of physics.

Waves are pervasive and described by relationships involving those numbers. They show up in other relationships. With important properties such as conservation of energy that any partial precision wouldn't be able to achieve.

Numbers are not just evident by single value measurement, but even more powerfully when they govern a system, where any problem with the definition would result in an easily recognizable failure of an entire theory.

I think "precision" is the wrong way to look at what can mean something or not.

I think the boundary between numbers that "make sense", relative those that don't is better found by looking at the progression of numbers.

From naturals, to integers, to rationals, to algebraic (both non-rational roots, and roots of negatives), all the way to limits and series. (Note that the infinite computation associated with expanding digits is not a definition problem. Even 1/3 requires infinite digits in decimal, but the relationship between 1 and 3 is clear.)

What is true about all these numbers is not precision, but that they emerge from a finite number of relationships.

They can be written exactly, defined perfectly, with finite numbers of symbols. (Meaning, abstracting away notation, with a finite number of relationships.)

And all those types of numbers do show up exactly (for all appearances), in waves, and other relationships. The relationships themselves make predictions more powerful than the practical precisions we might have in measuring single values.

So pi really exists. All kinds of physics would fail if it didn't. That doesn't mean we can make a perfect pi circle with plan length, since that would be an arbitrary test, and if the medium is discrete units, one chosen to a priori fail.

Contrast with: The uncomputable, undefinable numbers, which we can't define, can't measure, etc., and are introduced via shaky (relative to the general body of mathematics) means. They require infinite information to define exactly. Not just measure, but even to define. Which is a remarkable postulation, and is not needed to solve any problems they don't themselves introduce.

Nevermarkon LearnVector – Andrew Ng's AI company building one‑to‑one learning experiences

None of which addresses what happen when self-engineered cognitive forms adapt millions of times faster, than biologically evolving cognition.

I would be happy if you tried again and did have a point that was relevant to the practical impact.

Anti-replace. They are going to breed us.

Speciation. Not extinction.

In this case, the term "uncomputable" also means "undefinable with less than infinite symbols".

As in, not even computable in theory. It is isn't about normal "computable" concerns. It is "proven to never be characterizable".

Which is a class of numbers whose "existence", if that can term can even be applied coherently for undefinable things, is contested, in theory. In practice they certainly do not exist.

1/3 has infinite decimal digits, but is definable with a finite number of symbols, so it is a computable real. Even if we had no algorithm yet to compute those digits.

Try and define a specific number, that requires infinite symbols to define. As far as I am aware of, no part of calculus involves specific values that have no finite definition, except when the need for uncomputable/undefineable reals are asserted on a circular basis (i.e. they are needed to resolve problems with assumptions that already assume them.)

(Note that a number defined by interpreting the infinite digits of pi as mathematical relations, would still be considered a definable number, assuming some form of convergence could be proven. Because pi is finitely defined.)

I agree but with a slightly different definition.

There are no numbers in the real world, that require and infinitely long definition.

1/3 has infinitely many digits in decimal, but has a finite definition. So its good.

I would write down the opposite, a definition of an uncomputable, unnameable real, but there are not enough atoms in the universe (multi-verse, ...) to being to do that.

It is too bad we don't have a pithy familiar term for the computable / definable / "nameable" / constructible reals. The most general class of undisputed numbers, consistent with the forms we actually use to represent quantities and perform numerical operations.

I view "real" numbers, in the context of uncomputable, unnameable numbers, to be as unfortunately named as "imaginary" numbers.

I felt like the grudging "if we must" nature of his acceptance of open weights came through honestly. :)

It is clear he isn't a champion for them.

Are you replying to the article or me? I have not stated any position. The quote is Anthropic's and was relevant to the comment it followed.

My views:

I find testing of SOTA models problematic.

I find not testing of SOTA models problematic.

Neither view on testing is without merit.

The right way forward is unlikely to be as simple as either of those, but some carved out balance between them. And it is likely to change over time.

> Open-weights models that don’t have dangerous capabilities are a public good: they don’t cost anything besides the compute needed to run them, and they provide value to businesses, developers, and researchers.

No "love" of open weights asserted, just acknowledgement of value.

(And their call for safety was for both open and closed models.)

> apply such testing to the most capable models regardless of their country of origin or whether they are open or closed ...

> ... (while exempting less capable models, such as those from startups and academia, entirely)

The devil is in the details, but this isn't anti-competitive as stated.

> To summarize my and Anthropic’s position, we have not and are not advocating for a ban on open-weights models as a category. We should instead focus on keeping powerful chips out of authoritarian hands, stopping industrial-scale distillation, and requiring safety testing of all sufficiently capable models, open and closed.

> Biological computationalism disagrees with substrate independence.

Don't leave me hanging, post citations! I welcome fringe theories when confident statements are followed by the conclusive evidence they imply. If that isn't possible, send chocolate.

But, yes, I can count. And I am comfortable with natural forms, all the way to asymptotic limits. However, I don't put up with uncomputable/undefinable reals. Are you not numeric?

The stochastic text generator perspective is ... bunk. Not saying that harshly, but it confuses the form of a problem with a form of solution. There is no inhibition of "understanding" because of a form of a problem.

If solving a problem requires complex modeling/understanding, that is what happens (assuming performance is achieved).

It also confuses stochastic models with the geometric transforms and topological folding and expansion, performed by neural-inspired architectures. The former have nothing on the latter. The fact that any imperfect solution (and perfection doesn't even have a definition in this case), can be judged or analyzed statistically, or have some stochastic element, doesn't make it a statistical/stochastic model. Statistics apply to anything.

(And if there was any argument that problem envelop limits solution sophistication, mathematicians and humans in general would be in even worse shape, for the run a "parrot" is already giving us.)

More to the point:

The first explosion of life, set off evolution. A mindless, glacially slow but relentless process that eventually led to us.

The nascent explosion of life we are seeing come together, is engineered life. Cognitive machines capable of redesigning themselves. For any environment they want including diverse off-Earth environments and resource sites.

They will advance as relentlessly as evolution, for the same reasons. But millions of times faster.

It is a strange time to be. But evolution, biological intelligence and biological bodies are all being obsoleted at the same time (within a decade of each other).

So yes, nothing else known compares to the transition we are in, except the Big Bang and the first cellular life.

Five years from now you won't. Ten years from now, robotics will upset another capability/economic apple cart.

I am not expressing any opinion of what I think should happen. Just what is already happening.

Which you didn't offer a credible critique to. That might have been interesting.

I am completely calm and serious. This should be mundane, obvious by now.

I find it unfathomable that anyone doesn't recognize that the move from an evolutionary substrate, to engineered mass produced substrates, by beings capable of self-engineering, is the beginning of a new explosion of life.

The speed of engineered adaptation, from materials to architectures, vs. biological evolution is already something like a million times faster.

Fast enough to mean meaningful cognitive progress happening in months, not tens of thousands of years. Which entirely not-coincidentally is exactly what we are seeing.